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UBC Theses and Dissertations

Variable bit rate video transmission for code-division multiple-access systems in wideband fading channels Iskander, Cyril-Daniel 2003

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VARIABLE BIT RATE VIDEO TRANSMISSION FOR CODE-DIVISION MULTIPLE-ACCESS SYSTEMS IN WIDEBAND FADING CHANNELS by CYRIL-DANIEL ISKANDER B.Sc , Universite Laval, Quebec City, 1998 M . S c , Universite Laval, Quebec City, 1999 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES Department of Electrical and Computer Engineering D O C T O R O F PHILOSOPHY in T H E UNIVERSITY OF BRITISH C O L U M B I A September 2003 © Cyril-Daniel Iskander, 2003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of i=-l-fct*1"1 C C « - \ p - > H ^ t ^ m - f € K , The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T Efficient real-time transmission of video data over bandwidth-constrained wireless channels is challenging in several ways: in particular, due to the underlying compression algorithms, the source rate can vary in bursts, which complicates the resource allocation problem, isolated channel errors can totally corrupt a video frame if sensitive information is affected, and errors in earlier frames can cause damage to later frames due to error propagation. This thesis wil l dear in particular with the effect of source rate variability on current and future cellular systems which employ code-division as the multiple-access strategy, such as IS-95B and IS-2000 systems. The problem wil l be approached from a physical-layer perspective: hence issues relating to the channel- and cellular-level perfor-mances wil l be addressed in detail, and then integrated into the system-level performance. This nonconventional cross-layer approach allows us to obtain additional insights over studies which tackle the issue mainly or exclusively at the higher system layers. In the first part of this thesis, several contributions are made to the theory of wide-band fading channels, which wil l be considered as the physical channel model throughout the thesis. We derive the analytical level-crossing rates, average fade durations, enve-lope autocorrelations and baseband spectra of several channel models for some common diversity techniques. Based on some of the previously derived properties we design a fast wideband Nakagami channel simulator. We then derive the exact analytical error probabilities of several linear modulation schemes with diversity in correlated Nakagami channels, and validate them through simulation. In a second part, we derive accurate analytical or semi-analytical error probability expressions for the multicode and multirate configurations used in the physical layers of both the uplink and downlink of IS-95B and IS-2000 systems, in the presence of wideband fading. It is demonstrated that the effect of the multicode interference must be precisely taken into account to obtain reliable error statistics in wideband channels, especially for cellular systems with a low number of users. To this end, the fading dependence across i i multiple codes of a given user must be taken into account in the analysis, whereas for single-code systems this situation didn't occur. We consider systems which employ either maximal-ratio or equal-gain combining. The proposed methodology places no restrictions on the type of fading distribution, and examples are given for the cases of Rayleigh, Rice, Nakagami and lognormal fading, for both independent and correlated diversity branches. For the IS-95B uplink, the analysis is extended to deal with closed-loop power control using the inverse update algorithm, successive interference cancellation, and multicell systems. A l l analytical results are thoroughly validated through numerous entire system simulations, for different values of several transceiver and channel parameters. In the final part of this thesis, we demonstrate the benefits of employing rate smooth-ing for variable bit rate video applications in D S / C D M A cellular systems, and present and evaluate practical algorithms to achieve these gains. To support our exposition, a generic rate smoothing algorithm is developed, whose goal is to minimize the degrada-tion caused by source bursts in such systems. Its performance in terms of decoded video quality is compared to that of a popular algorithm which was developed in the context of wireline communications, and which serves as a benchmark. It is shown that for systems subject to certain practical constraints, in particular concerning the granularity of the transmission rates, the proposed algorithm can offer an improved decoded video quality with respect to the benchmark algorithm. The influence of smoothing-related parameters such as the startup buffering delay and sliding window length are quantified. In addition, the effects of some transceiver and channel parameters on the decoded video quality are presented. To carry out these performance evaluations, a flexible software platform has been developed which emulates the transmission of video data at the physical/link layers in IS-95B and IS-2000 cellular systems with wideband fading, and allows the user to objectively measure the decoded video quality directly at the application layer. i i i T A B L E O F C O N T E N T S A B S T R A C T i i T A B L E O F C O N T E N T S iv LIST O F F I G U R E S v i i i LIST O F T A B L E S xix LIST O F A B B R E V I A T I O N S xx A C K N O W L E D G E M E N T S xxv 1 I N T R O D U C T I O N 1 1.1 Introduction to Wireless Video Communications 1 1.2 Wideband Fading Channels 2 1.3 D S / C D M A Cellular Networks 5 1.4 Video Communications: Characteristics and Methods 6 1.4.1 Requirements of Video Transmission 6 1.4.2 Impairments to Video Transmission 7 1.4.3 Rate Control for V B R Video 8 1.5 Thesis Contributions and Organization 15 1.5.1 Statistics and Simulation of Wideband Fading Channels 15 1.5.2 Analysis of Multicode and Multirate D S / C D M A Systems in Wide-band Fading Channels 16 1.5.3 V B R Video Transmission for Multicode and Multirate D S / C D M A Systems in Wideband Fading Channels 17 1.6 Publications 18 iv 2 S T A T I S T I C S A N D S I M U L A T I O N O F W I D E B A N D F A D I N G C H A N N E L S 21 2.1 Introduction 21 2.2 A Review of Common Fading Models and Distributions 21 2.2.1 Rayleigh Fading 21 2.2.2 Rician Fading 24 2.2.3 Nakagami Fading . . '. 25 2.3 Analytical Second-Order Statistics of Fading Channels with Diversity . . 28 2.3.1 Analytical Level Crossing Rates and Average Fade Durations of Diversity-Combined Nakagami Fading Signals 28 2.3.2 Analytical Envelope Correlation and Spectrum of Maximal-Ratio Combined Nakagami and Rician Fading Signals 50 2.4 Simulation of Wideband Correlated Nakagami Fading Channels 62 2.4.1 Continuous Channel Simulation 62 2.4.2 Discrete Channel Simulation 71 2.5 Analytical Symbol Error Rates for Diversity Techniques in Correlated Nak-agami Fading Channels 92 2.5.1 Introduction 92 2.5.2 General Approach 93 2.5.3 Symbol Error Rate for Equal-Gain Combining 96 2.5.4 Symbol Error Rate for Maximal-Ratio Combining 107 2.5.5 Performance Evaluation Results 109 2.5.6 Conclusions 115 2.6 Conclusions 115 3 A N A L Y S I S O F M U L T I C O D E A N D M U L T I R A T E D S / C D M A S Y S T E M S IN W I D E B A N D F A D I N G C H A N N E L S 117 3.1 Introduction 117 3.1.1 Multicode Transmission 117 3.1.2 Multirate Transmission 119 v 3.2 Reverse Link Performance with Noncoherent M-a ry Orthogonal Modula-tion and Real Spreading Sequences 119 3.2.1 Introduction 119 3.2.2 Error Probability Analysis 121 3.2.3 Performance Evaluation Results and Discussion . 146 3.2.4 Conclusions 164 3.3 Reverse Link Performance with Coherent B P S K Modulation and Complex Spreading Sequences 169 3.3.1 Introduction 169 3.3.2 Error Probability Analysis , 170 3.3.3 Performance Evaluation Results and Discussion 184 3.3.4 Conclusions 186 3.4 Forward Link Performance with Coherent Q P S K Modulation and Real Spreading Sequences 196 3.4.1 Introduction 196 3.4.2 Error Probability Analysis 197 3.4.3 Performance Evaluation Results and Discussion 206 3.4.4 Conclusions 208 3.5 Forward Link Performance with Coherent Q P S K Modulation and Complex Spreading Sequences 212 3.5.1 Introduction 212 3.5.2 Error Probability Analysis 212 3.6 Conclusions 223 V B R V I D E O T R A N S M I S S I O N F O R M U L T I C O D E A N D M U L T I R A T E D S / C D M A S Y S T E M S IN W I D E B A N D F A D I N G C H A N N E L S . . . 224 4.1 Introduction 224 4.2 System Description 225 4.2.1 Overview of a Video Communication System for IS-95B/IS-2000 Networks 225 4.2.2 H.26x Video Coding and Bitstream Generation 231 vi 4.2.3 H.324/H.223 Packetization and Multiplexing 235 4.2.4 IS-95B Forward Link 241 4.2.5 IS-95B Reverse Link 248 4.2.6 IS-2000 Forward Link 251 4.2.7 IS-2000 Reverse Link 259 4.3 Smoothing for V B R Video over Multicode and Multirate D S / C D M A . . 264 4.3.1 Introduction 264 4.3.2 Motivation for Smoothing 267 4.3.3 V B R Video Constraints 271 4.3.4 Smoothing Algorithms for V B R Video over D S / C D M A 274 4.3.5 Performance Evaluation Results and Discussion 282 4.3.6 Conclusions 314 C O N C L U S I O N S A N D T O P I C S F O R F U T U R E R E S E A R C H . . . . 315 5.1 Contributions 315 5.2 Topics for Future Research 317 A P P E N D I X A : Special Functions, Series Expansions and Definite Integrals 320 A P P E N D I X B: Alternative Derivation of the L C R for SC 326 A P P E N D I X C: Calculation of the Generalized Marcum-Q Function 327 A P P E N D I X D: Derivation of Signal and Interference Terms for Multicode D S / C D M A with Noncoherent M-ary Orthogonal Modulation . . . . 328 A P P E N D I X E : Derivation of Symbol Error Probabilities of Noncoherent M-ary Orthogonal Modulation with E G C in Fading Channels . . . . 337 A P P E N D I X F: M A P Decoding Algorithm 340 R E F E R E N C E S 343 vii LIST OF FIGURES 2.1 LCR's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = l);m = 0.6 42 2.2 LCR's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L — 1); m = 1.3 42 2.3 LCR's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 3.0 43 2.4 AFD's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L=l);m = 0.6 43 2.5 AFD's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 1.3 44 2.6 AFD's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 3.0 44 2.7 L C R for SC with different diversity orders and m — 1.3 45 2.8 L C R for M R C with different diversity orders and m = 1.3 45 2.9 L C R for SC with L = 2 correlated branches, m — 1.0, p = 0.6425. — : upper bound; : exact, a — 0; - . - .: exact, a — n/2 49 2.10 L C R for M R C with L = 2 correlated branches, m — 1.0, p = 0.6425. — : upper bound; : exact, a = 0; - . - .: exact, a = TT/2 49 2.11 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rayleigh channel with L = 3 60 2.12 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rayleigh channel 60 2.13 Spectrum of the baseband envelope versus frequency for the Rayleigh channel. 61 2.14 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rice channel with L — 3 61 vi i i 2.15 B E R for SC: (a) m = 1, (b) m = 2, (c) m = 3. F S M C simulation: ++ ; theory: — 83 2.16 B E R for M R C : (a) m = 1, (b) TO = 2, (c) TO = 3. F S M C simulation: ++ ; theory: — 84 2.17 B E R for E G C : (a) m = 1, (b) TO = 2. FSMC simulation: ++ ; theory: — . . . 84 2.18 L C R for SC: (a) m = 1, (b) m = 2, (c) TO = 3. F S M C simulation: ++ ; theory: — 85 2.19 L C R for M R C : (a) m = 1, (b) TO = 2, (c) m = 3. F S M C simulation: ++ ; theory: — 86 2.20 L C R for E G C : (a) m = 1, (b) m = 2, (c) m = 3. F S M C simulation: ++ ; theory: — 86 2.21 Coded B E R with v = 6 km/h, TO = 1, and L = 1 89 2.22 Coded B E R with M R C and v = 6 km/h, TO = 1, and L = 2 89 2.23 Coded B E R with M R C and v = 6 km/h, m = 1, and L = 3 90 2.24 Coded B E R with v = 50 km/h, m = 1, and L = 1 90 2.25 Coded B E R with M R C and v = 50 km/h, TO = 1, and L — 2 91 2.26 Coded B E R with M R C and v = 50 km/h, TO = 1, and L = 3 91 2.27 Probability density function of the sum of two correlated Nakagami-m variables: exact (through Eq. (2.205)); approximation (using Eqs. (2.208)-(2.209)) 100 2.28 Cumulative distribution function of the sum of two correlated Nakagami-m vari-ables: exact (Eq. (2.207)); - - - - approximation (using Eqs. (2.208)-(2.209)) 100 2.29 SER of M - P S K with dual-branch E G C and m = 2, for different p's 110 2.30 SER of M - P S K with dual-branch M R C and TO = 2, for different p's 110 2.31 SER of 2-PSK with dual-branch E G C and M R C , for p = 0.5 and different m's. I l l 2.32 SER of 4-PSK with dual-branch E G C and M R C , for p = 0.5 and different m's. I l l 2.33 SER of 8-PSK with dual-branch E G C and M R C , for p = 0.5 and different m's. 112 2.34 SER of 4-PSK and 8-PSK with dual-branch E G C and m = 2, p = 0.5, for different {f i i ,n 2 } 1 1 2 ix 2.35 SER of 8-PSK and 16-QAM with dual-branch E G C and m = 2, {Sli = 1.0, fi2 = 0.5}, for different p's 113 2.36 SER of 2-PSK and 8-PSK with dual-branch E G C and m x = 2.0, m 2 = 1.0, for different {f2i,fi 2}, and p = 0 113 2.37 SER of 4-QAM and 16-QAM with dual-branch E G C and mi = 2.0, m 2 = 1.0, for different {f2i, fi2}, a n ( l P = 0. 114 2.38 SER of DE-2-PSK and DE-4-PSK with dual-branch E G C and mi = 2.0, m 2 = 1.0, for different {f i i , f22}, and p = 0 114 3.1 Transmitter for cth code of user 1 123 3.2 Demodulator for nth Rake finger of user 1 and code c 125 3.3 Simplified loglinear closed-loop power control model [205] 136 3.4 Cellular environment 144 3.5 B E R vs K for m = 1, L = 1, = 8. - : Eq. (3.40); - - : Eq. (3.42); + : simulation 148 3.6 B E R vs K for m = 1, L = 2, = 8. - : Eq. (3.40); - - : Eq. (3.42); + : simulation 149 3.7 B E R vs K for m = 1, L = 3, Art1) = 8. - : Eq. (3.40); - - : Eq. (3.42); + : simulation 149 3.8 B E R vs K for m = 1, L = 4, = 8. - : Eq., (3.40); - - : Eq. (3.42); + : simulation 150 3.9 B E R vs K for m = 1, L = 5, M1) = 8. - : Eq. (3.40); - - : Eq. (3.42); + : simulation 150 3.10 B E R vs K for m = 1, = 4. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3. 151 3.11 B E R vs K for m = 1, = 8. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3. 151 3.12 B E R vs K for m = 1, L = 1. — (+): = 1; - . - (*): JV*1) = 4; - - (o): JVW = 8. . 152 3.13 B E R vs K for m = 1, L = 2. — (+): = 1; - . - (*): AK 1) = 4; - - (o): Ard)=8 152 3.14 B E R vs K for m = 1, L = 3. — (+): = 1; - . - (*): A^ 1 ) = 4; - - (o): JVW =8 153 x 3.15 B E R vs K for TO = 1, = 1, = 4. — (+): L = 1; - . - (*): L = 2; - -(o): L = 3 153 3.16 B E R vs K for m = 1, = 1, iV(2) = 8. — (+): L = 1; - . - (*): L = 2; - -(o): L = 3 154 3.17 B E R vs K for L = 1, JVW = 8. — (+): TO = 1; - . - (*): m = 2; - - (o): m = 3 155 3.18 B E R vs K for L = 2, / V ^ = 8. — (+): m = 1; - . - (*): m = 2; - - (o): TO = 3 155 3.19 B E R vs K for L = 3, = 8. — (+): m = 1; - . - (*):. m = 2; - - (o): m = 3 156 3.20 B E R vs K for TO = 2, TV^ = 8. — (+): L = 1; - . - (*): L = 2; — (o): L = 3. 156 3.21 B E R vs K for m = 3, = 8. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3. 157 3.22 B E R vs K for m = 2, L = 2, p = 0.5. — (+): TV^1) = 1; - . - (*): iV^1) = 4; -- (o): NW =8. 157 3.23 B E R vs K for m = 2, L = 2, = 8. (+): p = 0.0; - . - (*): p = 0.3; - -(o): p = 0.5; • • • (o): p = 0.7 158 3.24 B E R vs K for a Rician channel with A = 1.0, i V ( 1 ) = 8. — (+): L = 1; - . -(*): L = 2; - - (o): L = 3 159 3.25 B E R vs K for a Rician channel with A = 2.0, = 8. — (+): L = 1; - . -(*): L = 2 ; - - (o): L = 3 160 3.26 B E R vs K for lognormal fading, L = 1, i V ( 1 ) = 8, m L A r = 0. — (+): a L N = 1.0; - - (*): a L i V = 0.5; - . - (o): aLN = 0.2 160 3.27 B E R vs K for m = 1, L = 2, and C L P C with kD = 4. — (+): TV^1) = 1; - -(*): NM =4; - . - (o): =8 161 3.28 B E R vs K for m = 1, L = 3, and C L P C with kD = 4. — (+): = 1; - -(*): JV(1) = 4; - . - (o): JVC1) = 8 161 3.29 B E R vs i f for TO = 1, L = 2, TV^) = 4 and C L P C for different kD. — (+): kD = 1; - - {*): kD = 2; - . - (o): kD = 4 162 3.30 B E R vs K for m = 1, L = 2, = 8 and C L P C for different kD. — (+): fco = 1; - - (*): kD = 2;-.- (o): fcp = 4 162 x i 3.31 B E R vs K for m = 1, L = 1, and SIC for different N™. — (+): A^ 1 ) = 1; - . _ (*): JVW = 4 ; - - ( o ) : JVW = 8 163 3.32 B E R vs K for m = 1, L = 2, and SIC for different A^ 1 ) . — (+): A^ 1 ) = 1; - . _( * ) : i \ r ( i ) = 4 ; _ _ ( o ) : ^ = 8 163 3.33 Multicell system: B E R vs K for m = 1, = 4. — (+): L = 1; - . - (*): L = 2 ; - - ( o ) : Z = 3 165 3.34 Multicell system: B E R vs K for m = 1, J V ^ = 8. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3 165 3.35 Multicell system: B E R vs K for L = 1, A^ 1 ) = 8. — (+): m = 1; - . - (*): m = 2; - - (o): m = 3 166 3.36 Multicell system: B E R vs K for L = 2, A^ 1 ) = 8. — (+): m = 1; - . - (*): m = 2 ; - - ( o ) : m = 3 166 3.37 Multicell system: B E R vs K for L = 3, A^ 1 ) = 8. — (+): m = 1; - . - (*): m = 2; - - (o): m = 3 167 3.38 Multicell system: B E R vs K for lognormal fading, L = 1, CTLAT = 0.2, m^w = 0. — (+); j\r(i) = 1; - - (*): iV*1) = 4; - . - (o): =8 167 3.39 Multicell system: B E R vs K for lognormal fading, L = 1, CTLJV = 0.5, mr,N =.0. — (+) : 7V(i) = 1; - - (*): M 1 ) - 4; - . - (o): =8 168 3.40 Multicell system: B E R vs K for lognormal fading, L = 1, <JLN — 1-0, raz,jv = 0. — (+); AT(i) = i ; _ _ (*): jv(i) = 4; - . - (o): = 8. 168 3.41 Channelization/Spreading/Modulation subsystem 172 3.42 Demodulator for the nth Rake finger 173 3.43 B E R vs K for A^ 1 ) = 8. + + + : £ = 1; * * * : £ = 2; o o o : £ = 3 187 3.44 B E R vs K for =4. + + + : £ = 1; * * * : £ = 2; o o o : L = 3 187 3.45 B E R vs K for = 8, L = 2. — : Eq. (3.140); - - : approximation. . . . 188 3.46 B E R vs K for A^ 1 ) = 8, L = 3. — : Eq. (3.140); - - : approximation. . . . 188 3.47 B E R vs K for = 8, and m = 2.0. — (+): L = 1; - . - (*):• L = 2; — (o): L = 3 189 3.48 B E R vs K for AK 1) = 4, and m = 2.0. — (+): L = 1; - . - (*): L = 2; — (o): L = 3 189 xi i 3.49 B E R vs K for = 1, and m = 2.0. — (+): L = 1; - . - (*): L — 2; — (o): L = 3 190 3.50 B E R vs K for TVt1) = 8, and L = 1. — (+): m = 1.0; - . - (*): m = 2.0; - -(o): m = 3.0 190 3.51 B E R vs K for = 8, and L = 2. — (+): m = 1.0; - . - (*): m = 2.0; - -(o): m = 3.0 191 3.52 B E R vs K for JVC1) = 8, and L = 3. — (+): m = 1.0; - . - (*): m = 2.0; - -(o): m = 3.0 191 3.53 B E R vs K for = 8, and L = 1. — (+): A = 1.0; - . - (*): A = 2.0; - -(o): A = 3.0 192 3.54 B E R vs K for = 8, and L = 2. — (+): A = 1.0; - . - (*): A = 2.0; - -(o): A = 3.0 192 3.55 B E R vs K for L = 2 correlated branches with p = 0.5 and m = 1.0. — (+): /VW = 1; - . - (*): JV*1) =4; - - (o): =8 193 3.56 B E R vs K for L = 2 correlated branches and m = 2.0, with iV"W = 8. — (+): p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (©): p = 0.7 193 3.57 B E R vs K for L = 2 correlated branches and m = 2.0, with = 4. — (+): p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7 194 3.58 B E R vs K for E G C with L = 2 and m = 1.0. — (+): = 1; - . - (*): JVC1) = 4; - - (o): T^ 1) =8 194 3.59 B E R vs K for E G C with L = 2, m = 2.0 and / V ^ = 8. — (+): p = 0.0; - . -(*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7 195 3.60 B E R vs K for E G C with L = 2, m = 2.0 and J V ^ = 4. — (+): p = 0.0; - . -(*): p = 0.3; - - (o): p = 0.5; • • • (©): p = 0.7 195 3.61 Transmitter for cth code of user 1 199 3.62 B E R vs K for M R C and m = 1.0. — (+): L =T; - . - (*): L = 2; - - (o): L = 3 208 3.63 B E R vs K for M R C and L = 2. — (+): m = 1.0; - . - (*): m = 2.0; - - (o): m = 3.0 209 3.64 B E R vs K for E G C and L = 2. — (+): m = 1.0; - . - (*): m = 2.0; - - (o): m = 3.0 209 xiii 3.65 B E R vs K for M R C and L = 2. — (+): A = 1.0; - . - (*): A = 2.0; - - (o): A = 3.0 210 3.66 B E R vs K for E G C and L = 2. — (+): A = 1.0; - . - (*): A = 2.0; - - (o): A = 3.0 : 210 3.67 B E R vs K for L = 2 correlated branches and m = 2.0, with M R C . — (+): p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7 211 3.68 B E R vs K for L = 2 correlated branches and m = 2.0, with E G C . — (+): p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7 211 3.69 Channelization/Spreading/Modulation subsystem 213 3.70 Demodulator for the nth Rake finger 214 4.1 Video communication system for cellular network 226 4.2 Generic real-time video encoder 228 4.3 cdma2000 wireless network model 230 4.4 H.263 picture structure for QCIF resolution 233 4.5 H.26x video encoder 234 4.6 H.223 protocol stack and data structures 237 4.7 A L - P D U for AL3 with optional control field (CF) 238 4.8 A L - P D U for A L 3 M (Annexes C and D) with control field (CF) 239 4.9 Structure for A L 3 M of Annex C 241 4.10 Format of H.223 M U X - P D U 242 4.11 Format of H.223 Annex A M U X - P D U 242 4.12 Format of H.223 Annex B M U X - P D U 242 4.13 Air interface for the IS-95B forward link traffic channels [177] 243 4.14 Generation of PNr and its offsets [177] 247 4.15 Generation of PNQ and its offsets [177] 248 4.16 Air interface for the IS-95B reverse link traffic channels [177] 249 4.17 Generation of PNL and its offsets [177] 250 4.18 Air interface for the IS-2000 forward link traffic channels [14] 253 4.19 Turbo encoder for the IS-2000 forward link [14] 256 xiv 4.20 Air interface for the IS-2000 reverse link Fundamental and Supplemental Chan-nels [14] 261 4.21 Video trace of Miss America, with 1 out of 132 I frames 269 4.22 Video trace of Miss America, with 1 out of 20 I frames 269 4.23 Video trace of Foreman, with 1 out of 132 I frames 270 4.24 Video trace of Foreman, with 1 out of 20 I frames 270 4.25 Rate adaptation framework 272 4.26 IS-95B or IS-2000-based system simulation model 282 4.27 IS-2000 peak and mean transmission rates against Nd/Np for SLWIN and SLWIN2 with TV/ = 20, Nw = 10, B^ax = 64k, for the Miss America bitstream. 289 4.28 IS-2000 variance of the transmission rates against Nd/NF for SLWIN and SLWIN2 with N[ = 20, Nw = 10, B^ax = 64k, for the Miss America bitstream 290 4.29 IS-2000 peak and mean transmission rates against Nw for SLWIN and SLWIN2 with Nj = 20, Nd = 9, B^ax = 64k, for the Miss America bitstream 290 4.30 IS-2000 variance of the transmission rates against Nw for SLWIN and SLWIN2 with Ni = 20, Nd = 9, B^ax = 64k, for the Miss America bitstream 291 4.31 IS-2000 transmission rate schedule for SLWIN and SLWIN2 with Nr = 20, Nd = 9, Nw = 1, B^ax = 64k, for the Miss America bitstream 291 4.32 IS-2000 transmission rate schedules for SLWIN and SLWIN2 with Nr = 20, Nd = 9, Nw = 10, B^ax = 64k, for the Miss America bitstream 292 4.33 IS-2000 transmission rate schedules for SLWIN and SLWIN2 with Nr = 20, Nd = 120, Nw = 1, B^ax = 8k, for the Miss America bitstream 292 4.34 IS-2000 transmission rates for SLWIN2 (i.e. Method 2) and SLWIN2' (i.e. Method 1) with TV/ = 20, Nd = 9, Nw = 1, B^ax = 64k, for the Miss America bitstream . 293 4.35 IS-2000 transmission rates for SLWIN2 (i.e. Method 2) and SLWIN2' (i.e. Method 1) with Nf = 20, Nd = 120, Nw = 1, B^ax = 64k, for the Miss America bitstream 293 4.36 IS-2000 peak transmission rates against Nd for SLWIN and SLWIN2 with Nr = 20, Nw = 10, B^ax = 64k, for the Foreman bitstream 294 xv 4.37 IS-2000 variance of the transmission rates against Nd for SLWIN and SLWIN2 with Nj = 20, Nw = 10, NF = 120 and B^ax = 64k, for the Foreman bitstream. 294 4.38 IS-2000 mean transmission rates against Nd for SLWIN and SLWIN2 with Nr = 20, Nw = 10, NF = 120 and B^ax = 64k, for the Foreman bitstream 295 4.39 IS-2000 variance of the transmission rates against Nw for SLWIN and SLWIN2 with A/"/ = 20, Nd = 12, B^ax = 64k, for the Foreman bitstream 295 4.40 IS-2000 mean transmission rates against Nw for SLWIN and SLWIN2 with AT/ = 20, Nd = 12, B^ax = 64k, for the Foreman bitstream 296 4.41 IS-2000 transmission rate schedules for SLWIN and SLWIN2 with AT/ = 20, Nd = 12, Nw = 10, B^ax = 64/c, for the Foreman bitstream 296 4.42 IS-95B peak transmission rates against Nd/NF for SLWIN and SLWIN2 with Ni - 20, Nw = 10, B^ax = 64k, for the Miss America bitstream 298 4.43 IS-95B variance of the transmission rates against Nd/NF for SLWIN and SLWIN2 with Nj — 20, Nw = 10, B^ax = 64k, for the Miss America bitstream 298 4.44 IS-95B transmission rates for SLWIN2 (i.e. Method 2) and SLWIN2' (i.e. Method 1), with AT7 = 20, Nd = 9, Nw = 1, B^ax = 64k, for the Miss America bitstream 299 4.45 IS-95B transmission rates for SLWIN2 (i.e. Method 2) and SLWIN2' (i.e. Method 1), with Nr = 20, Nd = 120, Nw = 1, B^ax = 64k, for the Miss America bitstream 299 4.46 IS-95B transmission rate schedules for SLWIN and SLWIN2 with A / = 132, Nd = 60, Nw = 10, B^ax = 64fc, for the Foreman bitstream 300 4.47 IS-95B transmission rate schedules for SLWIN2 with AT7 = 132, Nd = 60, Bmax — ^^fc, and different AT^'s, for the Foreman bitstream 300 4.48 Average PSNR for JV> = 20, Nd = 9, B^ax = 64k, for SLWIN, SLWIN2, and SLWIN2', in the IS-2000 uplink case, for the Miss America bitstream 302 4.49 Average PSNR for Nj = 132, Nd = 9, B^ax = 64k, for SLWIN, SLWIN2, and SLWIN2', in the IS-2000 uplink case, for the Miss America bitstream 302 4.50 Average PSNR for A> = 20, Nw = 1, 5 ^ a x = 64k and various Nd% for SLWIN2 in the IS-2000 uplink case, for the Miss America bitstream 303 xvi 4.51 Average PSNR for Nd = 9,NW = 1, B?nax = 64k and various JV}'s, for SLWIN2 in the IS-2000 uplink case, for the Miss America bitstream 303 4.52 Average PSNR for TV, = 20, Nw = 1, B^ax = 64k, for SLWIN and SLWIN2, in the IS-2000 downlink case, for the Foreman bitstream 305 4.53 Average PSNR for Nj = 20, Nw = 1, B^ax — 64k, 1 turbo iteration, and various NbdS, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream. . 306 4.54 Average PSNR for 7Y/ = 20, Nbd = 9, Nw = 1, B^ax = 64k, 6 turbo itera-tions, and various numbers of diversity branches Lr, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream 306 4.55 Average PSNR for JV> = 20, Nbd = 9, Nw = 1, B^ax = 64k, 6 turbo iterations, and various values of the mobile speed v, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream 307 4.56 Average PSNR for JVj = 20, Nbd = 9, Nw = 1, B^ax = 64k, 6 turbo iterations, and different code rates R, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream 307 4.57 Average PSNR for JVj = 132, Nw = 1, B^ax = 64k and various Nds, for SLWIN and SLWIN2 in the IS-95B uplink case, for the Miss America bitstream. . . . 308 4.58 Average PSNR for JV> = 132, Nd = 9, B^ax = 64k and various Nw's, for SLWIN in the IS-95B uplink case, for the Miss America bitstream 309 4.59 Average PSNR for JVj = 132, Nd = 9, B^ax = 64k and various Nw's, for SLWIN2 in the IS-95B uplink case, for the Miss America bitstream 309 4.60 Average PSNR for JVj = 20, Nw = 10, B^ax = 64k and different AVs , for SLWIN2 in the IS-95B uplink case, for the Miss America bitstream 310 4.61 Average PSNR for 7Y/ = 132, Nd = 9, Nw = 1, B^ax = 64k and various numbers of diversity branches Lr, for SLWIN2 in the IS-95B uplink case, for the Miss America bitstream 310 4.62 Average PSNR for Nr = 132, Nd = 9, Nw = 1, B^ax = 64k and various values of the mobile speed v, for SLWIN2 in the IS-95B uplink case, for the Miss America bitstream 311 4.63 Average PSNR for NT = 132, Nd = 60, Nw = 1, B^ax = 64k, for SLWIN and SLWIN2 in the IS-95B downlink case, for the Foreman bitstream 312 xvi i 4.64 Average PSNR for Nj = 20, Nd = 120, Nw = 1, B^ax = 64k, for SLWIN and SLWIN2 in the IS-95B downlink case, for the Foreman bitstream 312 4.65 Average PSNR for NT = 132, Nd = 60, B^ax = 64k and different A V s , for SLWIN2 in the IS-95B downlink case, for the Foreman bitstream 313 4.66 Average PSNR for A 7 = 132, Nd = 60, B^ax = 64k and different Nw% for SLWIN in the IS-95B downlink case, for the Foreman bitstream. 313 xvi i i LIST OF TABLES 2.1 Comparison in terms of B E R of methods for generating Nakagami random variables with m > 1.0: m — 2.0 65 2.2 Comparison in terms of B E R of methods for generating Nakagami random variables with m < 1.0: m = 0.6 66 2.3 Parameters associated with the SER's of M-a ry modulations [167], [171]. 94 4.1 Frame structure of the IS-95B forward link [14] 244 4.2 Parameters of the IS-95B forward link interleaver [14] 246 4.3 Radio configurations (RC) of the IS-2000 forward link [14] 251 4.4 Frame structure of RC3 of the IS-2000 forward link [14] 254 4.5 Frame structure of RC4 of the IS-2000 forward link [14] 254 4.6 Frame structure of RC6 of the IS-2000 forward link [14] 255 4.7 Frame structure of RC7 of the IS-2000 forward link [14] 255 4.8 Parameter n of the IS-2000 turbo interleaver [14] 256 4.9 Table lookup for the IS-2000 turbo interleaver [14] 257 4.10 Puncturing patterns for the IS-2000 turbo encoder [14] 258 4.11 Radio Configurations of the IS-2000 reverse link [14] 260 4.12 Frame structure of RC3 of the IS-2000 reverse link [14] 262 4.13 Frame structure of RC5 of the IS-2000 reverse link [14] 262 4.14 Parameters of the IS-2000 reverse link interleaver [14] 263 xix LIST OF ABBREVIATIONS A F Activi ty Factor A F D Average Fade Duration A L Adaptation Layer A R Autoregressive A R Q Automatic Repeat Request A T M Asynchronous Transfer Mode A W G N Additive White Gaussian Noise BS Base Station B E R Bi t Error Rate B F S K Binary Frequency-Shift Keying B H Basic Header bps Bits per second B P S K Binary Phase-Shift Keying cdf Cumulative Distribution Function C D M A Code-Division Multiple-Access C F Control Field CIF Common Intermediate Format C L P C Closed-Loop Power Control C R C Cyclic Redundancy Check D C C H Dedicated Control Channel D E - B P S K Differentially Encoded Binary Phase-Shift Keying D E - M - P S K Differentially Encoded M-ary Phase-Shift Keying D E - Q P S K Differentially Encoded Quadrature Phase-Shift Keying DS Direct-Sequence D S P Digital Signal Processor xx E A F Envelope Autocorrelation Function E G C Equal-Gain Combining EI Error Indicator E P M Equal Probability Method F C H Fundamental Channel F E C Forward Error Correction F E R Frame Error Rate F I R Finite Impulse Response FQI Frame Quality Indicator F S M C Finite-State Markov Channel G A Gaussian Approximation GHz GigaHertz G O B Group Of Blocks H E C Header Error Control Hz Hertz I D F T Inverse Discrete Fourier Transform I F F T Inverse Fast Fourier Transform IP Internet Protocol IS Interim Standard ISI Intersymbol Interference ISO International Standards Organization I T U International Telecommunications Union I U A Inverse Update Algorithm IWA Interworking Adapter kbps Kilobits per second kcps Kilochips per second ksps Kilosymbols per second k m / h Kilometers per hour L A C Link Access Control L C N Logical Channel Number L C R Level Crossing Rate xxi LOS Line-of-Sight M A C Medium Access Control M A P Maximum A Posteriori M B Macroblock M C Multiplex Code Mcps Megachips per second M C U Multipoint Control Unit M P Multiplex Packet M P E G Moving Picture Experts Group M P L Multiplex Payload Length M - P S K M-ary Phase-Shift Keying M - Q A M M-ary Quadrature Amplitude Modulation M R C Maximal-Ratio Combining MS Mobile Station M S C Mobile Switching Center M U X Multiplex M V Motion Vector N A K Negative Acknowledgment N C - M - F S K Noncoherent M-ary Frequency-Shift Keying O H Optional Header O - Q P S K Offset-Quadrature Phase-Shift Keying O T D Orthogonal Transmit Diversity O V S F Orthogonal Variable Spreading Factor P A R Peak-to-Average Ratio P C F Packet Control Function P C H Pilot Channel P D N Packet Data Network P D S N Packet Data Serving Node P D U Protocol Data Unit pdf Probability Density Function P M Packet Marker xxi i P N Pseudonoise P P P Point-to-Point Protocol P S D Power Spectral Density P S N R Peak Signal-to-Noise Ratio P S T N Public Switched Telephone Network Q A M Quadrature Amplitude Modulation Q C I F Quarter Common Intermediate Format QoS Quality-of-Service Q P S K Quadrature Phase-Shift Keying R A N Radio Access Network R C Radio Configuration R C P C Rate Compatible Punctured Convolutional R T P Real-Time Protocol SC Selection Combining S C H Supplemental Channel S D U Service Data Unit S E R Symbol Error Rate SF Synchronization Flag SIC Successive Interference Cancellation SLWIN Sliding Window Algorithm SLWIN2 Sliding Window Algorithm 2 S N R Signal-to-Noise Ratio SR Source Rate / Spreading Rate SRC Source Rate Control S-TRC Source and Transmission Rate Control T B Transmission Buffer / Tail Bits T R Transmission Rate T R C Transmission Rate Control T D M A Time-Division Multiple-Access T M N Test Model Near-term V B R Variable Bi t Rate xxi i i V C E G Video Coding Experts Group V L B R • Very Low B i t Rate V L C Variable Length Coding V P Video Packet V S G Variable Spreading Gain W - C D M A Wideband C D M A xxiv A C K N O W L E D G M E N T S I acknowledge first of all my parents, brother and sister, and grandparents Ines and Vittorio Zamuner for their unconditional support. I am especially thankful to Sonia for her patience and commitment through all of my graduate studies and beyond. It has been a great pleasure working with my supervisor Takis Mathiopoulos, who never runs short on words of encouragement, who has been constantly supportive of all my academic endeavours, both inside and outside of U B C , and who has run great lengths to help me in achieving my objectives. I am hopeful that we wi l l continue collaborating in the near and distant future. I am grateful to several professors for reviewing my work and providing constructive suggestions, in particular: my Committee Members Cyr i l Leung and Victor Leung, the Departmental Examiners Robert Donaldson and Robert Schober, the University Exam-iners Vikram Krishnamurthy and Son Vuong (Department of Computer Science), and the External Examiner Wi to ld A . Krzymien from the Department of Electrical and Computer Engineering Department at the University of Alberta. I acknowledge many current and former colleagues within the Communications Group, including Farshid Agharebparast, Lawrence Chen, Vaibhav Dinesh, Sarina Feng, Mah-noosh Mobasseri, Hong Nie, Joo-Han Song, Hansen Wang, Emre Yavuz, Fei Y u , and several others, for help and good times inside and outside the lab. I appreciate the financial support contributed by the Natural Sciences and Engineering Research Council of Canada, the Science Council of Brit ish Columbia, the Communica-tions Research Centre (CRC) , and the Advanced Systems Institute of Brit ish Columbia. I am particularly grateful to John Belrose, John Lodge, Michael Sablatash and others from the C R C , and Kar im Lakhani from Advanced Interactive Canada Inc. I am also grateful to Salvatore Morgera, Valentine Aalo and others from Florida Atlantic University, for allowing me a smooth transition from U B C . xxv C H A P T E R 1 INTRODUCTION 1.1 Introduction to Wireless Video Communications Wireless mobile communications can truly be described as multimedia when they include a video component in addition to voice and data. The rapid advances in digital video compression [1], [2], matched with techniques for increasing the capacity of wireless systems [3], are likely within the next decade to permit the widespread delivery of real-time video over mobile cellular networks, which has in fact recently begun in certain countries. While digital video codecs were originally designed for high-bandwidth, high-quality wirelines, the latest set of codecs were conceived with low bit-rate and error-prone channels in mind. The H.263 [4] and M P E G - 4 [5] standards build on techniques which produce very low bit-rates (in the order of tens of kbps), and have a number of modes allowing them to cope with errors. These characteristics make them more suitable for next-generation wireless networks. However, as time wil l pass by, it is expected that higher-quality video, necessitating higher data rates and fewer errors, wi l l become in demand. Thus it wi l l not suffice to simply transmit the video stream the same way voice or data are currently delivered. Techniques which make use of the capabilities of both the video codecs and the transceivers are to be developed to maximize the quality of the received video. Early papers on wireless video can be traced back to a decade ago [6], [7], however the bulk of the research was conducted in the last six years or so. Contributions are made not only by academic institutions, but also by research and development firms. For example, Texas Instruments is developing digital signal processors (DSP's) customized for wireless M P E G - 4 applications [8], and a number of large or small firms have patented technologies relating to wireless video [9]. 1 In this thesis we are considering the transmission of video over current and next-generation direct-sequence code-division multiple-access ( D S / C D M A ) cellular networks [10], [11], [12], [13], based on standards such as IS-95B and IS-2000, which are part of the cdma2000 family [14]. IS-95B systems have already been deployed in North America, and currently support voice and data, while IS-2000 systems are expected to be deployed on a large scale in the near future. Such systems experience wideband fading (i.e. mul-tipath fading where several components are resolved at the receiver) and multiple-access interference from other users: these are the two main factors which make the study of video transmission over wireless C D M A networks much different than that of wireline video systems. The wireless link is indeed the bottleneck in future global communica-tion networks, since the fading and interference limit the transmission rates over large ranges. Our goal is to develop rate control methods to allow the reliable transmission of a variable bit rate video stream by a C D M A system with rate and power constraints, for delay-sensitive applications. We wil l approach the problem from a physical/link layer perspective. Hence, before tackling the rate control problem, a detailed understanding of the impairements caused by wideband fading and multiple-access interference is nec-essary: we wil l thus consider the analysis and simulation of variable rate IS-95B/IS-2000 transmission over wideband channels. In this introduction, we begin by giving a brief general review of wideband fad-ing channels and D S / C D M A systems in Sections 1.2 and 1.3, respectively. Section 1.4 presents some important characteristics of video applications and low bit-rate video cod-ing, together with rate control methods needed in bandwidth-constrained environments. Then, in Section 1.5 we summarize our contributions in the areas of fading channel theory, D S / C D M A systems, and wireless video communications, and outline the organi-zation of the thesis. Finally, in Section 1.6 we present for reference purposes a list of the publications which have resulted from the research reported in this thesis. 1.2 Wideband Fading Channels In wireless mobile communications, the transmitted signal is subject to path loss, large-scale fading (shadowing) and small-scale fading ([15], Chaps. 1-2, [16], Chaps. 3-2 4). While accurate power control can mitigate the effects of the first two phenomena, which usually vary slowly in time, the small-scale fading can vary too rapidly in time to be accurately tracked by a power control mechanism, particularly for mobiles moving at high speed. Hence other mechanisms are necessary to counter the effect of the fading: diversity techniques ([15], Chaps. 5-6, [16], Chap. 6), which strive to capture the maximum energy from a transmitted signal, are by far the most popular due to their relative simplicity. The type of diversity circuit which should be used for maximum efficiency depends on the particular nature of the fading which is expected, as discussed below. Small-scale fading is due to spreading (distortion) of the transmitted signal in both the time and frequency domains, or either one ([16], Chap. 4). Temporal distortion is due to the signal traveling on multiple propagation paths (multipath fading), due to reflections from surrounding terrain features or objects, which causes scaled replicas of the signal to arrive at the receiver at different time instants. The difference between the arrival times of the first and last components is called the delay spread. The inverse of the delay spread is proportional to the coherence bandwidth, which is the range of frequencies over which the channel passes all spectral components with approximately equal gain and linear phase. Frequency distortion is attribuable to the random frequency modulation phenomenon caused by the motion of a mobile user, or by the motion of objects moving near a fixed user: the energy of the received signal is contained in a continuum of frequencies alongside the carrier frequency. The difference between the maximum and minimum frequencies (on one side of the carrier frequency) where the signal energy is non-zero is called the Doppler spread, or alternatively the Doppler shift. The inverse of the Doppler shift is proportional to the coherence time, which is the time duration over which the channel impulse response is mostly invariant. Among other classifications, the small-scale fading that affects a transmitted signal can be roughly categorized as either frequency-nonselective (or flat) or frequency-selective, although there is no clear-cut demarcation between both attributes, and rules of thumb are used instead [16]. Flat fading, also termed narrowband fading, occurs when the bandwidth of the signal is smaller than the coherence bandwidth of the channel, and the delay spread of the channel is smaller than the symbol period of the signal [17], [16]. The multipath components combine constructively and destructively at the receiver, which cannot separate them in the time domain. Frequency-selective fading, 3 also termed wideband fading, occurs when the bandwidth of the signal is larger than the coherence bandwidth of the channel, and the delay spread of the channel is larger than the symbol period of the signal [17], [16]. In this case, the receiver can resolve some of the multipath components. The small-scale fading can also be categorized as either fast or slow. Fast fading corresponds to a coherence time smaller than the symbol period (hence a high Doppler spread) and channel variations faster than baseband signal variations. Slow fading corresponds to a coherence time larger than the symbol period (hence a low Doppler spread) and channel variations slower than baseband signal variations. First and second-generation (1G and 2G) cellular systems with low data rates typ-ically had to deal with narrowband fading (except for spread-spectrum IS-95 systems). In this case, antenna spatial diversity techniques were useful in collecting energy from multipath components received at different spatial locations, at about the same times. Third-generation (3G) cellular systems based on spread-spectrum technology (so as their ancestors such as IS-95, and other high data-rate systems) typically experience wideband fading instead. Spatial antenna diversity techniques can still be used, but in addition these systems can now make use of temporal diversity: Rake receivers [18] with resolu-tions as low as one chip interval can be used to capture multipath components which are closely-spaced in the time domain. High-rate systems wil l typically experience slow fad-ing, especially if the mobile speed is low: this thesis wil l hence essentially be concerned with frequency-selective slow fading channels. The use of diversity is crucial in 3G systems in order to obtain a higher signal-to-noise ratio at the receiver end, and satisfy the higher quality-of-service required by certain new applications such as video telephony. In fact, the IS-2000 standard specifies the support of transmit diversity techniques [19], such as orthogonal transmit diversity (OTD) . Hence our simulation models for video communications over 2.5G (IS-95B) and 3G (IS-2000) D S / C D M A systems wil l include diversity techniques. While extensive research has been carried out for decades in the field of diversity techniques [20], [21], [22], there are still many open issues, and there wil l continue to be as long as new methods are proposed and more accurate characterizations in different fading environments are required. Part of this thesis wil l tackle several issues related to the statistics, simulation and effect on communication performance of wideband Nakagami fading channels [23]. The Nakagami 4 distribution has been chosen due to its generality (it reverts to the Rayleigh distribution as a special case), and its reported better fit to urban and suburban fading conditions [24]. In particular, the effect of correlation between diversity branches wil l be examined. Some of the results obtained for single-user diversity systems wil l prove useful in our work on multiuser systems. 1.3 D S / C D M A Cellular Networks IS-95 and IS-2000 D S / C D M A cellular systems use spread-spectrum modulation, which uses a bandwidth much wider than that of the information signal, due to spreading by pseudorandom sequences with high chip rates ([25], [26], Chap. 13). Spread-spectrum modulation has several advantages over narrowband modulation, in particular enhanced interference rejection capability, resistance to multipath fading, and increased communi-cation privacy ([16], Chap. 5). In a D S / C D M A system multiple users transmit simulta-neously in the same bandwidth and time slots: the user separation is realized in the code domain, i.e. each user is assigned a particular code ([26], Chap. 15). However, in most conditions this code domain separation is not perfect, and hence users wil l experience multiple-access interference from users in the same cell and other cells. Since the seminal work of [27] on how to approximately determine the error rate of a user in a D S / C D M A system, a large amount of studies have been carried out to either refine or improve the results of [27] (e.g. [28], [29], [30], [31], [32]), or to extend them to deal with fading channels (e.g. [33],' [34], [35], [36], [37], [38], [39], [40]) and a plethora of other conditions. In order to support high-data rate transmissions, e.g. for video telephony, multicode and multirate transmission have been recently introduced. These wil l be described in detail in Section 3.1. Part of this thesis (Chapter 3) wil l tackle the problem of accurately determining the error rate of multicode/multirate systems in wideband fading channels, for the main system configurations encountered in IS-95 and IS-2000 systems. 5 1.4 Video Communications: Characteristics and Methods 1.4.1 Requirements of Video Transmission From a communication viewpoint, video signals have characteristics which make their treatment different from that of voice and data signals. These characteristics are pre-sented below. 1.4.1.1 Bandwidth Requirements H.263 video (and its later versions such as H.263+ [41], H.263++) necessitates higher rates than voice (typically around 64 kbps for low-motion video, compared to 9.6 kbps for compressed speech in IS-95). However, the rates specified are usually average rates, and much higher peak rates need to be dealt with. Indeed, due to the coding mechanisms involved (c.f. Section 4.2.2), H.263 produces a variable bit rate ( V B R ) video stream. This variability can be short-term (at the individual frame level), middle-term (at the scene level) or long-term (between scenes). Short-term variations are due mainly to variable-length coding ( V L C ) and the possible different modes used to encode blocks of video data. The type of frames used (I versus P or B , c.f. Section 4.2.2) is essentially responsible for middle-term variations. Long-term variations are caused by changes in scene contents: motion (inside the scene, or by the camera) wil l significantly increase the required bit rate compared to a quasi-still scene. While variations on a short time scale could be handled by smoothing while keeping a low delay, middle and long-term variations wil l in most cases require changes in the bandwidth allocated (unless an arbitrary long delay is permitted). 1.4.1.2 Delay Requirements Different applications can tolerate different amounts of end-to-end delay. Real-time interactive video telephony cannot withstand more than a few hundred msec of delay (typically 150 to 400 msec one-way [42]), in order to avoid annoying periods of stalling in the conversation. Other real-time but non-conversational applications, such as streaming video, can tolerate a higher initial delay: an end-user logging on to a live one-way video-conference would be willing to wait a few (tens of) seconds of startup time in order to 6 have a higher-quality stream. The downloading of pre-compressed stored video can afford a much longer delay, in order to make sure that the connection isn't broken up due to in-sufficient resources. Hence, the amount of delay that can be supported wil l dictate in part the bandwidth requirements discussed above: bursty live interactive video wil l require higher peak rates, and a continous allocation of bandwidth, while stored video (whose treatment is closer to that of data) can be handled with discontinuous transmission and lower, smoother rates. 1.4.1.3 Error Requirements Like voice, video can tolerate a certain level of degradation (such that the user hardly notices it), however too many errors can have a catastrophic impact ([43], Chap. 8). Moreover, the video stream is unevenly sensitive to errors: transmission errors in segment headers can cause the loss of a whole picture, while those affecting texture coefficients and motion vectors wi l l usually have a limited and containable impact ([43], Chap. 6, [44]). This has lead to research on robust video coding, which resulted for example in the error-resilient modes used in H.263+ [45], [46], [47]. 1.4.2 Impairments to Video Transmission The V B R nature of compressed video can result in frame losses due to transmission buffer overflow or receiver buffer underflow ([43], Chap. 9). Transmission buffer overflow occurs when the rate at which the transmitter processes frames is consistently lower than the rate at which these frames are delivered to it from the source. This same situation can lead to receiver buffer underflow, that is when the video decoder at the receiving end processes frames at a rate higher than the rate at which it receives them. The techniques which allow increased video compression also lead to high vulnerability of the resulting stream to random, burst and erasure errors. The following impairments can be caused by such transmission errors: • Loss of video frame or segment. Errors in segment headers can cause the whole segment to be discarded, depending on the bits which are affected. 7 • Loss of synchronization. Because of variable length coding, when the decoder en-counters an error in a codeword, it doesn't know where to restart the decoding, and can possibly lose synchronization. • Temporal and spatial error propagation. Due to motion-compensated predictive coding, errors occuring in the current frames can carry over to the following frames, until the sequence is refreshed by an I-frame (or intra-coded macroblocks), or the effect of the errors is neutralized by a leaky prediction mechanism. The distor-tion can also propagate spatially in future frames, especially when high motion is present. 1.4.3 Rate Control for V B R Video In a V B R video communication system, the source encoder outputs the generated bitstream to a transmission buffer (TB) ([43], Chap. 9). The latter delivers its contents to the transmitter, according to a certain algorithm. The transmitter then performs on the bitstream the necessary operations specified by the corresponding standard (such as framing, error-control coding, interleaving and demultiplexing), and modulates it onto a signal suitable for transmission. The source rate (SR) is constrained by the maximal transmission rate (TR) that can be supported. The maximal T R is in turn determined by the transmitter specifications, i.e. the maximum rate supported by the transceiver, and the maximal bandwith allocated at a given time to the transmitter in a wireless network, which is influenced by the network load. The channel conditions can further impose time-varying limitations on the T R . Hence a certain control must be exercised on the SR seen by the transmitter or the T R made available to the source, in order to allow the V B R video to be transported adequately. Three scenarios can be envisioned for the rate-controlled transport of V B R video for such a communication system. In the first scenario, there is no feedback from the transmitter to the source. The source has no information about the maximum T R supported at any time. Hence it cannot adapt its rate according to constraints posed by the transmitter and network. The transmitter must thus handle by itself the variable bit rate produced by the source: 8 this can usually be done either by transcoding [48], smoothing [49], rate adaptation, or by a combination of all of these. This situation arises, for example, in the case of pre-recorded pre-compressed video. It is also common to certain systems supporting live or streaming video in which no feedback path is present from the transmitter to the source, for complexity or system design reasons. We wil l call this scenario the transmission rate control (TRC) case, to denote the fact that the rate control is carried out by the transmitter. Note that we are dealing here with the control of the actual transmission rate. In the second scenario, there is an interaction between the source and the transmitter. The source receives feedback from the transmitter concerning the maximal T R that can be supported (explicit rate feedback [50]), or elements that determine this rate (non-explicit rate feedback). These feedback messages can be periodic or occur upon certain events, e.g. when there is a change in the maximum T R . If the source is performing live encoding, it can therefore adjust its output rate in order to match the T R . The transmitter does not need to adapt its T R to the V B R source which is fed to it, even though it can still perform rate adaptation related to varying channel or network conditions. We wil l call this scenario the source rate control (SRC) case, to emphasize the fact that the rate control is performed by the source encoder. We are thus concerned here with the control of the source output rate. The third possibility is to combine the feedback capability of the second scenario with the transmitter rate control of the first scenario. In this case both source and transmitter strive to optimize the video quality under given rate constraints. For example, if after being rate-controlled by the source encoder, the video stream has a rate which is still too high, the transmitter can perform T R C in order to attempt the adequate delivery of the stream. Since source rate control and transmission rate control can be jointly performed, we wil l call this scenario the source and transmission rate control (S-TRC) case. It generalizes the previous two cases. In all of the previous cases, the receiver can play a role in the video delivery process. A certain number of video frames are stored in the receiver buffer before video decoding starts. This delay between the reception of frames and their playout is called the look-ahead interval [51]: it alleviates the delay constraints placed on the transmission of 9 video, and hence gives the source and/or transmitter more flexibility in performing rate adaptation. In the following we present an overview of the most common rate control approaches used in each of these scenarios. 1.4.3.1 Source Rate Control (SRC) Indicators for Rate Adaptation There are three ways that the source can be informed that a change in its output rate is desirable: by direct feedback from the receiver (through a return path), by examination of the buffer(s) present at the transmitter, or by a combination of the latter two approaches. Different indicators can be used for each approach. In the first case, indications about the current network or channel conditions are transmitted back to the source encoder via a control channel (or piggybacked on a traffic channel). Some examples of indicators used in previous studies are: • The channel state information which is determined at the receiver. In [52], the channel is determined to be in one of a fixed number of states. The index of the state serves as an indicator of the channel quality. Estimates of the signal-to-noise-ratio (SNR) at the receiver or of the fading coefficient(s) can also be used. • The number of negative acknowledgements (NAK's ) , in a system using automatic-' repeat request (ARQ) , as in [53]. Some systems cannot support a return channel needed for receiver feedback: this could be the case for example for a large multicast configuration. Moreover, certain systems would prefer an approach which is not feedback-based for one or more of the following reasons: • The need for feedback requires additional signalling on the reverse channel: this increases bandwidth requirements and complexity. • The feedback information is returned to the transmitter with a certain delay. If this delay is too large compared to the rapidity of changes in network conditions, the information can be outdated and lead to sub-optimal or incorrect source adaptation. 10 This brings us to the second case, where the source adapts its rate only from information available at the transmitter. The following indicators were used previously: • The transmitter buffer fullness. In [54], if the number of video bits waiting to be transmitted gets too large, e.g. because of a mismatch between SR and T R , or bad channel conditions, the source decreases its output rate. • The A R Q buffer fullness. In [55], a separate buffer is maintained to store the frames which haven't been positively acknowledged yet. The status of this buffer determines the source's rate adaptation. The transmitter and A R Q buffers could also be combined into a single larger buffer, and therefore the two previous indicators would merge. Some proposals use a combination of feedback and transmitter-based adaptations, which constitutes the third case of source rate adaptation. More reliable results are expected, since indicators from independent sources are used in deciding the type of rate adaptation to adopt. This translates of course in increased complexity, due to the additional resources required and the need for a more advanced decision process which encompasses all the information sources. In [56], the proposed algorithm takes into account both the channel state information and the A R Q buffer fullness. Note that rate adaptation based on transmitter information only can also be subject to the problem of outdated information, as in the feedback-based case. Indeed, by the time buffer congestion builds up due to bad network conditions or S R / T R mismatch, the latter can improve: thus the decrease in source rate which is requested wil l not be useful anymore. A n alternative to the observation of current (or recent) conditions would be to try to predict the future network conditions or source rate. Numerous papers have been published on the topic of source rate prediction, for various video coders [57], [58], [59]. They most often base their prediction scheme on a statistical or dynamic model devel-oped for the source of concern. However, the goal of such predictions was in most cases to obtain traffic management algorithms needed for middle/long-term traffic variations. A new topic of research would be on how to use source prediction algorithms to regulate the source or transmission rates on a short-time basis. Fading channel prediction is a 11 fairly recent research area [60], which could also be exploited for S R / T R adaptation. Source Rate Adaptation Techniques The following mechanisms are available to a video encoder in order to vary its output rate: • Adjustment of quantization step size. By chosing a larger quantization step size, the video sequence wil l be more coarsely encoded and thus use up less bits, at the ex-pense of a lower visual quality. The step size can be selected from a pre-determined set, as in H.263. Some authors opt for a simple "embedded quantization" scheme [55], where the least significant bit is abandonned in the event of a request for rate decrease. • Selection of coding modes. Since there are various coding options available for encoders such as H.263+ and M P E G - 4 , these options can be varied during the encoding process to maintain the target source rate. For example, the interval between intra-coded frames is varied adaptively in [61]. The number of intra-coded macroblocks (MB's) is adapted to the maximal T R in [54]. • Frame skipping. When the buffer occupancy reaches a certain level, certain frames belonging to the original raw (uncoded) video sequence are skipped in the encoding process, in order to decrease the total bitrate. While the number of transmitted frames is indeed lower, the number of bits carried by the latter is increased: the differentially-encoded motion vectors and/or texture coefficients wi l l be larger be-cause of an increase in the prediction range. The video sequence wil l also appear more jerky if too much frame skipping is used. The rate control algorithm used in the Test Model Near-term version 8 (TMN8) of H.263 uses frame skipping to com-plement quantization step size control. A modification of this algorithm presented in [56], and the framework of [53] also use frame skipping for wireless channels. • Layer dropping. In scalable video coders, enhancement layers can be dropped dur-ing periods of buffer congestion. This can be seen as a form of frame skipping, since all the frames belonging to a certain enhancement layer are dropped. Such a mechanism was exploited in [55]. 12 The rate control algorithm used in T M N 8 of H.263 [62], [63] performs in a first step frame-layer rate control (the total number of bits per frame is determined), then follows on with macro-block-layer rate control (the bit budget is split between the MB' s ) . Several other authors have opted for such an approach [54], [56]. 1.4.3.2 Transmission Rate Control ( T R C ) Very different approaches can be used and combined to assure the timely transmis-sion of the video stream delivered by the source encoder. They are individually reviewed below. Video Transcoding Video transcoding consists in recoding the video stream produced by the original encoder, in order to make it suitable for transmission or adapt it to the capabilities of the user. A clear drawback is the extra complexity required and the associated delay. It was nonetheless privileged in the following works: • In [48], the authors use transcoding in order to adapt the transmitted video sequence to the capabilities of the end-user and the network conditions. The transcoding is carried out at a proxy, which serves as an interface between the video server and the air interface of the network. • In [64], transcoding is also performed by a proxy to match the bit rate to the wireless link capacity. • In [65], transcoding for error-resilient purposes was reported. Smoothing To avoid dealing with the source coding mechanisms as in source rate control or transcod-ing, one can perform smoothing of the input source traffic, which consists in scheduling the future transmission rates in a manner that the rate variations are minimized, so that the source appears close to a constant bit rate one. To this end, a certain amount of buffering must be introduced either at the transmitter or the receiver. This wi l l how-ever inevitably result in a certain delay in the decoding. Methods to achieve smoothing 13 depend on whether the video frames are available progressively to the transmitter, as in real-time interactive or streaming video, or they are all available beforehand, as in stored video. In the first case (the online case), the transmitter can introduce a delay of a few frames in order to obtain a certain margin for smoothing. It has to determine the best sequence of transmission rates (or schedule) in a progressive way, either on a frame-by-frame basis, or based on a sliding window of frames. In the second case (the offline case), since the transmitter knows all of the frame lengths in advance, it can determine the optimal schedule before it initiates transmission, and then keeps it frozen. Several studies have been carried out for both offline and online smoothing algorithms, but mainly with wireline networks in mind [66], [49], [67], [68], [69], [51]. In Section 4.3.1, we give more details about some of the algorithms, and explain what are the new con-straints and their implications when dealing with power and bandwidth-limited cellular C D M A systems. Physical and Link-Layer Techniques for Rate Adaptation For time-division multiple-access ( T D M A ) systems, many studies have been carried out on variable modulation for video [70], [71], [72]. Comprehensive reviews are available in [73] and [74]. Essentially, the transmitter chooses from a set of predefined Quadrature Amplitude Modulation ( Q A M ) constellations the modulation format which best matches the channel conditions. However, most studies seem to overlook the impact of the V B R source on the adaptation schemes, or consider transmission rates which are well above those required for low-bit rate video. -For C D M A systems, as wil l be detailed in Section 3.1, multicode and multirate trans-mission are available to the transmitter in order to adapt its output rate. Some studies have considered multicode transmission for video [75], [76], [77], however the results are usually preliminary. We wil l show how to combine smoothing and link-layer rate adap-tation techniques in Section 4.3. 14 1.5 Thesis Contributions and Organization 1.5.1 Statistics and Simulation of Wideband Fading Channels Since our work targets transmission over wireless wideband channels, it is necessary to be able to understand the properties of such channels, so as to efficiently simulate them, and their effect on the performance of communication systems. In Chapter 2, we make several novel contributions to the theory of wideband fading channels. The Nakagami distribution, which is a generalization of the Rayleigh distribution (c.f. Section 2.2), wi l l be used to model the envelope of the fading process. A first objective consists in designing an efficient Nakagami fading simulator. To do so, we start by deriving several analytical expressions for the most common second-order statistics of Nakagami fading channels. In particular, in Section 2.3 we obtain the level crossing rates, average fade durations, envelope correlation and baseband spectrum of channels with diversity combining, which provide us with some insights into the behavior of such channels. The obtained theoretical level crossing rates are then used to construct in Section 2.4.2 a finite-state Markov channel simulator for slow fading, whose advantages and limitations are evaluated quantitatively. In Section 2.4.1 we also present a review of previous methods used to generate spatially or temporally correlated Nakagami ran-dom variables. We give some tricks on how to improve some of the spatially correlated simulators, and discuss the limitations of current temporally correlated simulators. A second objective is to obtain generic error probability formulas for different modu-lation schemes in correlated Nakagami fading channels, for both maximal-ratio or equal-gain combining. In Section 2.5, we consider the special but widespread case of two diver-sity branches, and derive in a unified manner several novel expressions in single-integral form. These expressions simplify or complement previous work on diversity combining in correlated channels. In some special cases, closed-form solutions (as infinite series summations) can be found. The results of Chapter 2 are of very general scope, and as such this chapter could stand on its own. In fact several of the analytical results and simulation methods presented here wil l be useful in Chapter 3, when we consider the performance of multiple-access systems in wideband fading channels. 15 1.5.2 Analysis of Multicode and Multirate D S / C D M A Systems in Wideband Fading Channels Since we are interested in V B R video transmission over multicode/multirate C D M A systems, it is necessary to be able to simulate such systems, by deriving the decision metrics which are used at the receiver, and to predict their performance in fading chan-nels. Chapter 3 presents a detailed and accurate analysis of the multicode/multirate configurations used in the physical layers of IS-95B and IS-2000 systems, when diversity receivers are used to improve performance in fading channels. The methodology takes into account in a precise manner the effect of the fading (which was overlooked or mis-understood in previous studies), and leads to theoretical bit error rates very close to the actual (simulated) ones. The analysis is carried out separately for: • Reverse link multicode D S / C D M A with M-ary orthogonal noncoherent modulation and equal-gain combining (as used in IS-95B), in Section 3.2; • Reverse link multicode D S / C D M A with binary coherent modulation and maximal-• ratio/equal-gain combining, and complex spreading sequences (as used in IS-2000), in Section 3.3; • Forward link multicode D S / C D M A with quaternary coherent modulation and maximal-ratio/equal-gain combining, and real spreading sequences (as used in IS-95B), in Section 3.4; • Forward link multicode D S / C D M A with quaternary coherent modulation and maximal-ratio/equal-gain combining, and complex spreading sequences (as used in IS-2000), in Section 3.5. The proposed methodology allows us to use any fading distribution: examples wil l be given for Rayleigh, Nakagami, Rician and lognormal fading. Moreover, correlated diver-sity branches (which represent a more realistic fading scenario) are easily integrated in the analysis. Some extensions to systems using advanced techniques are also presented in Section 3.2. 16 1.5.3 V B R Video Transmission for Multicode and Multirate D S / C D M A Systems in Wideband Fading Channels As stated in Section 1.1, we are interested in providing algorithms to support the efficient and delay-conscious transmission of variable bit rate sources over multicode and multirate D S / C D M A systems. Chapter 4 takes the IS-95B and IS-2000 cellular systems as case examples to support our research: it is stressed that our work is intended to be more general in scope, and not specific to particular standards, however to obtain more realistic results we have abided by many of the parameters specified in current and future D S / C D M A physical layer standards. In this chapter, we start by describing in Section 4.2 a software implementation of an end-to-end video communication system, which comprises the video coder, the pack-etizer/multiplexer, and the physical layer components of the reverse link (mobile user to base station, or uplink) and forward link (base station to mobile user, or downlink) of IS-95B and IS-2000 systems (which include error-control encoders, interleaver, spreading sequences and modulator). The system is subject to multipath fading and multiple-access interference, the latter being obtained by simulating all of the interfering users or codes. This platform wil l be used in the performance evaluations of the rest of the chapter. We then propose in Section 4.3 a smoothing algorithm optimized for multicode and multirate D S / C D M A systems, and compare it to a previous benchmark algorithm which we adapted to deal with the system at hand. End-to-end simulations are carried out using the previously described software platform, for both the uplink and downlink of IS-95B and IS-2000 systems, and the performance of both smoothing algorithms is assessed in terms of the decoded video quality. The effects of the startup buffering delay and of the length of the sliding window are investigated in detail. 17 1.6 Publications Based on the research reported in this thesis, the following publications have resulted: Refereed Papers in Journals J l . C . D. Iskander and P. T. Mathiopoulos, "Analytical level-crossing rates and average fade durations for diversity techniques in Nakagami fading channels", IEEE Trans, on Communications, vol. 50, no. 8, pp. 1301-1309, Aug. 2002. J2. C. D. Iskander and P. T. Mathiopoulos, "Performance of M - Q A M with coherent equal-gain combining in correlated Nakagami-m fading", IEE Electronics Letters, vol. 39, no. 1, pp. 141-142, 9th Jan. 2003. J3. C. D. Iskander and P. T. Mathiopoulos, "Performance of dual-branch coherent equal-gain combining in correlated Nakagami-m fading", IEE Electronics Letters, vol. 39, no. 1, pp. 1152-1154, 24th July 2003. J4. C. D. Iskander and P. T. Mathiopoulos, "Fast simulation of diversity Nakagami channels using finite-state Markov chains", to appear in IEEE Trans, on Broad-casting, Sept. 2003 (8 journal pages). J5. C. D. Iskander and P. T. Mathiopoulos, "Performance of multicode D S / C D M A with noncoherent M-ary orthogonal modulation in multipath fading channels", accepted for publication in IEEE Trans, on Wireless Communications, Dec. 2002 (14 journal pages). J6. C. D. Iskander and P. T. Mathiopoulos, "Online smoothing of H.263 video for the cdma2000 and IS-95B uplinks", accepted for publication in IEEE Trans, on Multimedia, Dec. 2002 (12 journal pages). Refereed Papers in Conference Proceedings C l . C. D. Iskander and P. T. Mathiopoulos, "Reverse link analysis and performance evaluation of H.263 video transmission for cellular D S / C D M A systems in frequency-selective lognormal-Nakagami fading", Proc. IEEE 53rd VTC Spring, vol. 3, pp. 2041-2045, May 6-9, 2001. 18 C2. C. D. Iskander and P. T. Mathiopoulos, "Finite-state Markov modeling of diversity Nakagami channels", Proc. Seventh Canadian Workshop on Info. Theory, pp. 76-79, June 3-6, 2001. C3. C. D. Iskander and P. T. Mathiopoulos, "Efficient H.263 video communication for 3G cdma2000 systems in frequency-selective Nakagami fading", Proc. 13th Int. Conf. on Wireless Comm. (Wireless), vol. 1, pp. 10-17, July 9-11, 2001. C4. C. D. Iskander and P. T. Mathiopoulos, "Analytical level-crossing rates and average fade durations for diversity techniques in Nakagami fading channels", Proc. 55th IEEE VTC Spring, vol. 4, pp. 1598-1602, May 6-9, 2002. C5. C. D. Iskander and P. T. Mathiopoulos, "Performance of multicode D S / C D M A with noncoherent M-ary orthogonal modulation in multipath fading channels", Proc. IEEE 55th VTC Spring, vol. 3, pp. 1210-1214, May 6-9, 2002. C6. C. D. Iskander and P. T. Mathiopoulos, "Rate-adaptive transmission of H.263 video for multicode D S / C D M A cellular systems in multipath fading", Proc. IEEE 55th VTC Spring, vol. 1, pp. 473-477, May 6-9, 2002. C7. C. D. Iskander and P. T. Mathiopoulos, " A joint smoothing and rate adaptation framework for the transmission of V B R H.263 video on the cdma2000 uplink", Proc. IEEE/IEE ICT, vol. 2, pp. 452-457, June 2002. C8. C. D. Iskander and P. T. Mathiopoulos, "Performance of multicode D S / C D M A with coherent detection and complex spreading sequences in multipath fading channels", Proc. IEEE/IEE ICT, vol. 2, pp. 458-462, June 2002. C9. C. D. Iskander and P. T. Mathiopoulos, "Comparison of standards and tech-niques for circuit-switched versus packet-switched H.26x video communications over C D M A mobile networks", Proc. 14-th Int. Conf. on Wireless Comm. (Wireless), vol. 2, pp. 474-489, July 8-10, 2002. C10. C. D. Iskander and P. T. Mathiopoulos, "Multicell uplink performance of multicode D S / C D M A with noncoherent M-ary orthogonal modulation in multipath fading 19 channels", Proc. 14th Int. Conf. on Wireless Comm. (Wireless), vol. 2, pp. 451-463, July 8-10, 2002. C l l . C. D. Iskander and P. T. Mathiopoulos, "Turbo-coded transmission of smoothed H.263 video for the cdma2000 downlink", Conf. Record of the 36th Asilomar Conf. on Signals, Systems, and Computers, vol. 2, pp. 1325-1329, Nov 3-6, 2002. C12. C. D. Iskander and P. T. Mathiopoulos, "Performance analysis of M - P S K , D E -M - P S K and M - Q A M with dual-branch coherent equal-gain and maximal-ratio combining in correlated Nakagami-m fading", Proc. IEEE CCECE, pp. 1683-1686, May 4-7, 2003. C13. C. D. Iskander and P. T. Mathiopoulos, "Analytical envelope correlation and spec-trum of maximal-ratio combined fading signals", Proc. IEEE PACRIM, pp. 446-449, Aug. 28-30, 2003. C14. C . D. Iskander and P. T. Mathiopoulos, "Performance of M-ary modulations with dual-branch coherent equal-gain combining in independent and correlated Nakagami-m fading", Proc. IEEE PACRIM, pp. 537-540, Aug. 28-30, 2003. 20 C H A P T E R 2 STATISTICS A N D SIMULATION OF WIDEBAND FADING C H A N N E L S 2.1 Introduction This chapter makes several novel contributions to the theory of wideband fading channels. Section 2.2 briefly reviews three types of fading channels commonly used in the study of mobile radio, namely Rayleigh, Rician and Nakagami channels. Section 2.3 analytically derives some important second order statistics of such channels, when diversity is used. Making use of some of the presented first- and second-order statistics of Nakagami channels, Section 2.4 proposes a new wideband Nakagami fading channel simulator, and also reviews and discusses existing ones. Then Section 2.5 derives new analytical expressions for the error rates of several common modulation techniques, in a wideband Nakagami fading environment with correlation between the diversity-received signals; these theoretical results are checked against computer simulation results obtained by implementing some of the previously described channel simulators. 2.2 A Review of Common Fading Models and Distributions 2.2.1 Rayleigh Fading The most common model used in the radio communications literature to describe flat fading in urban/suburban environments is Clarke's model [78]. It assumes a fixed transmitter with a vertically polarized antenna, and a mobile terminal; however it has also been used (with possible modifications) in several other scenarios which do not necessarily conform to these assumptions in a strict sense. 21 Clarke's original model considers an electric field incident on a mobile antenna. This field consists of N azimuthal plane waves (also often called scatterers), with: • Arbitrary carrier phases (f>n; • Arbitrary azimuthal angles of arrivals an; • Equal average amplitudes. Mathematically, this field can be expressed as: N N Ez{t) = ^2En{t) = J2EnCncos{27rfct + en) 71=1 71=1 N ' = E ^ Cn cos(2n fct + 9n) 71 = 1 = Tc{t)cos{2nfct) - r s ( i ) s i n ( 2 7 r / c i ) (2.1) where: N Tc{t) = EY,Cncos{9n) (2.2) 71 = 1 N Ts(t) = Ej2Cnsm(9n) (2.3) 71=1 are the uncorrelated in-phase and quadrature components of the electric field, respec-tively, E is a constant, Cn are mutually independent normalized random variables repre-senting the fading on each path, such that ^[52n=i C^] — 1> where E[-] denotes statistical expectation, and: 9n = 27Tfnt + 4>n (2.4) are the phases, with / „ = (v/X) co sa„ the Doppler shift of the nth wave, given a mobile speed v (m/s), a carrier frequency fc (Hz) and a carrier wavelength A = c / / c (c ~ 3.108 m/s is the speed of light in free space). fm = v/X is called the maximum Doppler shift. For a large number of waves N, as per the central limit theorem ([79], p. 266), samples 22 Tc and Ts of the in-phase and quadrature processes Tc(t) and Ts(t) become normally distributed with zero means and variances E[T2} — E[T2] = E2/2 = Q./2 = a2. Hence the envelope of a sample Ez of the electric field Ez(t): r = \Ez\ = y/ffTT! (2.5) 195) to be Rayleigh-(2.6) and cumulative distribution function (cdf): FR{r) = l - e " £ (2.7) where Q = 2a2 = E[r2]. The moments of the Rayleigh distribution are given by [26]: E[rk] = ftirQ + 1^ (2.8) where T(x) = J*0°° e _ t i I _ 1 < i t is the gamma function (Eq. 8.310.1 of [80]). The mean of r is hence given by nr — E[r] = O^-K/2 and its variance by a2 = E[r2} — E2[r] = a 2 (2 —7r/2). The quantity t a n - 1 (Ts/Tc) can be shown ([79], pp. 200-201) to be independent of r and uniformly distributed over [—7r/2,7r/2]. By periodicity, the phase of Ez, 9, is then independent of r and uniformly distributed over [0,2-7r]: Pe(0) = O<0<2TT. (2.9) Based on Clarke's model, Gans [81] developed a power spectral theory for the mobile radio channel. Assuming that N is very large, that the orn's are uniformly distributed over [0, 2TT], and that the signal is transmitted by a omnidirectional vertical A/4 antenna with gain G(a) — 1.5, the power spectrum of the electric field Eq . (2.12) was shown to where | • | denotes the absolute value of •, can be shown ([79], p. distributed, i.e. with probability density function (pdf): 2r PR{T) = yye « , r > 0 23 be (c.f. [15], Eq. 1.2-11): S(f) = 7 ^ = 7 = - ^ 2 ^ 1 - ( ^ ) 2 The power spectrum of the baseband envelope Eq. (2.5) was further shown to be (c.f. [15], Eq. (1.3-27) p. 29, or [16], Eq. (4.79)): where K(-) is the complete elliptic integral of the first kind [80]. 2.2.2 Rician Fading If a dominant stationary wave E0{t) — E0Co cos(27r/ct + 60) of constant amplitude A = E0CQ is included in the received signal (e.g. a nonfading line-of-sight component), the expression for the electric field becomes: N Ez(t) = E0(t) + ^EnCncos{2irfct + 0n) 71 = 1 N = Ej2CnCOs{2irfct + 6n) 71=0 = T c(t) cos(27r/ct) - r a ( t ) s in (27r / c t ) (2.12) where now T c and Ts are Gaussian with means J4cos(«?!»o), Asm(<j>o) (taking / 0 = 0), and powers E[T2] = A2 cos2(</>0) + a2, E[T2} = A 2 s i n 2 ( ^ 0 ) + o2. The envelope r can then be shown ([79], pp. 196-197) to be Rician-distributed, i.e. with pdf: Mr) - ^ . m . r > 0 (2,3, and cdf: 24 where I0(-) is the modified Bessel function of order 0, and Qi(-) is the Marcum-Q function of order 1. K = 101og 1 0(^4 2/fi) (dB) is termed the Rician factor. When A = 0 (i.e. K —> — oo), the Rice pdf reverts to the Rayleigh pdf. Note that the Rice pdf was also independently discovered by Nakagami [23] in his studies of mobile radio propagation, which he termed the n-distribution. Hence the Rice pdf is also called the Nakagami-n or Nakagami-Rice pdf. The moments of the Rice distribution are given by [26]: m = • ( - ! ; ! ; - £ ) ( 2 ' 1 5 ) where $(a;b;x) is the confluent hypergeometric function ([80], Eq . 9.210.1). The mean of r is hence given by / i r = E[r) = ^y/Qn<S> ^—|; 1; — and its variance by a2 = A2 + Q, - n2r. The phase of Ez can be shown ([79], pp. 499-501) to be given by (0O = 0): a2 -4 _ _ _ _ _ r dnnat(i\c 5^ 3 I / d rnal H\\ \ 0 < 9 < 27T (2.16) e 5^ Acos(9)e PM = -Z7T- + 2TT 2a\/_7r 1 + 2 e r f ( _ ^ ( _ where erf(jj) = 2/y/TT J Q X e~z2dz is the error function. The power spectrum of the electric field is similar to that of the Rayleigh case, but with an impulse at the frequency fc. 2.2.3 Nakagami Fading While the Rayleigh and Rice distributions can indeed be used to model the envelope of fading channels in many cases of interest, it has been found experimentally [23] that the Nakagami distribution offers a better fit for a wider range of fading conditions. The Nakagami distribution was proposed in the early forties for characterizing urban and suburban fading channels, and was originally deduced from a series of experiments. It was later shown that it constitutes an approximation to the pdf of the amplitude of a sum of phasors with random moduli and phases [23], [82]. Contrarily to the Rice pdf, it doesn't assume a line-of-sight (LOS) condition. Hence, while the Rice distribution can only describe better-than-Rayleigh fading conditions, the Nakagami pdf with parameter m < 1 models worse-than-Rayleigh conditions. Moreover, for m — 1, the Nakagami pdf reduces to the Rayleigh pdf, and can thus be seen as a generalization of the latter. It 25 was verified in several other independent experimental researches that the Nakagami-m pdf could indeed accurately represent the wide range of commonly encountered fading conditions [24], [83], [84]. As a result, it has been adopted in some software and hardware fading channel simulators for 3rd generation (3G) cellular networks [85], [86]. It is also increasingly used in the analysis and modeling of wideband channels, in particular for C D M A systems in frequency-selective fading [39]; this is also encouraged by the fact that its analytical form is more amenable to manipulations, compared to the Rice pdf, which contains a modified Bessel function of the first kind. We briefly review the theoretical development leading to the Nakagami pdf, along with the approximations made [23]. Using a complex notation for the plane waves, the sample envelope of the received signal can be written as: N r = ] T \Ene>e» | = Z(Eie>ei, E 2 e ^ E N e ^ N ) (2.17) 71 = 1 where no specific assumptions are made about the pdfs of the random variables En and 6n, n = 1, 2 , . . . , N, and £ = ^(E^01, E2ej92,... , ENej0N) is a positive definite function of N complex random vectors. The pdf of r can be written as: pR(r) = E[5(r-0] = E „v+l XJv(Xr)^dX o S = r"+1l ^ M w f r + i ) G O r ( * + i ) E . v+l roo dX (2.18) (2.19) where FV(X) = -) T(v + 1)E (2.20) In Eqs. (2.18) and (2.20), the expectations are taken with respect to the iV complex variables {Ene>6n}, 5{-) denotes the Dirac functional, v = N/2 — 1, and Jv(-) is the Bessel 26 function of order v > \. Using the infinite series expansion (A.5) for «/„(•): = ^ + 1 ) E r ( „ ; - , ; i ) t ! ( T ) E [ { e n (2 . 2 1 ) By replacing the Lyapunov inequality E[£2k+21] > E[£,2k]E[£21] by an equality, Nakagami obtains the approximation: F„(A) ~ e - ^ A 2 (2.22) where Q — E[r2] and is called the Nakagami fading figure. Using Eq . (2.22) in Eq . (2.19), and making use of Eq. (8.6:10) of [87], one obtains: " « « - s r ^ - 1 6 - * ' 1 ' r - ° ( 2 - 2 4 ) which is the Nakagami pdf. Note that while the above derivation was carried out with the assumption that m was an integer or half-integer, the pdf Eq. (2.24) holds for any real m > 0.5, as was also verified experimentally [23]. For m = 1, Eq. (2.24) reverts to Eq. (2.6). The moments of the Nakagami distribution are given by [26]: E[rk] = - V 2 ' ' • ' (2.25) \m ) 1 (m) The mean of r is hence given by = (^)2 F p 2 ^ ^ and its variance by: (2.26) a2 = n 1 - -r \ m r ( l + m) r(m) There was no phase pdf associated with the Nakagami distribution in the original paper [23]. While it has been argued on several occasions [88] that a uniform phase can be 27 assumed (as for the Rayleigh case), there is still no formal theoretical (or experimental) justification for this assumption. Determining the pdf (or proving the uniformity) of the phase is thus still an open problem. 2.3 Analytical Second-Order Statistics of Fading Channels with Diversity 2.3.1 Analytical Level Crossing Rates and Average Fade Dura-tions of Diversity-Combined Nakagami Fading Signals 2.3.1.1 Introduction The level crossing.rates and the average fade durations are two quantities which sta-tistically characterize a fading communication channel. The level crossing rate (LCR) is defined as the number of times per unit duration that the envelope of a fading channel crosses a given value in the negative direction [15]. The average duration of fades (AFD) corresponds to the average length of time the envelope remains under this value once it crosses it in the negative direction. These quantities reflect the correlation proper-ties, and thus the second-order statistics, of a fading channel. They provide a dynamic representation of the channel. They complement the pdf and cdf, which are first-order statistics, and can only be used to obtain static metrics associated with the channel, such as the bit error rate (BER) . The L C R and the A F D have found a variety of applications in the modeling and design of wireless communication systems, such as the finite-state Markov modeling of fading channels [89], the analysis of handoff algorithms [90] and the estimation of packet error rates [91]. Pioneering work on the subject was done by Rice [92], which examined LOS (Rician) channels. Much later, some expressions for the L C R and A F D of the combined envelope of diversity Rayleigh channels were published. For example, Lee [93] derived these quantities for equal-gain combining ( E G C ) . Adachi et al. [94] provided general expressions in the case of dual correlated channels with selection combining (SC), maximal-ratio combining ( M R C ) and E G C diversity; these expressions could be put in closed-form for independent channels. Recently, Yacoub et al. [95] 28 also presented expressions in the case of E G C and M R C with an arbitrary number of independent channels. In [96], using results from [97], the authors derived the L C R and A F D for Nakagami channels without diversity, and for a special case of M R C diversity. They also presented an approximate result for E G C , relying on Eqs. (81)-(83) of [23]. The L C R and A F D of non-diversity Nakagami channels with isotropic scattering were also later obtained in [98] using a different approach, which relied on the decomposition of the distribution. Note that the L C R had been previously obtained in an intuitive manner in the original paper of Nakagami ([23], Eqs. (23)-(25)), where they were denoted by "fineness". Field trials were carried out in [99] and [100] for the non-diversity case, and good agreements were reported between the experimental L C R and the analytical expressions. In this section, we present a very general approach which can be used to analyti-cally evaluate the L C R and A F D for Nakagami channels with diversity reception. Our methodology is the following: we straightforwardly rearrange the expression for the L C R , such that it can be expressed as the product of the probability density function of the received signal and an integral involving the conditional pdf of the derivative of this signal. Depending on the cases, the first term either can be found in the literature or has to be derived. The conditional pdf in the second term is found by examining the expression for the derivative of the received signal. It should be noted that the proposed methodology does not have limitations nor makes any simplifying assumptions. How-ever, we shall only present the cases where a simple analytical closed-form solution for the L C R and A F D is possible, which generally requires the diversity channels to fade independently. It wi l l also be shown that our general analytical expressions for Nakagami fading reduce to previously known results. In contrast to this work, earlier derivations of the L C R and A F D for channels with diversity reception were usually specific to a par-ticular channel/diversity pair, and can't always be used in obtaining the same quantities for different situations. For example, in [96] the L C R were obtained by finding a closed-form expression for the joint pdf of the received signal and its derivative for the case of Nakagami fading, which is not always possible; whereas in [94] and [95], the analytical derivations were conducted for specific situations only (dual-diversity for the former) or under certain assumptions (identical channels for both). 29 The organization of this section is as follows. After this introduction, our analytical approach and the steps needed to apply it to Nakagami channels (using the physical insights of [98]) are presented in Section 2.3.1.2. These results are used to obtain the L C R and A F D for Nakagami channels with SC, M R C and E G C diversity reception in the following subsections. The analytical expressions obtained are evaluated numerically and discussed in Section 2.3.1.6. The last section summarizes our contributions and cites some applications. 2.3.1.2 General Expressions Let r be the sampled value of the diversity combined envelope R(t) of a fading channel. The L C R NR(r) and A F D rR(r) are defined as a function of r by: /•oo NR(r) = J rpR>R(r,r)dr, (2.27) Mr) = FR(r)/NR(r) (2.28) where (') denotes the derivation operator with respect to time, FR(r) = f r pR(a)da is the cdf of the fading channel, and pR(r) is the corresponding pdf. The L C R can be rewritten in terms of pR(r) and the conditional distribution pR{r\r) as /•OO NR{r) = / rpR(r\r)pR(r)dr Jo poo = PR(T) / rpR(r\r)dr. (2.29) Jo This generic expression for NR(r) wil l be the basis for all later derivations. It is indeed applicable to all forms of diversity, and can be used in conjunction with any fading distribution. It reduces directly, for example, to Eq. (6) of [93], (15) of [96], and (16) of [98] for the special cases treated in these papers. The output sampled envelope of an L-branch diversity combiner can be expressed in the generic form of r = f(rur2,...,rL) (2.30) 30 where r/, with I — 1 , 2 , . . . , L is the envelope of the Ith diversity channel seen by the receiver, and /('•) is a function which depends on the diversity technique used. In the case of Nakagami fading, the pdf of rt is mathematically expressed as PRMI) 2 - r 1 mi 2 m , - 1 rl fij J r(m,) e nir', r< > 0 (2.31) where Q; = E[rf] and m; are the average power and the fading figure of the Ith channel, respectively. The cdf of r; is given by FRiiri) r(m,) (2.32) where j(x, a) — /* e Hx 1dt is the incomplete gamma function of the first kind (Eq. 8.350.1 of [80]). By analogy with [98], when 2m; is an integer, the envelope of the Ith diversity channel can be written as m;-rt = < rf0 + ^ rft with 2m; odd i=i mi y~]rfj with 2m; even (2.33) where rfQ — x]0, r\ — x\ + yf{, and the xu's, yu's are Gaussian random variables with zero mean and variance of = fi;/(2m;). The derivatives of the r;'s can then be calculated using mi -n = < (norio + nihi)/n with 2m; odd i=i mi C^/iihij/n with 2m ; even. (2.34) i=i From [15], for isotropic scattering, the f^s are Gaussian-distributed with zero mean and variance cr2. = Var[rf ] = of27r 2/^. We let of = of2n2f%l to alleviate the notation. Since r; is a sum of zero-mean Gaussian variables, it is also zero-mean Gaussian, conditioned 31 on vi. Using Eqs. (2.34) and (2.33), its variance is found to be a 2 = Var[rj] = of, which is independent of rj. As asserted in [98], r; and ri are thus independent, so that p(n,n) = p(ri)p{n). Using the above, the analytical L C R and A F D for the diversity methods of concern wil l be derived in the next sections. 2.3.1.3 Selection Combining (SC) In [94], the authors present an expression for the L C R for dual SC in Rayleigh fad-ing. It is generalized in [101] for an arbitrary number of independent and identically distributed (i.i.d.) channels. Below, using Eq. (2.29), we derive an expression for the L C R of SC for L independent but not necessarily identical channels. We then apply it to the case of Nakagami fading. The channel envelope at the output of a SC diversity system is well-known to be given by: r is thus a Gaussian random variable when conditioned on the r;'s, with zero mean and variance: r = max I = 1, 2 , . . . , L} . (2.35) Its derivative is: r r), Tj - max{r t, 1 = 1,2,... ,L}. (2.36) G] if (r,- = max rAvi = r •> J 1=1,2,...,L (2.37) Consequently, ar is a discrete random variable with pdf: L L (2.38) 32 From Eq. (2.29), the L C R , conditional on ar, are given by: r°° 1 -4, NR{r\or) = pR{r) / r e 2*r dr Jo v 27rar = P * ( r ) 4 = - (2-39) V _7T Eq. (2.39) is averaged over the pdf for <5>, i.e. Eq. (2.38), to obtain: poo NR{r) = / NR{r\ar)ptr{ar)dar Jo L 1=1 (2.40) By taking into account the independence assumption, the term P(rj = max rt\rj = r) 1=1,2, . . . ,Z/ can be evaluated as: P(r,- = max rl\rj = r) = P(n < r h I = 1 , 2 , L , I / j\rj = r) J i = l , 2 , . . . , _ = P ( r ; < r , ; = i , 2 , . : . , L , i / i ) £ _ • ( = 1 J = l From Eqs. (2.40) and (2.41) the L C R can be expressed as: L L NR(r) = 5>^)^_n F ^ r ) - <2-42) j=l V 27T Substituting Eqs. (2.31) and (2.32) into (2.42), and using Oj = o^\pl^jm leads to the L C R for a Nakagami fading channel with arbitrary parameters and SC: For identical channel parameters, mi = m, Qt = Q,l = 1,2, . . . , L , Eq. (2.43) reverts to the expression given in [102], and when m = 1, to the one given in [101], Eq. (45) for 33 Rayleigh fading: NR(r) = L ^ / m - ^ e - ^ ( l - e " ^ ) L \ (2.44) It can be verified (c.f. Appendix B) that the approach taken in [101] for obtaining the L C R , when extended to include arbitrary parameters, also leads to Eq. (2.43) for Nakagami fading. Eq. (2.43) gives the average number of times per second that the output R(t) of a selection diversity combiner falls below a specified value r, given L, fm, and the channel parameters mi, Qi, I = 1,2,. . . , L. Hence, if the channel conditions can be estimated and a maximum speed for the mobile is assumed, based on Eq. (2.43), one can determine the number of diversity branches needed (L) so that the combined signal R(t) doesn't fall below a threshold rT more than a specified maximum number of times NT- This can be done by evaluating NR(rT) for increasing values of L, until NR(rT) < NT-The cdf for SC is given by: FR{r) = P ( n <r,r2<r,... ,rL<r) (2.45) which reduces for independent Nakagami channels to: i=i i=i v 11 The A F D for arbitrary Nakagami channels with SC can then be obtained straightfor-wardly by substituting Eqs. (2.43) and (2.46) in (2.28): A 7 ( ™ „ ^ 2 ) rR(r) = r ^ F l • ( 2 ' 4 7 ) 3 - f y t e r 2 ) c - ^ T T 7 ( m ' " r 2 ) 34 Eq. (2.47) gives the average time (in seconds) that the combined signal R(t) stays below a specified level r, once it has crossed it in the downward direction, again given L, fm and the channel parameters. Hence, one can again evaluate the required L so that, on average, the combined signal R(t) doesn't stay below a threshold TT more than a specified maximum period of time r m Q I . The quantity r m a x can correspond, for example, to the average period of time a receiver can demodulate a signal of amplitude rT, without going into outage or losing synchronization. Similar insights can be obtained for the cases of maximal-ratio and equal-gain diversity, thanks to the expressions for the L C R and A F D derived in the following sections. 2.3.1.4 Maximal-Ratio Combining ( M R C ) The output of a M R C diversity system is given by [15]: r = L(=i (2.48) and its derivative by: (2.49) As in the SC case, r is a Gaussian random variable when conditioned on the r;'s, with zero mean and variance: (2.50) (=i where the last equation was obtained using the independence assumption between the branches. If the diversity channels are identically distributed, E[fi ] = of = a 27r 2/^, 35 and Eq. (2.50) reduces to: a2r = o*2** fa. (2.51) In that case, Eq. (2.29) can be solved to give: NR(r) = Pn(r)^=. (2.52) V Z7T The pdf of r (again for the special case of i.i .d. channels) is known to be given by [23]: / s n f m T \ m T r2mT~l _ H r 2 P « W = 2{WT) f W e - . ' > < > (2.53) where mr = rnL, £lT = Q.L, and the cdf is given by: 7(mr, ^ r 2 ) F » ( r ) = ± n ^ r - ( 2 - 5 4 ) Using Eqs. (2.53) and (2.51) in Eq. (2.52) leads to the following result for the L C R , which was also derived in [96] using the approach mentioned in the introduction: y/2irfm (mT 2 \ m r * . m r , Substituting Eqs. (2.54) and (2.55) in Eq. (2.28) yields the A F D : 7 ( m T , ^r2)enr rR(r) = 7 1 Q t ' m \ - (2.56) For Rayleigh fading (m — 1), Eq. (2.55) reduces to the following expression 1 : : I t should be noted that it is similar to Eq. (13) of [95], but the latter has possibly a misprint in the exponential term. 36 2.3.1.5 Equal-Gain Combining ( E G C ) The output of a E G C diversity system is given by " 7 i | > ( 2 ' 5 8 ) whereas its derivative by [93] As opposed to the previous cases, r is now a Gaussian random variable independently of the rj's, with zero mean and variance * - 7 E ^ 2 ] = 2 7 r 2 ^ 7 E ^ 2 (2-6°) 2=1 2=1 where the last equation was obtained using the independence assumption between the branches. Solving (2.29) leads to NR(r) = pR(r)-%=. (2.61) V _7T It is similar to Eq. (2.52) in the previous section, however for the M R C case this equation required the i.i .d. assumption in order to be valid, while this is not the case for E G C . For i . i .d. channels, the cdf and pdf of r were presented in [20] and [95] respectively, in integral form, for an arbitrary L . For independent channels with arbitrary parameters, the pdf can be written as pR{r) = VL / . . . / Jo Jo Jo 1=2 1=2 dr2... drL. (2.62) For Rayleigh fading, a simple closed-form solution is available only for L < 2, and is implicitly presented in [94]. For Nakagami fading with identical parameters and L = 2, 37 (2.62) reduces to fV2r pR{r) = V2 pR(rV2 - r2)pR(r2)dr2. Jo Substituting Eq. (2.31) in Eq. (2.63): (2.63) The variable transformation y = ^ ( \ / _ r i — r) in Eq. (2.64) gives 2 ( t ) 5 /— r ° - r -y 2 m - l e ^dy. [r(m)] 22 2 Using Eq. 3.383.1 of [80] then leads to the following closed-form solution: ' m \ 2 » > 2 _ ( 2 m , | ) (2.65) ^ („)' [ r (m)] 2 2 2 m - 2 2 ' r 4 m - 1 e - 2 ^ 2 $ [ 2 m , 2 m + ^ ^ r 2 1 m 2' n (2.66) where B(x,y) is the beta function and <_(_, c, x) the confluent hypergeometric function, given by Eqs. 8.380.1 and 9.210.1 of [80], respectively. Substituting Eqs. (2.63) and (2.60) in Eq. (2.61) yields the following expression for the L C R of dual-branch E G C : NR(r) 2nfmB(2m, | ) /m 2\2m~i _2™r2 ( 1 m 2 , . . [r{m)]>v>»-> (nr) e V $ r + 2 ' Q r l - ( 2 - 6 7 ) The cdf is given by: FR(T) m \ 2 ™ 2 S ( 2 m , i ; jT a ^ - ' e - 2 ^ 2 ® (2m, 2 m ^ a 2 ) da.(2.68) nJ [r(m)] 2 2 2 ™- 2 y0 V 2 'ft Making the change of variable x = 2(m/Q.)a2, using the infinite series expansion r(c) ^ r ( a + k) xk (2.69) 38 and the relation B(x,y) = T{x)T(y)/T(x + y) (Eq. 8.384.1 of [80]) in Eq . (2.68) results in Mr) T(n + 2m) [r(m)] 22 4 T O" 2 ^ T(n + 2m + 2 ^ / ° e-xx2m+n-xdx. (2.70) Applying Eq . 8.350.1 of [80] to the integral above leads to the desired representation: /^TT ^ T(2m + n) 1 FR{r) = E [r(m)] 22 4 T O" 2 ^ T(2m + n + ±) i ) 2 " n ! 7 ( . m 2m + n, 2—?" 2 ) . (2.71) By substituting Eqs. (2.71) and (2.67) in Eq. (2.28) we obtain the A F D 2 a r 2 V ^ T(2m + n) 1 / m 2 \ e n > —r-^ h - r r7 2m + n, 2—r n = 0 2 / m 5 ( 2 m , | ) ( 2 f r 2 ) 2 — ^ (2m, 2m + §, f r 2 ) (2.72) For m = 1, Eq. (2.67) can be simplified using Eqs. 9.212.2, 9.212.4, 9.212.1 and 9.236.1 of [80] for $(a,c,x), in that order, and Eq . 8.384.1 of [80] for B(x,y). This reduces to the following expression for the L C R of dual-branch E G C and i.i .d. Rayleigh fading channels, which was also presented in [94]: NR{r) = v ^ / m e - f r ,2 x " e ~ " + f S 4 ) ^ e r f ( v ^ ) (2.73) For independent but non-identical Rayleigh fading channels, we make use of Eqs. (2.62), (2.60) and (2.61) to obtain a closed-form expression for the L C R of dual-diversity E G C : NR(r) = v ^ / m ^ e - ^ x Mr, ( V ^ i 2 e + V ^ 2 i e nm 1 + erf ( ^ 2 1 r\ + erf f ^ j 2 r to* (2.74) with f2 m = (Qi + n 2 ) / 2 , Q 2 1 = fi2/^i, Q 1 2 = Qi / f2 2 and £7P = £ 7 x ^ / 4 . 39 2.3.1.6 Numerical Results and Discussion The L C R and A F D expressions presented above are plotted in logarithmic scale against the normalized value of the combined received envelope, rn = r/y/Q, in dB. Figs. 2.1-2.3 compare the L C R (normalized by fm) for the diversity techniques presented above with L = 2 and the no-diversity (ND) case (L = 1), for three values of the m-parameter: m = 0.6 corresponds to severe fading (worse than Rayleigh), m = 1.3 to fading conditions slightly better than Rayleigh, and m = 3.0 to line-of-sight conditions. For all the curves, it is observed that the L C R for M R C are the lowest for low values of r n ' s , and the highest for high values of r n ' s , while the opposite is true for the N D case. Indeed, in the N D case, fades occur more frequently due to the absence of diversity. As a consequence, the signal crosses lower values of rn more often than when diversity is used (with the lowest number of crossings occurring for the optimal diversity scheme, i.e. M R C ) , whereas it crosses high values of rn less often. Also, from these curves, the output of an E G C receiver fades less frequently than that of an SC receiver. However, the differences in L C R between M R C , E G C and SC depend on the Nakagami-m parameter, and thus on the severity of the channel in terms of fading, and are commented below. For m = 0.6, the L C R curves for SC and E G C are nearly identical, but differ from those of M R C , for which the combined envelope exhibits less severe fading. As m is increased from 0.6 to 3.0, the L C R curve for E G C gets closer to that for M R C , and further away from that for SC. For m = 3.0 the L C R curves for M R C and E G C nearly overlap. This reflects the fact that, as the fading severity decreases (i.e. for higher values of m), the performance of E G C tends to approach that of M R C , while the performance margin between the latter two and SC increases. This could also be observed by comparing plots of the error probabilities for these diversity techniques and different m's. The results plotted in Figs. 2.4-2.6 present the A F D , normalized by l / / m , for the same cases as before. From these curves it is seen that the A F D for all three diversity techniques remain very close for values of rn less than about -5 dB. This means that once the combined signal has faded below this value, it remains below for nearly the same amount of time for all of M R C , E G C or SC. However, from our previous examination of the L C R , since a M R C signal crosses low values of r n less often than E G C and SC signals, on average it will spend less time into deep fades than the latter two. For higher 40 values of rn, it is observed that the A F D are lower for M R C than they are for E G C and SC: for each rn, the combined signals obtained with E G C and S C spend more time below this value than that obtained with M R C , which reflects the fact that on average a stronger signal results from the use of M R C . This agrees with the previous discussion on L C R , in which it was pointed out that a M R C signal is more often in the high end of the signal strength rn than E G C and SC. As before, we observe that for severe fading (m = 0.6), the behavior of E G C follows closely that of SC, while for milder fading (m = 3.0), it compares to that of M R C . Thus, for a fixed set of parameters, there isn't a one-to-one correspondence between the behavior of the L C R and that of the A F D (similar observations were reported in [95], for the case of Rayleigh fading): the L C R of M R C differed from that of SC (and E G C for low m's) over the whole range of r n ' s , while the A F D are nearly identical for low r'ns. This is due to the term Fn(r) which intervenes in the relation (2.28) between the latter quantities. Figs. 2.7 and 2.8 illustrate the L C R for S C and M R C , respectively, for a variable number of diversity branches L. In the case of SC, as L increases, the frequency at which the received signal crosses high values (e.g. at approximately rn > 0 dB) stays almost the same. Whereas in the case of M R C , Nn(rn) increases with L for high values of rn. Moreover, for low values of rn, the L C R decrease faster for M R C than for SC as more diversity branches are added: e.g., for rn = —20 dB, the decrease in the dB value of Nji(rn) is more than six-fold for M R C as L goes from 2 to 4, while it is less than five-fold in the case of SC. This parallels the observations made in [20], according to which the advantage of M R C and E G C over SC (i.e. the strength of the signal) gets more pronounced as the number of diversity branches increases (with E G C following the behavior of M R C ) . In summary, the numerical results support the assertion that the gain in performance made possible using M R C and E G C , as compared to using SC, gets more important as the fading gets less severe and the diversity order increases. 41 r n(dB) Figure 2.1 LCR's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 0.6. 10' r \ V\ \ \ \ N\ -• A A \\ . \ \\ w n '• > M / / / . \ • • \ • \ \ \ A : A • A-S ' / / / / \ u. •\\ '• \ ND - SC '}'. - MRC - - EGC )-5l I I I 1 1 I -20 -15 -10 - 5 0 5 10 r n(dB) Figure 2.2 LCR's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); TO = 1.3. 42 -10 -5 ,(<JB) 10 Figure 2.4 A F D ' s wi th S C , M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 0.6. 43 Figure 2.6 AFD's with SC, M R C , and E G C dual-diversity (L = 2) and without diversity (ND, L = 1); m = 3.0. 44 45 2.3.1.7 L C R for Diversity Techniques with Correlated Branches In the previous sections, the assumption was made that the diversity branches were independent. In the cases presented, the combined signal r was independent of its deriva-tive r, which allowed the derivation of closed-form solutions for the L C R . If the branches are correlated, this no longer holds. For the case L = 2, as shown in [94], r is now condi-tional on r (or alternatively, on r\ and r2) and the phase difference 9\2 — 92 — 9X, where 9i,i = 1,2 are the phases associated with diversity channels 1 and 2. The conditional pdf of r can be written as: PR(f\ri,r2,012) = J - „ exp V 27TOV (r — m r ) 2 2oT~ (2.75) where mr = E[r] and of are now dependent on r i , r2 and 9\2. As a result, the expression for the L C R in Eq. (2.29) wil l be written as function of r i , r2 and 9\2. N, rOO <{ri,r2,912) = pRuR2le12{ri,r2,912) / rpk{r\rl,r2,9l2)dr. (2.76) Jo From [94], the expressions for the L C R for SC, M R C and E G C as a function of r can be obtained as: j fipRuR^QV2 (n = r, r 2 , 912)dr2d912 -7! JO /TT rT j npRuR2tel2(r1,r2 = r,912)dr1d912 for SC, (2.77) •T  JO NR{r)= / / hrpRuR2tei3(r1=rcoa<j>,r2 = rsin(f>,012)d<f>dBi2 for M R C , (2.78) J-TX JO NR{r) = [ f ^hpRuRi,eia(ri,r2 = ry/2 - r1,912)dr1d912 for E G C , (2.79) J-TT JO 46 with: n = / rpR{r\rur2,9l2)dr Jo exp -mt m, r erfc mr 2a2 J ' v " y/2aT V V2a, In the case of Rayleigh fading, the following joint pdf is known ([103], p. 63): r\ + rl - 2rxr2y/pcos(912) (2.80) P f l i , f i 2 , e i 2 ( r i , r 2 ,M = nr2 2TT(74(1 - p) exp 2 a 2 ( l -p) (2.81) where p — Cov(r 2 , r2)/ ^ /Var ( r 2 ) , Var(r | ) is the correlation coefficient between r\ and r 2 in power terms, Cov(-, •) and Var(-) denote the covariance and variance, respectively. Moreover, expressions for rhr and a2 are obtained in [94]. The latter and Eqs. (2.80) and (2.81) were then substituted in Eqs. (2.77)-(2.79), and the L C R were evaluated by solving numerically the double integrals. Unlike for the independent fading case, the L C R were a function of the angle a between the antenna axis and the direction of vehicle motion. The L C R were minimum for a = 0, and maximum for a = 7r/2. In the case of Nakagami fading, the following joint pdf was proposed in [104], even though it was stated that it is not unique: PHi , f l2 ,e 1 2 ( r i .r2,0i2) = 4(m - l )(r- i r 2 ) T _!7 / 2^fprlr2z 2TT T(m)( l - \p\)p^Qm+l Jo Z m _ 1 V ^(1 - P) xexp r\+rl- 2rlr2{\ - z)^/pcos(912) n(l - p) dz (2.82) where Im(-) is the modified Bessel function of order m. However, expressions for mr and a 2 are not easily obtainable, because the development used in [94] can't be applied straightforwardly to the Nakagami case. Indeed, in the Rayleigh case, in order to derive such expressions, the authors of [94] made use of the fact that the complex signals z\ = rieJ0i,i = 1, 2, received on each branch are complex Gaussian variables. This is not the case for Nakagami fading, thus more work is needed on this issue. Nonetheless, we examined the L C R for correlated diversity with L = 2, when it is assumed that r and r are independent. Following the development obtained in the 47 previous sections, for the identically distributed case, the L C R can be obtained as NR(r) = pR(r)ar/V2TT, with oy given by Eqs. (2.37), (2.50) or (2.60) and: Mr) = r (m) mr n l-Qr, 2mp 2m Q(l - p) ' V Q ( l - p) fo rSC , (2.83) 2 V / 7 ? ( m / Q ) m + 1 / 2 r 2 m e _ / ^pmr2 . P * W = r U ( i - p ) ^ M f e ) > f o r M R C ' ( 2 " 8 4 ) n = — - ^ f V ( w r £ ( 2 ( m + fc),§) f m \ 2 ( m + f c ) P f i l r j T ( m ) e ' f^Q k\r{m + fc)22(™+*)"2 ^ f t ( l - p ) / x r 4(m+fc ) - i $ / " 2 ( m + ^ 2 ( m + fc) + ^ , ^ ) for E G C , (2.85) V 2 iZ(l p) J where Qm(a, j3) is the generalized Marcum Q-function (Eq. 2-1-22, p. 44, of [26]). Eqs. (2.83) and (2.84) are deduced from [105] and [23], respectively, while Eq. (2.85) is a novel contribution (c.f. Section 2.5.3.2). Figs. 2.9 and 2.10 plot Eqs. (2.77) and (2.78), respectively, for a = 0 and a = 7r/2, along with the L C R obtained by using Eqs. (2.83) and (2.84) with m = 1, for p = 0.6425. It can be seen that the latter equations give an upper bound on the L C R for low values of r. Similar results were obtained with different p's. Since it is the low range of r's which is mostly of interest from a L C R standpoint (since one wants to determine the rate at which the signal goes into fade), the approximate expressions given above can be used to conveniently bound the maximum L C R obtained regardless of the angle a (which varies with the movements of the mobile). 48 Figure 2.9 LCR for SC with L = 2 correlated branches, m = 1.0, p = 0.6425. bound; : exact, a = 0; - . - .: exact, a = n/2. upper Figure 2.10 LCR for MRC with L = 2 correlated branches, m = 1.0, p = 0.6425. bound; : exact, a = 0; - . - .: exact, a = TT/2. -: upper 49 2.3.1.8 Conclusions Starting from a common representation for the L C R , we derived generalizations of expressions for the L C R of a diversity received signal in Rayleigh fading, in order to handle the more general Nakagami fading distribution. Closed-form solutions were presented for arbitrary L in the case of SC and M R C , and for L = 2 in the case of E G C . The assumption of i . i .d. channels was made throughout Sections 2.3.1.3-2.3.1.6 (except for SC, where nonidentical parameters were allowed) in order to obtain these results in closed-form, however the methodology used is not limited by this assumption: the correlated case can be dealt with in the same manner (c.f. Section 2.3.1.7), but wil l require numerical evaluations of the L C R and A F D for mOst cases of interest. The material we presented can be used in designing finite-state channel simulators [102], analyzing error-correcting schemes for burst error channels [91], determining the minimum duration outages in fading channels [106], or determining the delay spread of frequency-selective channels [107]2. 2.3.2 Analytical Envelope Correlation and Spectrum of Maximal-Ratio Combined Nakagami and Rician Fading Signals 2.3.2.1 Introduction The temporal variations of a mobile fading channel can be described by its envelope autocorrelation (or autocovariance) function. The latter can be used to determine the minimum average time interval that is needed for two samples of the channel envelope to be uncorrelated, or to have a correlation coefficient below a certain threshold. This can prove useful in determining the size of an interleaver, choosing an appropriate error-correcting code, or selecting a robust packet length. The autocorrelation function of a complex Gaussian random process is given for example in [111], p. 62 and [112]. The envelope autocorrelation function (EAF) of a flat-fading Rayleigh channel can be deduced 2Since the time [108] was submitted on May 2nd, 2001 (and several months after many of the results of [108] initially appeared in [102]), other independent contributions dealing with the L C R and A F D of diversity Nakagami channels have been published [109], [110]. The material presented here differs from that of [109] and [110] in the methodology used, and/or the generality (non-identically versus identically distributed) or the representation (closed-form versus integral-form) of the original analytical results. 50 from it, by substituting an appropriate time-variant correlation coefficient which describes the variations of a mobile radio channel [113]. For systems with diversity combining, the variations of the combined received signal is now of interest, since it is this signal which is available to the rest of the transceiver. In [114], Lee derived an expression for the E A F of the combined signal for a receiver with E G C , for the general case where the diversity branches can be mutually correlated. However, to our knowledge no expression has been published in the open technical literature for the E A F of a combined signal for M R C . Drawing upon the theory of multidimensional Gaussian distributions, this section provides such expressions in closed-form or as an infinite series, for Rayleigh, Nakagami and Rician fading channels. For the special case of no diversity, these expressions revert to those presented for example in [113], p. 50, [23], and [115], respectively. The power spectrum of the combined envelope is also derived for the Rayleigh and Nakagami cases. 2.3.2.2 Rayleigh Fading Let r i ( i ) , 7*2(£),... , T*LC0 be the received signals on the L uncorrelated diversity branches. They are assumed to be Rayleigh-distributed and stationary. The maximal-ratio combined signal is given by (c.f. also Eq. (2.48), where sampled values where used): Let r a and rt2 be the values of r;(t) (I = 1,2,. . . , L) sampled at t = T and t = T + r , respectively. rn and rt2 are then Rayleigh-distributed random variables, which can be expressed as: L (2.86) (2.87) (2.88) 51 where xit\, xij2, yi,i and yip are zero-mean Gaussian random variables with common variance a2. The values of r(t) sampled at t = T and t = T + r are then given by: 2 1 2 L. f/ = r (T) = [r 2 , + r221 + ... + ~2 ' ~ 2 2) + {.£,1 + 4 2 ) + • • • + ( 4 , i + ^,2)]* \ . / 2 . 2 \ , . / 2 . . 2 \ i i « i + < 2 )  ( 4 i   4  5 , v = r ( r + r) = [(y12,1 + y 2 , 2) + (y22,i + y22,2) + --- + ( y i , i + yi,2) 2 ]^  (2.89) (2.90) f/ and V can thus be interpreted as the magnitudes of vectors and Z^2\ respectively, given by: ZW = [ X i , i _ l l 2 z l z2 J 2 ) _(2) z l 2^ SL,1 XL,2\ ( l ) l T Z 2 L Z 2 L (2.91) (2.92) (!) _ z 2 — Xl,2i (1) - Z2L — XL,2-where (-)T denotes the transpose of a vector, and z[^ U and V are generalized Rayleigh random variables ([116], Chap. 2), and Z^ and Z^ are 2L-dimensional Gaussian random vectors. Let pW = [zj 1^- 2^] 2", i.e. P ^ is a random vector made up of the ith (i = 1,...,2L) components of Z^\ j = 1,2. Its covariance is given by ([26], p. 50): a2 Jpo2 M. (2.93) The correlation coefficient p depends on the spectrum of the fading signal. For example, for land-mobile radio channels, the correlation coefficient p(r) = J$ (27r/mr) is often used [113], where r is the time separation between two envelope samples, and JQ(-) is the Bessel function of zero-order. For clarity of presentation, from now on in this section, the dependence of p on r is dropped. The inverse of the covariance matrix can be calculated as W = ( M ) " 1 = 1 a*(l-p) a 2 ( l - p ) ~y/P 1 <r»(l-p) a » ( l - p ) (2.94) 52 and its determinant as det(M) = cr 4 (l - p). Making use of the previous expressions for W and det(M), the joint pdf of U and V can be obtained from the corollary to Theorem 3 (Chap. 2, Section 2) on pp. 32-34 of [116]: Puy(u,v) = 4(uv)Le r(L)(n)*+i (p) -* - ( i -p) -IL-I 2UV VP (2.95) where Q = 2a2 is the average power received on each of the diversity branches. W i t h this pdf given, and taking into account the assumed stationarity of r(t), the determination of the E A F reduces to the evaluation of a two-dimensional integral (instead of a 2L dimensional integral, if the averaging was performed over the individual ry's) . The E A F is given by: R(T) = E[r(t)r(t + r)] = E[UV] noo uvpu,v{u, v)dudv. (2.96) Substituting Eq. (2.95) into Eq. (2.96), using Eq. 2.5.6 on p. 115 of [116] and Eq. 7.621.4 of [80] to perform the integration with respect to the u and v variables, respectively, the following expression is obtained: R(T) = r ( L + | ) r(L) (2.97) where 2 F \ ( a , 6; c; x) is the Gaussian hypergeometric function ([80], Eq. 9.100). Eq. (2.97) is in fact a special case of Eq. (4), p. 73 of [116] (c.f. also Eq . 1.9 of [117]), which has thus been applied in this section to obtain the envelope correlation of maximal-ratio combined Rayleigh fading signals. It is pointed out that Eq. (2.97) can also be written as (c.f. Eq. 4.3 of [117]) R(T) = n(i-p)L+1 d T(L) d(p) JL-1 L-l E{Jp) K{Jp) (1-p)2 2(1 -p)\ (2.98) where K(-) and E(-) are the complete elliptic integrals of the first and second kinds, respectively, and dl(-)/d(x)1 is the Ith order derivative with respect to x. For a system 53 with no diversity (L = 1), Eq. (2.97) simplifies to the following well-known expression ([113], p. 50): R(r) = l n 2 F 1 ( ~ ~ ] l - t p (2.99) The mean of the combined envelope r[t) is given by: T ( L + i ) - " E[r(t)} = T(L) (2.100) The autocovariance of r(t) is then given by: C ( r ) = R(T)-(MI = r ( L + |) T(L) iFi 1 1 . , - , (2.101) Proceeding in similar lines as in Section 1.3.2 of [15], if we make the expansion 3p 3 2 ^ 1 I _ I r L l l P I P I 2 » 2 ' 4L 32L(L + 1) 128L(L + l ) ( L + 2) + . . . , (2.102) retain only the first two terms and substitute them in (2.101), the following approximate expression for the autocovariance is obtained: C(r) r(£ + §) r(L) 4LP{T)-(2.103) For L = 1 , the above reduces to C ( r ) ~ 7rfi/16p(r) (c.f. [113], Eq. (2.55)). Considering Clarke's model with isotropic scattering [78], where the transmitting antenna is vertically polarized and the receiving antenna is a vertical monopole, the power spectrum of the passband signal is known to be given by S(f) = 2 ^ - [ l — {^f^)2]~1^ (where here, unlike in Eq. (2.10), the gain of the receiving antenna, e.g. 1.5 for a vertical whip antenna, is included in Q). The approximate spectrum of the baseband signal can be obtained using Eq. (2.103) and the development of Section 1.3.2 of [15]: Sb(f) r(L) n 2 (2.104) 54 Eq. (2.104) is a generalization of the case L = 1, for which Sb(f) is given by Eq. (2.11). 2.3.2.3 Nakagami fading To carry out the derivation, we assume the Nakagami-m parameter to be an integer. The Nakagami pdf with m real is equivalent to the pdf of the square-root of a gamma random variable G(ct,/3), with parameters a = m and j3 = m/Qnak, where Qnak = m2a2 = mQ. As a special case, the Nakagami distribution with m integer is equivalent to the distribution of the square-root of a chi-square random variable x 2 ( m , m/Qnak), with 2m degrees of freedom [98], which is also called a chi random variable [118]. Hence, the Nakagami-fading signal on the Ith branch, sampled at t = T can be written as 2 m .i=l (2.105) The combined envelope at the output of a M R C receiver, sampled at t = T can thus be expressed as: U = r{T) = .1=1 n L 2 m y~i E xh .1=1 i=l (2.106) As before, U and V = r(T + r) correspond to the magnitudes of vectors and Z^2\ respectively, given by: Z{1) = [ X i , i . . . XH2M Z^ = ,(1) J l ) , ( 2 ) _(2) z{l) 62mL z{2) ' i2mL XL,1 T xL,2m\ (2.107) (2.108) Z^ and Z^> are also Gaussian random vectors, but of dimension 2mL instead of 2L. In order to carry on the derivation, we make the assumption that the covariance matrix of pw = k a ) 4 2 ) ] T takes the same form as in the Rayleigh case, i.e. Eq . (2.93). While there are typical correlation coefficients available for the Rayleigh case, to the best of our knowledge there haven't been any yet proposed on a physical basis for the Nakagami 55 channel. However, with the assumption that the covariance matrix can be put in the previously mentioned form (which necessitates m to be integer, in order for (2.105) to be valid), the relations developed below wil l hold for arbitrary correlation coefficients. Following the same steps as in Section 2.3.2.2 leads to: R(r) = T{mL+\) Y{mL) ft nak m -; mL; p (2.109) It can be observed that Eq. (2.109) is a generalization of Eq. (2.97), in which m = 1. For L = 1, Eq. (2.109) reduces to a special case of Eq. (137) of [23] (in which fti = ft2 = ^ and n = I = 1), i.e.: R(r) = r ( m +1) T(m) ft, nak m (2.110) However, while Eq. (2.110) holds for arbitrary m > 1/2 and P W , certain restrictions on the latter quantities were needed to obtain our derivation of (2.109), as detailed above, even though we do not claim that these restrictions can't be relaxed. It is interesting to note that by replacing m by L in Eq. (2.110), Eq. (2.97) can be obtained. The mean of the combined envelope r(t) is given by: V{mL+\) T(mL) The autocovariance of r(t) is then given by: ft nak m (2.111) C(r) = Y{mL) ft nak m rn I 1 1 r 2 * 1 —Z,-«'>ML'>P (2.112) As in the previous section, considering Clarke's isotropic scattering model, the ap-proximate spectrum of the baseband signal can be obtained by making the approximation C(r) r(m_ + i) T(mi) 1M2 imL sb(f) P(r): r ( m L + | ) l 2 ftnqfc/m L r ( m L ) J 4 ^ 2 / m m L K J_ Vrr (2.113) 56 2.3.2.4 Rician Fading In the Rician case, the development follows the Rayleigh case through Eqs. (2.86)-(2.94), except that x^i, x^2 (and y ^ i , yit2) are now Gaussian variables with means a^i, alt2, such that At = y a ^ + a 2 2 . The joint pdf of U and V is obtained from Theorem 3 (Chapter 2, Section 2) on pp. 32-34 of [116]: m r , u v ) _ 4 r (L - 1)[Q(1 - , ) ] * - 3 ( l - pfw P u ' v { ' } ~ ( - A 2 t y P ) ^ ( i - y P ) 2 ^ ^ , s (2L + k-3\ (2uv(-Jp) U \ 2 L - 3 £1(1-p) (2uAt(l-yp)\ (2vAt(l-vTp)\ l L + k ' 1 { Q ( l - p ) ) h + k - 1 \ Q ( l - p ) ) ( 2 - U 4 ) n where A 2 = J2i=i-^h a n d ^ J * s t n e binomial coefficient. The E A F is obtained by substituting Eq. (2.114) in Eq. (2.96). One way to solve the double integral is to make the expansion I„(z) = £ ° = o xv+2>/(j\T(v + j - 1)) for the term IL+k^ fi^lffi), and then use twice using Eq. 2.5.6 on p. 115 of [116], to perform the integrations with respect to the u and v variables, which gives the following infinite series expression (c.f. [119], Sec. 6, for the case L = 1; c.f. [120], Sec. 3, for arbitrary L): R(T) = T(L - 1)H(1 - p ) ( L + 1 ) e " nd-p) ^(L + k-l) k=0 2L + k- 31 2L- 3 x m+k)}2 Ai(i - yPy nn - P) k oo E 3=0 ( V p ) « [ r (L + k + j + l)}' 3'- r(L + k + j) (2.115) For At = 0, Eq. (2.115) simplifies to Eq. (2.97): it can verified by nulling all terms for which k > 1 in the outer summation, and using the relations ([80], Eqs. 9.131.1 and 57 9.111): { L + 1 ) r ( L + j + | ) r ( L + j + |) r (L) (i - p)<™> 2 j = 0 r ( L + |) r ( L + |) r ( L + J ) j ! (2.116) .(2.117) For L = 1, using the relation $(a;6;a;) = e x $ (6 - a; 6; - _ ) ([80], Eq. 9.212.1), Eq. (2.115) reduces to: 2 OO J2(T) = Q{l-p)2e n{1+^Y2€k (VPY k r 4 2 fc=0 E i=o ( ^ • [ r ( f c + j + |)] j ! r(fc + j +1) (fc!)2 2 ^ ( 1 - y ^ ) L ^(1 + v/p) v 2 fi(i + vp)y (2.118) where ek = 1 for fc = 0, = 2 for > 1, which is similar to [119], Eq . (6.13), along with the corrections made in [115]. The mean of the combined envelope r(t) is given by [26]: fj,T = Vfte n 1 4 2 r(L) V 2' ' n (2.119) The autocovariance of r(t) is obtained as per Eq. (2.101). The calculation of the baseband power spectrum for the diversity case is more involved than for the Rayleigh case and is deferred for future research. For the case of no diversity, expressions can be found for example in [113] and [121]. 2.3.2.5 Numerical Results and Discussion We check the validity of Eq. (2.101) by comparing it against the simulation results obtained with a Rayleigh fading simulator. The latter uses the sum-of-sinusoids Monte Carlo technique described in [122] with m = 1, and intends to approximate the spectrum of a land-mobile radio channel with correlation coefficient J |(27r/ m r). The number of sinusoids was taken to be 250. The average power is normalized to Q = 1. The carrier frequency and mobile speed are taken to be fc = 2 GHz and vc = 100 km/h , respectively. This corresponds to a Doppler frequency of fm = 185.18 Hz. F ig . 2.11 plots Eq. (2.101) 58 versus r x Rb, for L = 3 and a bit rate Rb = 64 kbps, along with simulation results. As it can be observed, the theoretical and computer simulation curves are in very close agreement. The slight mismatches arise from the fact that the channel simulator is non-ideal. The same order of precision was observed for different diversity orders and bit rates. Fig. 2.12 compares Eq. (2.101) for L = 1 — 3, and the same parameters as before. We observe that the autocovariance between the outputs of the diversity combiner is higher for larger diversity orders, especially for time delays corresponding to the local peaks of the envelope autocovariance function. However, as L increases, the differences between the E A F curves are gradually reduced. F ig . 2.13 plots «S&(/) for L = 1 — 3. F ig . 2.14 plots the Rician envelope autocovariance versus r x Rb, for L = 3 and a bit rate Rb = 9.6 kbps, for different values of Ai = A, I = 1, 2 , . . . , L. The zeros are at the same locations as for the Rayleigh case, but the amplitudes of the local peaks increase with A. 2.3.2.6 Conclusions Based upon multidimensional Gaussian calculus, we have provided a simple closed-form expression for the envelope autocorrelation function of the output of a maximal-ratio diversity combiner, in the case of Rayleigh fading. Using the same method we provided such an expression for Rician fading and for Nakagami fading when the m-parameter is integer. It has been shown that for the non-diversity case, these expressions simplify to results obtained previously by Uhlenbeck [111], Middleton [119] and Nakagami [23]. It has been observed that the use of diversity slightly increases the autocovariance between receiver inputs, with the largest difference evidenced when the number of branches goes from one to two; in particular, for the land-mobile radio channel, the increase is most important around the peaks of the envelope autocovariance function. 59 2 0.15 50 100 150. 200 250 300 350 400 450 500 Normalized time delay Figure 2.11 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rayleigh channel wi th L — 3. 150 200 250 300 350 400 Normalized time delay 450 500 Figure 2.12 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rayleigh channel. 60 50 150 200 250 Frequency (Hz) 350 F i g u r e 2 .13 Spectrum of the baseband envelope versus frequency for the Rayleigh channel. F i g u r e 2 .14 Envelope autocovariance function versus time-delay (normalized by symbol pe-riod) for the Rice channel with L = 3. 61 2.4 Simulation of Wideband Correlated Nakagami Fading Chan-nels The simulation of Rayleigh and Rice fading channels is based on their underlying well-defined and widely accepted physical interpretations, which can be directly used to generate random variables with the desired statistics. In the case of Nakagami fading, the'pdf represents only an approximation to the actual physical process, and hence, no simulation models have been found yet which rely on the actual physical mechanism, while providing the desired statistics. Simulation methods for Nakagami fading have essentially borrowed from well-known techniques for random variable generation [123]. We consider two family of approaches for Nakagami fading channel generation. The first one generates continous random variables (r.v.'s), i.e. which can theoretically take any positive real value (the actual number of values is limited only by machine precision). The second one generates discrete r.v.'s, i.e which can only be drawn from a limited (but possibly very large) set, with each value corresponding to a certain state. We wil l denote these approaches by continous simulation models and discrete simulation models, respectively. 2.4.1 Continuous Channel Simulation 2.4.1.1 Introduction We further classify continuous simulation models into those which can be used in generating multiple sequences of mutually-correlated r.v.'s, and those used in generating a sequence of time-correlated r.v.'s. The first class is useful in applications where space diversity is used (i.e. an antenna array with mutual correlation between each or some of its branches). The second class is useful in generating consecutive samples of a single fading channel, in which there is a positive correlation from one sample to the next one. 62 2.4.1.2 Mutually (Spatially) Correlated Nakagami Random Variables We review simulation methods according to the number of mutualled correlated se-quences they can generate, which we denote by L (i.e. the number of diversity branches). L= 1 The simplest case consists in generating one sequence of independent Nakagami variables. As mentioned in Section 2.3.2.3, the Nakagami pdf with m real is equivalent to the pdf of the square-root of a gamma random variable G(a, /3), with parameters a = m and j3 = m/Q, where fl = m2a2. Indeed, the gamma pdf is given by: pr(x) = J^-x^e-**, x>0 (2.120) Hence the pdf of the square root of a gamma variable is obtained by simple transformation [26] as: PMX) = 2 f ^ ) x 2 Q " l e _ / i X 2 ' X ~ ° ( 2 ' 1 2 1 ) from which the equivalence with the Nakagami pdf can be immediately made. Hence, to generate a Nakagami variable r with parameters {m, Q}, one can generate a gamma variable g with parameters a — m and /3 = m/Q and take its square root: r = y/g. (2.122) Fortunately, several techniques are available to efficiently generate gamma variables. When m > 1.0, the following methods can be used [123]: • Envelope rejection methods: - Cauchy method ([124], [123] pp. 108-110) - Log-logistic method ([125], [123] pp. 110-111) - t-distribution method ([126], [123] pp. 111-113) • Ratio of uniforms method: ratio method ([123], p. 118-119) When m < 1.0, the following methods can be used [123]: 63 • Envelope rejection method: switching algorithm ([123] pp. 116-118, [124]) • Ratio of uniforms method: power transformation method ([123] pp. 115-116) • Beta method ([123] pp. 108-110) We've implemented and tested each of the above methods. Table 2.1 shows the bit error rates (BER) for Binary Phase Shift Keying (BPSK) modulation obtained using the first group of methods, i.e. the Cauchy (C), log-logistic (LL) , t-distribution (tD) and ratio (R) methods, for m = 2.0, with a fixed energy per symbol Eb = 1000 W , and a noise spectral density iVo varying from 1 to 951 W in steps of 50. In each case 10 million samples were generated. It can be seen that the numerical results obtained with the different methods agree by usually at least two representative digits. Table 2.2 shows similar results obtained using the second group of methods, i.e. the switching (S), power transformation (PT) and beta (B) methods, for m = 0.6. [123] finds that the log-logistic and t-distribution methods are the fastest in the first group if all the variables need to be reset for each iteration (otherwise the ratio method can be faster), while the switching method is the fastest in the second group. However, as we have verified and as detailed in [123], the execution times are all within the same order of magnitude within each group (and depend on the implementations), and hence there is no single method that greatly outperforms the others for the whole range of m. [127] proposes an algorithm to generate 2 correlated Nakagami r.v.'s with identical m-parameters but unequal powers {Qi , f t 2 }- Their method is based on a modification of the inverse transform method [123]. The algorithm is summarized as follows: 1) Generate a Nakagami r.v. as L = 2 (2.123) 64 N0 B E R : C B E R : L L B E R : tD B E R : R 1 0.000001 0.000001 0.000000 0.000001 51 0.001650 0.001651 0.001657 0.001651 101 0.005615 0.005642 0.005648 0.005574 151 0.010991 0.010988 0.010998 0.010997 201 0.017151 0.017129 0.017222 0.017158 251 0.023824 0.023824 0.023796 0.023788 301 0.030762 0.030742 0.030680 0.030737 351 0.037768 0.037620 0.037764 0.037765 401 0.044680 0.044674 0.044711 0.044639 451 0.051482 0.051462 0.051619 0.051509 501 0.058106 0.058152 0.058148 0.058206 551 0.064765 0.064725 0.064651 0.064774 601 0.071051 0.071139 0.071131 0.070957 651 0.077261 0.077294 0.077343 0.077153 701 0.083280 0.083109 0.083261 0.083149 751 0.088827 0.089031 0.089133 0.088980 801 0.094629 0.094606 0.094495 0.094701 851 0.100009 0.099882 0.099977 0.099928 901 0.105191 0.105238 0.105325 0.105229 951 0.110328 0.110424 0.110336 0.110287 Table 2.1 Comparison in terms of B E R of methods for generating Nakagami random variables with m > 1.0: m = 2.0. where Uj, j — 1,2,... , TO, are uniformly distributed r.v.'s over [0,1]. 2) Solve for r 2 in F{r2\r{). = u (2.124) 65 N0 B E R : S B E R : P T B E R : B 1 0.003530 0.003503 0.003526 51 0.036516 0.036558 0.036595 101 0.054494 0.054436 0.054440 151 0.068661 0.068613 0.068621 201 0.080593 0.080624 0.080458 251 0.091017 0.090969 0.090988 301 0.100364 0.100496 0.100349 351 0.108998 0.108932 0.109038 401 0.116741 0.116638 0.116595 451 0.123994 0.124278 0.124088 501 0.130805 0.130807 0.130764 551 0.137328 0.137448 0.137052 601 0.143009 0.143295 0.143240 651 0.149024 0.148831 0.148593 701 0.154081 0.154113 0.154121 751 0.159331 0.159095 0.159219 801 0.163945 0.164113 0.164147 851 0.168676 0.168739 0.168663 901 0.172953 0.172914 0.173033 951 0.177327 0.177198 0.177216 Table 2.2 Comparison in terms of B E R of methods for generating Nakagami random variables with m < 1.0: m = 0.6. where u is a uniformly distributed r.v. over [0,1], and pr-2 fT2 F{r2\ri) = / p{r2\ri)dr2 = Jo Jo P(ri,r2) p(n) dr2 -I 4 m T O + 1 ( r i r 2 ) T o T{m){n1n2)m^(l-p)pmf-1 Im-l 2m^/pr1r2 « ( 1 - P) x • 2 Jlm-1 rdr2 r ( m ) ' 2 l - Qm (n e " i n, ' i 2p ( i - p ) r r 2 v a - p ) t (2.125) Due to Eq. (2.123) the algorithm as presented in [127] is restricted to m integer. To extend it to m real, we can substitute Eq. (2.123) with Eq. (2.122). A n algorithm for the 66 efficient calculation of the generalized Marcum Q-function Qm(a, /3) is given in Appendix C. L arbitrary [128] proposes an algorithm to generate L correlated Nakagami r.v.'s with identical m-parameters but unequal powers. The method is based on a decomposition of the fad-ing process, by generating correlated Gaussian r.v.'s with a correlation matrix specified such that the Nakagami r.v.'s obtained by a transformation from these Gaussian r.v.'s have the desired correlation matrix. The method works for m integer, however only an approximation is proposed for m arbitrary real. The algorithm is summarized below, z = \z\Zi... ZL] is the vector of L mutually correlated Nakagami envelopes, with L x L covariance matrix C z = {/%}. 1) Determine Vij, i,j = l,2,...,L,i^j (vu = 1) by solving: Pij = r 2 ( m + | ) r (m)r (m + i ) - r 2 ( m + i ; 2) Determine the L x L covariance matrix C x as: 1 _ 1 _ 2 ' _ 2' m ' ' - 1 (2.126) Cvar[zi] i = j C,(ysx{zi}vQx{zj]vij)^ i^j (2.127) where: var[zj] = i r 2 ( m + p -m T 2 (m) a (2.128) C = 1 2m 1 -1 T 2 ( m + m T 2 (m) (2.129) 67 3) Determine xfc = [xkixk2 •.. xkL], k = 1,2,. . . , JV, with N = 2m if 2m is integer and 2m + 1 otherwise: xfc = L x e f e (2.130) where L x is obtained by the Cholesky decomposition of C x = L X L ^ ((-)H denotes the Hermitian transpose) and ek = [ek\ek2 ... eki], k = 1,2,. . . ,N, are Gaussian vectors with zero mean and covariance the N x N identity matrix I. 4) Determine y = [yxy2 ...yL] as: E J S i A 2 m integer a E L P xfc + £ X P+I otherwise where the squaring operation applies to each element of the vectors and: (2.131) 2pm + \/2pm{p + 1 — 2m) a = (2.132) p(p + l) /3 = 2m-pa (2.133) and p = |_2mJ. 5) Determine z as (the square-root operation applies to each element of the vector): z = y 5 . (2.134) Note that a decomposition method relying on the same principles was also presented in [129], for m integer. 2.4.1.3 Time-Correlated Nakagami Random Variables Rayleigh Channel Simulators Several methods are available to simulate temporally correlated Rayleigh fading channels, either in the time or frequency domain. Below, we give a brief overview of the main categories of simulators: 68 • Sum-of-sinusoids simulators: the fading distribution is obtained by taking the en-velope of a sum of cosines (or sines) with given individual amplitudes and phases. This approach is used for example in the Jakes simulator for Rayleigh channels [15], [130], and was inspired by the work in [92]. Modifications to the Jakes simulator are proposed in [131], [132]. Other sum-of-sinusoids simulators are given in [133], [134]. By adding a constant term to this sum, a Rice simulator can be obtained, as in [122]. • Frequency-domain filtering simulators: a computationally efficient approach con-sists of generating in-phase and quadrature Gaussian samples in the frequency domain, shaping them using the desired spectrum, and then obtaining the time-domain envelope using the Inverse Discrete Frequency Transform ( IDFT) . Simula-tors built according to this principle are described in [135], [16], [136], [137]. • Time-domain filtering simulators: a sequence of Gaussian variables is processed by a time-domain filter in order to obtain the desired spectrum [138]. • Autoregressive (AR) simulators: the fading process is fitted to a (high-order) A R process [139]. Nakagami Channel Simulators Several approaches have been used in the literature for simulating a temporally corre-lated Nakagami fading channel. However, due to the lack of a solid physical basis for the Nakagami fading channel, all of the proposed models need to make assumptions or approximations regarding the fading process. Below we briefly review some of the main approaches along with their limitations. • Approximating the Nakagami channel by a Rice channel: - One of the first proposals for a Nakagami simulator appeared in [122]. The au-thors approximated the Nakagami channel as a Rice channel, by matching selected moments. This approximation was initially pointed out in [23]. However, as noted in [140], [141], the approximation doesn't hold for the tails of the pdf, which are essential in the determination of the B E R . We verified that a simulator based on such an approach approximates well the B E R for low signal-to-noise ratios (SNR's), 69 but for high SNR's a large discrepancy is observed. - In. [134], the authors considered a sum-of-sinusoids simulator, and tried to deter-mine the optimal coefficients (amplitudes and phases of the cosines) to approximate a Nakagami distribution. Unfortunately, the resulting approximation also suffers from the same problem as the previous approach: there are large mismatches in the tails' areas. • Modeling the fading as an A R process: - [142] uses an A R approximation for the fading process, coupled with the inverse transform method, in order to simulate an arbitrary number of Nakagami fading channels correlated both temporally and spatially. The resulting simulator is quite complex and only approximates the fading process. - [128] also uses an A R approximation for the fading process, and uses the decompo-sition method described above to produce the correlation. The resulting simulator also gave unprecise results. • Modeling the fading as a Rayleigh process and mapping the Rayleigh amplitude to a Nakagami amplitude using the inversion method: [88] assumes that the phase of the Nakagami fading process has a uniform pdf (like the underlying Rayleigh process), however there is no strict theoretical or experimental justification for this assumption. Also, the envelope correlation is determined from the simulation model, and not the opposite. • Modeling the fading envelope as the product of a square-root beta process and a complex Gaussian process: the method of [143] is restricted to 0.5 < m < 1, and the implementation is quite complex. Hence it is not yet clear whether some of the assumptions or approximations used above can be considered valid or not, and more theoretical and/or experimental work is needed in this direction. 70 2.4.2 Discrete Channel Simulation 2.4.2.1 Introduction The assumption that a fading channel can be modeled as a Markov chain can be used to design various low-complexity simulators. Instead of generating the correlated envelope of the channel, which then needs to be processed to determine if a decision error is made, one can directly produce the error process seen at the output of the demodulator. For over half a century, researchers have tried to fit such discrete-time models to realistic radio channels [144]. These models range from the early Gilbert-Ell iott two-state channels [145] to more complex models such as those based on hidden Markov chains [146]. The finite state Markov channel (FSMC) proposed in [89] has attracted quite some attention due to its good balance between accuracy and complexity. It is based on the partitioning of the received S N R in a finite number of states, an approach which was also presented independently in [147]. The use of a first-order Markov process to model the envelope of a Rayleigh channel has been shown to be a good approximation in [148], using an information theoretic criterion, and was discussed recently in [149]. A Rician channel was also simulated using the same approach in [150]. A higher-order model was proposed to represent Nakagami channels in [151], however the complexity of the calculations (requiring the numerical evaluation of double integrals) make its use less attractive. Several variations of the F S M C were examined for example in [152], [153], [154]. In previous papers F S M C ' s were mostly designed to model flat fading channels. If one needs to simulate the multiple paths of a channel in order to take into account diversity, the execution time of a waveform simulator increases with the number of diversity branches. Instead, one can generate directly the error process seen at the output of the diversity receiver. This technique was used in [101] for SC and Rayleigh fading, and was slightly discussed in [151] for an approximation to E G C in Nakagami fading. This section tackles the design of F S M C ' s for SC, M R C and E G C , in a generalized fading (Nakagami) environment. Most of the Nakagami fading simulators proposed up to date are complex and time-consuming [142], [143], [128], and many are restricted to particular values of the m-parameter (i.e. m integer). Our goal is to design a Nakagami simulator which generates samples at a very high speed, while giving a satisfactory accu-71 racy: this simulator wi l l be useful in simulation studies requiring a very large number of samples, such as the performance evaluation of digital video signals over fading channels. Indeed, the use of F S M C ' s has proven popular in previous papers dealing with wireless video transmission [155], [101], however only Rayleigh statistics were usually used. This section extends the use of F S M C ' s to Nakagami fading channels with diversity. In the next sections we derive the parameters used by the F S M C , for the three diver-sity combining methods, in addition to the non-diversity case: the analytical steady state, transition and error probabilities. These parameters are integrated in a low-complexity simulator, whose first and second order statistics are compared with theoretical expres-sions in Section 2.4.2.6. 2.4.2.2 Review of F S M C Model We use the approach first proposed in [89] to construct a F S M C . Let r be the combined envelope of the channel at the output of the diversity receiver, and 7 = r2Es/N0 the postdetection S N R per symbol of the received signal, where Es = Eb\og2M is the average energy per symbol (with Eb the average energy per bit and M the constellation size) and iVo is the single-sided power spectral density. Let pr(j) and -^HT) = Jr] Pr{a)da be the pdf and cdf of 7. We define K partitions for 7 such that if Tk < 7 < Tk+\, A; = 0 , 1 , . . . , K — 1, then the F S M C is said to be in the state sk- The T^'s are the thresholds of the partition, with To = 0 and T^- —> 00. A simple way of choosing these thresholds consists in specifying that the steady-state probabilities Trk of each state be all equal, i.e.: for k = 0 , 1 , . . . , K — 1. The set of equations (2.135) must be solved numerically (or analytically if a closed-form solution exists) for the thresholds Tk, k = 1,2,... , K — 1. This equal probability method ( E P M ) was proposed in [89]. Optimization of the thresh-olds using least squares quantization and the Lloyd-Max algorithm was later suggested in [155]. However, the latter requires much more computations, and as the number of (2.135) 72 states increases the advantage in accuracy with respect to the E P M diminishes. We thus rely on the E P M throughout this work. Let 7 = E[y] = flEs/N0 be the average S N R per symbol of the received signal, with Q = E[r2]. The average S N R corresponding to the state k is then: 1 fVk+l 7/c = — / . apr(a)da. (2.136) 71"* Jrk In a first-order Markov model, transitions are possible only between adjacent states. In a slow fading environment, the variations in the received S N R during a symbol period are slow enough that we can consider only adjacent state transitions without incurring a significant penalty. Let tij denote the transition probability between states Sj and Sj. Following [89], these can be approximated as: tk>k+1~Nk+1/RW, k = 0,1,2,... ,K-2 (2.137) tk,k-i^Nk/RW, = 1,2,... ,K-1 (2.138) (k) where Nk is the theoretical L C R evaluated at Tk, and Rs = irkRs is the average number of symbols transmitted per second during which the S N R is in state sk, for a symbol rate Rs. The remaining probabilities are deduced using: ^0 ,0 = 1 — *0,1) IK-\,K-1 — 1 — * K - l , K - 2 , (2.139) tk,k = 1 - tfc,it-i - **,*+!. k = 1,2,... ,K-2. (2.140) The symbol error probability for each state is calculated as: Pk = — Ps(a)pr(a)da (2.141) TTfc Jrk where ^ ( 7 ) is the average symbol error probability for a nonquantized model, conditioned on the SNR. For coherent detection and B P S K : Ps(7) = Q(v^) (2.142) 73 where Q(x) = / x°° exp ( - a 2 /'2) da. Following the procedure given in the Appendix of [89], Pk can be written as: e 2 '>/2a V 2 7 T dx pr{a)da = — / / pr{a)da -j=dx + / / \/2r*+1 pr(o;)(ia; e 2 2ir dx Fr ( - ) - F r(r f e) e 2 2vr dx + [ F r ( r f e + 1 ) - F r ( r f c ) ] ^ = L d x x/WkTi V2-T y - f - ^ r ( r f c ) \Q(y/2Tk) - Q( + / + [ F r ( r f c + 1 ) - Fr(Tk)] Q(y/2T^)) TTk \ L J / — ( F r ( r f c + i ) Q ( v / 2 T W ) - F r C r ^ Q C v ^ ) + 7Tt. V (2.143) where V 2 r * + 1 / _ 2 \ e _ V (2.144) Eq. (2.143) can be rewritten as Pk — —(6s+i — (2.145) where: 6 = Fr(r,)Q( v / 2rT) + //=, ' (2.146) (2.147) 74 For noncoherent detection and M-ary orthogonal frequency-shift keying modulation Below, analytical expressions are provided for the parameters {Nk} and {h}, used in Eqs. (2.137)-(2.138) and Eqs. (2.145)-(2.146) respectively, for different diversity tech-niques in a Nakagami fading channel. Once the cdf of the S N R is known, the {Tk} are solved for numerically using Eq. (2.135). The obtained simulator needs only to com-pute these parameters at the beginning of a set of (multiple) simulations. After that, in the simulator's software implementation, only two (uniform) random numbers need to be generated for each iteration: one to determine if a state transition has taken place, and one to determine if an error has occurred. This drastically reduces the computa-tional complexity compared to waveform simulators, which need for example to compute a sum of sinusoids for each iteration (Jakes' method, [15]), or to perform the Inverse Fast Fourier Transform (IFFT) of a block of samples the size of the total number of iterations (Smith's method, [135]). 2.4.2.3 No Diversity The pdf and cdf of the received signal S N R for a Nakagami fading channel and no diversity are given by: ( N C - M - F S K ) [26]: (2.148) (2.149) 7(m, f 7) r(m) (2.150) 75 From Eqs. (2.135) and (2.150), the thresholds are obtained by solving numerically (e.g. using the bisection method [156]) the following equation for Fk, k — 1, 2 , . . . , K - 1: ( m \ ( m \ F(m) 7 (m , - r f c + 1 j - 7 [m, -^rk j = — knowing that r 0 = 0. The average SNR corresponding to state k is j/m Ik 7TI \ i TTt 7 ( m + 1, — Yk+l - 7 m + 1, — Tk 7 / V 7 7rfer(m) From [96] or [98] (c.f. Eq. (2.43) for L — 1), the L C R ' s can be obtained as: ^fm ( N, = r(m) \y e i BPSK Using the following infinite series expansion for j(a,x) in Eq . (2.150): n = 0 n\ a + n and substituting Eqs. (2.150) in (2.147) results in: 1 " (_l)n (|)™+" ^ h ~ f ( m ) ^ n ! ( m + n) ^ Making the change of variable y = x2/2 and using Eq. 8.350 of [80] leads to : ( \ m+n * J 7 ( m + n + | , r f c ) h = 20rT(m; E n = 0 n\(m + n) (2.151) (2.152) (2.153) (2.154) x 2{m+n) 2 e'^dx. (2.155) (2.156) 76 NC-M-FSK Substituting Eqs. (2.149) and (2.148) in Eq. (2.141) leads to f i M - l \ 1 1 Pk. = nkT(m) \ j m ) m M-l E( - ! ) m i = l ' 0 1 + 1 / _ J i_ rn i+1 " r 7 7 ™, + — r f c + 1 - 7 m, 1 + ™ i r , i + 1 7 (2.157) ' \i + 1 ' 7 2.4.2.4 Selection Combining (SC) Diversity In the rest of this section we consider only the case of independent diversity channels and identical fading parameters for every channel, for which closed-form expressions for the L C R have been found in Section 2.3.1. The pdf and cdf of the output S N R of a L-branch diversity combiner are: l(m, f 7) T(m) -I L - l ) ™ m—1 7 p 1 ' T(m) (2.158) ^ ( 7 ) = T K f r ) T(m) (2.159) From Eqs. (2.135) and (2.159), the thresholds are obtained by solving numerically the following equation for Fk, k — 1, 2, ,K-1: 7 m, — Tk+1 7 - L \ ( m-n M - 7 ^m, - r f c J (2.160) For m integer, the incomplete gamma function can be written as (Eq. 8.352.1 of [80]): 7(m, x) = (m — 1)! 1 - e " (m-l E *,n=0 n! (2.161) 77 Making use of Eq. (2.161) in Eq. (2.158), of the binomial expansion (Eq. 1.111 of [80]), and of the following multinomial expansion [157]: E ^ = E&*' U = 0 / k=0 (2.162) the pdf for m integer can be written as: L - l r(m) L - l , - C + 1 ) T T (=0 ( ( m - l ) E « fc=0 fc( 7 7m + f c - 1 . (2.163) In the above equations, f3ki are the coefficients of the multinomial expansion, evaluated using [157]: 'kl — _ _ ? i r rST / [o , ( i - i ) ( i - i ) ] (*) i - f c - ( L - l ) (k-i)V (2.164) with Poo = (30l = 1, Phi — Pu - I and I[a,b]{i) = 1 \i a < i < b and 0 otherwise. Similarly, the cdf for m integer can be written as: (=0 V ' / k=o v 7 7 (2.165) The average S N R corresponding to state k is then Lj/m ^ , L - l \ * / & ^ + m + n + l 1 ( m + n + l,(l + l )^r f c + 1 ^ - 7 (m + n + 1, {I + l)™ r*) • (2-166) The L C R ' s are obtained as (c.f. Eq. (2.43)): Ly/^frn (in, [T(m)r V7 -™r f c e T 7(m, ?r f c) 7 L - l (2.167) 78 BPSK Using Eq . (2.165) in Eq. (2.147), with the aid of Eq. 8.350 of [80] we obtain: El"1)' j \ l(m-l) / \ k / - \ k+k 2^ pa k=0 7 J \lm + j , , 1 Im + 7 ^ (2.168) Recall that this expression is valid only for m integer. For m arbitrary, there is no simple closed-form solution obtainable for Ik, and the e^'s must be calculated numerically via direct integration. NC-M-FSK Substituting Eqs. (2.163) and (2.148) in Eq. (2.141) leads to: L M-l ir*r(m) £ f i + l M - l \ 1 L-l X - 1 m+k i + 7 I m + k, (=0 i L _ 1 \ « ( ™ - 1 ) / m x m + f e _ + + r f e + 1 * + 1 7 / 7 777 7 ( m + fc,(4T + (i + l ) ^ ) r f c .7 + 1 7 (2.169) 2.4.2.5 Maximal Ratio Combining ( M R C ) Diversity W i t h the previous assumption of identical fading parameters, the pdf and cdf of the output S N R are [23]: p r ( 7 ) = ( W iW (2.170) 7 ( m r , ^ 7 ) (2.171) 79 with mT = Lm and yr = £ 7 . The desired quantities can be deduced from Eqs. (2.149)-(2.157) by substituting rn with mT and 7 with yr, i.e.: % = gkterjr*.-** (2.172) I ( m r ) \77' / and: 1 » ( - l ) n ( ^ J / 1 \ 4 = - ' , E F T T " " / m r + n+- , r f c . (2.173) 2.4.2.6 Equal G a i n Combining ( E G C ) Diversi ty We consider only the cast; of dual-branch diversity (L = 2), for which we. found in Section 2.3.1.5 the following closed-form and infinite series representations for the pdf and cdf, respectively: » W - pg^ g)^ .^**.*!.^ ). (2.174 ^r(7) = 00 Y ^ F(2m + n) 1 / m , / 0 1 7 - ^ 7 ~ 7 2m + n, 2—7 . (2.17o) [r(m)] 2 2 4 m - a ^ r(2m + 71 + ±) 2nn\ ' \ 7 From Eqs. (2.135) and (2.175), the thresholds are obtained by solving numerically the following equation for Ffc, k = 1,2,... , K — 1: r(2m + r i) '•Zt T{2rn + n~~+Y) 2»n! E n=0 771 7 I 2m + n, 2—r f c + 1 7 7 ( 2m + n, 2—Tk [r(m)]22 o4m—2 (2.176) 80 The average S N R corresponding to state k is 7fc y/nj/m T(n + 2m) 1 7r*[r(m)]224m-2 ^ T(n + 2m + | ) 2"n! , 2rn \ / , 2rn 7 2m + n + 1, — Tk+1 - j 2m + n + 1, — r f c 7 / V 7 (2.177) We previously obtained the L C R ' s as (c.f. Eq. (2.67)): 2irfmB{2m,\) fm 22 ( m - i ) [ r( m)]2 \y 2m- 1 m . e 7 2 m , 2 m + - , — r f c . (2.178) 2 7 Substituting Eqs. (2.175) in (2.147), and making use of Eq. (2.154) and Eq. 8.350 of [80] results in: h = E 3=0 1 A T(2m + n) 1 ] 2 9 4 m - l / > n=0 (2?) [ r (m) ] 2 2 4 r n - 1 ^ T(2m + n + §) 2 n n! 2m+n+j 1 j ! 2m + n + j V 2 (2.179) The previous doubly infinite series can become unstable when calculated using limited precision software, thus one might use direct numerical integration to evaluate I K . 2.4.2.7 Numerical Results and Discussion Uncoded BER The theoretical uncoded B E R ' s for B P S K for the three types of combining schemes, in i . i .d. Nakagami channels, are obtained as /•oo Ps = / Q(v^7)Pr(7)rf7 (2-180) Jo 81 where the pr{j) were given previously. Closed-form expressions for these B E R ' s have been obtained in previous literature and are given below. Selection Combining L - l (m >' 1=0 I l(m-l) , . /-, \ m+k 1^ Pkl-k=0 + m+k m + k - l + h \ (\ + l H h=0 h (2.181) where m is integer, and fj,t - + (I + l)m). Eq. (2.181) is similar to Eq. (18) of [158], however some small typos in the latter have been corrected. Maximal Ratio Combining 1 y/l/m P., r ( m + | ) 2 v ^ [ ( m + 7 ) / m ] m H r ( m + l) V 2 ™ + 7 Eq. (2.182) is given for example in [39]. For m integer, it reduces to [39]: Ps = 2 m—1 fe=0 2/fc k 1-fj2 2 \ (2.182) (2.183) where /i — \A)7(m + 7)-Equal Gain Combining, L — 2 1 fl T?2™ F(m + i ) = - - 2 A / - -2 V TT ( i + 2 r ? ) 2 m + i r(m) x F 2 (2M + I; 1,2m; ?, 2m + I ; ^ ^ | (2.184) 82 where r\ — m/j, and F2{a; bi, 62; C i , c 2; x, y) is the Appell hypergeometric function of two arguments ([80], Eq. (9.19)). Eq. (2.184) is derived in Section 2.5.3.2. For each diversity type, the B E R ' s obtained via the proposed simulator are compared to the theoretical values for different values of the m parameter. L — 2 branches are used in the results, however any number of branches can be accommodated (except for the E G C case). The fading power in each branch is normalized to 1.0. The F S M C has K = 16 states. The estimated B E R was averaged over 100 simulation runs, each one producing 10 6 samples. Figs. 2.15-2.17 illustrate the B E R versus the S N R per bit per diversity branch: it can be seen that the B E R ' s obtained from the simulator match very well the analytical curves. 10° E b / N Q per diversity branch (dB) Figure 2.15 B E R for SC: (a) m = 1, (b) m = 2, (c) m = 3. F S M C simulation: ++ ; theory: 83 E b / N Q per diversity branch (dB) Figure 2.16 B E R for M R C : (a) m = 1, (b) m = 2, (c) m = 3. F S M C simulation: ++ ; theory: — . E b / N Q per diversity branch (dB) Figure 2.17 B E R for E G C : (a) m = 1, (b) m = 2. F S M C simulation: ++ ; theory: — . 84 Level Crossing Rates Figs. 2.18-2.20 compare the normalized L C R ' s of the F S M C (Nn) with the cor-responding normalized theoretical expressions, where in both cases the normalization factor is l / / m . The L C R ' s are plotted as a function of the normalized received envelope rn = r/y/Q (in dB) . For the F S M C , the value of the channel envelope when the model is in state sk is computed as rk = y/lk/l- K — 64 states were used in order to get sufficient data points, and L — 2 diversity branches. As can be seen from the plots, the model generates level-crossing statistics very close to the theoretical ones. z -10 -8 -6 -4 -2 0 2 4 6 r n(cB) Figure 2.18 L C R for SC: (a) m = 1, (b) m = 2, (c) m = 3. F S M C simulation: ++ ; theory: 85 86 Coded BER Performance The previous two sections showed that the F S M C produces average L C R ' s (and con-sequently A F D ' s ) and uncoded B E R ' s very close to the theoretical ones: this is to be expected, since the F S M C dynamics are based on the theoretical L C R ' s , and the error rates in each state are derived from the theoretical B E R ' s (one could alternatively add white Gaussian noise to the generated envelope r in order to calculate the decision met-rics for any modulation scheme: this would add one more processing step, but wouldn't require the knowledge of the theoretical B E R ' s ) . However, the generated envelope does not necessarily possess the other higher order statistics of the theoretical envelope: for example, [149] shows that depending on the fading rate (i.e. the maximum Doppler shift fm times the symbol period T 5 ) , the generated envelope does not necessarily match the autocorrelation function and the probability distribution of the fade durations (contrary to the average fade durations). Nevertheless, it is shown below that for small fading rates, the coded B E R obtained by applying error correction to the output of the F S M C is reasonably close to the coded B E R obtained using a traditional waveform simulator. Indeed, the design of the F S M C was based on the slow fading assumption [89], and if we depart significantly from this assumption the behavior of the F S M C wil l also diverge from the theoretical one. In the following simulations, the bit rate is taken as 76.8 kbps, and groups of 1536 bits (including 8 added tail bits) form frames of 20ms. These frames are encoded with a rate 1/4 convolutional code, with generator functions (765), (671), (513) and (473) (in octal form). These parameters are taken from the cdma2000 standard [14], for medium-rate data or video transmission. The receiver performs hard-decision decoding using the Viterbi algorithm. Figs. 2.21-2.23 show the coded B E R ' s (as a function of the S N R per branch Eb/N0) obtained with both the F S M C and a waveform simulator (WS), for a mobile speed of v = 6 km/h , a fading parameter m = 1, and maximal-ratio combining with L — 1, 2 and 3 diversity branches. The coded B E R ' s are calculated using 160 frames, and are averaged over 100 and 20 simulation runs for the F S M C and WS, respectively. Results for the F S M C are shown for two numbers of states: K — 1024 and K — 2048. 87 It can be seen that the coded B E R ' s for the F S M C and the W S are reasonably close together, especially for lower diversity orders. The number of states needs to be chosen with care. For systems with lower bit error rates, a higher number of states needs to be chosen: indeed, the quantization of the S N R needs to be small enough to allow the existence of several states with very small error probabilities. Otherwise, if the quantization of the S N R is too coarse, the F S M C wil l not be able to produce very small B E R ' s . For example, in Fig. 2.23, for Eb/N0 = 7 dB the F S M C with 2048 states gives a B E R in the order of 10~ 6 - 10~ 5, while a B E R of 0 is obtained with K = 1024. However, there is a limit on the number of states: with K too large, the F S M C is not able to move quickly enough from one state to another (recall that in the current model, one state transition is allowed per symbol period), which leads to erroneous results. The higher the fading rate, the lower the maximum K: for the current simulations, it was found that for v = 100 km/h , K must be smaller than about 200, while K < 400 for v = 50 k m / h (i.e. it appears that the maximum K is directly proportional to the fading rate). Figs. 2.24-2.26 show the coded B E R ' s for the same parameters but with v — 50 km/h , i.e. a higher fading rate. The accuracy of the F S M C is not as good as in the case v = 6 k m / h (for the reasons given above), but overall the coded B E R ' s are roughly in the same range. 2.4.2.8 Conclusions This section presented the fast simulation of Nakagami fading channels using F S M C ' s . The simulators include the effect of diversity reception in order to directly simulate the envelope of the combined signal, thus avoiding the generation of multiple separate channels. It was verified that for slowly fading channels, the simulators can accurately reproduce the theoretical bit error rates and level crossing rates. The coded B E R ' s can be reasonably approximated by a careful choice of the number of states. The simulators developed are thus useful in evaluating the performance of broadband systems, such as wireless video distribution, where a large number of samples need to be generated for performance evaluation. 88 0 2 4 6 8 10 12 14 16 18 20 E b / N Q per diversity branch (dB) 89 Figure 2.23 Coded B E R with M R C and v = 6 km/h, m = 1, and L - 4 - WS - 0 - FSMC 128 slates FSMC 256 states FSMC 384 states E b / N Q per diversity branch (dB) Figure 2.24 Coded B E R with v = 50 km/h, m = 1, and L = 1. 90 91 2.5 Analytical Symbol Error Rates for Diversity Techniques in Correlated Nakagami Fading Channels 2.5.1 Introduction M R C is known to be the optimal combining technique in white Gaussian noise for mobile radio systems employing diversity, however its use requires the careful estimation of the channel-induced fading and phase-shift at the receiver. In contrast, E G C only requires the estimation of the phase-shift (for coherent modulation schemes), which leads to a simpler implementation. Several analyses of M R C and E G C in Nakagami fading have been obtained for systems where it is assumed that the signals received at each diversity branch fade independently [159], [160]. However, in practical mobile radio systems the antennas are close together due to space constraints, which typically leads to non-zero correlation coefficients between the signals, and thus voids the independence assumption. In the case of M R C , many analyses have considered correlated diversity branches with Nakagami fading. Without being exhaustive, [161] obtains the Symbol Error Rate (SER) in double integral form of M-ary Phase-Shift Keying ( M - P S K ) for two diversity branches (L — 2). For two special correlation models, Aalo [162] obtains closed-form expressions for Binary Phase-Shift Keying (BPSK) and Frequency Phase-Shift Keying ( B F S K ) , for a general number of branches L. For more general correlation models, one-dimensional integral expressions with general L are obtained in [163], [164], for B P S K / B F S K , and in [165], [166], [167] for M-a ry Quadrature Amplitude Modulation ( M - Q A M ) and M - P S K . In the case of E G C , much fewer analytical results are available in the literature. It is well accepted that the accurate analysis of E G C over fading channels is a more challenging task, due to the difficulty in finding a simple exact expression for the pdf of the combined signal: indeed, there is still no general exact closed-form expression for the pdf of the sum of L Rayleigh or Nakagami variables, for an arbitrary L. In the case of uncorrelated fading, in [168] the authors used the characteristic function method in order to obtain generic expressions for the S E R of E G C , for a general number of branches L. In the special case of B P S K , their approach leads to closed-form solutions for the SER. However, in the case of higher-order modulations such as M - P S K and M - Q A M , their approach leads to expressions containing double and single integrals, respectively. The 92 case of correlated fading has only been tackled for L = 2. The case of dual-diversity is an important one, since the greatest diversity gain is obtained when the number of antennas is increased from one to two: hence several commercial wireless systems (such as outdoor cellular and indoor cordless base stations) employ only two antennas. Previous work on dual-branch E G C in correlated fading include [161], which obtains the S E R of M - P S K in the form of a double integral, and [169] and [170], which obtain closed-form solutions for the S E R of B P S K in Rayleigh and Nakagami fading, respectively. In this section, we derive the exact theoretical S E R of the coherent detection of several M-ary modulation schemes such as M - P S K , Differentially Encoded M-ary P S K ( D E - M -P S K 3 ) and M - Q A M , with M R C and E G C , for two correlated diversity branches in a Nakagami-m fading environment. Several novel expressions are obtained, either in the form of a single-dimensional integral, or in closed-form as infinite series summations. In the case of M R C , closed-form solutions for the S E R can be derived by using an alternative expression for the pdf of the sum of the squares of two correlated Nakagami-m variables. In the case of E G C , the S E R is obtained by first deriving the pdf of the sum of two correlated Nakagami-m variables, for which, to our knowledge, no closed-form expression was previously available in the open technical literature on communications. We also derive the exact pdf of the sum of two independent Nakagami variables with unequal SNR's 7i,72 and fading parameters m i , 7722 on each branch, when the m^'s are arbitrary multiples of half integers. Using this pdf, expressions for the S E R are obtained containing only a single integral (as opposed to a double integral in [168] for certain modulations such as M - P S K ) . 2.5.2 General Approach Let r = / ( r i , r 2 ) be the normalized output of the diversity combiner, where /(•) depends on the diversity technique used. We use the classical approach to the derivation of error probabilities for fading channels [26], by first deriving an expression for the pdf Pn(r) of the combined signal r, and then directly averaging the conditional S E R Ps(j) over the pdf of the signal-to-noise ratio after combining, which leads to the unconditional 3 Not to be confused with noncoherently-detected M-ary Differential P S K (M-DPSK) [22]. 93 SER: /•oo Ps = / Ps(l)Pr(l)dj, (2.185) Jo where 7 = r2Es/N0 is the instantaneous S N R per symbol, with pdf p r ( 7 ) = ^ ' ^ ™ ' (2,86) and average value 7 = E[r2]Es/N0 = QES/N0. Es — Eb\og2M is the energy per symbol (with Eb the energy per bit and M the constellation size) and N0 is the single-sided power spectral density. The conditional S E R for several different M-a ry modulations, such as M - P S K , M-Q A M and M - P A M , can be expressed in a unified manner as [167]: ps(j) = £ [°l ai(9)e-^d9 (2.187) 1=1 J o where the parameters K, ai(9), (f>i(9), 0 ; are given in Table 2.3, in which f{9) — Sg§^ - 1 (c.f. [167], [171]). K a/(0).7r 0,(0). sin' 2 0 0,/TT M - P S K 1 1 1-k M - Q A M 2 3 2 (M - 1 ) 1/2 3 2 (M - 1 ) 1/4 M - P A M 1 2 ( 1 - ^ ) 3 M 2 - l 1/2 D E - B P S K 2 2 1 ' 1/2 -2 1 1/4 D E - Q P S K 4 ' 4 1/2 . 1 / 2 -8 1/2 1/4 ! c o s - V W 1/2 1/6 ^ ( T T - C O S - ' / W ) 1/2 TT \/3 Table 2.3 Parameters associated with the SER's of M-ary modulations [167], [171]. 94 Below, we also recall expressions for the individual SER's for M - P S K , B P S K , Q P S K , M - Q A M and D E - Q P S K , conditioned on 7 [22]. We also give alternative forms, obtained using Eq. (A.31), which wil l be helpful in obtaining expressions in fading channels. p M - P S K (2.188) ^ B P S K ( 7 ) = Jerfc(V7) V ^ ( l , | , 7 7T \ 2 (2.189) (2.190) p Q P S K ( 7 ) = erfc(V^ 2) - e^rfc2(y772) 3 4 2TT V 2 ' 2 / 2TT $ 1 3 1 2 ' 2 (2.191) (2.192) p M - Q A M ( 7 ) = 2 g e r f c ( ^ ) - g 2erfc 2( v/p7) 2 g - « 2 - 4 ( g - ( r l ) ^ e - ^ * ( l > | > p 7 -Aq 2 ^ 7 e - 2 p 7 7T * ( 1> yPl (2.193) (2.194) where q = l- 1 / V M , p = 3 / ( 2 ( M - 1)). p D E - Q P S K ( 7 ) = 2 e r f c f j 2 , _ 2 erfc erfc 3 7 _ 7 e 7 4 7T $ [ 1 l l 1 ' 2 ' 2 + T 2 2 _ 2 _ p - 2 7 9 7T erfc $ 1 3 I 2 ' 2 (2.195) .(2.196) Eq. (2.196) is obtained by using Eq. (A.31) for each term of (2.195), expanding the powers and simplifying. 95 2.5.3 Symbol Error Rate for Equal-Gain Combining 2.5.3.1 Unequal Branch SNR's, Identical m-Parameters (2m Integer) Let r = (r\ + r 2 ) / \ / 2 be the normalized amplitude of the signal at the output of the equal-gain combiner, r; being the envelope of the fading affecting the signal on the ith antenna. Then, the exact pdf of r is obtained as: rry/2 P R i , f l 2 ( r i ) r 2 = rV2 - ri)dri (2.197) Jo where PRuR2{fi,r2) is the joint pdf of r x and r 2 . For Nakagami correlated r.v.'s with different powers and identical m-parameters: where m > 0.5 is the Nakagami fading parameter, Qj = E[rf\ is the average fading power of each channel, p is the power correlation coefficient between r\ and r 2 . Hereafter we briefly summarize the main steps followed in solving (2.197). We first use the series expansion ([80], Eq. 8.445) ^ 1 /z\v+2k w = E m , + t + 1) {-,) ("») k=0 in Eq. (2.198). Upon substituting in Eq. (2.197), we complete the square in the exponential, make the change of variable y = \Jm/{\ — p)[ri\/(Qi + Q2)/(QiQ2) — r x / 2 ( Q i / f t 2 ) / ( ^ i + ^ 2 ) , use twice the series expansion (y + a)n = Y^H=O {^Vlan~l ([80], Eq. 1.111) (which necessitates 2m to be an integer), and solve for the integral using the incomplete gamma function j(a,x) = J* e~Ha~1dt. This results in the following novel expression for the pdf of the sum of two correlated Nakagami-m variables with arbitrary 96 powers and identical m-parameters (2m an integer): PR(T) = 2^/2 mh r(m) Q p ( l - p ) E 2fc X 2m-l+2fc E ( - l ) S ' 2 m - 1 + 2fe> fc=o m - i + f c - i 277J-1+2A; E ' k\Y(m + k) 2m - 1 + 2fc^  ' m f i 2 _ 2 . 1 - P i + j + 1 m f i i 2 1 - p r2 + ( - 1 ) ' 7 i + j + l mQ.2 ^2 2 ' l - / 9 r (2.200) where we have defined Vts = Qi + ^2, QP = ^1^2, 1^ = (^1/^2)/^) ^2 = ( f ^ / ^ i V ^ s The pdf of the combined S N R is then: Pr(7) = \[2 m5 r (m) ft™ - p) _ _ 2 m T . 1 ^—\ [•v/f2Dp/f2sl 2k X 2m-l+2fc E i=0 '2771-1+2^^ 2 — ^ - 7 1 - P A;=0 -1 m - i + * - 5 2m-l+2fc / k\T{m + k) 2m - 1 + 2fc> x 3=0 \ J ( - 1 ) 7 s , 2 - -7 + ( -1 ) J 7 o :T m Q 2 2 i T i 1 - p ' 1 - P (2.201) Substituting Eqs. (2.187) and (2.201) in Eq. (2.185) and using Eq. 6.455.2 of [80], the following expression is obtained: 2y/2 r(m) 7 f 2m-l+2*: 7P(1 - p) Is m - i 00 ? T(2(m + fc)) ^ fe!r(m + fc) 2*; E i=0 (2m - 1 + 2fc\ m i+fc-i 2 r \ ^ 2 f c - 1 + 2k\ m _ i + i t _ i 1 E (=1 (-ir a i+j+l 2 (ai + /3(0))2(m+ f c)' 2 F i l , 2 ( m + fc); i + J + 1 i + j + 3 a i i+j + l + ( - l ) J (a 2 + /3(0)) 2(m + f c) ' 2 Fx l , 2 ( m + fc); i + j + 3 « 2 2 ' a 2 + 0(0) old (2.202) where we have defined a, = 2m^i/(l — p), i = 1,2, /3(0) = ^(0) + 2m/(js(l - p)), 7s = 7i + 72, % = 7i72, 7i = (71/72)/7»> 7 2 = (72/71)/^, 7i = ^iEs/N0, i-1,2. 97 2.5.3.2 Identical Branch SNR's and m-Parameters For Nakagami correlated random variables with identical parameters = Q2 — ^) [23]: Substituting Eq. (2.203) into Eq. (2.197) results in «*> - i g ^ r ^ - ^ ( - " [ , t ^ n " ) In order to solve the integral in Eq. (2.204), we use the infinite series expansion Eq. (2.199) for / m _ i ( - ) , make the variable transformation y = m / ( 2 Q ( l - p))(y/2ri - r ) 2 , and use Eq. 3.383.1 of [80] to solve the integral, leading to the following expression for the exact pdf of the normalized sum of two correlated Nakagami variables with identical parameters: , s . * I 2mr2 \^pk(l-p)mB(2(m + k),± , 2(m+fc) r (m) F V ^ ( i - p ) / ^ fc!r(m-f-fc)22(m+*-1) V n ( ! - p ) . 2(m + fc),2(m + fc) + - , ^ 1 _ ^ j • (2.205) Note that an expression for the pdf of the sum of two correlated chi variables was given in [118] (recalling that the chi pdf is equivalent to the Nakagami pdf when the m-parameter is an integer), however it contained a one-dimensional integral. The cdf is given by FR(r) = J^pR(a)da, where pR(r) is expressed in Eq. (2.205). B y making the change of variable x = m/(Q(l — p))ct2, using the infinite series expansion 98 and using Eqs. 8.384.1 and 8.350.1 of [80] to solve the integral, the following exact expression is obtained: FO = y y Pk(l-P)m Y{2{m + k)+n) R [ r ) Y(m) ^ ^ k\n\2i(m+k)+n-2 T(m + Jfc)r(2(m + k) + n + \) X 7 ^ ( r a + i ) + n , _ _ j . (2.207) The truncation of the infinite series in Eqs. (2.205) and (2.207) to 50 terms gave an excellent accuracy in all the cases we tried. In the case of uncorrelated fading, i.e. p — 0, it can be easily verified that Eqs. (2.205) and (2.207) reduce to Eqs. (38) and (40), respectively, of [108]. The pdf of r' = y/2r = n + r 2 , obtained through Eq. (2.205), is plotted in Fig . 2.27 against the approximation of [172], for the two correlated cases presented in [172], i.e. m = 1.8, p = 0.7, and m = 3, p — 0.3. In [172], the pdf of the (un-normalized) sum of two correlated Nakagami variables with identical parameters {m, Q} is approximated by the Nakagami pdf with equivalent parameters: ^ 2 , , n 2 , c n 2 P t(fl\2T\m+\) ( „ ( 1 m m \m J r 4 (m) \ V 2 2 Q \ 2 r ( m + \)T(m+ | ) ( I _3_ m) VHm) ~ 2 i < 1 V~2' 2 5 Q 2 r 2 ( m + | ) ^ / 1 1 -8 ; 2>2FX \-----m-p m r2(m) \ 2 2 (2.209) While overall the exact and approximate curves appear to be close, the discrepancy for low values or r' can have a significant impact on the evaluation of the SER, since the latter depends strongly on the lower tail of the pdf. The same can be observed from Fig. 2.28, which plots Eq. (2.207) and the cdf of a Nakagami variable [23] with the parameters specified in [172]. 99 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Amplitude of the combined signal (r') Figure 2.27 Probability density function of the sum of two correlated Nakagami-m variables: exact (through Eq. (2.205)); approximation (using Eqs. (2.208)-(2.209)). Amplitude of the combined signal (r) Figure 2.28 Cumulative distribution function of the sum of two correlated Nakagami-m vari-ables: exact (Eq. (2.207)); approximation (using Eqs. (2.208)-(2.209)). 100 From Eq. (2.205) the pdf of the combined S N R is obtained as: **W = F 7 ^ T e x P - o T T — S E 0 0 nk(l _ ^ " I R Z - O C m J- lr\ I W ™ \ 2(m+fc) 2 m 7 \ ^ P f c ( l - p)mB(2(m + k),\ r(m) F V fc!r(m + fc)22("*+fc-1) ~ p) m X 7 2 ( m + f c ) - 1 $ ^ 2 ( m + 2 ( m + fc) + 1 ^ T ^ ) ) • ( 2 - 2 1 ° ) To obtain the SER, one can substitute Eqs. (2.210) and (2.187) in Eq . (2.185), leading to the following expression valid for an arbitrary real m > 0.5: 1 y> ~ p)mB(2(m + fc), |)r(2(m + fc)) 2 ( m + k ) r(m) ^ fc!r(m + fc)22(TO+fe-1) - ^ x ^ r 0 ' , ^ 2 ^ 1 ( 2 ^ + fc),2(m + fc);2(m + fc) + i ; ^ ) uc7 (2.211] E / a<w: [/3(^)]2(m+fc) where we have defined 77 = 771/(7(1 — p)), and now /3(9) = <f>i(9) + 2m/(7(1 — p)). Using the above derived pdf (Eq. (2.210)) we can also obtain several expressions specific to each modulation scheme, some of them which can be put in closed-form. M - P S K ' . . Substituting Eqs. (2.188) and (2.205) in Eq. (2.185), and using Eq . 7.621.4 of [80], the following expression is obtained: 1 ™p\l-p)mB{2{m + k),\)Y(2{m + k))( m \2("+*> P s ~ 7 r r ( m ) ^ fc!r(m + fc)22("l+fc-1) \y{l - p)) * /T jSp j j ^ 1 (2(m+*>•2<m+*>•2<m+k) + \-m-mp)m)M (2'212) where N(0) = ^ f f l + Eq. (2.212) gives the exact S E R of M - P S K with dual-branch E G C in correlated Nakagami fading. In the case of uncorrelated fading (p = 0), Eq. (2.212) reduces to 1 (rnV- B(2m|)r(2m) r-*J_^ / 2m + - , <ffl(2.213) 101 where = + M >• 1 s in - 1 8 f Special case: B P S K (M = 2) Substituting Eq. (2.190) in Eq. (2.185) gives Ps = \~ \Jl*~^{^\^Ml)dl. (2-214) The second term of Eq. (2.214) is 1 1 ~ pk(l- p)mB{2(m + k),\) nT(m) ^ k\Y{m + k)22^k)-2 m .7(1 -P). 2[m+k) X j f~2(m+*)-i e -( i+2^)7 $ ^ | ^ $ ^ 2 (m + fc), 2(m + fc) + i , ^l} -(2-215) The integral in Eq. (2.215) can be solved using Eq. (A.38) (c.f. Eq. 7.622.3 of [80] and [162], Eq. (A-12)). Further straighforward simplifications then lead to: T ^ p k ( l - p ) m ri2(m+Q rf 1 2 r ( m ) V 7 r ^ fc! [i + 2 r ? ] 2 ( m + f c )+l V 2 , x F 2 ^2(m + fc) + 1, 2(m + fc); ^ 2(m + fc) + ^ ; ^ - ^ ( 2 . 2 1 6 ) where we have defined 77 = 777/(7(1 — p)), and recall that F2(a; bi, b2; c i , c 2; z , y) is the Appell hypergeometric function of two arguments. In the case of uncorrelated fading, Eq. (2.216) reduces to p 1 0 [1 ' v2m r (m + | ) r* = - - 2W - -2 "V 7T (1 + 2 r ? ) 2 m + i r(m) x F l ( 2 m + i i l , 2 m i | , 2 m + i i I ^ , r L ) (2.217) where now 77 = m / 7 . Note that the S E R for B P S K with coherent E G C in uncorrelated and correlated Nakagami-m fading was also previously obtained in [168] and [170], re-spectively, for different parameters on each branch, using a different method based on the characteristic function. The results were also expressed in terms of the Appell hy-102 pergeometric function. Special case: Q P S K (M = 4) Substituting Eq. (2.191) in Eq. (2.185) gives PS = ^ T O erfc (V772 )pr (7)rf7 -^"erfc 2 (v^72 )pr (7)^7- (2-218) The first term in Eq. (2.218) is seen to be equal to A — PS,BPSK{J/2), where PS,BPSK{I) is given by Eq . (2.216). The second term in Eq. (2.218) can be expressed as: B 2 = i f [ 1 - # " ' * ( 1 ' ? 5 ) ] 1 r°° I 1 f00 ( 3 "y\ i r°° • 1 1 ' 2 ' 2 Pr(7)^7- (2.219) The first term of Eq. (2.219) readily evaluates to \ . The integrals of the second and third terms of Eq. (2.219) are solved using Eq. (A.38). Simplifications lead to: P = 3 2 1 y V ( l ~ P r A + f c ) 4 V ^ r ( m ) ^ fc! E r v",,,r/ ^OT+^r ( m + k + ^ | x J i . + 2 7 7 ] 2 ( » * + * ) + £ »7 x F 2 ( 2(m + fc) + ^ ; 1,2(m + fc); 12(m + fc) + l-\ + " r ( 2 ( m + " ) + ^ 3 ( 2 ( m + fc) + l ; l , l,2 ( m + f c ) ; [! + 2 r 7]2(m+*)+l p ( 2 ( m + fc) + I) 3 3 „, , , 1 -,-,2(m + fc) + - ; r + 27?' I + 277' l + 2r? (2.220) where F 3 ( a ; bi, 62,63; C i , c 2 , c 3; x, y, 2) is the Appell hypergeometric function of three ar-guments ([80], Eq . 9.19). The S E R for the uncorrelated case can be obtained by setting p = 0 in Eq. (2.220). 103 D E - Q P S K ( M = 4) Substituting Eqs. (2.195) and (2.205) in Eq. (2.185), the following expression is obtained: P„ 4 7rr(m' E fe=0 pk(l - p)mB(2(m + fc), i)r?2(m + /=) k\T(m + /t)22(m+ f c-!) " r(2(m + fe) + l ) (1 + 2r7) 2(m + f c)+ 1 ( , s , , 3 3 , , 1 1/2 1/2 77 \ F , (2 (m + k) + 1; 1,1,2(m + *)i ? ? 2(m + t) + - ; ^ ^ j + Tr ( 2 X ^ f ' ( 2 ( r o + *) + M . 1.1,2(m + * ) i f, | , | , | . 2 ( - + *) + 1/4 1/4 1/4 1/4 T?/2 I + 77' I + 77' I+77' 1 + 77' l-t-77 (2.221) where Fn(a; bi, b2, • • • , bn; c\, c2, • • • , cn; Xi, Xi,... , xn) is the Appel l hypergeometric func-tion of n arguments ([80], Eq. (9.19)). The SER's for D E - M - P S K with M > 4 can be similarly obtained, but the resulting expressions are increasingly complicated due to the larger number of terms. M - Q A M The unconditional S E R can be obtained following the same development as for the Q P S K case, leading to: 1-q [p + 2 7 ? ] 2 ( " l + f c )+5 ,F2 (^{m + k) + 1, 2(m + fc); ^ , 2(m + k) + i ; 1 p p + 277' p + 2rj + qy/* T(2(m + k) + + fc) + 1 ; x> ^  2 ( m + 11 2 ( m + fc) + 1. [2p + 277 ] 2 ( m + / j ) + 1 r(2(m + fc) + | ) P P »7 2p+ 277' 2p + 277' 2p + 2r]' (2.222) 104 For M = 4, Eq. (2.222) reduces to Eq. (2.220). In the case of uncorrected fading, i.e. p = 0, Eq. (2.222) reduces to l - q f 1 3 1 p r\_ [p + 2r7]2-+* 2 + 2 5 2 ' 2 m + 2 ; p T V p T V ' [2p + 2 ( 7 ] « + x r (2m + 1) „ / 1 1 i n 3 3 „ l p { T ( F 3 ( 2 m + l ; l , l , 2 m ; - , - ) 2 m + - ; ^ -P »7 (2.223) r (2m + ±)"° V " " ' - ' - ' - . - - > 2 ' 2 ' - - ' 2 '2^+27? '2p + 2r?'2p + 2 r ? 7 J ^ 2.5.3.3 Uncorrelated Fading, Unequal Branch SNR's and m-Parameters (2mj Integer) For Nakagami uncorrelated (p = 0) random variables with arbitrary parameters: / \ m i / \ m 2 4 Of) (%) r(m!)r(m 2 ) PRl,R2(ri,r2) = PRl(n)pR2{r2) = r2ra,-\r2mi-le-^r.-^r^ ^.224) Substituting Eq . (2.224) in Eq. (2.197) and following a development parallel to that presented in Section 2.5.3.1, the following pdf is obtained for the sum of two uncorrelated Nakagami random variables with arbitrary parameters, but 2m, integer: P f l (0 = ( \ mi / \ 7712 r ( m i ) r ( m 2 ) 2m " 1 1 + 1 1 2 - 2 , 2 '2mi - r m 2 r2 i / f t 2 2 4 — — r x 2 m i - l E i=0 \ ( _ i ) i 7 (i±i±l 2 m ^ i M r 2 m , 2 f m i - i - i 27712-1 E 3=0 '2m2 - 1 J 'Am\fl2/fli 2 4—-— r m i+j 2 2 m + ( - 1 y 7 ( ' + J 2 + 1 , 2 ' " ^ / n V ) (2.225) 105 where Q p = fiiQ2, rn = 7 B 1 Q 2 + m 2 f i i . The cdf can also be obtained as: Fair) = 9 I TUX. I I "12 I r ( m 1 ) r ( 7 7 i 2 ) 2m mi+rri2 — 7 r2mi - r 2 m i - l >< E i=0 x 2 m ^ ( - l ) f c 7 K + ma + § m i ~ 5 ~ 5 2m 2 -- l E j=0 2vTl2 - r 4 m ? g - , m 2 - i - i m /j 2771J7712 2 ) fc=0 fc! |-277117712 j "11+7712+3 ^j+j + 1 j , ^ / 2 m ^ \ ^ «+j+i +A; m (2.226) The pdf of the combined S N R is then: ( \ 7711 / \ "12 7711 \ I 7712 I , 1 l l ) \f2 ) f 7X) - | m i + m 2 - 2 _ 2 m 1 m g 7 Pr(7) r (mi)r (m 2  L2mJ 27711-1 E '2mi - r 4 ^ 7 i / 7 2 7 m m i _ l _ i 27712-1 f 2 m 2 _ £ E j=0 4 m | ^ / 7 i ^ m 2 2 2 2 2 X 7 ( - D ' 7 [ 2 ^ 7 ) + ( - D ' 7 C ^ t i , 2 ^ 7 m m (2.227) where m = m i72 + m 2 7 i -Substituting Eqs. (2.187) and (2.227) in Eq. (2.185) and using Eq . 6.455.2 of [80], the following expression is obtained: P* r (mi)r (m 2 ) V 7 J V 72 / ^2m. 27711-1 i=0 ( - I ) ' E j=o « i 2 (aj + / 9 ( 0 ) ) m i + m 2 [2e* 2 P 2F1 I l , m i + m 2 ! i 2 ^ 2 2 i + j + 3 a i + ( - l ) j j+j+i 2 (a2 + p(9))mi+m*-2Fi l , m i + m 2 ; i + 3 + 3 a 2 2 ' a 2 + "0(0) 2 V + /?(#) d# (2.228) 106 where we have now defined ax = 2J^l, a2 = p{0) = &(0) + ^ s p . 2.5.4 Symbol Error Rate for Maximal-Ratio Combining Let r = \fr\ + rl be the normalized amplitude of the signal at the output of the maximal-ratio combiner. Then for identical m-parameters on each branch it is known that ([23], Eq. (142)): PR{T) = m M 2 ( i -P) r 2 / r ^ m ^ w h e r e a s (m(ft 1 + Q 2 ) ) / ( 2 f t 1 Q 2 ( l - p ) ) and /32 = m 2 [ ( f t 1 - f t 2 ) 2 + 4 f t 1 f t 2 p ] / ( 4 ( f t 1 f t 2 ) 2 ( l -p) 2). Using Eq. (A.33), Eq. (2.229) can be alternatively written as: PR{r) = or, in terms of SNR: T(2m) m Pr(l) = T(2m) ftift2(l-p)J m L 7172(1 - p) r 4 m - i e - ( a + / V $ ( m i 2m, 2(3r2) (2.230) 7' 2 m - V ( a + / 3 ) 7 $ ( m , 2m, 2 0 7 ) (2.231) This form wil l facilitate the error probability derivations, which are carried out in the same way as for E G C . Substituting Eqs. (2.187) and (2.231) in (2.185), and using Eq . (7.621.4) of [80], the following general expression is obtained: {^h)Ytl eVW)]-2™^ ( 2 ^ , 2 ™ , ^ (2.232) where N(0) •= <j)i(0) + a +/3. Solutions specific to different modulation schemes are derived separately below. 107 M - P S K Using the parameters of Table 2.3, Eq. 2.232 reduces to: p*= K7^^)7*"S|NWr2m2Fi(2m'm'2m'i))ds (2-233) where now = + a + B. Special case: B P S K ( M = 2) Substituting Eqs. (2.190) and (2.231) in Eq. (2.185), the following expression is obtained (c.f. development for the E G C case): P = I _ fiT(2m+\) (7172T1-P)) 2 VTT T(2m) [1 + a + B)2m+* ( 1 3 1 23 \ 2 m + - ; l , m ; - l 2 m ; r T ^ , r T ^ j . (2.234) D E - Q P S K ( M = 4) Substituting Eqs. (2.196) and (2.231) in Eq . (2.185), the following expression is obtained (c.f. development for the E G C case): P, = - - -1 1 m T{2m + 1) 4 ^ r ( 2 m ) \7i72(i-/o)y U1 + a + P)2m+1 ( 3 3 1/2 1/2 F3 2m + l ; l , l , m ; - , - , 2 m ; x 2B 2 ' 2 ' ' l + a + /3' l + a + /5' l + a + B T(2m + 2) „ . 3 3 3 3 n — — i ^ — - F 5 2m + 2:1,1,1,1, m; - , 2m; 7r(2 + a + /?) 2 m + 2 v ' 2 ' 2 ' 2 ' 2 ' 1/2 1/2 1/2 1/2 2/3 + 2 + a + B'2 + a + Br2 + a + B'2 + a + B' 2 + a + B (2.235) M - Q A M The unconditional S E R can be obtained following the same development as for the E G C 108 case, leading to: Ps - 2a-q - { l - q ) ^ - r ( 2 m ) [p + a + p?mH>< ( n 1 n 3 _ _ P _ 20 \ 2 P r ( 2 m + 1) ( ^ f e ) ) F 2 2 m + - ; l , m ; - , 2 m ; — — 5 , — — r - q 4-2 ' ' ' 2 ' ' p + a + 0 'p + a + 0y * T  T(2m) [2p + a + 0]2to+1 F 3 f 2m + 1; 1,1, m; - , - , 2m; - , - , — ) . (2.236) V 2 ' 2 ' 2p + a + P' 2p + a + P 2p + a + ) y ' 2.5.5 Performance Evaluation Results Figs. 2.29 and 2.30 show the S E R of coherent 2-, 4- and 8-PSK with dual-branch E G C and M R C , respectively, for m = 2 and different correlation coefficients. Figs. 2.31-2.33 plot the S E R for 2-, 4- and 8-PSK, respectively, with E G C and M R C , for p = 0.5 and different Nakagami-m fading parameters. Simulation results, obtained using the method of [128] (c.f. Section 2.4.1.2), are seen to match very well the theoretical curves. Fig. 2.34 shows the S E R of coherent 4 -PSK and 8-PSK with dual-branch E G C , for m = 2, p = 0.5, and different sets of powers ft2}. As expected, the S E R performance degrades with a decrease in the total power captured by the diversity combiner. F ig . 2.35 illustrates the S E R of coherent 8-PSK and 1 6 - Q A M with dual-branch E G C , for m = 2, {Qi — 1.0,^2 = 0.5}, and different p's. Simulation results are included and are seen to validate very well the theoretical curves, which are obtained by truncating the infinite sum in (2.228) to only its first 11 terms. Fig. 2.36 shows the S E R for 2-PSK and 8-PSK with m1 = 2.0, m 2 = 1.0, and different sets of branch powers {Oi,02}. Similar results are shown in Fig . 2.37 in the cases of 4 - Q A M and 1 6 - Q A M , and in Fig. 2.38 in the cases of D E - 2 - P S K and D E - 4 - P S K . 109 0 2 4 6 8 10 12 14 16 18 20 E 3/N 0(dB) F i g u r e 2.29 SER of M - P S K with dual-branch E G C and m = 2, for different p's. E 8/N 0(dB) F i g u r e 2.30 SER of M - P S K with dual-branch M R C and m = 2, for different p's. 110 10° 1 (J"'* I I I I I 1 I 1 I I I 0 2 4 6 8 10 12 14 16 18 20 E a/N 0(dB) F i g u r e 2.31 SER of 2-PSK with dual-branch E G C and M R C , for p = 0.5 and different m's. I l l F i g u r e 2.34 SER of 4-PSK and 8-PSK with dual-branch E G C and m = 2, p = 0.5, for different {Q±, fl2}-112 8 10 12 14 16 18 20 E /N per diversity branch (dB) F i g u r e 2.35 SER of 8-PSK and 16-QAM with dual-branch E G C and m = 2, {fii = 1.0, Q 2 = 0.5}, for different p's. F i g u r e 2.36 SER of 2-PSK and 8-PSK with dual-branch E G C and mi = 2.0, m 2 = 1.0, for different {f i i , fi2}, and p = 0. 113 114 2.5.6 Conclusions We derived novel expressions for the pdf of the sum of two correlated Nakagami vari-ables with arbitrary powers and identical m-parameters, and for the pdf of the sum of two uncorrelated Nakagami variables with arbitrary powers and m-parameters, when the mj's are multiples of half-integers. In both cases of correlated and uncorrelated Nakagami fading, these pdf's were used to obtain unified S E R expressions for several M-a ry mod-ulations with dual-branch unbalanced equal-gain combining. If the branch powers and m-parameters are equal, an alternative expression was provided which is valid for any real m > 0.5. Closed-form expressions were also obtained for several M-a ry modulations with dual-branch unbalanced maximal-ratio combining in correlated and uncorrelated Nakagami fading. The SER's for Rayleigh fading are special cases which can be easily deduced from the previous general S E R expressions by setting m = 1 in the latter. 2.6 Conclusions This chapter presented several new results in the area of wideband fading channels, for the general Nakagami distribution (which includes Rayleigh as a special case). After an introduction, in a first part (Sections 2.3 and 2.4) we considered the char-acterization and simulation of wideband Nakagami fading channels. In particular, we derived in a unified manner the analytical level crossing rates and average fade durations of systems with diversity combining, and the envelope correlation and baseband spectrum of channels with maximal-ratio combining. The expressions for the level crossing rates were used to design a discrete Nakagami fading simulator based on a finite-state Markov chain. We further compared techniques to generate samples of independent Nakagami samples, and reviewed methods to generate spatially and temporally correlated Nak-agami channels. We pointed out some of the restrictive/unjustified assumptions which were previously made in designing continuous temporally correlated Nakagami channels. Since we consider the design of the latter to be still an open issue, in the rest of the thesis we wil l only use Nakagami simulators which produce independent or spatially correlated samples. The simulation of temporally correlated samples wil l be carried out 115 only for m = 1, which corresponds to Rayleigh fading, for which well-accepted simulation methods are available. In a second part (Section 2.5), -we considered the performance evaluation of some diversity techniques in Nakagami channels. In particular, we carried out an analysis of commonly used dual-branch diversity receivers, which employ either M R C or E G C , for the general case of correlated and/or unbalanced channels. The analytical results were presented in a unified manner for various M-ary modulation schemes as single-integral expressions, which generalized or simplified previous works. In some cases closed-form solutions were also obtained. The analytical expressions were thoroughly validated by comparing them against simulation results, obtained using the methods for generating spatially correlated Nakagami samples which were previously reviewed. Several of the results obtained in this chapter wil l be useful in the next chapter, where the analysis of multicode/multirate C D M A systems in general wideband fading channels wil l be tack-led. 116 C H A P T E R 3 ANALYSIS OF MULTICODE A N D M U L T I R A T E D S / C D M A SYSTEMS IN WIDEBAND FADING CHANNELS 3.1 Introduction As discussed in Chapter 1, the V B R nature of compressed video wil l require the transmitter to vary its rate in order to reduce bandwidth requirements, unless perfect smoothing is achieved. In C D M A systems, two main methods have been standardized for variable-rate transmission: multicode transmission and multirate (or variable spreading gain) transmission. The organization of this chapter is as follows. After this introduction, we first briefly introduce in Sections 3.1.1 and 3.1.2 these two approaches. Then, in the following sections, we present detailed theoretical analyses of the multicode/multirate configurations which are used by the forward and reverse links of IS-95B and IS-2000 systems. The analysis is intended to be more general than that specific to the IS-95B/IS-2000 standards. 3.1.1 Multicode Transmission In the multicode approach, one or several channels are assigned to each user. Each channel is spread by a code specific to it, whose purpose is to differentiate it from every other channel. Variations on this scheme were proposed in [173], [174], [175]. Ideally, the channels should be made orthogonal to each other by the spreading mechanism, in order to totally eliminate the interference between users. This process of assigning or-thogonal codes to the set of channels is called orthogonal covering [176]. In order to 117 remain orthogonal, the signals on each channel must be transmitted in a synchronous (time-aligned) fashion. For example, in the forward link of an IS-95 system, during each time (symbol) slot, the base station (BS) sends the information destined to each mobile station (MS), which renders the signals synchronous. This differs from the reverse link of an IS-95 system, where the MS's transmit asynchronously. While perfect orthogonal covering is theoretically possible in a flat-fading propagation channel with synchronous channels, this orthogonality is destroyed in wideband channels: indeed, the multipaths are asynchronous with the desired path, thus making a receiver vulnerable to multipath interference contributed by its own or other users' signals. In the forward link case, in or-der reduce the interference, the channels are further spread by an additional pseudonoise (PN) spreading sequence [177], with a period much longer than the symbol duration. This second stage of spreading reduces the correlation between asynchronous channel codes, and results in decreased multipath interference. The technique of spreading each channel by both a code member of an orthogonal set and a P N sequence is called concatenated spreading [178]. The multicode capability of a system can be exploited in different ways by each user. In one scenario, a user can assign a different service to each of its channels; the information rate of each of these services must be below or equal the maximum bandwidth supported by the corresponding channel. This is the case, for example, in the reverse link of an IS-2000 system, where up to 3 channels (the Fundamental Channel and two Supplemental Channels) can be used to transmit 3 types of services [14]. In a second scenario, only one type of service at a time can be supported. However, if the rate needed for a specific service exceeds the maximum bandwidth offered by the highest-rate channel, then two or more channels can be assigned to that service, in other to realize a rate aggregation. In that case, the information stream is split into several parallel sub-streams at the transmitter end, and each sub-stream is assigned to a separate channel. At the receiver, these sub-streams are multiplexed together in order to form the original information stream [174]. This is the method specified for the transport of high-rate data in the IS-95B standard, where all the channels (at most eight) can offer the same maximum information rate. A system could also operate according to a mixture of the first and second scenarios: certain low-rate services would be carried individually on a 118 set of channels (one service per channel), while a high-rate service would be split onto several of the remaining channels. 3.1.2 Multirate Transmission To increase the transmission rate, the data rate of a channel can be increased, while the number of codes assigned to this channel stays the same. W i t h the chip rate held constant, this results in a lower processing gain, hence the appellation variable spreading gain (VSG) [179]. If the power remains the same, a higher B E R wil l result. Hence, in order to maintain the same error rate on a high-rate channel, the power assigned to it must be increased. However, this wil l inevitably result in a higher level of interference seen by the other users in the cellular system, unless the high-rate user is isolated from the others by using, for example, adaptive antennas. Hence, a tradeoff must be made between the B E R sustained by the high-rate user and the requirements of the other users in the system. 3.2 Reverse Link Performance with Noncoherent M-ary Or-thogonal Modulation and Real Spreading Sequences 3.2.1 Introduction M-ary orthogonal modulation with noncoherent reception has been used successfully in the reverse link of IS-95 cellular systems, and is also specified in Radio Configurations 1 and 2 of the reverse link in the cdma2000 standard. Such an approach does not require the estimation of the phase of the received signal, which leads to reduced complexity implementations but also to a higher theoretical B E R with respect to coherent demodu-lation. Two early analyses of the performance of this scheme in a C D M A environment were given in [180] and [181] for an additive white Gaussian noise channel. Extensions to the case of a multipath fading channel were presented in [182] for the Rayleigh distri-bution and in [183] for a general fading distribution. Some closed-form solutions for the B E R in the case of Nakagami fading were presented in [184], [185] and [186] for M = 2, and in [187] for M arbitrary, where the interference analysis followed the same method-119 ology. A n upper bound on the coded error probability of a system with soft-decision Viterbi decoding and Rice-lognormal fading was presented in [188]. In these studies, the users transmit only one code each. However, for high-rate applications such as data or low-quality video [189], [190], the IS-95B standard allows users to transmit up to eight codes in parallel [191]. Such a scheme is termed multicode C D M A , and variations of it were proposed in [173], [174], [175]. A n analysis of multicode C D M A was given in [192] for coherent detection, in a reverse link scenario. Further results are available in [193], [194], also for the case of coherent detection. In [195] and [196], analytical and semi-analytical studies, respectively, of the capacity of a multicode C D M A system were also provided, but the effects of the modulation format and multipath fading were averaged out. [197] evaluates the performance of the IS-95B reverse link in terms of required S N R for a given Frame Error Rate (FER) , using computer simulations. Despite the many re-cent contributions, to our knowledge no detailed mathematical analysis (which considers in a precise manner the effect of multipath fading) of multicode C D M A with noncoherent M-ary orthogonal modulation has been published. The goal of this section is to tackle this issue. In the reverse link of IS-95B systems, i.e. the MS to BS communication link, the spreading sequences assigned to each code are not orthogonal, since they correspond to different offsets of a long P N sequence [177]. Hence, a correlation receiver is subject to multi-access interference from all codes different from the desired code, and to multipath interference from all users (including the desired user) if the channel is frequency-selective. In a first step, we express the interference terms as a function of aperiodic crosscorrelation functions [27], which was omitted in previous papers (except partly in [198]), and detail the statistics of these terms for multicode transmission. In [183] and [187] the B E R was calculated using the standard Gaussian approximation (GA) , by first replacing the values of all the fading coefficients in the interference terms by their expectations, and then either using Stirling's formula or averaging over a known fading distribution in order to reflect the effect of the fading. While this leads to a good approximation in the single-code case, our research wil l show that the same approach cannot be applied for multicode transmission. Indeed, if the desired user transmits many codes simultaneously, they all fade in unison. Therefore the interference resulting from each of the non-desired codes 120 is not independent from the desired signal. To obtain a better approximation, we derive the B E R conditioned on the fading coefficients affecting the desired user, then integrate it numerically over the pdf of the fading. The resulting B E R curves match the ones obtained by simulation of an IS-95B system better than if we use the previous methods: the latter become less and less accurate as the diversity order increases, or as the number of users decrease when no diversity is used, while our approach gives a good degree of accuracy for most situations. The organisation of this section is as follows. After this introduction, Section 3.2.2 de-tails the signal, channel and receiver models for a single-cell system. It gives the decision metrics of the receiver, which are used in deriving the probability of error for Nakagami, Rician and lognormal fading channels. Extensions to the closed-loop power control, suc-cessive interference cancellation, and multiple cells cases are detailed in Sections 3.2.2.6, 3.2.2.7 and 3.2.2.8, respectively. They are followed by illustrative numerical results in Section 3.2.3 and a concise conclusion in Section 3.2.4. 3.2.2 Error Probability Analysis 3.2.2.1 Signal Model ' The transmitter is a multicode Offset-QPSK (O-QPSK) M-ary orthogonal modulator, whose diagram is shown in Fig . 3.1. User k transmits streams in parallel, each one formed by the concatenation of a data stream with a spreading code. The stream of data symbols for code c (c = 0 , 1 , . . . , — 1) of user k is given in the time-domain by: oo W^(t) = £ Wi{kcj){t-JTw) (3.1) j = - o o where W^kcj)(t) is a Hadamard-Walsh function of dimension M and duration Tw, whose index i{kc,j) G [1, 2 , . . . , M ] depends on the indices of the user (k), code (c) and symbol sequence number (j). In order to alleviate the notation, unless otherwise noted, we wil l drop the dependence of i on these indices. The set of M Hadamard-Walsh functions are orthogonal to each other, i.e. JQTw Wm(t)Wn(t)dt = Tw5mn, where = 1 if i = j and 0 121 otherwise. We can further expand Eq. (3.1) as: oo M—1 W{kc){t) = ] T Y,w^PTw{t-jTw-rTw) (3.2) j=—oo r=0 where Wi>r, r = 0 , 1 , . . . , M — 1, form a sequence of M Walsh bits, which correspond to a particular M-a ry symbol Wi = [^ ,0^ ,1 , • • • ,witM-i] [177], and Prw(t) is a rectangular pulse of unit amplitude and duration Tw = Tw/M seconds. This symbol stream is spread by a long P N sequence a^c\t) = a S W c - JTe) with chip period Tc, which corresponds to a shifted version of a reference long sequence a r i i ( i ) . This assignment of different offsets to every code of each user is responsible for the multicode multi-access capability of the system 1 . The same stream is then mapped onto both the in-phase (J) and quadrature (Q) branches of the transmitter, with a delay of T 0 (of half a chip duration in IS-95) introduced in the Q branch in order to achieve O - Q P S K modulation. In the IS-95 and cdma2000 standards, two short spreading sequences asj(t) and a S )g(i) with the same chip period Tc further spread the signal on the / and Q branches, respectively. These sequences are identical for all the users of the system. To alleviate the notation, we can combine the long and short P N sequences into two / and Q long P N sequences afc\t) and a,QC\t). The transmitted signal for the kth user can then be written as: sW(t) = y/pW [W^c){t)afc)(t)cos{wct + ^kc)) c=0 + W ^ ( t - T 0 )ag c ) ( t - T 0 ) sm(wct + <f>M)] (3.3) where is the average power of user k which, without loss of generality, can be assumed identical for all users, i.e. P^ = P V/c. wc = 2nfc is the carrier frequency in radians per second. 1 This corresponds to the code aggregation scheme of [189], which is used here. The subcode concate-nation scheme of [189] further spreads each code stream of a same user by a different code taken from an orthogonal set: this cancels the multicode self-interference from the same path (but not the one from different paths, for which the orthogonality property is destroyed), and hence leads to a lower B E R than that obtained using the code aggregation scheme. 122 Information bits cos(wj.t) M-ary orthogonal modulator 0 <*HIEK*) a(^(t) s i n(wc?) Figure 3.1 Transmitter for cth code of user 1. 3.2.2.2 Channel Model The frequency-selective channel is modeled as a tapped delay line filter [26] with impulse response h(t, r ) = X^o" 1 ai(t)ejd'^S(T-Ti), where Lc is the number of multipath components, ai(t) and 9t(t) are the time-variant amplitude and phase, respectively, of the complex short-term fading coefficient of the Ith path, and r\ is the delay of the latter. It is assumed that either the delay spread of the channel Tmax can be upper-bounded by the symbol period T V , in order to avoid intersymbol interference (ISI), or that the latter is removed by an equalization mechanism. Also, ai(t) and 9i(t) are assumed constant during a symbol interval. Samples of cti(t) are not restricted to any particular distribution. For illustration purposes, we wil l consider in particular the Nakagami distribution, given by where m ( > 0.5 and = E[af] If m ; = 1, this density function reverts to the Rayleigh distribution. We wil l also consider the Rician distribution, given by: where 2a R t is the power in the random component and Ai is the amplitude of the non-random specular component [26] of the Ith multipath, and the lognormal distribution, [23]: (3.4) (3.5) 123 given by: P\al) = ~7== e x P ~- 7T2 > a i > 0 (3-6) V 27TCLNjCXl \ ZaLN,l J where rriLNj, and OLA^ are the mean and standard deviation of In (a/.), with ln(-) denoting the natural logarithm. The stationary complex additive Gaussian noise at the output of the received band-pass filter can be expressed mathematically as n(t) = nc(t) cos(wct)+ns(t) sin(io ci), where nc(t) and ns(t) are uncorrelated white low-pass Gaussian processes with two-sided power spectral density (PSD) N0. 3.2.2.3 Receiver Model and Decision Metrics Using the above mentioned system and considering that there are K users, the com-posite signal received at the output of the channel is: K I V W - 1 L C - 1 k=\ c=0 1=0 +WM(t - T 0 - i f V S C ) ( * - To - r/ f c ) ) sm(wct + ^kc))} + n(t) (3.7) with </?|fcc' = cj)(kc) + 6^ — WcT^. Note that all the received codes pertaining to a same user are affected by the same fading coefficients and delay, since they are transmitted via the same channel. This fact wil l be crucial in determining the error probability for a low number of interfering users. The phases, however (as specified in the IS-95B standard), can be different if a specific phase shift is assigned to each channel by the transmitter. The receiver consists of L Rake fingers and performs E G C . The schematic structure for the nih finger is illustrated in Fig. 3.2, and its operation is detailed in [177]. In the following, the metrics calculated refer to code 0 of user 1, which is assumed to be the desired stream. Let Z}" 1 '" ' be the output of the mth correlator of the nth Rake finger, in response to the received signal on the /-branch despread by a 7 1 0^(i). It can be expressed 124 (1.11) Figure 3.2 Demodulator for nth Hake finger of user 1 and code c. by: . * n ) = - ^ = / m ( r W c o s M J ^ ^ t - r W j W ^ C t - r W ) * (3.8) where ( - )LP denotes low-pass filtering, and the are perfectly estimated. Upon ex-panding terms in (3.8) and integrating, one obtains: 7{m,n) _ Q ( m , n ) T{m,n) T(m,n) r(m,n) j-(m,n) AT(.m,n) / q Z 7 7 - ^77 "+"J177 + J277 +-'377 + J 77 i V 7 7 \ 6 ^ ) where 5j™'n^ is the signal term, l[™j'n^ is the I/Q crosstalk interference, J^™'^ i s t n e multipath self-interference, /j™'"^ is the multicode self-interference, ij™'n^' is the other-user multi-access interference, and yVJ™'7^ is an integrated noise term. Let C = | y/P/Tw-125 Then these terms are identified as [183]: Si7'n) = / m a ^ t J H ^ J ^ ^ t - ^ ^ l t - T i 1 ) ) * , (3.10) rril)+Tw , / f t " 0 = C a£\t) s in (^ 1 0 >)W*°>( t - - T o )ag°>( t - - T 0 ) r(m,n) /"Tn -t-i W c +WM{t ~ r , ( 1 ) - T 0)ag°>(t - r/ 1* - T 0 ) s m ( ^ ) ] W m ( t - r ^ a ? 0 ^ - r W j ^ . W ) •/ TVi J 1 ; _ n c=l /=0 + W ( i c ) ( t _ T j(D _ T 0 )og e ) ( t - r / 1 } - T 0 ) s i n ( ^ l c ) ) ] W m ( t - r W ) o ? 0 ) ( t - rW)rft,(3.13) i\Tn) = cT ^ E E ^ W ^ H ^ - ^ V f ^ i - ^ ^ c o s ^ ) • ' T « fc=2 c=o ;=o +W™{t - rlk) - T 0 )ag c ) ( i - r ; ( f c ) - T 0 ) sin(rf= c ))]Wm(t - r^)4°\t - r ^ ) * , (3.14) Ni?'n) = / nc(t)Wm(t ~ r W ) a ? 0 ) ( t - r ^ J d * . (3.15) VP JTP The first term readily simplifies to: Si?'n) = a l 1 ) ^ c o s ( ^ ° ) ) 5 m i (3.16) 126 where we recall that i is the index of the desired Walsh data symbol in the current integration interval, and Ew = PTW. The interference terms can be expressed as a function of aperiodic crosscorrelation functions. Similarly to [199], let: n{kc) n \ _ ^xy,muv W N - l - l E M 3=0 N-l+l E w 3=0 0 (kc) • J x>3 m,[ (kc) (10) ' t \ a y , 3 + i (10) u , l W x j - I - N W m , l i } a v J 0<1<N-1 1 - N < I < 0 \T\>N (3.17) where N = Tw/Tc is the processing gain, h = N/M is the number of P N chips per Walsh bit, I — [{r^ — ITC)/TC\ (with [x\ denoting the largest integer smaller than or equal to x), and u and v are the indices of two consecutive Walsh symbols belonging to code 0 of user 1 and contained in the integration interval [ T n \ + Tw]- The indices {xy} are chosen from {11} ,{IQ} ,{QI} or {QQ}. The aperiodic crosscorrelation functions can be written in terms of the previous quantity as [27]: p(fcc) i \ _ . rWfcc) ( / . I j i - ™ i r n i v V / i xy,muv\ 1 "xy,mu  \ + {Cxt c) ,muv N)-(l-N)]Tc, C%%UV{1-N)](T-TTC) (3.18) +[ciky%uv(i)]T, (3.i9) The interference terms in Eqs. (3.11)-(3.15) can then be expressed as a function of these functions (where, for convenience of presentation, the dependence on m, u and v has 127 been dropped), which was not done explicitly in [183]: l[-'n) = Cog> sm(^)[R^(T0) + fig? (To)], _ (3.20) L c - l r(m,n) _ n „ (1) f(m,n),-. n N (=0 hn 2 ^ l^ai'. JIT (1c), (3.22) i V W - H c - l hi ~^2^ 2^ l^ai Jh' \kc) (3.23) fc=2 c=0 (=0 where JJfP'n)(kc) = C [ ( i ? f / ) ( r 2 ) ) + ^ S c ) t f ) ) ) c o s ( # c ) ) + + To) + ^ ( r ^ + To)) s in(^ f c c ) ) ] (3.24) and = r / ^ — r^. Similarly, signal and interference terms can be derived for Z ^ ' N \ Z ^ J ' ^ and Z^Qn\ and these quantities are listed in Appendix D for completeness. The final decision metrics are given by: U m = J2(lZn'n) + 4 T } ] 2 + ~ Z S ' n ) ] 2 ) ( 3 - 2 5 ) n=0 where m = 1, 2 , . . . , M. The receiver selects the symbol Wj whose index j corresponds to the maximum Uj of all C/m's. 3.2.2.4 Statistics of Decision Metrics As stated in [183], the noise terms N^,rC] can be easily shown to be mutually uncor-related zero-mean Gaussian random variables with variance: a2N = V a r [ A ^ > ] = i V 0 / 4 . • (3.26) 128 Using the standard G A as in [183], the interference terms of Eqs. (3.20)-(3.23) can be modeled as mutually uncorrelated zero-mean Gaussian random variables with given conditional variances. These variances are derived below,, and are conditional on the random variable s = Ylt^i01^)2- If T0 = 0, because of the orthogonality between the Walsh functions, the interference terms ij™'71^ vanish. Otherwise, Var[i?^°'(To) + Rxy\To)] = 2Tw/(3N); which results in a variance: °?i = V a r [ / £ ' n ) ] = ^^f, T0 ? 0. (3.27) Again if T 0 = 0, the interference terms Jj™'^ and I^^1 are chip synchronous with the desired user signal, resulting in Var[ ( /?^ c ) ( r^ ) ) + Rif (r^))] = Tfa/N. If T 0 ^ 0, Yar[R{xyC) {r^ + T0) + R{xyC) {rn]]+T0)} = 2T^/(W). This leads to the following variances: o2n = V a r [ / £ ' n ) ] = ^ E ( « , ( 1 ) ) 2 , ( 3 - 2 8 ) 1=0 4 , = V a r t / ^ l = ^ ( / V ( 1 ) " 1) E K ( 1 ) ) 2 , (3-29) (=0 where 57 = 1/4 if T 0 = 0 and n = 5/24 if T 0 # 0. Replacing (anl))2 with its expected value, we could approximate Eq. (3.28) by a22 = Var[/£SB'] * r, N E K( 1 ) ) 2 - w . 1=0 (3.30) in order to render the expression fully conditional on s. Alternatively, as in [39], we could also approximate it by E L ^ oJ2 = v<*[i!£n)]*v^'En\1) (3.3i) N 1=1 129 in order to remove the conditioning on the (aj^) 2 's. A comparison with numerical results shows that for similar fi^'s, the use of Eq. (3.31) leads to a better estimate of the B E R , especially for low values of L. Indeed, when performing the numerical integration of Section 3.2.2.5, for certain low values of the variance a2n given in Eq. (3.30) becomes negative (because of the term — Q n ^ ) , which doesn't make sense. Hence we wil l favour Eq. (3.31) in our calculations. Unlike with the previous variance aj2, the (a^)2,s in the expression for aj3 cannot be approximated by their expected values. Indeed, we must take into account the fact that the instantaneous fading coefficients affecting the multicode self-interference of user 1 are all equal, and equal to the fading coefficient which affects the desired signal. This results from all the codes of user 1 being transmitted via the same channel. Irrespective of T 0 , the interference terms Ix^'n^ are chip asynchronous with respect to the desired user signal, leading to V a r f ^ V n ? ) + R%c) {rnh)))] = 2T^/{3N), k ± 1, and a variance: j = Var[/g->] = H NW £ ^ (3-32) k=2 1=0 We notice that this variance does not depend on s. In the following, we wil l adopt T 0 = 0 (as in [183] and [187]), in order to ease the development. Thus, we.can neglect i^ '™' (which we can in fact do even in the case T 0 ^ 0). As sums of mutually uncorrelated Gaus-sian random variables, the terms Z\M,N^+ZQQ1^ and Z^Q'71^ — ZQ1^ are therefore Gaussian random variables with means a^y/Ew cos(y4 1 0^)5 T Oi and a^y/Ew sin(y4 1 0 ')<5mj, respec-tively, and common variance a2 conditioned on s: a2 = 2(aj2 + a23 + aj + a2N) N0 , Ew [3 | " , A r m l X _ , ^,(1) 2 + 3 A M 2 -l)s+ J2n 1 1=1 K Lc-1 + E j v ( f c ) E f i n - (3-33) k=2 1=0 In the following, we take the number of fingers of the Rake receiver, L, to be equal to the number of multipaths Lc (i.e. all the energy of the channel is collected), as was done in [183] and [187]. Let v = Ey/s. The quantity Um is thus a chi-square random variable 130 with 2L degrees of freedom, conditioned on v, whose pdf is given by: ^ ( 5 ) ^ e - ^ / L _ x ( ^ ) if m = i, Pum{u\v) = 3.2.2.5 Probability of Err or if m ^  i. (3.34) In this section, the method used for computing the probability of error of a multi-code system in multipath fading is presented. Assume that the Walsh function Wi(t) is transmitted, and again that code 0 of user 1 is the desired data steam. The probability of a correct symbol decision, conditioned on v, is given by: Pc{u) = P(U2<U1,U3<UU... ,UM <Ul\v) poo = / [P(U2 < u\U, = u))M-lpuMv)du Jo 1-e ^ M - 1 -L M-l = £ < - D =o / where the coefficients 0nr are given by [157]: L - l E n=0 ( u \n \2a2 > •  M-l Pu^u^du r=0 r ( L - l ) n=0 e xl u 2 + n e ^ i L _ i [ 2 C T 2 ] " + ^ V du (3.35) ^ = E f n - i V / [ 0 , ( , ' ~ 1 ) ( z ' ~ 1 ) 1 ^ i = n - ( L - l ) ^ (3.36) with /30o = /?or = 1, /5„i = 1/n!, / ? i r = and I[a,b](i) = 1 if a < i < 6 and 0 otherwise. The integral in Eq. (3.35) can be evaluated with the help of Eq . 6.643.2 of [80], yielding: I . _ / M - 1' r(L) E ( - Dr r=0 T ( L + n) (r + 1)L+"V r ( L - l ) ^ ] A i r n=0 e aT2"^ L + n, L , 2a 2 ( r + l ) (3.37) 131 The probability of symbol error, conditional on s (recalling that v = Ews), is thus: In the above it is recalled that the variance a2 is a function of s, as made explicit in Eq. (3.33). The probability of symbol error is then obtained by averaging Eq . (3.39) over the pdf of s: The probability of bit error is approximated as Pf> = ^-\PS [26]. In the case of multicode transmission, for most practical fading distributions, the dependence of a2 on s in Eq. (3.39), and therefore in Eq. (3.40), forbids a simple closed-form solution for Ps. Numerical integration is thus used to evaluate Eq. (3.40). In the case of single-code transmission, a2 does not depend on s, and closed-form expressions for Ps can be found in some special cases. In the following, the cases of Rayleigh, Nak-agami, Rician and lognormal fading are treated individually. As previously mentioned, the Rayleigh case is a special case of both Nakagami and Rician, but a separate treatment is included for it due to its widespread use. The details of the analytical derivations are given in Appendix E . Rayleigh Fading In the case of Rayleigh fading with identical powers fij1^ = f i o n each branch, the pdf of s is given by: Ps(s) = l-Pc(s) (3.38) (3.39) (3.40) s ,L-1 Ps(s) nLF(L) e n. (3.41) 132 No simple closed-form expression for Ps seems available for the general multicode case. In the case of single-code transmission, Eq. (3.40) can be put in closed-form by using Eqs. 7.621.4 and 9.121.1 of [80]: P,= M - l ( - i ) r+ l M - 1 r ( L - l ) BnrF(L + n) 71=0 1 + ilEw 2a2 r + 1 + 2a2 . (3.42) This expression is identical to Eq . 14.4.46 of [26]. Nakagami Fading In the case of Nakagami fading with identical fading parameters and powers on each branch, the pdf of s is given by: 771.L qmL-l S ms -e « T(mL) (3.43) It appears that there is no closed-form solution for Ps for multicode transmission, for the same reasons as the Rayleigh fading case. In the case of single-code transmission, the following expression can be obtained, again through the use of Eq . 7.621.4 of [80]: 1 ^ ( - i ) H - i M - l r ( L - l £ finr 7~^~TV7X^"2Pi \mL,L + n; L\ 71=0 (r + l)L+"' (3.44) This expression is identical to Eq. (36) of [187], and reduces to Eq . (3.42) for m = 1. The analysis can also be applied to the more general case of correlated diversity branches, if an expression for the pdf of s is known. For example, in the case of equally correlated diversity branches, it is known that [162], [187]: Ps(s) = m (1 - v ^ - ^ l - VP + Ly/p)mr{mL) -i - i (m \ m L - l xexp m s $ m ; m l ; Lm s) .(3.45) 133 For single-code transmission, a closed-form solution for the S E R of equi-correlated diver-sity in Nakagami fading has been found as (Eq. (46) of [187]): m 71 = 1 1=0 j=0 ^ + m / ~ t ^ t ^ (r + l )i + " n \ / i \ r ( m L + i ) r ( m + j) f H i -VP) r ( m L + j ) r ( L + i ) ^ ^ ( i - ^ ) + ( r + r ( l - v / P ) [ 7 T T i P 1 ( l - v / P ) + ml (3.46) A n approximate expression for exponentially correlated diversity branches has also been proposed in [162], [187], but it was shown in [200], [163] to be inaccurate. Rician Fading In the case of Rician fading with identical fading parameters and powers on each branch, the pdf of s is given by [26]:-Ps(s) = 1 2a\ L - l 2 \ + 3 e ^h-i (3.47) where A = LA2. In the case of single-code transmission, a closed-form expression can be obtained by first calculating the unconditional pdf, using Eq . (2.4.13), p. 115 of [116] (c.f. also [201], [202]) poo Pum{u) = / pUm(u\s)ps(s)ds Jo = < (ul(2olpT))^- „ 2<r2(l+/3) m = i if m ^ i (3.48) (3.49) 134 where B = 2er|f^- and pr = LA2fffi, and then substituting it into the third line of (3.35) and (3.38), which results in (after making use of Eq. 6.643.2 of [80]): _ ^ (-ir»e-*& fr-lY(f% T ( L + n) 8 " ^ [r(l +/?) +1]^  ^ r ) h  r^  , 1 + „ f ] \ { L + n,L,^l^Y (3.50) L r ( l + /3) + l J V ' [ r ( l + /5) + l ] ; k / This expression is identical to Eq. (41) of [201], and reduces to Eq . (3.42) for A = 0. Lognormal Fading In the case of lognormal fading with L = 1, the pdf of s is known to be given by: Ps(s) = TH= e x p - ^ l n ^ m L N > > . (3.51) 2\Z2TTOLNS 2<X 2 L i V However, for L > 1 an exact closed-form expression is difficult to obtain [203], although approximations can be made [204]. We wil l therefore consider only the flat-fading case, for which no exact closed-form solution is known either for Eq . (3.40) [22], for both the single- and multi-code cases. 3.2.2.6 Extension 1: Closed-Loop Power Control with the Inverse Update Algorithm Closed-loop power control ( C L P C ) is used in the reverse link of IS-95 systems, and in both the forward and reverse links of IS-2000 systems [14]. Its goal is to compensate for the small-scale fading which affects the received signal, and it works on top of the open-loop power control mechanism which compensates for large-scale fading (shadowing) and path loss. In the reverse link of IS-95B, the C L P C algorithm sets a target received power at the BS, which should allow a certain F E R to be attained while keeping to a minimum the interference to other users. The BS then instructs the MS to raise or lower its transmitted power in order to achieve this target, by sending it power control update commands at a certain rate (800 Hz for IS-95B and IS-2000). The MS adjusts its transmitted power in certain fixed size increments: for example, 1 dB , 0.5 dB and 0.25 dB 135 increments are allowed in IS-95B. There is a necessary delay between the instant the BS makes a decision on the power control update command and the instant this command is executed by the MS. Fig. 3.3 gives a schematic block diagram of a C L P C algorithm, using the loglinear model of [205] (in which all the given powers are expressed in dB). Estimation error PN(t)(dB) Desired received power P S< d B ) ' Sampling Received P ( t ) ( d B ) power Channel gain P c (t)(dB)—H3) Averaging filter l / ( k s T w ) Loop | k D T w delay Zero-order hold P T ( t - k F T w ) ( d B ) + + PD(0(dB) PT(t)(dB) Transmitted power Power control step-size F i g u r e 3 .3 Simplified loglinear closed-loop power control model [205]. A purely analytical determination of the B E R for a system using the previous C L P C algorithm is made difficult by the presence of nonlinearities (fixed size increments of the transmitted power) and delays. Hence, B E R and F E R studies of such a system have been based mostly on computer simulations [206], [207], [205], [208], [209]. Nevertheless, a simplified linear version of this algorithm, often denoted as the inverse update algorithm ( IUA), has been used in the analysis of systems with both coherent [210] and noncoherent detection [211]. In the algorithm studied in the previous references, the power increments have an infinite quantization precision, and in the case of [211], power control commands are issued by the BS at every symbol interval. Indeed, the MS mul-tiplies its transmitted signal by the inverse of the square-root of the BS-estimated total power contained in the frequency-selective channel envelope. However this algorithm still considers the delay between the BS decision on the power control update command and the execution of this command. The analysis of such a theoretical system gives a lower bound on the B E R which is attainable with a practical C L P C algorithm. If there was no power control updating delay in the system and the estimation process was perfect, the channel would revert to an A W G N channel. 136 Below, we extend the analysis given in [211] for the I U A to a multicode system. The BS issues power control commands updates at every symbol instant (k$ = 1 in Fig . 3.3), and these command bits are received error-free at the M S . There is a total loop delay of rd = kDTw seconds. It is assumed that the BS estimates perfectly the power of the frequency-selective channel envelope at time t—rd, given for user 1 by Yla=o l{af\t~Td))2 (T/v(t) = 0 in Fig . 3.3). A t time t, the MS (user 1) multiplies its transmitted signal by l / y E i = 0 1 ( a i 1 \ t - Td))2 (kF = kD in Fig . 3.3). Referring to the analysis given in previous sections, the {II} component of the desired received signal is now given by: -,(771,71) 'II w COS (v£0 )K (3.52) where afl is the sample of o^\t) taken rd seconds before af \ The variances of the mul-tipath, multicode, and other-user multi-access interference terms are given respectively by: 2 ° J 2 E I 2 J W \ " 77. (1 )A2 (3.53) On = L c - l > L ^ ^ L c - \ , (1) i=o l^v=o \ai',Td) (3.54) Oj = 6N k=2 1=0 LZ^'=o ynv,Td) . For i . i .d. Rayleigh fading multipaths, using Eq. 6, p. 73 of [116] (with m with Eqs. 9.131.1 and 9.100 of [80], it can be easily shown that: E .E"=o(4lf. L-P2{rd) L(L - 1) (3.55) 2) along (3.56) 137 for L = Lc > 1 (which is given also in [211]), where p(r) is the correlation coefficient of the channel. For example, for a land-mobile radio channel with isotropic scattering, it is recalled that p(r) = J0(2irfmT) [113], where fm is the maximum Doppler frequency. Again for the Rayleigh fading case, the pdf of s' = 1 ( a / 1 ^ ) 2 / YIVL'Q1 (ai^Td)2> ps'(s'); is obtained by a simple variable transformation of Eq . 1, p. 50 of [116] for i. i .d. channels (c.f. Eq . (8) of [211]), while its cdf is given by Eq . 44 of [210] for the general case of non-i.i.d. channels. The symbol error probability for this C L P C system can be obtained as before through Eq. (3.40), substituting s with s', and making use of Eqs. (3.53)-(3.56). It was shown in [211] that the pdf of s' can be approximated by a lognormal pdf (Eq. 3.6) with parameters m i N — 0 and a\N = ^ln[(L — p 2 ( r p ) ) / ( L — 1)] (obtained by equating the first moments of the exact pdf and the lognormal pdf). It was also shown that this approximation was very satisfactory for an IS-95 system with more than K = 8 users. Section 3.2.3 presents the numerical results obtained using this approximation (which facilitates the numerical integration) for a multicode system. 3.2.2.7 Extension 2: Successive Interference Cancellation Successive interference cancellation (SIC) is a multiuser detection technique [212] which consists in successively detecting each user, and substracting estimates of the previously detected users from the total signal-plus-interference signal received by the next user to be detected. Hence, in the ideal case, if all K — 1 users of a system with K users have been correctly detected and substracted, then from the standpoint of the Kth user the system wil l appear as an A W G N channel. A n early discussion of SIC applied to D S / C D M A , in terms of capacity achievable, was given by Viterbi in [213]. Other early papers are [214], in which SIC is used to cancel cochannel interference, and [215], in which SIC is performed in the frequency domain on a D S / C D M A system. In [216], the authors apply SIC to a system using noncoherent M-ary orthogonal modulation, similar to that used on the single-code IS-95 uplink, and perform a detailed B E R analysis. Implementation aspects are discussed in [217]. In this section we extend (and somewhat modify, as discussed below) the analysis of [216] to deal with a multicode system. 138 As in [216], at each iteration the correlation metrics (i.e. the signal and interference terms, instead of just the signal part, which would be difficult to extract from the total received composite signal-plus-interference) are used to remodulate the detected symbol which has been chosen in the previous iteration, and the resulting remodulated signal is substracted from the remaining composite signal-plus-interference. Hence, taking into account the possibility of incorrect decisions on previously detected symbols, the low-pass filtered received signal on the /-branch at the (h + l)th iteration can be mathematically expressed as (taking, for clarity, L = Lc): L-1 7{j{h),l),h-lUr f f chJW)) / . (hU (=0 - g Z%h^h-lW3[h){t - T 0 - rlhVQk°\t - To - r/fc>) 1=0 K / V W - I L - I = \ E E E v^'c) Kc)c - *i(fcV?c)(* - ^  co s(^c )) k=l c=0 1=0 keKr +W^(t - T 0 - r\k))a^\t - T0 - r / f c ) ) s i n ( ^ c ) ) ] + ^ 1=0 i=0 • +c%h)^Wm{t -To- T^)af\t -To- r , w ) ] (3.57) where r] LP{t) = rj^p(t) is the low-pass filtered version of the /-component of Eq . (3.7). h + l (h = 0 , 1 , . . . , K — 1) is the index of the user which is being detected, which we also choose, without loss of generality, to be equal to the iteration number (i.e. the users are successively detected according to their index). KT is the set of users which have not yet been decoded, or which have been incorrectly decoded in previous iterations. The index j(h) corresponds to the index of the symbol which has been chosen in iteration h (which can be either correct or incorrect). The c ^ 1 ^ ' 1 are interference terms due to imperfect cancellation of the correctly or incorrectly decoded users (since the correlation metrics zuw),h-^ z(m,D,h-^ i n g t e a d o f o n l y t h e d e s i r e d s i g n a l s s(m,D,k-^ s(m,i),k-^ a r e used for cancellation), and wil l be detailed below. 139 Sti l l at the (h + l)th iteration, the output of the mth correlator of the nth Rake finger, in response to the low-pass filtered received signal on the /-branch despread by af'+l'0\t), can be expressed as (c.f. Eq. (3.8)): (ft + l) , rp z\7n)Ml = - 7 = / r J + M O ^ 1 , 0 ^ * - ^ ) ^ ^ - ^ 1 5 ) * _ c ( m , n ) , / i + l T(m,n),h+1 . T(m,n),h+l T(m,n),h+1 T(m,n),h+1 j.j(m,n),h+l where: 'II T J l / J ^ J 2 / / T - * 3 J / ^ / J ^ J , / 7 V " LUl'). 'J .<w....^ _ J ' l u W f t _ ;=o i= i +cSg i W ) , i W J - ( 0 ( t - T 0 - r / l ) )ag 0 ) ( t - T 0 - r « ) = s(m,n),h+l + c[m,n),h+l (3.58) 5 ( m , n ) l f c + l = ^ 1 ) ^ ^ ( ^ 1 , 0 ) ) ^ ^ ^ ( 3 . 5 9 ) Jrn,n),h+\ ^11 „(m,n), l -7 / = I (m,n),/»+l . ,-(m,n),/!.+l 7-(ra,n),/i+l . T-•+• i 9 r r -f- -t I III L2II h (m,n),h+l _|_ jy(m,n),h+l II "II w L-l EE[< -IQ =0 i = l >(»), +c^'-",(i2g|;(rS) + To) + + To)) r ( m , n ) , l , r ( m , n ) , l , r J l / 7 ' 211 ~ r J 3 / / ( m , n ) , l _ j _ j ( m , n ) , l jy(m,n),l (3.60) (3.61) (m,n),/t+l ( m , n ) , l (m,n),/t+l _^ j(m,n),h+l j(m,n),h+l j_ j-(m,n),h+l w + I2IQ L-l h E E 1=0 i=l + iV (m,n),ft+l ,0(0.0.' -II 3IQ ^ ± I Q ^ l y I Q p(*°),v(0 (R%\T%-T0) + iW(T»>-T0)) .(0 + c / ^ T O ( r ^ ) + ^ ( r ^ + T o ) ) .(0 («) /^(») = J. ( m , n ) , l j ( m , n ) , l _,_ j-(m,n), l HQ 2IQ + 1. 3/Q r ( m , n ) , l , AT{m,n),l - r J I Q + l y l Q 1 (3.62) (3.63) 140 i f t " ) M 1 - ^ f ^ + 1 ) 8 i n ( ^ - 0 ) ) [ < + 1 " 0 ) ( r o ) + < + 1 ' 0 ) C T o ) ] , J. ( m , 7 i ) , / i + l 2 / 7 J JV; L - I E ( / i + l ) r ( m , n ) , / i + l ; 1n (ft+ 1,0), T-(m,n),f t+l ' 3 7 / ~ _ _ iv( h + 1 )—1 L - 1 • f 5 1 E E ^ ^ r ^ + M , ( m , n ) , h + l _ II ~ o c = l (=0 7f / v C O - l L - 1 E E £<•!"'. fc=l c = 0 (=0 fce/fr,/c^/i+i (fe) T ( m , n ) , / i + l (M (3.64) (3.65) (3.66) (3.67) (m ,n ) , f t .+ l II • : i /-rA + T W — I nc(t)a\h+1>°\t - r^)Wm(t - r^)dt, (3.68) where ^[(RfiC\r^) + Rir\rnk)))cos(^) + {R[Qi\rnk) + To) + P}*?{r% + T 0 )) sin^r')] (3-69) 5(fcc) / (fc) and 7-W = r/ f c ) - r ^ + 1 ) . The terms { i $ c ) ( r ) , ^ ( r ) } are given by Eqs. (3.18)-(3.19), but now with: n{kc) m ^xy,rnuv\ I N-l-l E % , L i J ° . (fee) ( M - 1 , 0 ) J = 0 N-i+i 0 < Z < TV - 1 E ^ L ^ j C « - A . L i J * 1 , 0 ) l - i V < ^ " < 0 > N (3.70) with p = N/M. The term 7 ( m , n ) , / i + i c a n ^ e ^ j - ^ g j . broken down into: j-(m,n),h+l Il,und II,inc 2 &W (^fc) j{m,n),h+l ^ fe=/j+2 c = 0 i = 0 h J v W - l i - l (fe), E E (fc) ^( r r i jn) . / ! - ! - ! (/CC) fc=l c = 0 (=0 (3.71) (3.72) 141 where / /™^<f + 1 is the interference corresponding to the users which haven't been decoded yet, and IIr^'^/l+l is the interference due to the users which have been incorrectly decoded and hence haven't been cancelled (belonging to the set Kinc). Similarly, signal and interference terms can be derived for ^(™>")>'H-I^ ^(m,n),/H-i a n ( ^ ZQQn^'h+1, as was done in Appendix D, with: (m,n),h+l j-(m,n),h+l r(m,n),h+l , r(m,n),h+\ j{m,n),h+l ™ — 11QQ + 22QQ J 3 Q Q ' 1QQ L-l h -QQ J1™ i=o i=i L ,(;W,!),'/p(«o) "' •QI y^oi AT(m,n),h+l + i V Q Q (m,n),l r(m,n), l W ^ - T o J + ^ C r W - r o ) / 7-(m,n),l , r(m,n),l , r(m,n), l , »r(m,n),l " l " J 2 Q Q i _ i 3 Q Q "r J Q Q i V Q Q ' (3.73) (3.74) (m,n),h+l „(m,n),l T(m,n),h+1 r-(m,n),h+l , j{m,n),h+l j{m,n),h+l » r i llQI 3QI "T" J Q / "T" J , Q 7 (*0) (m,n),/i+l 2Q7 -^=EE [ ^ ' ^ V i ? + T 0 ) + R%>(T»> +T0)) (=0 i=l r(m,n), l , r(m,n), l , r(m,n), l 7-(rn,n),l »T 1iqi J 2Q7 J 3Q7 J Q 7 ~*~ J V Q 7 (m,n),l (3.75) (3.76) The variances of the different interference terms can be obtained as (following Section 3.2.2.4): 2 aN Jl °I2 a 73 I,und a I,inc Yav[N^'n)'h+1] = No/A, L-l N i=i = Var[J&">- f c + 1] = V ^ ( N ^ - D E ^ ) L - l Ci+1)\2 iV (=0 = V a r [ / g : ^ + 1 ] ^ 6~JV E K L-l £ ^ E ^ /c=h+2 /=0 (3.77) (3.78) (3.79) (3.80) (3.81) 142 where 77 = 1/4 if T 0 = 0 and 77 = 5/24 if T 0 ^ 0. In Eq . (3.81), P s f c denotes the probability of symbol error at the kth iteration. It is included in order to account for the effect of error propagation due to incorrect decisions during previous iterations. This serves only as an approximation, and its validity/limitations wil l be examined in Section 3.2.3. As mentioned in Section 3.2.2.4, Var[ ( i?S c ) ( r^ ) ) + R%c){T^))] = 2Tw/{3N),k ± h + \. Using Eqs. (3.77)-(3.81) and Eq. (3.60), the total variance of zfi1^'^1 + Z$%n)M1 is: <£u = Var[C/7'^+1] + V a r [ c ^ + 1 ] = T + 2 A ^ ( ^ + 1 ) - 1 ) S + 2 ^ E ^ + 1 ) 1=1 £ N m £ "S" + § X > W £ Wtf k=h+2 1=0 k=l 1=0 + W £ jy^®™*] + Var [c^ M ] + Var[cg«' ( )1 + Var[cg«^' <]](3.82) 1=0 i=l where ^ E ^ o 1 ^ ) 2 -The error probability at iteration h+l is thus obtained by first successively computing the error probabilities Pk, k = 1,.. . , h, substituting these values in Eq . (3.82), and then making use of Eq. (3.40). 3.2.2.8 Extension 3: Multiple-Cells System with Hard Handoff The analysis in the previous sections only took into account the effect of interference from users of the same cell as that of the desired one. In this section we extend this analysis to the multiple-cells case, in order to reflect a more realistic cellular environment. The cellular environment considered is sketched in Fig. 3.4. We assume a cluster of seven hexagonal equal-size cells, corresponding to the home cell (0) and the first tier of interfering cells (1-6). More than one tier can be considered, but the insights gained are small compared to the increase in simulation time. We employ the commonly used model where the mobile stations are uniformly distributed over each cell. The emitted signals are subject to path loss, with a commonly chosen path loss exponent 7 = 4 [218], and lognormal shadowing with standard deviation aL = 8 dB. We make the assumption 143 Figure 3 . 4 Cellular environment. that each MS is power-controlled by the BS of the cell in which it is physically located: we leave the effects of soft handoff and BS macrodiversity for future consideration. As a result of the reverse-link open-loop power control mechanism, the path loss and lognormal fading are compensated for in the case of the MS's of cell 0, as seen by the home cell BS, while they must be taken into account when characterizing the interference from the first-tier cells. Closed-loop power control is not considered, but can be straightforwardly integrated as per the previous section. Let user {q, k} denote the kth user of cell q, for k = 1, 2 , . . . , K, q = 0 , 1 , . . . , Q' — 1 {Q' — The composite signal received at BS 0 can be mathematically expressed as: Q'-l K W(« f c ) -1 L c - 1 r(i) = n(t)+J2/Z E E v W ^ W q=0 k=l c=0 (=0 x [w^kc\t - T{9k))afc)(t - rlqk)) cos(wct + ^qkc)) +W^kc\t -To- riqk))afc\t -T0 - r\qk)) sin(wct + <p{qkc))} (3.83) 144 where g(qk^ = (r0qk^) y10^" ' / 1 0 is the combined path loss and lognormal attenuation function for the path between user {q, k} and BS 0, and C(?fc) = ( r ^ n O " ^ " 0 / 1 0 is the open-loop power control function applied to user {q, k} by its home BS q. rtf^ and r\qk^ are the distances between user {q, k} and BS's 0 and q, respectively. £ ^ and are Gaussian random variables with standard deviation O~L = 8 dB, and correlation coefficient E[£q9k^£,o9kS>]/°~L = 1/2 [219]. The presence of-the subscript q on the remaining terms and quantities means that the latter are related to cell q. A n outer-cell interference term 1^™'^ is now added to the decision metric of Eq . (3.9). This term is defined as: I'nf = c E E E Ev^^w^^-^rvr^-rr) •*Tn q=l k=l C=0 (=0 x[W^kc\t - r^qk))a{qkc)(t - T[qk)) cos(^ ( 9 f e c ) ) +W^kc\t - rlqk) - %)a[qkc\t - r / 9 f c ) - T0) sm(ip[qkc))]dt (3.84) where: o \ 1 ,-.51! 52 10 10 (3.85) Its variance is given by: Q'-I /<: L c - i (qfc) 9=1 k=l 1=0 (3.86) with p = E{p(qk)\ = E r_q Jik) 71 . „ ! Q Z5« 10 i o (3.87) by assuming that the quantities £qQk^ are independent of rqqk\ The first expectation on the right-hand side of Eq. (3.87) can be computed as in Chap. 10 of [177] or [220], which 145 assume circular cells, and results in: E 3 19 121n ~ 0.11558. 2 4 The second expectation can be evaluated as in [218], which leads to: E Aqk)_ (qk) 10" 10 = g 2 V°i- 10 I Hence p ~ 0.6304. The new variance of the total interference is then: (3.88) (3.89) <c = 2(pi2 + al3 + o-i + ai>0 + a2N). (3.90) Thus in the multiple-cells environment with hard handoff, the B E R is obtained by using the same equations as before, but with a2 replaced by a2 r2 . mc 3.2.3 Performance Evaluation Results and Discussion We implemented a multi-cell IS-95B software simulator in order to check the validity of our equations. The B E R is evaluated through Monte Carlo error counting. The P N sequences are as specified in [14]. Without loss of generality, we have used M — 8 instead of M — 64 (as used in the standard) in order to speed up the evaluation of Eq. (3.40). W i t h a bit rate Rb = 28.8 kbps and a chip rate Rc — 1.228800 Mbps, this results in a processing gain of iV = (log 2 M)RC/Rb = 128. Unless stated otherwise, the channel is Rayleigh fading with = 1 for all Z, k, and M f c ) = 1, k = 2, 3 , . . . , K, while i V ^ can take different values. A t first we consider a system with A^ 1 ) = 8 codes. We compare the results obtained in two cases: i) Through the use of Eq. (3.40), which is the proposed approach; ii) Through the use of Eq. (3.42), where 2 N0 Ew ° = T + 3 A ( / V ( i ) _ 1 } £ + £ n (i) 1=0 1=1 K L c - 1 + EIV{FC)E^))<3-91) k=2 1=0 146 The second approach has been used in the past for example in [75] and [193], but for the case of coherent reception. The B E R curves obtained using both methods are plotted for L — 1,2, 3, 4, 5, along with the simulation results, in Figs. 3.5-3.9. From these it is apparent that the use of Eq. (3.42) is not appropriate for the eval-uation of the B E R in a multicode C D M A system. In the case L = 1, i.e. when no multipath interference is present, the approximation provided by Eq . (3.42) gets worse as the number of users in the sytem decreases. Indeed, this equation assumes that the fading coefficients affecting the multicode self-interference of the desired user are independent of the fading experienced by the latter, which is obviously incorrect since all the codes of this user fade in unison. As a consequence, it does not take into account the fact that when the desired user's received signal is low due to deep fading, the multicode self-interference wi l l also be decreased proportionally, leading to a better B E R than that predicted by Eq. (3.42). When multipath interference is present, we observe that Eq. (3.42) differs from the simulated results for most values of K. For higher values of L, we notice that for values of K greater than a certain breakpoint, Kb, Eq. (3.42) is optimistic compared to the simulation results. As L increases, Kb becomes smaller and the gap between the two curves gets more important. For L — 5, Eq. (3.42) underestimates the B E R for all values of K. In contrast to the previous observations, Eq. (3.40) leads to values of the B E R very close to the simulated ones over the whole range of K. Indeed it captures the dependence between the fading coefficients of all of the desired users's codes, which is necessary for an accurate B E R evaluation. Next, F ig . 3.10 illustrates the use of Eq. (3.40) in evaluating the B E R of a system with = 4, for L — 1,2,3. For low values of interference (smaller i f ' s ) , the use of diversity produces a small improvement, with the largest performance gain obtained by going from one diversity branch to two, which is a well-known fact [20]. However, after a certain value of K, the use of diversity slightly increases the B E R : indeed, the level of interference becomes such that the use of E G C worsens the situation, which is due to the noncoherent combining loss described for example in [26], Section 12.1.1, or [177], Section 9.3.2.1. The same type of results are plotted for M 1 ' = 8 in Fig . 3.11. As before, for a high level of interference the use of diversity actually increases the B E R . However, in this case the use of L = 3 does not bring any improvement over L — 2, and 147 even slightly worsens the performance for most of the AT's: because of the high number of parallel codes transmitted by the desired user, the degradation due to the increase in multicode self-interference offsets the gain realized by diversity. Figs. 3.12-3.14 illustrate the influence of the number of codes J V ^ on the system B E R , for L = 1,2, 3. When L — l, the difference in B E R for the cases i V ( 1 ) = 1, 4 and 8 is relatively small. As the number of branches grows (L = 3), the gap in B E R increases. As before, this can be explained by the larger amount of multicode self-interference due to the multipaths, since the multiple-access interference from other users remains the same. If > 1 for any user k > 1, the extra interference produced by the additional codes of this user is accounted for through Eq. (3.32). W i t h respect to the desired user, the interference produced by each supplemental code of another same-cell multi-code user is equivalent to the interference of a separate same-cell single-code user. F ig . 3.15 illustrates the B E R of the desired user (with = 1), when user 2 transmits = 4 codes (for L — 1,2,3). Fig. 3.16 illustrates similar results when = 8. 10° L I I I I I -I 10" 1 0 - s | 1 1 : 1 - i 1 0 10 20 30 40 50 60 Number of users K F i g u r e 3.5 B E R vs K for m = 1, L = 1, = 8. - : Eq. (3.40); - - : Eq. (3.42); + : simulation. 148 Number of users, K Figure 3.6 B E R vs K for m = 1, L = 2, = 8. - : E q . (3.40); - - : E q . (3.42); + simulation. 10° Number ot users, K Figure 3.7 B E R vs K for m = 1, L = 3, T V ^ = 8. - : E q . (3.40); - - : E q . (3.42); + simulation. 149 II I I I I I I 0 10 20 30 40 50 60 Number of users K F i g u r e 3.8 B E R vs K for m = 1, L = 4, = 8. - : Eq. (3.40); - - : Eq. (3.42); + simulation. 10° 1 0-*l 1 1 1 1 ' 0 10 20 30 40 50 60 Number of users, K F i g u r e 3 .9 B E R vs K for m = 1, L = 5, = 8. - : Eq. (3.40); - - : Eq. (3.42); + simulation. 150 10 20 30 40 Number of users K Figure 3.10 B E R vs K for m = 1, = 4. — (+): L = 1; - . - (*): L = 2; — (o): L = 3. 30 Number of users K Figure 3.11 B E R vs K for m = 1, /VW = 8. — (+): £ = 1; - . - (*): L = 2; - - (o): L = 3. 151 20 30 40 Number of users K 60 Figure 3.12 B E R vs K for m = 1, L = 1. — (+): = 1; NW = 8. (*): 10 20 30 40 Number of users K Figure 3.13 B E R vs K for m = 1, L = 2. — (+): = 1; = 8. 152 10° 10"' 10" •10 20 30 40 Number of users K 50 F i g u r e 3.14 B E R vs K for m = 1, L = 3. — (+): = i ; _ . _ (*): Ar(i) = 4; - - (o) = 8. 20 30 40 Number of users K F i g u r e 3.15 B E R vs K for m = 1, = 1, = 4. — (+): L = 1; - . - (*): L = 2; (o): L = 3. 153 10" 0 10 20 30 40 50 60 Number of users K F i g u r e 3 .16 B E R vs K for m = 1, = 1, = 8. — (+): L = 1; - . - (*): L — 2; — (o): L = 3. Figs. 3.17-3.19 show the effect of the Nakagami-m parameter on the system, for L = 1,2,3 and = 8. As expected, it is noticed that as the m-parameter increases, the performance improves. However, this gain in performance becomes smaller with the number of diversity branches: for example, with L = 3, there is little difference between systems with m = 2 and m — 3. Indeed, while a high value for the m-parameter increases the desired signal, it also results in larger multicode self-interference, which grows with the number of multipaths. Figs. 3.20 and 3.21 show the effect of the number of diversity branches on systems with m — 2 and 3, respectively. Fig. 3.22 plots the B E R of a system with 2 correlated diversity branches (p — 0.5) in Nakagami fading with m = 2, for different numbers of codes. F ig . 3.23 shows the effect of the correlation coefficient p on a similar system with — 8: for p smaller than 0.7, the correlation between branches doesn't significantly affect the B E R . Simulation results were obtained using the method of 2.4.1.2 (L arbitrary) and are seen to match very well the theoretical curves. 154 10 20 30 40 Number of users K F i g u r e 3.17 B E R vs K for L = 1, = 8. — (+): m = 1; - . - (*): m = 2; - - (o) m — 3. 30 Number of users K F i g u r e 3.18 B E R vs K for L = 2, JVW = 8. m = 3. — (+): m = 1; - . - (*): m = 2; - - (o) 155 10" 10" 20 30 . 40 Number of users K 50 60 Figure 3.19 B E R vs K for L = 3, = 8. — (+): m = 1; - . - (*): m = 2; - - (o) m — 3. 10 rr .2 UJ 10 m 10" 10 30 Number of users K 40 50 Figure 3.20 B E R vs K for m = 2, = 8. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3. 156 10 20 30 . 40 Number of users K 50 60 F i g u r e 3.21 B E R vs K for m = 3, = 8. — (+): L = 1; - . - (*): L = 2; - - (o): L = 3. 30 40 Number of users K F i g u r e 3.22 B E R vs K for m = 2, L = 2, p = 0.5. — (+): AK 1) = 1; - (o): = 8. - (*): JV(i) =4; 157 Figs. 3.24 and 3.25 plot the B E R for a system with Rician fading and A = 1.0,2.0, respectively, for L = 1,2,3 and = 8. 'It is seen that the B E R increases with the number of diversity branches for large i C s , which is explained by a larger noncoherent combining loss due to the non-fading component. 0 10 20 30 40 50 60 . Number of users K F i g u r e 3.23 B E R vs K for m = 2, L = 2, JVW = 8. — (+): p = 0.0; - . - (*): p = 0.3; — (o): p = 0.5; • • • (o): p = 0.7. Fig. 3.26 shows results for the lognormal fading case with L — 1, J V ^ = 8, mLN = 0 and aLN = 0.2,0.5,1.0. Slight discrepancies (less than a factor of 2) are noticeable for aLN = 1 in the range 2 < K < 15. The value of aLN has an important impact on the B E R , especially for low K's. Figs. 3.27 and 3.28 plot the B E R for a system using the C L P C inverse update algorithm of Section 3.2.2.6, for L = 2 and 3, respectively, with a loop delay kD = 4 and different number of codes N^K The channel fading is Rayleigh-distributed. It can be seen that the lognormal approximation which is used in the analysis fares better for larger J V ^ ' s , i.e. for a larger value of the total interference, which is consistent with the observations of [211]. Hence, while this approximation is accurate for a single-code when K is at least 8 or so, for a multicode system it is accurate for a lower number of 158 0 10 20 30 40 50 60 Number of users, K F i g u r e 3 .24 B E R vs K for a Rician channel with A = 1.0, M 1 ) = 8. — (+): L — 1; - . -(*): Z = 2 ; - - ( o ) : L = 3. interfering users, depending on the value of (e.g. for = 8, it is accurate for all K). Figs. 3.29 and 3.30 illustrates the effect of the loop delay ko on the B E R , for L = 2, and A^ 1 ) — 4 and 8, respectively. A performance degradation is noticed for higher values of kp, however it is less important for higher values of L. Figs. 3.31 and 3.32 show results for a system of K users using SIC, where the B E R is that of the last user to be decoded (user K), given that all previous K — 1 users have been tentatively cancelled. A l l cancelled users transmit only one code (N^ = 1, k = 1,. . . , K — 1), while the last one transmits codes. From both the theoretical and simulation results it can be seen that as more codes are used (a higher N^), more incorrect decisions are made in the earlier cancellation iterations, leading to error propagation and a higher B E R . The theoretical results in general overestimate the B E R , especially in the low-user region for L > 1: the discrepancies are imputable to the approximation which had to be made in the analysis in order to include detection errors and error progagation, the effect of which is very difficult to account for in an exact manner. 159 10° 10-l 1 1 1 1 1 1 0 10 ' 2 0 30 40 50 60 Number of users, K Figure 3.25 B E R vs K for a Rician channel with A = 2.0, = 8. — (+): L = 1; - . -(*): L = 2 ; - - ( o ) : L = 3. 10 1 .. I —-(•' • / + /+:::::::::: :' *• -.- .*"*. :::.JO'': +' - * - - 4 - * — * - -u . ^ „ . _ . . . : . . 4 . 6 . . . . , . ; o . i i . -:••£> .,/ A» : -' C -*/ O' ' ::,:::::9-::::::::: -'.:::<?:::::::::::: *' q ' ! " " • -1 " , / i-i / . : : : : : : : : : : : : : : : : : : • 1 J 1 1 1 0 10 20 30 40 50 60 Number of users, K Figure 3.26 B E R vs K for lognormal fading, L = 1, = 8, m L i v = 0. — (+): aLN = 1.0; - - (*'): crLN = 0.5; - . - (o): aLN = 0.2. 160 10° Number of users K F i g u r e 3.27 B E R vs K for m = 1, L = 2, and C L P C with kD = 4. — (+): = 1; (*): = 4; - . - (o): = 8. F i g u r e 3.28 B E R vs K for m = 1, L = 3, and C L P C with kD = 4. — (+): i V ^ = 1; (*): N<U =4; - . - (o): =8. 161 10° Number of users K Figure 3.29 B E R vs K for ro = 1, L = 2, A ^ 1 ) = 4 and C L P C for different kD. kD = 1; 7 - (*): kD = 2; - . - (o): kD = 4. Number of users K Figure 3.30 B E R vs K for m = 1, L = 2, A^1) = 8 and C L P C for different fcD. feo = 1; - - (*): kD = 2; - . - (o): kD = 4.. 162 10° Number of users K Figure 3.31 B E R vs K for m = 1, L = 1, and SIC for different — (+): = 1; - (*): 7V(i) = 4 ; - - (o): =8. : : : : : : : : : : : : : : : : : ! : : : : : : : : : ! : : : : : : : : : : : : : : : : ! : : : : : : : : : : 1 I:::::::::::::: :l: : : : : : : : : : : : : : : :: ::::::::::! M: : : :m: : : : :U : , - : ^mOf ^l|:!!ii!!!!!!!n —. ,-rn • - .*.^~^rr. „ : : - ' i . . y * r ^ \ -* • + : :::::::••::•(•:::•:::::: * -- *+ :::: L i-;:::::::::::::::: 1 1 1 1 0 10 20 30 40 50 60 Number of users K Figure 3.32 B E R vs K for m = 1, L = 2, and SIC for different — (+): = 1; -'(*): JVd) = 4 ; - - (o): =8. 163 Results for "a multi-cell system are illustrated in Figs. 3.33 and 3.34, with = 4 and J V ^ = 8, respectively. Similar remarks as for Figs. 3.10 and 3.11 can be made: in particular, the use of more than two diversity branches does not improve the B E R for all values of K, especially that the multiple-access interference is now higher because of the interference contributions of other cells. Figs. 3.35-3.37 further illustrate the effect of the m-parameter for Nakagami fading, for L = 1, 2, 3 and = 8. Figs. 3.38-3.40 plot the B E R as a function of the number of codes for a flat-fading (L = 1) lognormal channel and different variances a IN-3.2.4 Conclusions The goal of this section was to provide an accurate analysis of reverse link multicode C D M A systems with noncoherent M-ary modulation and equal-gain combining in mul-tipath fading channels. We provided convenient expressions for the interference terms as a function of the aperiodic crosscorrelation functions. The derivation of the error probability relied on the fact that the multicode self-interference is dependent on the desired received signal, because both are subject to the same level of fading. This led to a considerable improvement in accuracy as compared to a simpler method. Using this expression for the B E R , we verified that the use of E G C allows improvement only for a certain range of values of the total interference seen at the receiver. In particular, when the number of users is too large, the B E R is slightly increased by the use of E G C . Also, for users with a large number of codes, the use of diversity increases considerably the multicode self-interference, which can lead to reduced performance compared to a system with little or no diversity. The analysis applies to any type of fading, and results were given for the common cases of Rayleigh, Nakagami, Rician and lognormal fading. The case of correlated diversity branches was also treated within the same framework. The analysis was extended to deal with closed-loop power control using the inverse update algorithm, successive interference cancellation, and multi-cells systems. 164 10 10 20 30 40 Number of users, K F i g u r e 3.33 Multicell system: B E R vs K for m = 1, = 4. — (+): L L = 2; - - (o): £ = 3. a 10 20 30 40 Number of users, K 50 F i g u r e 3.34 Multicell system: B E R vs K for m = 1, i V ( 1 ) = 8. L = 2; — (o): L = 3. - (+)= £ 165 Number of users, K F i g u r e 3 .35 Mul t ice l l system: B E R vs K for L = 1, = 8. — (+): m = 1; - . - (*): m = 2; — (o): m = 3. 10° Number of users, K F i g u r e 3 .36 Mul t ice l l system: B E R vs K for L = 2, i V ^ = 8. — (+): m = 1; - . - (*): m = 2; — (o): m = 3. 166 10" 10 20 30 40 Number of users, K 50 Figure 3.37 Multicell system: B E R vs K for L = 3, = 8. — (+): m = 1; - . - (*): m — 2; — (o): m = 3. 30 Number of users K Figure 3.38 Multicell system: B E R vs K for lognormal fading, L = 1, OLN = 0.2, TTILN = 0. _ (+): JVt1) = 1; - - (*): - 4; - . - (o): J V ^ = 8. 167 10"' ! I - " . - " . - I I I ^ F ^ l l . . -, . . . -<f ..ii i •. : : : n •71 *i '•+(• i • i •• 1 1 1 • i 1 1 -1 Number of users K F i g u r e 3.39 Multicell system: B E R vs K for lognormal fading, L = 1, GLN — 0.5, mL^ = 0. — (+): = 1; - - (*): iVd) = 4; - . - (o): = 8. 10° Number of users, K F i g u r e 3.40 Multicell system: B E R vs K for lognormal fading, L = 1, aLN = 1.0, TTILN = 0. . — (+): JVU) = 1; - - (*): ATd) = 4; - . - (o): =8. 168 3.3 Reverse Link Performance with Coherent BPSK Modula-tion and Complex Spreading Sequences 3.3.1 Introduction The reverse link of an IS-2000 system supports simultaneously a number of control or data channels, each one spread by a certain Walsh code in order to maintain orthogonal-ity. As such, we can view its operation as a form of multicode transmission. In addition, some of the channels (such as the Supplemental Channels) support multiple spreading rates, leading to multirate transmission. Unlike the IS-95 systems from which it partially evolved, the IS-2000 system makes use of complex spreading sequences in order to limit the power imbalance between its in-phase (J) and quadrature (Q) branches. Also, coher-ent modulation is used on both forward and reverse links. A n analysis of a IS-2000-type uplink with coherent modulation was provided in [221]. However, it assumed only one channel per I/Q branch (the pilot channel on the / , and the data channel on the Q). Other earlier works which derived the S E R of C D M A systems with complex sequences for various modulation schemes include [222], [223], [224], [225]. A l l considered single-code transmission. In this section, we consider the general situation where multiple channels are trans-mitted on each branch, and derive the B E R performance of such a multicode C D M A system with complex spreading sequences, in a multipath fading environment. In such a system, the fading which affects the multicode self-interference and the multipath self-interference is not independent of the desired code: indeed, since all codes of a same user are transmitted through the same physical channel, they fade in unison. Therefore, unlike most recent analyses of coherent multicode C D M A which assumed independent fading, we take into account the dependence between the fading of self-interference and that of the desired code. We wil l show that this is necessary to obtain reliable results, es-pecially for the case of few interfering users. We derive an expression for the B E R where the self-interference is conditional on the fading coefficients affecting the desired code. We then integrate numerically this expression over the pdf of the fading. This method was also used in the previous section for the case of noncoherent M-a ry modulation with equal-gain combining. 169 The organization of this section is as follows. Section 3.3.2.1 describes the signal model, and Section 3.3.2.2 details the receiver model and the decision metrics, whose statistics are derived in Section 3.3.2.3. Section 3.3.2.4 presents the error probability analysis, which is validated against simulation results in Section 3.3.3. 3.3.2 Error Probability Analysis 3.3.2.1 Signal and Channel Models The transmitter uses B P S K modulation with quadrature branches, in which different channels are mapped to each of the / and Q branches. For example, in the reverse link of a cdma2000 system, the Pilot Channel (PCH) and the Supplemental Channel 2 (SCH2) are mapped onto the /-branch, while the Fundamental Channel (FCH) , the Dedicated Control Channel (DCCH) and the Supplemental Channel 1 (SCH1) are mapped onto the Q-branch. Let Nj^ and NQ^ be the number of channels (and thus codes) assigned to branches / and Q of user k, where k = 1,2,. . . , / C We denote by {kc} code c (c = 0 , 1 , . . . , Nj^ — 1 or NQ^ — 1) of user k, for either the / or Q-branch. The streams of binary data symbols for code {kc} of branches / and Q are given in the time-domain by: oo i=-oo oo j=-oo where Tb is the symbol (bit) period. The transmitter uses concatenated spreading, as defined in the introduction. Walsh sequences are used for orthogonal covering, as in the cdma2000 standard. Each Walsh sequence corresponds to a row of a Hadamard matrix of size M ([177], Ch.5), and thus consists of M elements taking values of 1 or -1 (mapped from symbols 0 and 1, respectively, of the Hadamard matrix). The Walsh sequence 170 assigned to code {kc} of the /-branch, = [wffiwffl ... I<4*M-I]> satisfies: twM.wM) = \Mii{kc) = {jd) \ 0 if(kc)*(jd) where (x • y) is the scalar product operation between vectors x and y. The periodic orthogonal covering sequence can be expressed in the time domain as: Wikp\t) = ]T Wjkc\t-hTw) h=—oo oo M—1 = E Y,w^)PTwc{t-hTw-iTwc) (3.92) where Tw is the duration of WJkc\t) and period of WJhp(t). Prwc(t) is a rectangular pulse of unit amplitude and duration Twc = TW/M seconds. The Walsh sequence assigned to code {kc} of the Q-branch, W Q C \ is defined in a similar manner. After the orthogonal covering phase, the data on the / and Q-branches are spread by the same complex spreading sequence a^k\t) = a^{t) + jaq\t), where j = and: oo a?\t) = 4 > T c ( i - i T c ) , 6 [-1,1] (3.93) j'=-oo oo « 0 } ( * ) = E ^Mt-JTC), a g . G [ - l , l ] (3.94) j=-oo where T c is the P N chip duration. The chip durations of the Walsh and P N sequences are not necessarily equal. For example, in cdma2000 systems Twc can be a multiple of Tc. After complex spreading, the real and imaginary parts of the output are separated and modulated onto orthogonal carriers, as illustrated in Fig . 3.41. 171 (l,c) Wj (t) tfj c)(t)—<X a^(t)+ja^(t) cos(wct) ke() -* lm() -s(]c)(t) (l,c) w 0 (t) sin(wct) Figure 3.41 Channelization/Spreading/Modulation subsystem. The transmitted signal of the kth user can be expressed as: < > - i s?Xt) cos(wct + <f>?c)) - ]T s^(t) cos(wct + ^c)) c=0 c=0 J2 s%(t)sm(wct + <J>?c))+ £ s%(t)sin(wct + ^ c)) (3.95) c=0 c=0 where = [ v ^ ^ c ) w ^ f c ) W ] 4 f c ) w , s%(t) = [Jp^bfc){t)WJkc) „(*) (*) = [ ^ f ' W ^ W ^ ' C ) and P J f c c ) = ^ * c ) T 6 ( * c ) is the.average power of code {A;c} of branch a; (x = / ' o r Q). The channel model is the same as that used in the previous section. 3.3.2.2 Receiver Model and Decision Metrics The composite signal received at the output of the channel is: K Lc-l (3.96) k=l 1=0 172 n{t) is A W G N with double-sided power spectral density NQ/2. Note that all the received codes pertaining to a same user are affected by the same fading coefficients and delay, since they are transmitted via the same channel. The phases, however, can be different if a specific phase shift is assigned to each channel by the transmitter (as in IS-95B). The receiver consists of L Rake fingers and performs either M R C or E G C . One finger of the receiver is illustrated in Fig. 3.42. In the following, the metrics calculated refer to code 0 of user 1 of branch / , which is assumed to be the desired stream. They further correspond to finger n of the Rake receiver. Without any loss of generality, we assume that N \ K ) = N { Q ] = N ( K ) VA; in order to ease the notation. F i g u r e 3.42 Demodulator for the nth Rake finger. The quantities uy' it) and UQ (t) at the output of the lowpass filters are given by: 4n)(t) = (r(t)cos(wct + ^ ) ) L P + (n(t)cos(wct + ^ ) ) L P (3.97) u%\t) = (r(t)sm(wct + ^WJ))LP K J V W - l t c - l = ^ E E £«! f c ) w(4?(*^ + (n(t)sm(wct + ^ ) ) L P (3.98) 173 where (-)Z-P denotes low-pass filtering, ipxkf Akc) n(k) (fc) , (kc) _ ( f cc )_ (10) : > J — <px i-f; , ana <px<nl — tpx<l y I < n . The output of the complex spreading operation is given by: = U +j[-uP(t)a$(t - r^) + rtiWPit - T ^ ) ] (3.99) where (•)* stands for complex conjugate. From Fig . 3.42, the decision metric for Rake finger n is thus given by: {n) , ( i ) + r ( irj ) = r i J Re[U^(t)}wi10\t-r^)dt fc=l c=0 /=0 J T n p ( f c c ) , ( f c c Q °Q + p(kc)^(k<. I p ( fcc ) Akc \JFQ bQ yfpj^bf6-I p ( f c c ) , (fcc \jFQ bQ ( t - r W ) W ^ ( t - r W {t-r^)Wlkc\t-r^ ( t - r ( * ) ) W ( * « ) ( t - r W (t-Tlk))WikC\t-T^ (t-r^)W^(t-rik) {t-rlk))W\kc\t-rlk) (t - T^)wgc\t - r/ f e ) ag'ct-(k Ti )wf0 ) ( t - C 0 S (vg.nl) (fc) r ; ) W f0 ) ( i - s i n ( v © 4 f e )(i- (fc) Ti )Wf 0 ) ( t - s i n (^gS) (fc )Wf 0 ) (*-sin(^2) (fc Ti )wf 0 ) (*-a ? ( * -(fc r j )W\w\t- cos(v?2) a?>( t - (fc )WJw\t- C0S(^g,n()] _ c(") o w i r(n) °I "r 7 M P , / , r(") ^ J C 5 ' , / (n) M C , / ^ MA.I An) 1N,I (3.100) where Re[-] is the real part operator. is the signal term, I^PJ IS the multipath self-interference, I'cl i is the complex spreading self-interference, I^ci is the multicode self-interference, i l s the other-user multi-access interference, and is an integrated noise term, all referring to the nth finger. In order to obtain closed-form expressions for 174 the previous signal and interference terms, we use the following aperiodic crosscorrelation functions: Rg'Ur) = [Ciky%z(l + 1-N)- C%%{1- N)]{r - lTe) + [C%%{1- N)]T&.m) RifUr) = [Cgl(f + 1) - CMS{T)]{T - lTe) + [CJg%MTc- (3-102) with: N-l-l ^xy,wz(^) ~ \ E (fcc) 3=0 N-l+l x i w 0<1<N-1 Yw ,^,a{kc) T nW.i.al1"' 1-N<1<0 / J w,[J x,j-l-N z,L£J y,3 — 3=0 0 \l\ > N Am (3.103) and where N = Tb/Tc is the processing gain, h = Twc/Tc is the number of P N chips per Walsh chip, I — [ ( T ^ — ITC)/TC\. The indices {xy} and {wz} are chosen from {II} , {IQ} , {QI} or {QQ}. After expanding the expression for U\n\ the terms in Eq. (3.100) can be written as follows: (3.104) r(") -lN,I (i) [(n(t)cos(wct + ^ ) ) L P a ? ( t - r ^ ) +(n(t) s inKt + ^ n ) ) i P ^ \ t - rP)]dt, (3.105) L , - l An) 1MP,I [ 0 ^ > ( 1 ) ) + *SS>(1))) + + « / / 0 - ( 1 ) ) ) P - i o 6 ) 175 (3,107) NW-I L c - 1 = E E aW'( 1. c). (3-108) c = i ;=o = E E E a W M , ( 3 - 1 0 9 ) fe=2 c=0 (=0 where: + C*S^(fc))) s i nO +v^(C<W(-(fe))+c<w-(fc)))cos(^^)] (3-n°) 176 and where {u,v} = {-1,0} and = rnk) if rnh) > 0, and {u,v} = {0,1} and r<fc> = Tni + T& if T ^ . < 0. Notice that for I^CI the component for which indice I = n has been removed. Indeed, when the collected multipath components are synchronous with the desired signal (which happens when I = n), the interference produced by these components is null due to the orthogonal covering of the spreading sequences. A similar development can be carried out for the decision metrics on the Q branch, leading to the following: U r(n) Q , r(") _L /•(") 4- r(n) + rO ^ 1CS,Q ^ 1MC,Q ^ MA,Q ' A (3.111) where Im[-] is the imaginary part operator, and: ,(n) Q (3.112) + (n(t) s i n K i + ^ ^ ( i - r ^ ) ] * , (3.113) [C(«',«(- ( i ))+*a,<T"»))+©cn+« w( T ( , )))p .n4) 177 m,Q = 5 E • n - w d - " ) + « » ( * © 1=0 l#n + V ^ ( C < W - ( 1 ) ) + C<W- ( 1 ) )) sin(*C) + > / ^ C ? < ! r g ( ^ ( 1 ) ) + C^/Q^15)) (3-115) NW-I L c - 1 WCQ = E E a W ( l , c ) , (3-116) c=l 1=0 l^n K A f ( f c ) -1L C -1 = E E E«W*.C>- <3-17» fc=2 c=0 1=0 where: + v ^ ( « < W - ( f c ) ) + C ^ S U ^ ) ) c o s O + V ^ ( 6 W / 2 g j ) / g ( r W ) + b^R^JQ(r^)) s i n ( ^ ) - v^c^k^)+cc)#/:ig(-(fc))) sin(^i) 178 3.3.2.3 Statistics of Decision Metrics Maximal-Ratio Combining Let Eb = E^p and Tb = T ^ . W i t h M R C , the expected value of the desired received signal 5/ = J2t=o a n ) 5 { n ) is obtained from Eq. (3.104): n= 0 The noise term IN>I — Yln=o a " ^jv,/ c a n be easily shown to be a zero-mean Gaussian random variables with variance: 4 / = V a r [ / A , , / ] = ^ ^ k 1 ) ] 2 . (3.120) n=0 Using the standard Gaussian approximation, we model the interference terms Eqs. (3.106)-(3.109) as mutually uncorrelated zero-mean Gaussian random variables with given con-ditional variances. These variances are derived below, and are conditional on the ran-dom variables I = 1,2, . . . , L . The interference terms IMP,I = E « = o A « ^ M P , / ' Icsj = E n = d anl)lcs,i a n d JMC,I = E ^ = o an)L(MC,i a r e c h i P synchronous with the de-sired user signal, since they originate from signals which are transmitted synchronously on> the reverse link. Therefore T$ = TN)] for all ft, n and I. This results in V a r ^ L f r ^ ) + Rxyljzir^)} = Tb2/N (with l^ri). However, the interference term IMA,I = E n = o " " ^ M A , / is chip asynchronous with the desired user signal, from which Var[i?i^i 2 (r^ ')+^iy < ; l ( , 2 :(r ) [^)] 2T 6 2 / (3iV), k = 2 , 3 , . . . ,K. From these observations, the following variances are ob-tained: p ( 1 0 ) T 2 L-1 Lc-1 = V a r f W ^ ^ ^ ^ k 1 ) ] 2 ^ ^ ] 2 , (3.121) n=0 1=0 = V a r f e , ] = J2[an^(Pr E ^ + < ° } E rf'H. ( 3 - 1 2 2 ) L-1 CSJ 4N n=0 1=0 1=0 179 L-l (p(lc) p ( l c ) \ T 2 L c - l °1UCJ = var|/MC,,] = 2><'f £ 1 ' \FNQ ' ' E t f ' l ' n=0 c=l 1=0 Jjfn L - l /Y JVW-1 /p(fcc) , p ( f c c K T 2 L c - l n=0 A;=2 c=0 1=0 In Eqs. (3.121), (3.122) and (3.123), the variances.are seen to be conditional on the term X]f=o 1[ Q !/ 1^] 2- Indeed, the self-interference terms IMPI, ICS I and IMC i are affected l^n by the same coefficients af \ I — 0 , 1 , . . . ,L — 1 as the desired signal Si, since they all fade in unison. Thus these interference terms are correlated with the signal term, and this fact must be taken into account when determining the error probability. In contrast, the other-user multiple-access interference term IMAJ is affected by the fading coefficients a[k\ I = 0 , 1 , . . . , L — 1, k = 2 , 3 , . . . ,K, which are indedependent from af \ I — 0 , 1 , . . . , L — 1: indeed, the signals from each user are assumed to travel along different paths to the base station, and hence experience independent fading. As a consequence, the squared fading coefficients (a| f c ' ) 2 , I = 0 , 1 , . . . , L — 1, k = 2, .3, . . . , K can be replaced by their expected values — E[(a[h^)2], as was done in most analyses using the Gaussian approximation [39]. The total variance from the interference and thermal noise sums up to: °I = alMP,l + UlMC,I + ° / c S , / + °\iA,I + ANJ- (3-125) Equal-Gain Combining W i t h E G C , the signal and interference terms are now 5/ = X3n=o ^i^-> = En=o -^ v,/> T \-^L—1 T(TI) r v-'L—1 An) T v - ^L -1 An) j T rr^L—1 An) 1MP,I - 2^n=0 1MP,D 1CS,I - Z^n=0 iCS,V 1MC,I ~ 2^n=0 1MC,D A N C L 1MA,I - 2_,n=0 MA,I' Proceeding as in the previous section, the signal term is now: L - l 5/ = VWtYan] (3-126) n=0 180 and the variances of the (Gaussian-modeled) interference terms: ,(10)^2 Lc-1 ajMpj = V a r f e ^ ^ — A L ^ h ^ ] 2 , (3.127) 4N 1=0 ajCSJ = V a r [ / C S i / ] = ^ L ^ ^ a ™ ] * + I P ^ ^ f ) , (3-128) <c, = V a r [ / M C , , ] = L £ ' ' 1 ' E K 1 " ] 2 . (3.129) c=l (=0 K NW-1 (p(kc) p{kc)srp2Lc-l a?.,, = V a r | W ] = L E £ ' £ " E"i"- («*» fc=2 c=0 1=0 The total variance from the interference and thermal noise is obtained as per Eq. (3.125). 3.3.2.4 Probability of Error In order to clarify the presentation of the B E R derivation, but without loss of gener-ality, we take the number of fingers of the Rake receiver, L , to be equal to the number of multipaths L C (i.e. all the energy of the channel is collected). If some paths are not collected (i.e. L C > L), then the interference resulting from these paths can be considered as additional independent Gaussian noise, and incorporated in the term IMA,I, without having an impact on the validity of the B E R derivation: indeed, the fading coefficients af\ I = L , . . . , L C — 1 are independent of the desired signal, because in this case they don't appear in the expression for S[, i.e. Eq. (3.119). It is considered unlikely to have L C < L , because in this case the Rake receiver would simply switch off its fingers which aren't collecting any signal, instead of adding background noise to the received signal. Moreover, if the powers in each multipath are similar, we can replace the term 181 E f i o W 5 ] 2 b y Td^W?? i n o r d e r t 0 e a s e t h e analysis, as in [39]. Maximal-Ratio Combining From Eqs. (3.119), (3.125) and (3.121)-(3.124), and using the previous assumptions, the conditional output S N R at the receiver, SNR0 = S/faj, takes the following form: = - 7 £ = i — r ° / ^ - i — x — ( 3 - 1 3 1 ) ^ ( E ^ 1 ) ] 2 J + S ( E [ - ! 1 ) ] 2 J + C where the terms A, B and C are constants given by: A = 4 ^ ' ( 3 - 1 3 2 ) S 4iV 1 f 1 + 9 ^ ^ L ) + — V f + l (3 133Y C K j V W - 1 / Trp(kc) p{kc) \ Lc-1 ° i V fc=2 c=0 \ 6 J 6 , / 6 Jb,Q J 1=0 N, 0 (3.134) 2Eb If all the energies and bit periods are the same, the expressions for B and C reduce to: B = — + —(NW - l ) , ' (3.135) c = -Ly y YQW + IOL. ( 3 . I 3 6 ) k=2 c=0 1=0 182 Moreover, if no data is transmitted on the Q-branch, i.e. B P S K modulation is used, these expressions become: B = — + —(NW-1), (3.137) ' 4N 2NK h K ' K j V ( * ) - n , c - l C 3N ^ ^ ^ 1 2Eb k=2 c=0 1=0 It can be seen that SNRo, as given by Eq . (3.131), is a ratio of sums of weighted gamma variables, with some of these variables present in both the numerator and the denominator. A general expression for the pdf of such a ratio is given on pp. 155-156 of [226]. It is everything but simple. Instead, we decided to try out an approximation. If we replace the second term in the denominator of the right-hand side of (3.131) by B IZiio1^^]2 ( i - e - the indice / = 0 is included in the summation), and use the previous assumption that L — L C , the approximate output S N R becomes: S N B ° = ( A T W T C <3'139' where s = ^ n = o [ a « ^ ] 2 - The pdf of s was given for example in Eqs. (3.41) and (3.43) for independent Rayleigh and Nakagami fading, respectively, and in Eq . (2.231) for Nakagami fading with 2 correlated branches. The B E R can then be written as: P. = Jf 1 ^ ( A + B)s + c)^s)iS- (3'140) Since we haven't found any closed-form solution for (3.140), we evaluated it numerically. 183 Equal-Gain Combining From Eqs. (3.126)-(3.130), the conditional output S N R at the receiver is now: SNR0 'L-l 2 EA»> IL _n=0 'Lc-1 1=0 (3.141) where the terms A, B and C are as given previously. In the above, the numerator term s = E n = o an^]/L is the weighted square of the sum of the fading coefficients, while the terms 2^=o~ ^ " P ^ 2 a n ^ X ^ i " 1 ^ 1 ' ] 2 i n the denominator are sums of the squares of the fading coefficients. Hence, contrary to the M R C case where by replacing B J2i=i1[a[1'1]2 with B Y^i=o 1 [ Q ! P^] 2 ^ w a s possible to obtain an expression for SNRQ conditional solely on Yln=\\[a^}2i here no such simplification is possible. Nevertheless, considering the case L — 2 for which a closed-form expression for the pdf of s is known (c.f. Eq . (2.201), with p = 0 for uncorrelated channels), we attempted to replace the terms X)j!=o ^ l ^ ] 2 a n ( i X ^ r 1 ^ / 1 ' ] 2 w i t h s. As shown in the next section, satisfactory results were obtained for L — 2 using this approximation. 3.3.3 Performance Evaluation Results and Discussion We implemented an IS-2000 software simulator (c.f. Section 4.2.7) in order to check the validity of our equations, with the parameters specified in [14]. Throughout our simulations, we assume that NW = 1, k = 2 , 3 , . . . ,K, while N<M can take different values. Also, with no loss in generality, we run simulations only for the case where no data is transmitted on the Q-branch, and the power and symbol periods are equal for each channel (although our simulation model is general enough to include data on both / and Q-branches, and different powers and symbol periods). Simulation and analytical results are plotted in Fig . 3.43 and 3.44, for iV^1) = 8 and = 4, respectively, and L' = 1, 2, 3, for a Rayleigh fading channel with = 1 for al l l,k. In the case of 8 codes, Eq . (3.140) gives a very good approximation to the actual B E R . However, for a low number of codes (e.g. for iV^1) = 4), there are slight 184 discrepancies when L > 1, especially when the number of users is small. This is due to the fact that the Gaussian approximation is used, in conjunction with the approximation made in deriving Eq. (3.131). Figs. 3.45 and 3.46 compare the B E R obtained by using our method with that resulting from overlooking the dependence between the fading affecting the codes of a same user, for N1-1^ = 8 and L = 2,3, respectively: a significant improvement in accuracy can be noted, especially for a low number of users. Figs. 3.47-3.49 illustrate results for = 8, = 4 and = 1, respectively, and L — 1,2, 3, for a Nakagami fading channel with m = 2.0 and Q,\k^ — 1 for all I, k. As for the case m = 1.0, it is seen that the accuracy of the approximation increases with the number of multicodes. In all cases the results are more accurate (or equal for — 1) than if we made the independence assumption between the codes of a same user. The effect of the m-parameter is shown in Figs. 3.50-3.52 for diversity orders L = 1,2,3, respectively, with = 8. For increasing L, it is seen that the effect of the m-parameter on the B E R is reduced: indeed, with a higher diversity order the combined signal is less likely to be in a deep fade, and is thus less sensitive to the severity of the fading determined by m. Figs. 3.53 and 3.54 present results for a Rician fading channel with L — 1 and L = 2, respectively, for different values of the Rician parameter A. For L — 1 there is no multicode interference, and the match between theory and simulation is very good for all K. For L — 2 the theoretical curves become more accurate as K increases, especially for higher A's: for K > 10 the match is very good in all cases. As in the Nakagami case, when a higher diversity order is used, the effect of the parameter A on the B E R is reduced, since the combined signal becomes less sensitive to the severity of the fading. Next, results are presented for a system with two correlated diversity branches. In Fig. 3.55 a correlation coefficient p — 0.5 is used, and the channel is Rayleigh fading. The theoretical B E R curves are close together for different numbers of codes A ^ 1 ' = 1,4,8, and agree reasonably well with the simulation results, especially for a higher . Figs. 3.56 and 3.57 illustrate the effect of the correlation coefficient p for a Nakagami fading channel with m = 2.0, for A/W — 8 and A/W = 4, respectively. As before, the accuracy of the analysis is slightly better for a higher N^. 185 Fig. 3.58 compares the approximate analytical error probability of E G C of Section 3.3.2.4 with simulation results, for a Rayleigh fading channel, L = 2, and different J V ^ ' s . There is good agreement between both, especially for — 8, despite the rough ap-proximations which needed to be made in the analysis. Figs. 3.59 and 3.60 show results obtained with E G C and two correlated diversity branches, in Nakagami fading with m = 2, for different correlation coefficients. Again due to the approximation in the anal-ysis for E G C , the theoretical curves underestimate the simulated SER's , with a better match seen for the case A W = 8. 3.3.4 Conclusions This section provided an analysis of reverse link multicode C D M A systems with co-herent B P S K modulation and maximal-ratio or equal-gain combining in multipath fading channels. As in the previous section, the analysis took into account the fact that both the multicode interference and the desired signal were affected by the identical fading pro-cess. Using some approximations to facilitate the analysis, good matches were obtained between the theoretical curves produced by the method and the simulation results. The analytical results were seen to be especially accurate for a higher number of codes (e.g. 8 codes). The analysis applies to any type of fading, and results were illustrated for the cases of Rayleigh and Nakagami fading, for both independent and correlated diversity branches. 186 Number of users K F i g u r e 3.43 B E R vs K for =8. + + + : £ = 1; * * * : £ = 2; o o o : L = 3. Number of users K F i g u r e 3.44 B E R vs K for = 4, + + + : L = 1; * * * : L = 2; o o o : L = 3. 187 188 10 20 30 Number of users K F i g u r e 3.47 B E R vs K for = 8, and ro = 2.0. — (+): L = 1; - . - (*): L = 2; - - (o) L = 3. 20 30 40 Number of users K F i g u r e 3.48 B E R vs K for = 4, and m = 2.0. — (+): L = 1; L = 3. *): L = 2 ; - - ( o ) 189 20 30 40 Number of users K F i g u r e 3.49 B E R vs K for = 1, and m = 2.0. — (+): L = 1; L = 3. (*•): L 20 30 40 Number of users K F i g u r e 3.50 B E R vs K for i V ^ = 8, and L = 1. — (+): m = 1.0; (o): m = 3.0. (*): 190 10"' Number of users K Figure 3.51 B E R vs K for = 8, and L = 2. — (+) : m = 1.0; - . - (*): m = 2.0; (o): m = 3.0. Number of users K Figure 3.52 B E R vs K for = 8, and L = 3. — (+): m = 1.0; - . - (*): m = 2.0; (o): m = 3.0. 191 Number of users K F i g u r e 3.53 B E R vs K for A^ 1 ) = 8, and L = 1. — (+): A = 1.0; - . - (*): A = 2.0"; (o): A = 3.0. F i g u r e 3.54 B E R vs K for M 1 ) = 8, and L = 2. — (+): A = 1.0; - . - (*): A = 2.0; (o): A = 3.0. 192 10° Number of users K F i g u r e 3 .55 B E R vs K for L = 2 correlated branches with p = 0.5 and m = 1.0. — (+) iVW = 1; - . - (*): = 4; - - (o): J V « = 8. 10"' 10-=l , 1 J 1 1 1 0 10 20 30 40 50 60 Number of users K F i g u r e 3 .56 B E R vs K for L = 2 correlated branches and m = 2.0, with = 8. — (+) p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7. 193 0 10 20 30 40 50 60 Number of users K Figure 3.57 B E R vs K for L = 2 correlated branches and m = 2.0, with = 4. — (+) p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (©): p = 0.7. 10° io- 5 lO"6LeJ 1 ' > ' ' 1 0 10 20 30 40 50 60 Number of users K Figure 3.58 B E R vs K for E G C with L = 2 and m = 1.0. — (+): = 1; - . - (*) JV(i) = 4 ; - - (o): =8. 194 Number of users K Figure 3.59 B E R vs K for E G C wi th L = 2, m = 2.0 and = 8 (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7. 0 10 20 30 40 50 60 Number of users K Figure 3.60 B E R vs K for E G C with L = 2, m = 2.0 and = 4. — (+): p = 0.0; (*): 0 = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7. 195 3.4 Forward Link Performance with Coherent QPSK Modula-tion and Real Spreading Sequences 3.4.1 Introduction As mentioned in Section 3.1.1, the forward link of IS-95B uses concatenated W a l s h / P N spreading. Since the BS transmissions to each mobile user are chip synchronous, in the absence of multipath there won't be any intra-cell interference (due to the orthogonality of the Walsh sequences), but in a multipath environment a mobile user wi l l suffer from the interference due to the multipaths of the signals intended for other users (which are no longer orthogonal). In a forward link configuration the interference affecting a mobile user travels on the same propagation path as the desired signal, hence the signal and interference from all users fade in unison, unlike in a reverse link configuration where only the multicode interference fades in unison. Most analyses of the C D M A forward link B E R considered asynchronous transmission and non-concatenated spreading [227], [228], [229], [230], [220], [231], [232], [233]. A few studies have considered synchronous transmission and concatenated W a l s h / P N spreading [178], [234], but either the theory and simulation weren't compared ([178], possibly due to approximations used in the analysis, which can cause a significant departure from the exact B E R values), or they didn't match ([234], possibly due to a mistake in the analysis). In this section, we provide a detailed analysis of the IS-95B forward link, using the actual modulation scheme (while other studies often used a simplified version), and derive the B E R in a semi-analytical manner. We show that the Gaussian approximation works quite well for a wide range of values, which is seen by a good match between theory and simulation. Moreover, we obtain results for the general case of correlated diversity branches, for Nakagami fading with M R C or E G C , thus extending the scope of previous work which only considered independent Rayleigh fading with M R C [178], [234]. 196 3.4.2 Error Probability Analysis 3.4.2.1 Signal Model The transmitter uses the type of Q P S K modulation specified in IS-95, i.e. the same information sequence is mapped onto both its / and Q branches, as opposed to the traditional Q P S K scheme where consecutive symbols are mapped alternatively onto each branch. This leads to quadrature spreading, i.e. the information sequence is spread by two sequences. Let be the number of channels (and thus codes) assigned to user k, where k = 1, 2 , . . . , K. We denote by {kc} code c (c = 0 , 1 , . . . , 7V ( f c )) of user k. The stream of binary data symbols for code {kc} is given in the time-domain by: oo &(*<=)(*) = Y bfc\t-jTb), bfc) e [-1,1] (3.142) j=-oo where Tb (= Ts) is the bit (symbol) period. The transmitter uses concatenated spreading, with Walsh sequences (c.f. previous section) used for orthogonal covering. The Walsh sequence assigned to code {kc}, = [w0kc^w[kc"1.. . w £ ] , satisfies: M if(fcc) = (jd), 0 if (Ac) ^ ( j d ) . The periodic orthogonal covering sequence can be expressed in the time domain as: oo W[kc\t) = 53 W{kc){t-hTw) h=—oo oo M - l = E J2wikC)PT„At-hTw-iTwc) (3.143) h=—oo i=0 where Tw is the duration of (t) and period of Wpkd) (t), prwc (t) is a rectangular pulse of unit amplitude and duration Twc = T-wjM seconds. Two different P N sequences, ar(t) and OQ(£), are used on the I and Q branches of the transmitter, respectively. These same sequences are used to spread all of the information channels associated with the users of one cell, in a forward-link configuration. They may 197 be expressed as: oo 0/(t) = Y ai,jPTc(t- JTC), aId € [-1,1], (3.144) j=-oo oo j ' = - o o where Tc is the P N chip duration. In the following, for simplicity, we take the chip durations of the Walsh and P N sequences to be equal, i.e. Twc = Tc. This is the case for IS-95, but not necessarily for cdma2000, where Twc can be a multiple of Tc. In the latter case, one simply needs to replicate Nu = Twc/Tc times each chip of the Walsh sequence W^kc\ in order to obtain a new upsampled sequence Wikc) of length NUM: the new associated Walsh function Wikc\t) now has chip duration Tc. The analysis, however, wil l remain the same. The concatenated W a l s h / P N sequences are given by: aj{t)WPkc\t) oo E a^PTc{t ~ JTC), a\k% 6 [-1,1], (3.146) j=—oo aQ{t)W{kc\t) oo E a Q ^ ( * - i T c ) , ag^  € [-1,1]. (3.147) J = - o o The transmitted signal of the kth user can be expressed as: cos(wct + c=0 +6 { f c c )(t)ag i ) c(t) sin(™ c£.+ 4>(fcc))] (3.148) where = E{kc)/Tb is the average power of code c of user k, which we assume identical for all users without loss of generality ( P ( f c c ) = P for all {k, c}). The IS-95B forward link transmitter is illustrated in Fig . 3.61. a\k?(t) = 198 aQ(t) sin(w<P Figure 3.61 Transmitter for cth code of user 1. 3.4.2.2 Receiver Model and Decision Metrics The composite signal received at the output of the channel is: K N^l-l LC-1 fc=l c=0 (=0 +bM(t - r/fc))aig(* - rh s in ( W c * + rf=c))] + n{t) (3.149) with ip[kc^ = 4>(HC^ + 0\k>> — WcT^. n(t) is A W G N with double-sided power spectral density NQ/2. AS in the reverse link case, note that all the received codes pertaining to a same user are affected by the same fading coefficients and delay, since they are transmitted via the same channel, while the phases can be different if a specific phase shift is assigned to each channel by the transmitter. The receiver consists of L Rake fingers and performs either M R C or E G C . In the following, the metrics calculated refer to code 0 of user 1, which is assumed to be the desired stream. The decision metric at the output of the receiver, U is the sum of the decision variables at the output of the / branch, Uj and the Q branch, UQ. The former is given for M R C by Ut = Ylll dPuP and for E G C by Ux = E ^ o ^ w i t h : U\n) = / a > ( r ( t ) c o s K t + ^ 1 0 ) ) ) i p a ( i 0 ) ( t - r , l 1 0 ) ) ^ . (3.150) 199 Upon expanding terms in Eq. (3.150) and integrating, one obtains: ^ / - . ^ 7 ' MPJ ' MC,I ' MAJ ' N,I ^ . I D I J where the terms in Eq. (3.151) are defined as in the previous section. Let C = \\/P-The signal component (corresponding to the terms in Eq. (3.150) for which k = 1, c = 0 and I = n) is given by: 5 W = C 6 i 1 0 ) T b a W . (3.152) The interference terms are given as follows: it/ = c m E *?W10Ht - T p w y - TP) cos^T) Jrn 1=0 +6<«»(t - TpVgit - TP) sin(^%\f(t - TM)dtt (3.153) r T W + T B N W - l L C - 1 4"cv .= C / " E E « f 1 ) [ 6 ( 1 C ) ( * - ^ 1 ) ) ^ H * - r / 1 ) ) c o s ( ^ e ) ) Jrn c = i ;=o +6( l c)(t - r f V g ^ t - r / 1 } ) sin(#)]4;c0)(i - r^)dt, (3.154) I [ M \ : = C 7 / ( i ) E E E ^ ^ H ^ ^ V f f ^ - ^ ^ c o s ^ f ) + & ^ > ( i - T / * V ^ (3-155) = -jpjrW- n(t)a^(t-r^)dt. (3.156) They can be expressed as a function of the following aperiodic crosscorrelation functions: Rikyc)(r) = [CJM(l + l - N ) - C g c > ( f - N)](r - ITC) + [C<g\T- N)]Te, (3.157) 200 R[S\r) = [Cxkyc\l + l)-C^mr-m + [C{kc)mTc (3.158) where cSC)(0 = Nyaxkfa{% 0<1<N-1 j=0 /v-i+r V 0(*<=) _ J 1 0 ) j=o 0 IT! > TV (3.159) and N = Tb/Tc is the processing gain, I = |_(r/fe) - ITC)/TC\, and the indices {xy} are chosen from {II}, {IQ}, {QI} or {QQ}. The interference terms in (3.153)-(3.155) can then be expressed through these functions: 4 2 v = C E « i ( 1 ) [ ( ^ 0 ) «f t 0 ) ( r ( 1 ) ) + ^ f f V 0 ) ) c o s ( ^ ) 1=0 +(6( 1 «»/2g?(rW) + &l 1 0 ) i?g/V ( 1 ) )) s i n ( ^ 0 ) ) ] , ' (3.160) r(») 1MC,I c E T,"^^^ c=l (=0 + ( & l l c ) < C ) ( r ( 1 ) ) + e ' f l&V 1 ' ) ) sinfotf*)], (3.161) MA,I K N^-lLc-l CH E E ^ W C ) * ^ fe=2 c=o ;=o + {b^R[kf(r^) + ^ / ^ ( r ^ ) ) s i n ( # ) ] (3.162) where {u,v} = {-1,0} and = r^ f if > 0, and {u ,«} = {0,1} and = +Tb if < 0. Notice that for IMCJ and IMAJ, the components for which indice I = n have been removed. Indeed, when the collected multipath components are synchronous with the desired signal (which happens when I = n), the interference produced by these 201 components is nil due to the orthogonal covering of the spreading sequences. Signal and interference terms can be derived in a similar fashion for UQ, and are given as follows: Sg° = Cb[w)Tba^\ (3.163) 1=0 + (e )< )(- ( 1 )) + e^ggV 1 5)) s i n ^ H ] , (3.164) 4% = C E E ^ t ^ ^ ^ ^ + e 5 ^ ^ 1 5 ) ) ^ ^ ) c=l (=0 + (b£c) RQQ ) + R$g (rW)) s i n ( r f c ) ) ] , (3.165) 4% = ^ E E E ^ K ^ ^ ^ ^ + e 5 ^ ^ ) ) ^ ^ ) fc=2 c=0 (=0 + (e ) ^ Q ) (^ ) ) + e , ^ g ) ( r ( f c ) ) ) s i n ( ^ c ) ) ] . (3.166) 3.4.2.3 Statistics of Decision Metrics Maximal-Ratio Combining The conditional expected value of the desired received signal 5 = SJ+SQ = Ylt=o Oin\Sjn^ + S Q n ) ) . is obtained from Eqs. (3.152) and (3.163): 202 The noise term IN = INj + IN<Q = Ylt=o a^(^Nj + ^ Q ) c a n be easily shown to be a zero-mean Gaussian random variable with conditional variance: al = Var[I„] = ^ Y}^?- (3-168) n=0 Using the standard Gaussian approximation, we model the interference terms Eqs. (3.160)-(3.162) as mutually uncorrelated zero-mean Gaussian random variables with given condi-tional variances. These variances are derived next. The interference terms IMP = IMP,I + i-MP,Q - 2_.n=0 A N \2MP,I + 1MP,Q)^ LMC - *MC,I + J M C , Q - Z^n=0 A™ U M C , / + Y M C , Q J and IMA = IMA,I + IMA,Q = ]Cn=d ""^ 4^ ,/ + -^M^C?) a r e a 1 1 c h l P synchronous with the desired user signal, since they originate from signals which are transmitted syn-chronously on the forward link. Therefore rff = T$ for all k, n and I. This results in V a r ^ V i ? ) + Rxy\rnk))} = Tb2/N (with I ^ n if {kc} = {10}), and in the following variances: a]up = V a r [ / M P ] = ^  f > W ] 2 £ [of >]2, (3.169) 71=0 i = 0 a ? M C = V a r [ / M C ] = ^ ( i V ( 1 ) - l ) D^XX']2. (3-170) n=0 /=0 « ? „ = V a r [ / M , ] = EK") 2 E ^  EVP1]2-n=0 &=2 i=0 Since all channels are transmitted via the same physical propagation paths, we have that Oi\^ = for all k and I. The total variance from the interference and thermal noise 203 sums up to: -I MP ' ~'MC L-l + o2N EbTb ( n=0 (3.172) l^n k=l Equal-Gain Combining W i t h E G C , the signal and interference terms are now Si = YlnZoi^i^ + SQ^)> IN,I = v ^ L - l / r ( n ) , An) \ T _\-L-l(T(n) An) N J _ TT^L-1 , An) An) N , IMA,I = Y^tZo(^MA,i + IMA,Q)- Proceeding as in the previous section, the signal term is now: 5 = EbTb L-l n=0 and the variances of the (Gaussian-modeled) interference terms: TbN0 °~N = -2 °IMP 2 °IMC 2 GIMA Var[ / N ] = V a r [ / M p ] V a r [ / M C ] V a r [ / M A ] EbTb AN L-l Lc-1 ££h( n=0 1=0 Ijtn (1)12 EbTb 4JV L-l L C - 1 (1)12 EbTb 47V L-l K n=0 1=0 E E ^ ' E i - : (1)12 n=0 k=2 1=0 l^n (3.173) (3.174) (3.175) (3.176) (3.177) The total variance from the interference and thermal noise sums up to: a2 = EbTb K L-l L C - 1 k=l n=0 1=0 Ijtn J (3.178) 204 3.4.2.4 Probability of Error Maximal-Ratio Combining From Eqs. (3.167) and (3.172), and using (without loss of generality) the assumption that Lc = L, the conditional output S N R at the receiver then takes the form 2 : SNR0 = -,—— —. (3.179) L-1 IK1' 71=0 / K L-1 i l l . _L ^ ( f c ) \ ^ r ^ , ( 1 ) i 2 2Eb 2N \ " k=l 1=0 It is seen that the fading coefficients [ojj1^]2, 1 = 1,... , L — l appear both in the numerator and denominator of the SNR. Indeed, the interference terms are subject to the same fading as is the received desired signal. Therefore the terms [a^]2 in the denominator cannot be replaced by their expected values, like it can be done in the single-code reverse-link scenario. The expression above is thus the ratio of a combination of products and weighted sums of correlated random variables, and a closed-form solution for the pdf of the S N R is thus difficult to find, even for low diversity orders involving only a few terms. To circumvent this difficulty, we have used a semi-analytical approach to obtain the theoretical error probability. The conditional B E R (conditioned on the set {a^\l = 0 , . . . , L - 1}), is given for B P S K as: Ps({a\1]}) = Q(y/SNRo) - (3.180) with SNR0 as given above. The unconditional B E R is found by Monte Carlo integration [156], i.e. by averaging the above conditional B E R over a set of randomly generated 2 This expression is similar to [178], Eqs. (44) and (47), and [234], Eqs. (36) and (38), for a single-cell environment with Lc = L, with the exception that [178] and [234] use a factor 1/(3N) for the multipath interference, which corresponds to multipath which is not chip synchronous with the desired signal, while Eq. (3.179) uses a factor 1/(2N), which corresponds to chip synchronous multipath. However, [178] further makes an approximation in its Eq. (44) to derive the B E R , but doesn't check its accuracy by simulation, while the theoretical and simulation results of [234] are curiously very far apart. 205 {a, ( 1 ) ,Z = 0 ,L-iy. i Niter P s . = iter EP«(W 1 , , <}) (3.181) where NiteT is the number of generated sets of {a[ }. Methods for generating indepen-dent or correlated sets of {a\ } for Rayleigh and Nakagami pdfs have been reviewed in Section 2.4.1.2. Equal-Gain Combining The conditional output S N R at the receiver now takes the form: The B E R can be obtained following the same semi-analytical Monte Carlo integration approach outlined in the previous section. 3.4.3 Performance Evaluation Results and Discussion We simulated the forward link of a cellular system employing coherent Q P S K Mod-ulation and real spreading sequences. The orthogonal Walsh sequences and the short code spreading sequences are as specified in the IS-95 standard [177]. The theoretical B E R is obtained by numerical Monte Carlo integration as explained above: 10 million samples were used in each case. The B E R ' s obtained via simulation are averaged over 1 million samples in order to maintain a reasonable simulation time (especially when the number of users is large): hence the accuracy is lower than that obtained theoretically, and simulation results wi l l typically show a good fit for above 20 users, but looser fits (or even outliers) for a lower number of interfering users, where the B E R becomes very low. Fig . 3.62 plots the B E R for M R C in Rayleigh fading, for different diversity orders. In the case L = 1, since all the users are synchronous and there is no multipath, there SNRn = (3.182) K L-l L-l 206 is no multiuser interference due to the orthogonality of the Walsh spreading sequences. Hence only A W G N is present, and the B E R is constant across the range of numbers of users. In the cases L = 2 and L = 3, there is now multipath interference from all the users (including the desired one). It can be seen that for certain values of the number of users (e.g. for K > 25 in the case L = 2), the use of diversity actually increases the B E R : indeed the gain made possible by diversity is now offset by the higher level of interference present (due to the nonorthogonality between multipaths). In fact, for L = 3 the performance is seen to be worse than for L = 2: this is imputable to the larger number of multipaths, each contributing additional interference. There is a very good match between the simulated results and the theoretical curves obtained through Monte Carlo integration, especially for K > 20. For lower values of K the large number of samples needed for averaging (given the very low BER's ) make it more difficult to obtain reliable values for the simulation results, although the general tendency still remains. Figs. 3.63 and 3.64 plot the B E R for M R C and E G C , respectively, with L = 2 in Nakagami fading, for different values of the m-parameter. In the case of M R C , a larger m (less fading) improves the B E R for a low number of users K < 15, but leads to a higher B E R for larger AT's. It can be explained by the fact that for a larger m, the power in the interfering multipaths also becomes stronger, and hence for a large number of users (and hence multipaths) the degradation caused by the total interference offsets the benefit of having less fading of the desired signal. However, in the case of E G C , a larger m actually improves the B E R across all K: a possible explanation is that E G C doesn't amplify the interference as much as M R C , which assigns higher weights to strong multipaths, and hence to strong interference. The same phenomenon can be observed in the case of Rician fading. Figs. 3.65 and 3.66 plot the B E R for M R C and E G C , respectively, with L = 2, for different values of the Rician parameter A. As in the Nakagami case, it can be seen that as the fading gets less severe (higher A), the B E R increases for M R C , but decreases for E G C . The effect of correlation between L = 2 diversity branches is shown in Figs. 3.67 and 3.68 for M R C and E G C , respectively, where the channel is Nakagami fading with m = 2.0. In the case of M R C , the presence of correlation slightly increases the B E R , while for E G C , the opposite is true. The outliers which can' be seen for lower numbers 207 of users are imputable to the finite number of samples used (given the very high number of samples which would be needed to obtain accurate results for very low B E R ' s ) . 3.4.4 Conclusions This section analyzed a forward link multicode C D M A system with coherent B P S K modulation and maximal-ratio or equal-gain combining in multipath fading channels. A l l users fade in unison since they are transmitted along the same propagation path. Taking into account this fact was crucial in obtaining accurate analytical results. Due to the difficulty of finding general closed-form solutions (if they do exist), Monte Carlo integra-tion was used to obtain the semi-analytical B E R ' s , which matched closely the simulation results. The method of analysis was applied to Rayleigh, Rice and Nakagami channels (although any fading distribution can be used), for both independent and correlated diversity branches. :;:;::::::!::::::::::(::::::::: !::::::: ::!:::::::: ::!::::::: ; : ! : : : : : : : : : : ! _ -: a < a - ' ii *\ '. .^*"' ' ' .: O : T : : : -j ; : : : : : : : V.r.*T~.~. «-•j*-!t!'!" • • : • •' | ' (• ;.+ . : • •' _ :::::::::::>::::::::::/::::::::: 7 : / : / • • : •* • • • • / • • : n ^ : : : h : ^ : m -/ : : : : : : : :::: • / : ' • ' / ' :::::::<:::::::::::::::::::::::: / : : . . . / ; : 1 1 10"' 10^ h CC _4 S 1 0 10" 10 15 20 25 30 35 40 Number of users K 50 F i g u r e 3.62 B E R vs K for M R C and m = 1.0. — (+): L = 1; L = 3. • (*): L = 2; (o): 208 10 15 20 25 30 35 40 45 50 55 Number of users K Figure 3.63 B E R vs K for M R C and L = 2. — (+): m = 1.0; - . - (*): m = 2.0; - - (o) m = 3.0. 209 F i g u r e 3.65 B E R vs K for M R C and L A = 3.0. - (+): A = 1.0; - . - (*): A -8' •9./ 7 : : : 10 15 20 25 30 35 40 Number of users K 45 50 55 F i g u r e 3.66 B E R vs K for E G C and L = 2. — (+): A = 1.0; - . - (*): A A = 3.0. 210 10"' 1 0 - ' l 1 1 1 1 1 1 1 1 10 15 20 25 30 35 40 45 50 55 Number of users K F i g u r e 3 .67 B E R vs K for L = 2 correlated branches and m = 2.0, with M R C p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7. 10"' CC --A ///•' -A' III: //<•' r /,'• : * A' <>/"••' ' 'ii-' ' f 10 15 20 25 30 35 40 45 50 55 Number of users K F i g u r e 3 .68 B E R vs K for L = 2 correlated branches and m = 2.0, with E G C p = 0.0; - . - (*): p = 0.3; - - (o): p = 0.5; • • • (o): p = 0.7. 211 3.5 Forward Link Performance with Coherent QPSK Modula-tion and Complex Spreading Sequences 3.5.1 Introduction The physical layers of the IS-2000 and IS-95B forward links share essential features: both use concatenated W a l s h / P N spreading and coherent demodulation. There are how-ever two major differences between the systems (at the spreading/modulation levels): • IS-95B uses a form of Q P S K where the same bit is mapped to both / and Q branches, while IS-2000 uses "true" Q P S K , as detailed below. • IS-95B uses real short-code spreading sequences, while IS-2000 employs complex short-code spreading sequences in order to reduce power imbalances on the I and Q branches. In this section we present a detailed analysis of the IS-2000 downlink, and show that under the same conditions the B E R is theoretically similar (although not exactly identical) to that of the IS-95B downlink. In practice however IS-2000 systems are expected to perform better than their IS-95B counterparts, as they support advanced communication and signal processing techniques (e.g. transmit diversity, fast closed-loop power control). 3.5.2 Error Probability Analysis 3.5.2.1 Signal and Channel Models The transmitter uses Q P S K modulation in the conventional sense, i.e. in which alternating bits from a same channel are mapped to the / and Q branches. Let be the number of channels (and thus codes) assigned to both branches / and Q of user k, where k = 1, 2 , . . . ,K. We denote by {kc} code c (c = 0 , 1 , . . . , / V { f c ) - 1) of user k. The streams of complex data symbols for code {kc} are given in the time-domain by: b^(t) = bfc){t)+jb{QC){t) (3.183) 212 where b[kc\t) = j r ft^ct-jTi), e [-1,1] i=-oo O O j ' = -oo and Tb is the bit period, and Ts = 2Tb is the symbol period. The transmitter uses concatenated spreading, and the Walsh sequences used for orthogonal covering were described in the previous section. After the orthogonal covering phase, the data symbols from each user are spread by the same complex spreading sequence a(t) = aj(t) + jag( i ) , where ai(t) and ag(t) are given by Eqs. (3.144) and (3.145), respectively. The real and imaginary parts of the output are then separated and modulated onto orthogonal carriers, as illustrated in Fig . 3.69. tq c ) ( tr tf|f(t)-(l,c) W (t) X (l,c) W (t) cos(w,t) ke() -* ImU -* sin(wct) s('c)(t) Figure 3.69 Channelization/Spreading/Modulation subsystem. The transmitted signal of the kth user can be expressed as: JV(*0_1 sw(t) = I [4?(t) cos(w;ct + 4kc)) - s<$(t) cos(wct + <t>[kc)) c=0 +8%(t) sm(wct + <#c)) + s in ( W c t + 4>™)\ (fc) (3.184) 213 where ,(*) ,(*) (*) = Kit) = (t) = [y/pW)bfc\t)W^c)(t)]ar(t), [y/pl^b[kc)(t)W[kc)(t)}aQ(t), [y/pW)b{*c\t)W{kc)(t)}ar(t) and P(kc^ = /TskcS> is the average power of code {kc} on each branch J and Q. 3.5.2.2 Receiver Model and Decision Metrics If there are K users in the system, the composite signal received at the output of the channel is: K Lc-1 r(t) = J2J2aik\t)sW(t-r^)+n(t) (3.185) k=l 1=0 where n(t) is A W G N with double-sided power spectral density N0/2. The receiver consists of L Rake fingers and performs either M R C or E G C . One finger of the receiver is illustrated in Fig. 3.70. In the following, the metrics calculated refer to code 0 of user 1, which is assumed to be the desired stream. They further correspond to finger n of the Rake receiver. (l,c) W (t) r(t) (l.c) W (t) Figure 3.70 Demodulator for the nth Rake finger. 214 The quantities (t) and UQ ^ (t) at the output of the lowpass filters are given by: «<»>(*) = (r(t)cos(wct + ^ ) ) L P K A f ( f c ) -1L C -1 2 k=l c=0 1=0 +s%{t - r/*>) sin(^l) + - r/fc>) sinfoffl) + (n(r)cos( l « c r + ^ ) ) ) i p (3.186) = (r(i)sinKt + ^ ) ) ) L p K" J V W - 1 L C - 1 ^ E E E ^ w ^ X ^ - V V s ( ^ 2 ) + - r<*>) c o s f o © ) + (n(t)sinM + ^ ) ) ) L P (3.187) where ^ f c ; c ) = 4fcc) + 0Jfc) - wcr\k\ and = <p<*f - The output of the complex spreading operation is given by: <«>(*) = (uf\t)+ju%\t))a(t-T^y = [u[p\t)aI{t-r^) + u^{t)aQ{t-r^)} +j[-u[p\t)aQ{t - rBW) + t i ^ M * - r^)]. (3.188) 215 From Fig . 3.42, the decision metric on branch / for Rake finger n is thus given by: (n) - r ( i ) + T a o ) Re{UM{t)}W^°\t-r^)dt K A f W - H c - l ( D + r ( i o ) fc=l c=0 (=0 , 7 r " [6 f c ) ( i - T™)WM(t - rlh))aj{t - T?])WW{t - T^)aT(t - r ^ ' c o s ^ (ioh + 6 f c ) ( t - r / * ) ) ^ f c ' + 6 g c ) ( t - r / f c ) ) ^ f c < +&£ c ) ( t -T i ( * ) )W<* < + & / f c c ) ( i - r / f c ) ) W A f c ' + 6 g c ) ( t - r / f c ) ) ^ f c < (fc) (* " ^ (*• ag(t - T,w)W< 1 0>(t - T (D) f l / (t - r ^ ) s i n ( < ® a,(* - r ( ( f c ) ) ^ ( 1 0 ) ( i - ^ ) a / ( t - r^)M^%) - r ; ( f c ) ) W ( 1 0 ) ( t - ^ W(t - r ^ ) s in (v® ao(t - r / f c ) ) ^ 1 0 ' ( t - rW)a Q ( t - r ^ ) s i n ( ^ ( ) a 0 ( * " r<fc))W<10>(t - r^)aQ(t - r^) c o s f o © a,(t - T<fc))W<10)(t - T n 1 })ao(t - r ^ J j c o B ^ ) ] (n) q(n) , An) An) An) An) An) °I "r 1MP,I J C S , 7 J M C , / ' IMA,I i J / V , 7 (3.189) where the terms in Eq. (3.189) are as defined in Section 3.3. After expanding the expression for u\n\ the terms in Eq. (3.189) can be written as follows: 5 = a (3.190) An) (1) - ( 1 0 ) Tn i ± s (i) [(n(t) cos (« ; c t+ ^ ) ) ) ^ ° / ( * - < r » 1 ) ) + (n(t) sin(w ct + ^ ° ) ) ) i p a Q ( t - r ^ ) ] ^ , (3.191) 216 ^ a i 1 ) v ^ ) c o s ( ^ | ) x 1=0 [C(4f(- ( 1 )) + C(r ( 1 ))) + C W V ^ ) + R^(r^))], (3.192) + ( C < V ( 1 ) ) + C < ( - ( 1 ) ) ) |=0 + ( C < } ( r ( 1 ) ) + C ^ 0 ) ( - ( 1 ) ) ) ^ ( O NW-1 Lc-1 4 t / = E E^ ( 1 ) /!n )(i^), (3.194) c = i ;=o K N ^ - l L c - l 41/ = E E EW(M), (3.195) fc=2 c=o ;=o 217 where: -Xb^R&irW) + b^PSkf{r^)) c o s ( ^ ) + (C C ) < C ) ( r ( f c ) ) + C ^ Q / V f c ) ) ) s i n ( ^ ) + (C4/CV ( f c )) + s i n f c O -Cfl!?^ ') + C ^ V ^ ) ) SIN(^£!) +(C C )CV ( , S )) + C X Q ^ ) ) «»(<4$) + ( C < C ) ^ f c ) ) + C ^ c ? ^ ) ) cos^gS,)] (3-196) and where {u:v} = {-1,0} and r<fc> = T$ if > 0, and {u,v} = {0,1} and r<*> = r^f + T 6 if r^f < 0. The terms R%C){T) are as defined in Section 3.4, but with N = Ts/Tc = 2Tb/Tc. Notice that for I^ci and I^AJ * n e component for which indice I = n has been removed. Indeed, when the collected multipath components are synchronous with the desired signal (which happens when I = n), the interference produced by these components is null due to the orthogonal covering of the spreading sequences. A similar development can be carried out for the decision metrics on the Q branch, leading to the following: T ( i ) + r ( i o ) = / " S Im[U^(t)]WW(t-TnV)dt — C?(n) _L J"'7 1' i An) i An) i r(™) i An) In 1 Q7\ — £>Q "I" J-MP,Q 2CS,Q 1MC,Q "t" 1MA,Q J W , Q { O . i y f ; where: 5 W = a ^ v ^ C T i 1 0 ) , (3.198) 218 X [ (nWcosN + ^ J W a / l t - T i 1 ) ) + (n(t) s in(w c i + ^ U P M * - r ^ ) ] ^ , (3.199) (=0 C ( < V ( 1 ) ) + i2g§(rW)) + C(Ai i 0 ) (TW) + C V ( 1 ) ) ) L (3-200) + ( C ) < v i ) ) + c o ) < ° v ( i ) ) ) cos(^i)] 1=0 # n -(C<V ( 1 ) ) + C ^ V 1 5 ) ) - ( C ) < H r ( 1 ) ) - r C ^ ° ) ( r ( 1 ) ) ) s i n ( ^ ° ! ) +(C<}(-(1)) + C*g/V1})) sin(C)]> (3-201) J V W - l L c - 1 4 c , Q = E E ^ M , (3.202) c = l (=0 #71 4 % = E E E W t M , (3-203) fc=2 c=0 (=0 #71 219 where: + C ^ V f c ) ) ) c o s ( ^ S ) - ( f t K X g ^ ) + C ^ w V ( f c ) ) ) s in (^ ) - ( C < C V ( f c ) ) + C ^ V * ' ) ) s i n ( ^ ) - (6£ , J?5i c , ( rW) + aft 5A£V*>)) s in (^ ) +(C*8/Vfc)) + C^ cV ( f e ))) sin(^S) 3.5.2.3 Statistics of Decision Metrics Let Eb = E^P and Tb — T^T°\ Considering M R C , the conditional expected value of the desired received signal Sj = Ylt=o &n^Sjn^ is obtained from (3.190): L-l Sr = y/WsJ2[^]?. • (3.205) 71=0 The noise term IN>R = J2n=o c a n be easily shown to be a zero-mean Gaussian random variables with variance: 4 , = Varf/^] = ^ ] [ > < ^ (3.206) 71 = 0 Using the standard Gaussian approximation, we model the interference terms (3.192)-(3.195) as mutually uncorrelated zero-mean Gaussian random variables with given con-ditional variances. These variances are derived below, and are conditional on the ran-dom variables I = 1,2,... , L . The interference terms IMP,I = Ylt=o ttn^Mp/i T sr^L-1 (1) T-(n) T (1) T(n) i r sr^L-1 (1) An) ICS,I = En=0 an ICS,D ^ C , / = 22n=0an JMC,D A N D LMA,I = E n=0 4 /4 , / a r e a 1 1 chip synchronous with the desired user signal, since they originate from signals which are transmitted synchronously on the forward link. Therefore rffl = for all k, n and 220 I. This results in Va,v[{R^c){r^) + R^irff))] = T 2 / 7V (with I ^ n). The following variances are then obtained: L - l > L C - 1 n=0 p(10)T2' V a r t / M P , ^ ^ ^ ) ] 2 ^ ^ ^ : ^ 1 ' ] 2 , 1=0 (3.207) L _ 1 p(10)7i2 "Is, = • V a r [ / C S i / ] = £ [ a ( i > l » £ l ^ 71=0 47V L c - 1 L c - 1 £ [ a ^ ] 2 + 2 £ [ c ^ ] 2 , (3.208) (=0 (=0 L - l _2 °IMCA = V a r [ / M C , / ] = E ^ ^ ] 2 E 71=0 C = l p(ic)j>: TV . L c - 1 E (=0 (1)12 (3.209) I MA,I L-l K NW-1 p ( f c c ) T 2 i c - l V a r ^ ^ ^ k ^ E E T E K ( (1)12 71 = 0 k=2 c=0 1=0 (3.210) £ c - l u , ( l ) 1 5 In Eqs. (3.207)-(3.210), the variances are seen to be conditional on the term X^ =o ia] Indeed, the interference terms are affected by the same coefficients af\ 1 = 0,... , L—l as the desired signal 5/, since they all fade in unison. Thus these interference terms are cor-related with the signal term, and this fact must be taken into account when determining the error probability. Assuming equal powers = P = Eb/Ts for all k and c, the total variance from the interference and thermal noise sums up to: Or = 2 I 2 IMP,I IMC,I L - l = £ k ' >i2 71 = 0 'N,I 1=0 k=l N 47V (3.211) 221 3.5.2.4 Probability of Error From Eqs. (3.205) and (3.211), and using our assumption that LC = L, the conditional output S N R at the /-branch of the receiver for M R C then takes the form: SNRQJ = j ^ ( 3 - 2 1 2 ) E^ 2 n=0 2Eb N ^  ^ L 1 J 4 A l \ 0 fc=l (=0 \ # n As in Section 3.4.2.4, it is seen that the fading coefficients [ o f ] 2 , I = 1,. . . , L — 1 appear both in the numerator and denominator of the SNR, since the interference terms are subject to the same fading as is the received desired signal. Neglecting the term 1 UW12 4 i V a}i ] 2 due to the complex spreading self-interference, Eq . (3.212) is seen to be similar to Eq. (3.179) (recalling that the processing gain N = Ts/Tc = 2Tb/Tc in (3.212) is twice the processing gain used in Eq. (3.179)), which is to be expected. Letting Pbj = Q(y/SNR0,i) arid PBTQ = Q(^/SNR0,Q) be the conditional probabilities of bit error on the i - and Q-branches, respectively, the conditional probability of symbol error is obtained as: P.({a{1}}) = 1 - (1 - PbJ)(l - Pb,Q) = 2Q{y/SNR0) + [Q(VSNRo)]2 (3.213) where SNR0 = SNR0,i = SNR0,Q due to symmetry. The conditional B E R can be approximated as [26] Pb{{a\1]}) ~ lPs{{a\1]}) ~ Q(^/SNR0), which is similar to Eq. (3.181). The unconditional B E R is then obtained in a semi-analytical fashion as in Section 3.4.2.4, through Monte Carlo integration. The numerical results wil l be similar to those presented in Section 3.4.3, and hence are not presented again to avoid repetition. 222 3.6 Conclusions This chapter presented the analysis of multicode D S / C D M A systems in wideband fading channels, which hadn't been tackled yet in a precise manner in previous works. By taking into account the effect of dependent fading between the multicode or multiuser interference and the desired signal, accurate analytical or semi-analytical theoretical re-sults were obtained for the B E R , and interesting insights into the behavior of multicode systems were given. Analyses were provided for the four main types of configurations encountered in IS-95B and IS-2000 systems: reverse link with noncoherent M-a ry mod-ulation and real spreading, reverse link with B P S K modulation and complex spreading, forward link with Q P S K modulation and real spreading, and forward link with Q P S K modulation and complex spreading. In the first case, the system employed non-orthogonal long spreading sequences to differentiate the codes of a same user, while in the other cases the systems used orthogonal Walsh covering. The analysis was general enough to be ap-plicable to different fading distributions, to systems with correlated diversity branches, to multi-cell systems, and was extended to deal with advanced techniques such as closed-loop power control and multiuser detection. In all cases, the theoretical results were thoroughly validated using entire system simulations. The analysis presented in the present chapter was essential in obtaining the re-sults reported in the next chapter, which considers V B R video transmission over multi-code/multirate IS-95B and IS-2000 D S / C D M A systems, given that: • It provided the decision metrics used in the simulation implementations. • It derived B E R expressions which were used to validate the simulations. While this analysis considered several different fading environments, the next chapter wil l mainly make the assumption of Rayleigh fading and uncorrelated diversity branches, which is commonly used in order to simplify simulations. The Rayleigh distribution represents the worst case of the Rice and Nakagami (for m > 1; the case m < 1 is more rare) distributions, which wil l hence allow us to observe to the full extent the degradations caused by the channel fading. The Rayleigh fading model (along with its methods of simulation) is also more widely accepted than the Nakagami model. 223 C H A P T E R 4 V B R VIDEO TRANSMISSION FOR M U L T I C O D E A N D M U L T I R A T E D S / C D M A SYSTEMS IN WIDEBAND FADING C H A N N E L S 4.1 Introduction In the previous chapters, we provided detailed analyses of the performance of multi-code and multirate D S / C D M A systems used to support high-rate services (in our case streaming or interactive video), in the presence of wideband fading and multiple-access in-terference. The systems were evaluated in terms of B E R at the modulation/demodulation level, i.e. at the lowest level of the physical layer. This was helpful in understand-ing the effects of the channel parameters (e.g. severity of fading, correlation between branches) and of the transmitter/receiver parameters (e.g. number of assigned parallel codes, data rate and processing gain, number of diversity branches). In this chapter, we move further up the transmission chain to include the error-control coding/decoding, interleaving/deinterleaving, framing, packetizing and video coding/decoding operations. While both analysis and simulation were consistently used in Chapters 2 and 3, in this chapter we wil l essentially rely on simulation results, due to the high complexity of the full transmission system, and the resulting difficulty in finding exact or even accurate analytical solutions. Indeed, our main performance criterion wil l be the peak signal-to-noise ratio (PSNR), which is an objective measure of the decoded video quality, and is hence more representative of the actual user-perceived performance than measures such as the bit and frame error rates. The P S N R is a highly nonlinear function ,of the B E R and the distribution of the errors, which themselves depend in a nonlinear fashion on the peak, mean and variance of the rate (as wil l be discussed in Section 4.3). Most previously 224 published papers which try to obtain relations between the P S N R and the channel char-acteristics usually assume a simplified time-invariant channel model (such as a Markov chain), and yet the analysis remains approximate [235], [236]. In our case, we are further dealing with a time-variant channel (due to the changing rates and thus varying pro-cessing gain or number of multicodes), which includes sophisticated elements such as a channel coder/decoder, an interleaver/deinterleaver, and which is additionnally subject to not-necessarily-Gaussian noise (the multiple-access interference from other users). We thus believe that a formal analysis in terms of P S N R of such an end-to-end system re-mains beyond the current state-of-the-art of the analysis of video communication systems (which is in its early stages). For our purposes simulations are much more useful. As a result we implemented a sophisticated customizable software platform which includes the video coder/decoder, the packetizer/depacketizer, and the physical layer components of the IS-95B and IS-2000 systems, for both uplink and downlink. The organization of this chapter is as follows. In Section 4.2, we describe in detail the characteristics of all of the elements of the software platform, according to the corre-sponding industry standards. This is necessary to appreciate under which conditions the simulation results were obtained, and in order to be able to reproduce these conditions if the need arises. In Section 4.3, we tackle the issue of transmission rate control for V B R video over multicode/multirate D S / C D M A . In particular, we compare the performances of a proposed smoothing algorithm and of a benchmark one, which are both adapted to deal with transmission formats compatible with IS-95B/IS-2000 systems. To evalu-ate the P S N R in the presence of wideband fading and multiple-access interference, the smoothing algorithms are used in conjunction with the previously described simulation platform. 4.2 System Description 4.2.1 Overview of a Video Communication System for IS-95B/IS-2000 Networks Fig. 4.1 gives a high-level representation of a video communication system for 2 .5G/3G cellular networks. The representation is based on the elements which compose the sys-225 tern: a video source, a transport coder, a network and a radio link. Each of these entities wil l be briefly reviewed separately. Application Transport layer layer Video Transport source coder H.223 RTP/UDP Network layer Circuit-switched data Packet-switched data (IP) Link + Physical layers L| Mobile station Figure 4.1 Video communication system for cellular network. 4.2.1.1 Video Services Video Applications A wide range of video applications can be conceived for wireless networks. From a delay viewpoint, they can be roughly categorized in the following three groups: • Low-latency video conversational applications, e.g. video telephony, video telecon-ferencing. The video encoding is done in real-time, and the end-to-end allowable delay must be very limited (at most within a few hundred milliseconds); • Low-to-medium-latency video streaming applications, e.g. video-on-demand, video broadcast, video surveillance. The allowable end-to-end delay is variable, but in the order of seconds or tens of seconds; • High-latency video messaging applications, e.g. video e-mail. There is no hard constraint on the delay, which must however be reasonably bounded. From an encoding viewpoint, these applications can also be categorized in the follow-ing two groups: • Live video applications, e.g. video telephony, video teleconferencing, video surveil-lance. The video encoding is done in real-time; 226 • Prerecorded video applications, e.g. video-on-demand, video e-mail. The video encoding has already been done before the beginning of transmission. This chapter wil l mainly deal with conversational and streaming video, for both live and prerecorded applications. Indeed, messaging applications can use protocols which are used for conventional data applications. Video Encoding Architecture In F ig . 4.1, at the sender end, the video bitstream can be produced in real-time by compressing a captured video source, such as for live conversational or broadcast video. It can also have been produced earlier and stored in digital format, and be retrieved from its storage location and streamed to a mobile user, such as for video-on-demand. If the video application is duplex, such as conversational video, the sender also needs to be able to decode an incoming video bitstream and display it in real-time. Fig . 4.2 illustrates the main functions performed by a real-time video encoder: image capture, compression, bitstream generation and rate control. The image capture (or frame grabbing) step consists in converting a sequence of analog pictures into a sequence of digital images which can be processed by the video compressor. The goal of the compression step is to reduce a very large quantity of image data to a much smaller number of bits, while minimizing the effect that the loss of information due to lossy coding wil l have on the reconstructed video quality. The ratio between the uncompressed and compressed rates (i.e. the compression efficiency) depends on the attributes of the video sequence (such as the amount of motion, the number of scene changes, the scene complexity, the level of details, etc.); for H.263 and H.264 it can be higher than 100 for some video streams. The goal of the bitstream generation step is to organize the information produced by the compression step in a bit sequence which can be handled efficiently by the transport coder/decoder and the video decoder, while lending itself to error resilient techniques. A l l of the previous operations can be viewed as belonging to the application layer of the ISO (International Standards Organization) layering model [237]. 227 H.26x Standard Capture Video Compression Bitstream Generation Rate Control F i g u r e 4.2 Generic real-time video encoder. Standards for (Very) Low Bit Rate (VLBR) Video Coding Currently, H.263/H.264 and M P E G - 4 are the most popular video codecs for V L B R mobile applications: they have been recommended as the video components for some mobile multimedia terminals [238], [239]. H.263 is the I T U - T (International Telecommunications Union - Telecom) standard for V L B R coding. It evolved from another I T U - T standard, H.261 [240], originally proposed for videoconferencing over the public switched telephone network (PSTN) . The first H.263 standard was released in 1996 [241]. Revision 2 (also termed H.263+) was issued in 1998 [41], and revision 3 (or H.263+-f-) in 2000. A detailed description of H.263 can be found in book Chapters 11 of [242] (version 1) and 1 of [43] (version 2), for example, and a short description wil l be given in Section 4.2.2. By H.263, we wil l denote the latest revision. The H.263 standard describes the syntax and the functional operation of the decoder, and does not mandate any particular algorithm to be used at the encoder side, as long as the bitstream produced by the encoder complies with the syntax expected at the decoder for proper operation. The image capture procedure and the rate control strategy are outside the scope of the H.263 standard. The standard consists of the main body for baseline syntax and decoder specification and of a set of Annexes (A through X in H.263+-T-) which describe optional modes and some additional specifications. A set of levels and profiles (Annex X ) recommend combinations of these optional modes as a function of the application and/or type of network connection. While allowing different levels of operation for different systems, the large number of optional modes in H.263 also has its share of disadvantages, such as implementation complexity and overhead. In 1999, opting for a "back-to-basics" approach, the working group in charge of H.263 standardization, I T U - T Study Group (SG) 16/ Video Coding Experts Group ( V C E G ) , Question 6 (Q.6, formerly Q.15), started working on a new V L B R coding standard, 228 H.264 (also previously denoted as H.26L or J V T ) , whose main objectives were better compression efficiency and network friendliness [243]. Since the operations of H.263 and H.264 codecs have many fundamental similarities, the description in Section 4.2.2 will apply to both codecs, denoted by H.26x. 4.2.1.2 Transport and Network Services Transport and Network Layer Functions The video data generated at the application layer is passed on to the transport and network layers. The specifications of the transport and network layers describe how the packetized video stream is to be transported over a circuit-switched or packet-switched network connection, in order to be delivered to/from the cellular BS serving the MS. The main tasks supported by the transport layer are the following: • Segmentation and packetization of the video stream in a format suitable for trans-mission over the network layer; • Multiplexing of the packets of the video connection with packets from other con-nections/applications; • Error control at the transport layer, either by error detection, Forward Error Cor-rection ( F E C ) , retransmission A R Q ) , or combinations of these; • Flow control (if there is a need for it at this stage). The main tasks supported by the network layer are the following: • Routing of the packets delivered by the transport layer across the core wireline network and the radio access network ( R A N ) , to/from the BS; • Congestion (or flow) control. Fig . 4.3 gives a simplified view of the architecture of a cdma2000 radio access network, which supports both circuit-switched and packet-switched traffic, according to [244]. Considering the MS-to-network link, the circuit-switched data is sent from the BS to a 229 mobile switching center (MSC), which then directs it to the P S T N , or to a packet data network (PDN) such as the Internet. The packet-switched data is first processed by a packet control function (PCF) , which relays it to a packet data serving node (PDSN), whose functions are to establish, maintain and terminate link layer sessions to mobile stations. The data is then routed to a P D N . C S D M S J If BS M S C PSD P S T N P C F P D S N Gateway P D N MS: Mobile Station M S C : Mobile Switching Center PSTN: Public-Switched Telephone Networ BS: Base Station PCF: Packet Control Function PDN: Packet Data Network CSD: Circuit-Switched Data P D S N : P a c k e t D a l a Service Node PSD: Packet-Switched Data F i g u r e 4 . 3 cdma2000 wireless network model. Standards for Transport and Network Protocols The importance and relevance of the tasks describe above differ according to whether the network is circuit-switched or packet-switched: therefore, different protocols are defined for each case. In particular, the functions performed at the network layer are more relevant to packet-switched networks than they are to circuit-switched networks, since the latter benefit from a dedicated connection and are therefore less concerned about packet routing. For circuit-switched networks, H.324/M [245] is the leading standard defining the operation of mobile multimedia terminals. It mandates in particular the use of the following set of standards: • H.261 [240] and H.263 [246] for video coding, G.723.1 for audio coding (optionally G.728 and G.729); • H.223 [247] for multimedia multiplex and synchronization; • H.245 [248] for system control. 230 Other standards are also recommended (T.120 for data, H.226 for multilink operation). The main features of H.223, which wil l be used in our circuit-switched simulation model, are summarized in Section 4.2.3. 4.2.1.3 Radio Link Services The radio link is reponsible for the communication between the BS and MS. It covers both the link layer and the physical layer. The tasks of the link layer are to: • Provide link access control ( L A C ) , which includes management of signaling control information; • Provide medium access control ( M A C ) , which includes optional best effort delivery (retransmission, using the Radio Link Protocol, R L P ) , optional Quality-of-Service (QoS) control, resource allocation between services and multiplexing of the traffic channels onto the physical layer channels. The point-to-point protocol (PPP) [249] is used to interface between the L A C layer and the IP layer. The physical layer specifies the wireless transmission technology. In our simulation model we wil l be using the IS-95B and IS-2000 standards, whose main features of interest to us are described in Sections 4.2.4-4.2.7. 4.2.2 H.26x Video Coding and Bitstream Generation This section first describes the picture formats used by H26x coders and the organi-zation of pictures into smaller structures. It then reviews the general coding principles used by these coders. The principles behind the bitstream generation process are also summarized. 4.2.2.1 Picture Format and Organization Different picture formats corresponding to different resolutions can be used by the codec. Five of them are explicitly specified in H.263: C I F (common intermediate format), Q C I F (quarter-CIF), sub-QCIF, 4CIF and 16CIF. Custom picture formats can also be 231 negotiated by the encoder. However only the Q C I F and sub-QCIF are mandatory for an H.263 decoder, and the encoder only needs to support one of them. These formats are obtained by subsampling P A L / S E C A M or N T S C video signals, the conversion process being outside the scope of the H.263 standard. A Q C I F picture consists of 144 lines and 176 pels (or pixels) per line. Such a picture is divided into a number of groups of blocks (GOB's) . Each G O B spans one or more rows of the picture, with each row consisting of 11 macroblocks (MB's) . In Fig . 4.4, the picture is divided in 9 G O B ' s , with each G O B containing one row of the picture. M B ' s are composed of 6 blocks, 4 of which are luminance (Y) blocks and 2 chrominance (Cb and Cr) blocks. Blocks are square and consist of 8 by 8 pixels. While blocks are the basic structure used for many encoding/decoding operations in H.263, smaller structures (e.g. groups of 4 x 4 pixels) are preferred in H.264 for certain operations. A slice is a grouping of an arbitrary number of M B ' s (within a picture): the use of slices is optional in H.263 (Annex K : Slice Structured mode), but is part of the baseline coder in H.264. 4.2.2.2 Overview of the Core Video Coding Process Fig. 4.5 depicts the schematic of an H.26x encoder, whose operation is based on hybrid differential/transform encoding, and is a combination of lossy and lossless coding. There are two fundamental modes which are jointly used for maximum compression efficiency: the intra and inter modes. Different types of frames correspond to these modes. In the intra mode,.the contents of a video frame are first processed by a transform for energy compaction and thus bit-rate reduction. The resulting coefficients are quantized with a chosen quantizer step size, thus leading to a loss of information. The quantized coefficients are encoded using a certain entropy coding strategy, scanned across the pic-ture (often using a zig-zag strategy), and delivered to an encoder buffer. A n optional rate control algorithm, which probes the buffer fullness, is used to adapt the quantizer step size or decide the number of frames to be skipped. In the inter mode, the same operations are applied to the motion-predicted difference between the current frame and the previous (or earlier) frame, instead of to the frame itself. To this end a motion estimation algorithm is applied to the input frame, and the extracted motion information (in the form of motion vectors, M V ' s ) is used in predicting 232 Picture Frame Group of Blocks (GOB) 176 pels -GOB 1 GOB 2 GOB 3 GOB 4 GOB 5 GOB 6 GOB 8 MB 1 MB 2 MB 3 MB 4 MB 5 MB 6 MB 7 MB 8 MB 9 MB 10 MB 11 Macroblock (MB) Y l Y2 Y3 Y4 Cb Cr t j J ... \*_ 8 lines 1 -8 pels—*• Figure 4 . 4 H.263 picture structure for Q C I F resolution. the following frame(s), through a motion-compensation circuit. In order to avoid a drift between the encoder and decoder due to motion prediction, the motion compensation circuit needs to use a locally reconstructed version of the compressed frame being sent: this explains the presence of an inverse quantizer and an inverse transform in the feed-back loop. The M V ' s are differentially encoded in order to realize bit rate savings. A deblocking filter can also be included in the motion-compensation loop in order to reduce visual artefacts. The intra mode produces an intra frame or I-frame. This type of frame is needed for the decoder to have a reference for prediction. I-frames should also be transmitted at a certain frequency (the H.263 standard specifies at least one I-frame for every 132 frames) in order to refresh the prediction, or when considerable motion is detected in the sequence, which can forbid any accurate prediction. However, I-frames use a large number 233 Coding control Input frame —*k EC Quantizer Motion compensation Motion estimation EC Inverse qn:i IT Frame memory Motion vectors EC Bitstream multiplexing T; Transform IT: Inverse Transform EC: Entropy Coding Figure 4.5 H.26x video encoder. of bits, so that they should be used sparingly in low bit-rate applications. The inter mode produces prediction frames or P-frames, which can be predicted from I-frames or other P-frames. These in general use considerably less bits than I-frames, and are responsible for the large compression gain. The inter mode can also produce bidirectionnally predicted frames, or B-frames: these are predicted from two pictures (which are allowed to serve as reference pictures), one being temporally precedent and the other temporally subsequent. B-frames are not used as reference pictures for other frames. In H.263, B-frames are either coded jointly with a P-frame as a PB-frame (Annex G: PB-frames mode) or Improved PB-frame (Annex M : Improved PB-frames mode), or coded separately as a standalone ("true") B-frame if the temporal scalability mode of Annex 0 (temporal, S N R and spatial scalability mode) is used. In H.264, B-frames are coded separately. 234 Other frame types are used for scalability or error resilience purposes. Annex 0 of H.263+ allows the encoding of enhancement frames, such as EI (Enhanced Intra) and E P (Enhanced Predictive) frames, in addition to true B frames. H.264 allows for interstream transitional SI and SP-frames to be encoded. 4.2.2.3 Bitstream Syntax and Generation A n H.26x-compliant bitstream can be characterized in terms of a number of syntax layers, which are organized in the following top-to-bottom hierarchy: • Picture layer; • Slice or G O B layer; • M B layer; • Block layer; A n element from each layer contains a group of elements from the previous layer, along with a layer-specific header. For example, a picture consists of a picture header followed by a group of slices or GOB' s . The picture header contains information on how to decode elements from the s l i ce /GOB layer, and possibly from the lower layers. The headers belonging to the higher layer elements are more important than those of the lower layer elements: for example a lost or corrupted picture header wi l l likely render unusable the data received for all of the lower layer elements, since the decoder relies on this picture header to process the next syntax elements. 4.2.3 H.324/H.223 Packetization and Multiplexing This section describes how units from the video layer are packetized and multiplexed in order to be transported over the underlying network. The first step, which is the task of the application layer, is to map the bitstream into a sequence of service data units (SDU). Each S D U is defined as the logical unit of information used in the transfer from one protocol layer entity to the peer protocol layer entity [247]. For video applications a S D U consists of a video packet. In H.263 and 235 H.264, a video packet contains either a picture, a G O B (for H.263), a slice or a slice data partition. The resulting SDU's are then handed over from the application layer to the lower layers, for transport-layer and network-layer multiplexing and packetization. A t these layers, the protocols used for packetization and multiplexing are I T U - T H.223 (Mult i -plexing protocol for low bit rate multimedia communication) for H.324-based circuit-switched networks, and R T P / U D P / I P for packet-switched networks. The packetization and multiplexing aspects for H.324/H.223 are described below. 4.2.3.1 H.324/H.223 Packetization and Multiplexing for Circuit-Switched Networks H.223 is a packet-oriented multiplexing protocol which can be used between either two low bit rate multimedia terminals, or a low bit rate multimedia terminal and an interworking adapter (IWA) or multipoint control unit ( M C U ) . Its role is to multiplex audio, video and data information streams over a single communication link using pack-ets. It can optionally add sequence numbering information and perform error-control procedures. The control of H.223 is handled by the I T U - T H.245 recommendation [248]. Each information stream handled by H.223 can correspond to one or many logical channels. Each logical channel is assigned a logical channel number (LCN) , an integer between 0 and 65535, with LCNO being the H.245 control channel. In order to keep the delay and packetization overhead low, H.223 uses segmentation and reassembly, and can combine information from different logical channels into a single packet [247]. The baseline H.223 protocol specification is complemented by a set of Annexes which provide optional features in the case of communication over error-prone environments: • H.223 Annex A [250] is a multiplexing protocol for low error-prone channels; • H.223 Annex B [251] is a multiplexing protocol for moderate error-prone channels; • H.223 Annex C [252] is a multiplexing protocol for highly error-prone channels; • H.223 Annex D [253] is an optional multiplexing protocol for highly error-prone channels; 236 The protocol stack and data structures of H.223 are illustrated in Fig . 4.6. H.223 consists of two layers: the adaptation layer (AL) and the multiplex ( M U X ) layer. The functions performed by each layer are reviewed below. Bitstream Application Layer A L - S D U A L - S D U A L - S D U * A L - S D U H.223 Adaptation Layer (AL) C F A L - S D U * E D / E C Bits A L - P D U M U X - S D U M U X - P D U M U X - P D U M U X - P D U M U X - P D U S ; II Payload H.223 Multiplex ( M U X Layer C F : Control Field E D / E C : Error Detection/ Error Correction S: Syncrhonization flag H: Header : Optional field F i g u r e 4 .6 H.223 protocol stack and data structures. Adaptation Layer (AL) The A L receives a S D U from the application layer, which is then called an A L - S D U and consists of an integer number of bytes. If the A L - S D U is not segmentable, it is conveyed as a single A L - P D U (protocol data unit) to the M U X layer. If the A L - S D U 237 is segmentable, then it can be broken down and conveyed as multiple A L - P D U ' s to the M U X layer. Depending on the type of A L used, bytes can be added to each A L - P D U for error detection (for audio and video), and optionally for sequence numbering (for audio and video) and retransmision (for video). There are three types of A L ' s in baseline H.223: • A L 1 is geared towards the transfer of data and control information. It doesn't support error control, and works in either framed (the application layer sequence is framed) or unframed (the application layer sequence is unframed) transfer mode; • A L 2 is geared towards the transfer of digital audio. It provides error detection (1-byte cyclic redundancy check, or C R C ) and optional sequence numbering (1 byte sequence number); • A L 3 is geared towards the transfer of digital video. It provides error detection (2-byte C R C ) , optional sequence numbering and retransmission. The format of an A L - P D U for A L 3 is given in Fig. 4.7, along with the syntax for the optional control field. S N S N (optional) PT A L - P D U payload (>= 1 byte) C R C field (2 bytes) PT: Payload Type (1 bit) P T = 1 for A L - P D U ' s PT = 0 for S - P D U ' s (supervisory message) S N : Sequence Number (7 or 15 bits) F i g u r e 4.7 A L - P D U for AL3 with optional control field (CF). The same A L ' s are defined for H.223 Annexes A and B . However, distinct A L ' s are defined for Annexes C and D: in both Annexes they are denoted by A L 1 M (data, control), 238 A L 2 M (audio) and A L 3 M (video). F ig . 4.8 illustrates the format of an A L - P D U for the A L 3 M of both Annexes C and D, along with the syntax for the optional control field. Control field (2 or 3 bytes) A L - P D U payload (>= 0 byte) C R C field L (Annex C: 4, 12, 20 or 28 b i t s ^ B (Annex D : 8, 16 or 32 bits) \ PI X R N SN5 SN4 SN3 SN2 SN1 P9 P8 P7 P6 P5 P4 P3 P2 O R SN8 SN7 SN6 SN5 SN4 SN3 SN2 SN1 P4 P3 P2 PI X R N SN10 SN9 P12 P l l P10 P9 P8 P7 P6 P5 R C P C or RS parity bytes (variable length) T B : Tail Bits (4) (Annex C only) R C P C : Rate Compatible Puntured Convolutional code (Annex C) RS: Reed-Solomon code (Annex D) S N : Sequence Number (5 or 10 bits) R N : Receive Number (1 bit) X : Odd (X=l)/Even(X=0) parity of length (in bytes) of non-empty A L - P D U (1 bit) P: Control Error Code (CEC) Parity for S N , R N and X fields: - 9 bits: Systematic Extended Bose-Chaudhuri-Hocquenghem (SEBCH) (16,7,6) code - 12 bits: Extended (E) Golay (24,12,8) code Figure 4 .8 A L - P D U for A L 3 M (Annexes C and D) wi th control field (CF) . Fig. 4.9 illustrates the structure of the A L 3 M of Annex C. It uses rate compatible punctured convolutional ( R C P C ) coding [254] and interleaving to protect the contents of the A L - S D U , and allows optional retransmission procedures (hybrid A R Q types I and II [255]). The following steps are used for encoding a (possibly partial) A L - S D U to an A L - P D U : • Calculate the length of the A L - P D U ; • A d d a C R C of length 4, 12, 20 or 28 bits; • A d d 4 tail bits (TB); • Perform convolutional encoding using a systematic recursive convolutional code; • Puncture the output of the encoder, while maintaining rate compatibility, and fill it in a linear buffer according to a given mapping procedure; 239 • Output the contents of the linear buffer to the A L - P D U payload field; • A d d a C F if the retransmission mode is used; • Perform block interleaving of the whole A L - P D U . The details of the above steps can be found in [252]. A t the receiver side, the decoding of the A L - P D U in order to reconstruct an A L - S D U follows the inverse steps. If the received A L - P D U is invalid (the number of bytes of the A L - P D U is outside the range of authorized values or is not an integer, or the number of bytes of the reconstructed A L - S D U is not an integer), it is discarded. If it is valid and there is no C R C error, the A L - P D U is delivered to the A L 1 M along with an error indicator (EI) set to 0. If it is valid but there is a C R C error, the presence of a C F field is checked. If there is an error-free C F , then one of two retransmission procedures, A R Q I and A R Q I I , can be used. If there is no C F or it is errored, then the A L - P D U is delivered to the A L 1 M with an EI set to 1. The encoding/procedures for Annex D parallel those of Annex C, except that Reed-Solomon encoding is now used instead of R C P C coding. Only A R Q I is supported. Multiplex (MUX) Layer The A L - P D U ' s are transferred to the M U X layer as M U X - S D U ' s , where each M U X - S D U contains data from a single logical channel and consists of an integer number of octets. The M U X layer allows to multiplex several segmentable logical channels contained in M U X - S D U ' s into a packet called M U X - P D U , whose size is typically smaller than that of a M U X - S D U (e.g. 254 bytes). Each logical channel is assigned a certain.pattern of slots. This assignment is made through a 4-bit multiplex code (MC) , which references an entry from a multiplex table containing 16 possible patterns. The entries of this multiplex table are controlled by the transmitter. In baseline H.223, the M C is included in a 1-byte header prepended to the M U X - P D U payload, along with a 1-bit packet marker (PM) field, which marks the end of M U X - S D U ' s of segmentable logical channels, and a 3-bit header error control (HEC) field, used for error detection over the M C . A n 8-bit synchronization flag is prepended to the concatenation of the header and the M U X - P D U payload, leading to the M U X - P D U format given in Fig . 4.10. Bi t stuffing is performed in 240 From Application Layer A L - S D U Convolutional encoding To Application Layer Convolutional decoding A L - S D U AL-SDU AL-SDU* CRC TB AL-SDU* CRC Transmit punctured data Insert punctured data CF AL-PDU Payload Block interleaving Block deinterleaving AL -PDU To MUX Layer From MUX Layer I 1 : Optional field F i g u r e 4.9 Structure for A L 3 M of Annex C. the M U X - P D U payload in order to prevent the emulation of the synch flag, by inserting a 0 bit after a sequence of five consecutive 1 bits. The M U X - P D U formats for H.223 Annexes A and B are given in Figs. 4.11 and 4.12, respectively. Annex B also contains an optional header mode, in which 5 header bits ( P M , M C ) from the previous M U X - P D U are prepended to the current header, and protected with a 3-bit H E C . The M U X - P D U formats for H.223 Annexes C and D are similar to that of Annex B , except for its stuffing mode (used when no information is available). 4.2.4 IS-95B Forward Link Channels The IS-95B forward link consists of up to 64 code-division multiplexed channels, each 241 Synch Flag H E C j M C P M Bit-stuffed payload 01111110 H E C : Header Error Control (3 bits) M C : Multiplex Code (4 bits) P M : Packet Marker (1 bit) Figure 4.10 Format of H.223 M U X - P D U . Synch Flag HEC i MC : I'M Non bit-stuffed payload 0100110111100001 HEC: Header Error Control (3 bits) MC: Multiplex Code (4 bits) PM: Packet Marker (1 bit) Figure 4.11 Format of H.223 Annex A M U X - P D U . channel being assigned a different orthogonal Hadamard-Walsh sequence taken from the set { IFf 4 } , i = 0 , . . . ,63. These channels are divided among the following: - A pilot channel (VFQ 6 4 ): serves as a coherent phase reference for demodulating the other channels, and doesn't carry any data modulation; - A synchronization (synch) channel (W^)- used to continuously broadcast to all mobile users the synch channel message, which contains information allowing the mobiles to synchronize to the system clock; - Up to 7 paging channels (W^-W^): used to alert mobiles of incoming calls, convey channel assigments, and transmit system overhead information. They can also be as-signed as traffic channels in the case of a heavily loaded system; - Up to 55 traffic channels {W^-W^, W^-W^): carry the digital information to the Synch Flag P VIPI. ; M C Non bit-stuffed payload 0100110111100001 M C : Multiplex Code (4 bits) M P L : Multiplex Payload Length (8 bits) P: Parity bits (12 bits), Extended (E) Golay (24,12,8) code Figure 4.12 Format of H.223 Annex B M U X - P D U . 242 mobile. Several data rates are supported on a traffic channel: 0.8, 2.0, 4.0 and 8.6 kbps. For IS-95B systems, up to 8 traffic channels can be assigned in parallel to a single user, re-sulting in a maximum data rate of Rb = 8.6 x 8 = 68.8 kbps per user. In the following, specifications are given only for the traffic channels. A block diagram of the air interface for the IS-95B forward link traffic channels is given in Fig . 4.13. The air interface for other channels is detailed in Chapter 4 of [177]. Information bits 8.6 kbps 4.0 kbps 2.0 kbps 0.8 kbps Add frame quality bit indicators (CRC) for 9.6 and 4.8 kbps Add 8-bit encoder tail 9.2 kbps 4.4 kbps 2.0 kbps 0.8 kbps 9.6 kbps 4.8 kbps 2.4 kbps 1.2 kbps Convolutional encoder R=l/2, K=9 Code symbols 19.2 ksps 9.6 ksps 4.8 ksps 2.4 ksps Symbol repetition 19.2: xl 9.6: x2 4.8: x4 2.4: x8 19.2 ksps Block interleaver 24 x 16 = 384-symbol array Modulation symbols 19.2 ksps 42-stage long-code PN generator Decimator: sample every 64th symbol Power control bits 800 bps ~~ Decimation for M U X timing control M U X 800 bps 1.2288 Mcps 19.2 ksps 15-stage short code PNj 4 Baseband filter Baseband filter 15-stage short code P N r sin(wct) e-F i g u r e 4.13 Air interface for the IS-95B forward link traffic channels [177]. Framing A n IS-95 forward link traffic frame spans 20ms. It is formed by: 243 - Grouping Udata data bits together; - Appending a C R C of UCRC bits, which serves as a frame quality indicator (FQI), for the two highest rates (4.0 and 8.6 kbps); - Appending an 8-bit convolutional encoder tail; - Convolutionally encoding the concatenation of the previous fields, using a rate R = 1/2 encoder; - Repeating n T e p times the encoded symbols. Table 4.1 details the constitution of the traffic frames for the different data rates, in which Rb is the data rate, n^ata is the number of data bits, UCRC is the number of C R C bits, ritaii is the number of tail bits, n r e p is the repetition factor, and n s y m is the total number of symbols in a frame. Rb (kbps) nCRC R 0.8 16 0 8 1/2 8 384 2.0 40 0 8 1/2 4 384 4.0 80 8 8 1/2 2 384 8.6 172 12 8 1/2 1 384 Table 4.1 Frame structure of the IS-95B forward link [14]. Coding The 8.6 and 4.0 data rates use the following generator polynomials to compute the 12 and 8-bit C R C ' s , respectively: g(x) = l + x+.x* + x* + x9+x10 + xn+x12, 8.6 kbps (4.1) g(x) = l - r - x + a;3 + x 4 - r - a ; 7 - r - a ; 8 , 4.0 kbps (4.2) The convolutional encoder has rate 1/2 and constraint length K = 9. Its generation functions are, in octal form, g0 = (753) and g1 = (561). The resulting transmission rate for the 8.6 kbps data rate is 19.2 kbps. 244 Block Interleaving The block interleaver corresponds to a matrix consisting of 64 rows and 6 columns. The coded bits are serially input into the matrix by column, starting with the upper element of the leftmost column. Once the matrix has been filled with N = 64 x 6 = 384 bits, its rows are permuted by bit reversing the indice of each row (also called the bit reversal technique). The bits are then read out by rows, starting with the leftmost element of the first row. This set of operations can be mathematically expressed as follows: where A\ is the address from which input symbol i (i = 0,1,... ,N — 1) is read out, B R O m ( x ) denotes the bit-reversed m-bit value of x, and m and J are 2 integer parame-ters which are determined by the size of the interleaver array (total number of elements N) as given by Table 4.2. The block deinterleaver simply reverses the previous operation. Interleaving and deinterleaving cause a total delay of 2 x 20 ms = 40 ms. Scrambling and Power Control The interleaved sequence is scrambled by the decimated version (1 out of every 64 sym-bols) of a user-specific phase offset of a long-code P N sequence running at 1.2288 Mcps. This P N sequence, PNL, is generated by a 42-stage shift register of period 2 4 2 — 1, with characteristic polynomial [177]: (4.3) f{x) = 1 + x7 + x9 + x11 + x15 + xie + x17 + x20 + x21 + x .23 + x ™ + x ™ + ^26 + x32 + ^35 + ^36 + x 3 7 + ^ .39 .42 (4.4) or, equivalently, reciprocal characteristic polynomial (= xnf(x : ) ) [14]: P{x) = 1 + x + x2 + x3 + x5 + x6 + x7 + x 1 0 + x16 + x 17 +x18 + x19 + x21 + x22 + x25 + x2& + x27 + x: .31 + r + x™ + x .42 (4.5) 245 Block Size N m J 48 4 3 96 5 3 192 6 3 384 6 6 768 6 12 1536 6 24 3072 6 48 6144 7 48 12288 7 96 144 4 9 288 5 9 576 5 18 1152 6 18 2304 6 36 4608 7 36 9216 7 72 18432 8 72 36864 8 144 128 7 1 Tab le 4.2 Parameters of the IS-95B forward link interleaver [14]. The scrambled sequence is punctured at a rate of 800 bps in order to insert power control bits. Orthogonal (Walsh) Covering Each channel is spread by a distinct sequence consisting of the repetition of a 64-chips Walsh sequence running at a rate of 1.2288 Mcps. There are hence 1228.8/19.2 = 64 Walsh chips per coded bit. Each Walsh sequence WfA, i = 0 , 1 , . . . , 63, of length 64, corresponds to row i of a Hadamard matrix # 6 4 of size 64 x 64 (with the mapping 0 —> +1, 1—>•—!), which is generated according to the following: Hon - (4.6) 246 where n = 1, 2,4,8, 32, H0 = 0, and (7) denotes the complement (such that H0 = 1). The rows of the Hadamard matrix are orthogonal to each other, which allows the channels to be differentiated at the receiver. The assignment of the W^'s to the different channels was described previously. Quadrature Spreading The same Walsh-spread sequence is passed to both the / and Q branches of the Q P S K transmitter, where it is further spread by different short-code P N sequences PNi and PNQ on branches / and Q, respectively. PNi and PNQ are maximal-length sequences generated by 15-stage shift registers, and lengthened by 1 chip period: their period is thus (2 1 5 — 1) + 1 = 32768. Each base station assigns a different offset (a multiple of 64 chips) to PNT and PNQ; the latter are the same for each user corresponding to a same base station. PNi and PNQ are generated using the following characteristic polynomials, respectively [177]: fi(x) = 1+x2 + x& + x7 + x8 + x10 + x15 (4.7) fQ(x) = l + x3+ xA+ x5+ x9+ x10+ xn+x12+ x15 (4.8) The previous can be expressed as fy(x) = x15Py(x~1), y € {I,Q}, where PI,PQ are the reciprocal polynomials of the P N sequences, as given in the IS-95 standard. Modular shift registers used to generate PNi and PNQ along with their offsets are illustrated in Figs. 4.14 and 4.15. 1 2 3 4 5 15-bit mask 8 -9 - ^ ® + 10 11 12 13 ^ \ Ti l-branch PN code offset F i g u r e 4.14 Generation of PNi and its offsets [177]. 247 F i g u r e 4.15 Generation of PNQ and its offsets [177]. Pulse Shaping and Modulation The PN-spread sequences on each branch are shaped using a baseband finite impulse response (FIR) filter, in order to constrain the spectral bandwidth of the transmitted signal, and to minimize the ISI. The resulting shaped sequences are modulated by / and Q carriers (cos(27r/ ci) and sin(27r/ c i) , respectively), combined and transmitted. 4.2.5 IS-95B Reverse Link Channels The IS-95B reverse link consists of up to 64 code-division multiplexed channels, each channel being assigned a different offset of the long-code P N sequence of (4.4): - Up to 32 access channels per forward link paging channel: used to initiate communica-tion with the BS, or to respond to messages received on the paging channel; - Up to 62 traffic channels (equal to the number of forward traffic channels). Similarly to the forward link, data rates of rates 0.8, 2.0, 4.0 and 8.6 kbps are supported on each traffic channel, and up to 8 traffic channels can be assigned in parallel to a single user. In the following, specifications are again given only for the traffic channels. A block diagram of the air interface for the IS-95B reverse link traffic channels is given in Fig . 4.16. Framing and Coding Traffic frames are generated similarly as on the forward link, with the difference that the convolutional coder has a rate of 1/3. The coder has constraint length K = 9 and generation functions (in octal form) g0 = (557), g\ = (663), and g2 = (711). For the 8.6 kbps data rate, the resulting transmission rate is 28.8 kbps, and frames consist of 576 248 Information bits 8.6 kbps 4.0 kbps 2.0 kbps 0.8 kbps Add frame quality bit indicators (CRC) for 9.6 and 4.8 kbps Add 8-bit encoder tail 9.2 kbps 9.6 kbps Convolutional | encoder R=l/3, K=9 Code symbols 4.4 kbps 2.0 kbps 0.8 kbps 4.8 kbps 2.4 kbps 1.2 kbps 28.8 ksps 14.4 ksps 7.2 ksps 3.6 ksps Symbol repetition 28.8: xl 14.4: x2 7.2: x4 3.6: x8 Block interleaver 32 x 18 = 576-symbol array Code symbols W(64,6) Walsh orthogonal modulator Modulation symbols Data burst randomizer 28.8 ksps 28.8 ksps 4.8 ksps Frame data rate- Control bits 42-bit long code mask * 1.2288 Mcps 42-stage long-code PN generator 15-stage short code PN| (vfct) Baseband filter 1/2 chip| delay Baseband filter 15-stage short code P N r sin(w t) Figure 4.16 A i r interface for the IS-95B reverse link traffic channels [177]. bits, divided into 172 data bits, 12 C R C bits, 8 encoder tail bits, and 384 redundancy bits. Block Interleaving The block interleaver corresponds to a matrix consisting of 32 rows and 18 columns. The coded bits are serially input into the matrix by column, starting with the upper element of the leftmost column. Once the matrix has been filled with N = 32 x 18 = 576 bits, the bits are then read out by rows, starting with the leftmost element of the first row. The block deinterleaver simply reverses the previous operation. Interleaving and dein-terleaving cause a total delay of 2 x 20ms = 40 ms. 249 Orthogonal (Walsh) Modulation Every group of 6 binary code symbols { 6 0 , 6 i , . . . ,65} is mapped into one of 2 6 = 64 Walsh sequences WfA, by the relation: i = b0 + 2&i + 462 + 8c 3 + 16c4 + 32c 5 (4.9) This leads to a Walsh symbol rate of 28.8/6 = 4.8 ksps, or alternatively a Walsh chip rate of 4.8 x 64 = 307.2 kcps. Note that Walsh sequences are used here for modulation, and not for orthogonal covering (as on the forward link). Long-code P N Spreading The Walsh modulation symbols are spread by a phase offset of the long P N sequence PNL of (4.4). Each code channel of each user employs a different phase offset: this is what permits the BS to differentiate between channels. F ig . 4.17 illustrates the modular shift register used to generate PNL along with its offsets. 42-bit mask 33 K E M 341 3 5 ! - * © - • [ 0 36 37 58 39 < M) 41 42 Long PN code offset F i g u r e 4.17 Generation of PNL and its offsets [177]. Quadrature Spreading The quadrature spreading operation is the same as that used on the forward link. Pulse Shaping and Modulation The PN-spread sequence on the Q-branch is delayed by half a chip: this allows to im-plement a form of O - Q P S K , in order to have a more constant envelope. The PN-spread 250 sequences on each branch are then shaped and modulated as on the forward link. 4.2.6 IS-2000 Forward Link Multiple radio configurations (RC) are specified in the IS-2000 forward link, each one supporting a certain set of data rates. Each R C is assigned a spreading rate: either spreading rate 1 (SRI, 1.2288 Mcps) or spreading rate 3 (SR3, 3 x 1.2288 = 3.6864 Mcps). The modes of operation, types of channels, coding methods and rates, spreading sequences, orthogonal covering sequences and modulation schemes depend on the spread-ing rates and radio configurations, and are presented separately below. Radio Configurations and Channels Table 4.3 presents the different radio configurations (RC1-9), along with the sets of supported data rates Rb and coding rates R. R C l and RC2 are similar to the IS-95B forward link, and are included in IS-2000 for backwards compatibility. Hence in the following we wil l only describe the operations associated with RC3-9. R C SR Rb (kbps) R 1 1 1.2 2.4 4.8 9.6 1/2 2 1 1.8 3.6 7.2 14.4 1/2 3 1 1.5 2.7 4.8 9.6 19.2 38.4 76.8 153.6 1/4 4 1 1.5 2.7 4.8 9.6 19.2 38.4 76.8 153.6 307.2 1/2 5 1 1.8 3.6 7.2 14.4 28.8 57.6 115.2 230.4 1/4 6 3 1.5 2.7 4.8 9.6 19.2 38.4 76.8 153.6 307.2 1/6 7 3 1.5 2.7 4.8 9.6 19.2 38.4 76.8 153.6 307.2 614.4 1/3 '8 3 1.8 3.6 7.2 14.4 28.8 57.6 115.2 230.4 460.8 1/4 9 3 1.8 3.6 7.2 14.4 28.8 57.6 115.2 230.4 460.8 1036.8 1/2 Tab le 4.3 Radio configurations (RC) of the IS-2000 forward link [14]. The channels supported by the IS-2000 forward link are listed below, along with their maximum numbers (NS means no maximum number is specified, while N A means not available). Their functions can be found in the previous sections or in [14]. Unless noted otherwise, they can be used for both SRI and SR3: 251 - Common Assignment Channel ( C A C H ) (NS) - Common Power Control Subchannels ( C P C S C H ) (NS) - Pilot Channels: - Forward Pilot Channel (PCH) (1) - Transmit Diversity Channel (TDCH) (1) - Auxil iary Pilot Channel ( A P C H ) (NS) - Auxil iary Transmit Diversity Pilot Channel ( A T D P C H ) (NS) - Common Control Channels (CCCH) (NS) - Synch Channel (SCH) (1) - Traffic Channels: - Dedicated Control Channel (DCCH) (1) - Fundamental Channel (FCH) (1) - Power Control Subchannels (PCSCH) (1) - Supplemental Channels (SCH) (2) (RC 3-9) - Supplemental Code Channels (SCCH) (7) (RC 1-2) - Broadcast Channels (BCH) (NS) - Paging Channels (PGCH) (7 for S R I , N A for SR3) - Quick Paging Channels ( Q P G C H ) (3 for S R I , NS for SR3) In the following we wil l only describe in detail the air interfaces of the Fundamental and Supplemental Channels (which are similar). A block diagram of the air interface for the IS-2000 forward link traffic channels is given in Fig . 4.18. The air interface of the Supplemental Code Channels is similar to that of the IS-95B forward link. Framing Tables 4.4 and 4.5 illustrate the frame structures associated with R C 3 and RC4, respec-tively, which both use S R I . npunc denotes the number of punctured symbols. Tables 4.6 and 4.7 illustrate the frame structures associated with RC6 and R C 7 , respectively, which both use SR3. These are the RC's which wil l be used in our simulation models. Further details about the other RC's (i.e. RC5, RC8-9) can be obtained from [14]. 252 Information bits Add frame quality bit indicators (CRC) Add 8 reserved/ encoder tail bits Convolutional/ turbo encoder Code symbols Symbol repetition Symbol puncture Modulation symbols Block interleaver 42-stage long-code PN generator I/Q Demux Decimator Power control bits. 800 bps MUX 1.2288 xNMcps Decimation for MUX timing control 800 bps 15-stage short code PN| Baseband filter cos(wt) i Baseband filter 15-stage short code P N r sin(w t) Figure 4.18 Air interface for the IS-2000 forward link traffic channels [14]. Coding The Quick Paging and Common Power Control Channels are not coded. A l l other chan-nels use convolutional codes, with rates depending on the SR and the R C . The Forward Supplemental Channels can use turbo codes for high data rates (Rs > 19.2 kbps). Convolutional Coding The convolutional encoders have the following generation functions (in octal form): - R = 1/6 - R = 1/4 - R= 1/3 -R=l/2 go = (457), 9 l = (755), g2 = (551), g3 = (637), g4 = (625), g5 - (727). 9o = (765), 9 l = (671), g2 = (513), g3 = (473). 5o = (557), 9 l = (663), g2 = (711). go = (753), gr = (561). 253 Rb (kbps) ^data ncRC ntail R Tlrep Tlpunc 0.8 16 6 8 1/4 8 1 of 5 768 2.0 40 6 8 1/4 4 1 of 9 768 4.0 80 8 8 1/4 2 0 768 8.6 172 12 8 1/4 1 0 768 18.0 360 16 '8 1/4 1 0 1536 37.0 , 744 16 8 1/4 1 0_ 3072 75.6 1512 16 8, 1/4 1 0 6144 152.4 3048 16 8 1/4 1 0 12288 Table 4 .4 Frame structure of RC3 of the IS-2000 forward link [14]. Rb (kbps) ncRC R Tlrep 0.8 16 6 8 1/4 8 1 of 5 384 2.0 40 6 8 1/4 4 1 of 9 384 4.0 80 8 8 1/4 2 0 384 8.6 172 12 8 1/4 1 0 384 18.0 360 16 8 1/4 1 0 768 37.0 744 16 8 1/4 1 0 1536 75.6 1512 16 8 1/4 1 0 3072 152.4 3048 16 8 1/4 1 0 6144 306.0 6120 16 8 1/4 1 0 12288 Tab le 4 . 5 Frame structure of RC4 of the IS-2000 forward link [14]. Turbo Coding The turbo encoder is illustrated in Fig. 4.19. The transfer function of the constituent code is the same for both upper and lower recursive systematic convolutional encoders, and is given by: G(D) = n0{D) m(£>)' d(D) d{D) where d(D) = 1 + D2 + D3, n0{D) = 1 + D + D3 and m(D) = 1 + D + D2 + D3. (4.10) The algorithm for the turbo interleaver is given as follows: 1) Write consecutive values of the input stream into an array of 2 5 rows by 2 n columns, 254 Rb (kbps) ncRC ntail R Tlpunc risym 0.8 16 6 8 1/6 8 1 of 5 1152 2.0 40 6 8 1/6 4 1 of 9 1152 4.0 80 8 8 1/6 2 0 1152 8.6 172 12 8 1/6 1 0 1152 18.0 360 16 8 1/6 1 0 2304 37.0 744 16 8 1/6 1 0 4608 75.6 1512 16 8 1/6 1 0 9216 152.4 3048 16 8 1/6 1 0 18432 306.0 6120 16 8 1/6 1 0 36864 Tab le 4.6 Frame structure of RC6 of the IS-2000 forward link [14]. Rb (kbps) ncRC Html R Tlrep 0.8 16 6 8 1/3 8 1 of 5 576 2.0 40 6 8 1/3 4 1 of 9 576 4.0 80 8 8 1/3 2 0 576 8.6 172 12 8 1/3 1 0 576 18.0 360 16 8 1/3 1 0 1152 37.0 744 16 8 1/3 1 0 2304 75.6 1512 16 8 1/3- 1 0 4608 152.4 3048 16 8 1/3 1 0 9216 306.0 6120 16 8 1/3 1 0 18432 613.2 12264 16 8 1/3 1 0 36864 Tab le 4.7 Frame structure of R C 7 of the IS-2000 forward link [14]. in a row-by-row fashion. The parameter n is given in Table 4.8. 2) Permute the elements within each row according to the following row-specific linear congruential sequence rule: x[i + l] = (x[i] + c)mod2n, where x[0] = c and c is determined from n and the row number according to the table lookup of Table 4.9. 3) Permute the rows by bit-reversing the row numbers. 4) Output the elements of the array in a column-by-column fashion. A n alternative formulation of the above algorithm can be found in [14]. 255 N information bits (input) Symbol Puncture and Repetition (N+6)/R Code Symbols (output) Control F i g u r e 4.19 Turbo encoder for the IS-2000 forward link [14]. Block Size n 378 4 570 5 762 5 1146 6 1530 6 2298 7 3066 7 4602 8 6138 8 9210 9 12282 9 20730 10 Tab le 4.8 Parameter n of the IS-2000 turbo interleaver [14]. 256 Index n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 0 5 27 3 15 3 13 1 1 15 3 27 127 1 335 . 349 2 5 1 15 89 5 87 303 3 15 15 13 1 83 15 721 4 1 13 29 31 19 15 973 5 9 17 5 15 179 1 703 6 9 23 1 61 19 333 761 7 15 13 31 47 99 11 327 8 13 9 3 127 23 13 453 9 15 3 9 17 1 1 95 10 7 15 15 119 3 121 241 11 11 3 31 15 13 155 187 12 15 13 17 57 13 1 497 13 3 1 5 123 3 175 909 14 15 13 39 95 17 421 769 15 5 29 1 5 1 5 349 16 13 21 19 85 63 509 71 17 15 19 27 17 131 215 557 18 9 1 15 55 17 47 197 19 3 3 13 57 131 425 499 20 1 29 45 15 211 295 409 21 3 17 5 41 173 229 259 22 •15 25 33 93 231 < 427 335 23 1 29 15 87 171 83 253 24 13 9 13 63 23 409 677 25 1 13 • 9 15 147 387 717 26 9 23 15 13 243 193 313 27 15 13 31 15 213 57 757 28 11 13 17 81 189 501 189 29 3 1 5 57 51 313 15 30 15 13 15 31 15 489 75 31 5 13 33 69 67 391 163 Tab le 4.9 Table lookup for the IS-2000 turbo interleaver [14]. 257 In order to support different code rates, symbols can be punctured prior to trans-mission. The puncturing patterns for the different code rates are given in Table 4.10. Output R = l/2 R=l/3 R = 1/4 5,1 Xk 11 11 11 xk 10 11 11 p,12 Xk 00 . 00 10 . s,2 Xk 00 00 00 xk 01 11 01 k 00 00 11 Tab le 4.10 Puncturing patterns for the IS-2000 turbo encoder [14]. Block Interleaving The forward link block interleaving can be mathematically expressed as follows, with a block size of N: f 2 ™ [ i m o d J ] + B R O m ( L ^ J ) i = 0, 2 , . . . , N - 2 ' M =- < , / N i i i \ (4-11) I 2™ [(TV - *±±) mod J] + B R O m ( | A T H j ' * = M , . . . , AT - 1 v where the symbols in the above equation are as defined previously, and m and J are determined as in Table 4.2. Scrambling and Power Control The interleaved sequence is scrambled by the decimated version of a user-specific phase offset of a long-code P N sequence running at 1.2288 Mcps (c.f. Eq . (4.4)). The scrambled sequence is punctured at a rate of 800 bps in order to insert power control bits. Orthogonal (Walsh) Covering Each traffic channel is spread by a different Walsh sequence Wf*, taken from a N-axy or-thogonal set, where ./V depends on the R C and the data rate. For R C 3 , N = 4,8,16, 32, 64 are used for Rb = 152.4, 75.6, 37.0,18.0,8.6 kbps, respectively. For R C 4 and RC6, N = 4, 8,16, 32, 64,128 are used for Rb = 306.0,152.4, 75.6, 37.0,18.0, 8.6 kbps, respectively. 258 For RC7, N = 4, 8,16, 32, 64,128, 256 are used for Rb = 613.2, 306.0,152.4, 75.6, 37.0,18.0, 8.6 kbps, respectively. The Walsh sequences run at rates of 1.2288 Mcps for SRI and 3.6864 Mcps for SR3, and with periods of A/1.2288 fjs (SRI) and N/3.6864 (SR3). In order to be mutually orthogonal, the Walsh sequences of different lengths, denoted as orthogonal variable spreading factor (OVSF) sequences, can be generated according to the method described in [256]. Modulation, Complex Spreading and Pulse Shaping After the orthogonal covering, each channel is split into I and Q branches, with consecu-tive bits being mapped to different branches in alternation, leading to QPSK-modulated symbols. The resulting I/Q channel is spread by a complex spreading sequence. For S R I , the real and imaginary parts of the complex spreading sequence have characteristic polynomials given by Eqs. (4.7) and (4.8), respectively. For SR3, the real and imaginary parts of the complex spreading sequence are the first 3 x 2 1 5 chips of the offset versions (by 19 and 19 + 2 1 9 chips, respectively) of the P N sequence with characteristic polynomial: f(x) = l + x n + x15 + x17 + x20. (4.12) The PN-spread sequences on each branch are then shaped using a baseband F I R filter, modulated by / and Q carriers, combined and transmitted 4.2.7 IS-2000 Reverse Link Radio Configurations and Channels Table 4.11 presents the different radio configurations (RC1-6), along with the sets of supported data and coding rates. R C l and RC2 are similar to the IS-95B reverse link, and are included in IS2000 for backwards compatibility. Hence in the following we wil l only describe the operations associated with RC3-6. The channels supported by the IS-2000 reverse link are listed below, along with their maximum numbers. Unless noted otherwise, they can be used for both S R I and SR3: - Access Channel (ACH) (1 for S R I , N A for SR3) - Enhanced Access Channel ( E A C H ) (1) 259 R C SR Rb (kbps) R 1 1 1.2 2.4 4.8 9.6 1/3 2 1 1.8 3.6 7.2 14.4 1/2 3 1 1.35 1.5 2.7 4.8 9.6 19.2 38.4 76.8 153.6 307.2* 1/4 4 1 1.8 3.6 7.2 14.4 28.8 57.6 115.2 230.4 1/4 5 3 1.2 1.35 1.5 2.4 2.7 4.8 9.6 19.2 38.4 76.8 153.6 307.2+ 614.4+ 1/6 6 3 1.8 3.6 7.2 14.4 28.8 57.6 115.2 230.4 460.8 1036.8* 1/4 *: R = 1/2; +: R = 1/3 Table 4.11 Radio Configurations of the IS-2000 reverse link [14]. - Reverse Common Control Channel ( R C C C H ) (1) - Traffic Channels: - Pilot Channel (PCH) (1) - Dedicated Control Channel (DCCH) (1) - Fundamental Channel (FCH) (1) - Power Control Subchannel (PCCH) (1) -Supplemental Channels (SCH) (2) (RC 3-6) - Supplemental Code Channels (SCCH) (7) (RC 1-2) In the following we wil l only describe in detail the air interfaces of the Fundamental and Supplemental Channels (which are similar). A block diagram of the air interface for the IS-2000 reverse link Fundamental and Supplemental Channels is given in Fig . 4.20. The air interface of the Supplemental Code Channels is similar to that of the IS-95B reverse link. Framing Table 4.12 illustrates the frame structures associated with R C 3 , which uses SRI . Table 4.13 illustrates the frame structures associated with R C 5 , which uses SR3. These are the RC's which wil l be used in our simulation models. Futher details about the other RC's (i.e. RC4, RC6) can be obtained from [14]. 260 Information bits Add frame quality bit indicators (CRC) Add 8 reserved/ encoder tail bits Convolutional/ turbo encoder Code symbols Symbol repetition Symbol puncture Block interleaver Modulation symbols 15-stage short code PNi (+ + —) or (+ + + +) SCH2 • PCH 4-(++++++++-DCCH PCH •e--> (±> (+ + + + + + + + -SCHl T (RCCCH, EACH) f + _j o r ( + + _ f 0 f S C H 1 (or (+ + - - + + - - ) for RCCCH and EACH) 42-stage long-code PN generator os(\^ ,t) Baseband filter •e-Baseband filter siii(wct) Decimator by factor of 2 1—chip delay 1.2288 x N Mcps -0 15-stage short code P N r E A C H : Enhanced Access Channel RCCCH: Reverse Common Control Channel PCH: Pilot Channel DCCH: Dedicated Control Channel FCH: Fundamental Channel SCHl: Supplemental Channel 1 SCH2: Supplemental Channel 2 Figure 4.20 Air interface for the IS-2000 reverse link Fundamental and Supplemental Chan-nels [14]. Coding The convolutional and turbo encoders are the same as those on the forward link. Block Interleaving The reverse link block interleaving can be mathematically expressed as follows, with a block size of TV: Ai = 2 m [i mod J] + B R O m ^LjJ^ i = 0,1,... ,N — 1 (4.13) where the symbols in the above equation are as defined previously, and m and J are determined as in Table 4.14. Orthogonal (Walsh) Covering Each channel is spread by a distinct Walsh sequence Wf*, which corresponds to the ith row of a Hadamard matrix of size N x N. A l l Walsh sequences used are mutually or-261 Rb (kbps) Tldata ncRC n-tail R Tlpunc ^sym 0.8 16 6 8 1/4 16 1 of 5 1536 2.0 40 6 8 1/4 8 1 of 9 1536 4.0 80 8 8 1/4 4 0 1536 8.6 172 12 8 1/4 2 0 1536 18.0 360 16 8 1/4 1 0 1536 37.0 744 16 8 1/4 1 0 3072 75.6 1512 16 8 1/4 1 0 6144 152.4 3048 16 8 1/4 1 0 12288 306.0 6120 16 8 1/2 1 0 12288 Tab le 4.12 Frame structure of RC3 of the IS-2000 reverse link [14] Rb (kbps) ncRC n-tail R Tirep 0.8 16 6 8 1/4 16 1 of 5 1536 2.0 40 6 8 1/4 8 1 of 9 1536 4.0 80 8 8 1/4 4 0 1536 8.6 172 12 8 1/4, 2 0 1536 18.0 360 16 8 1/4 1 0 1536 37.0 744 16 8 1/4 1 0 3072 75.6 1512 16 8 1/4 1 0 6144 152.4 3048 16 8 1/4 1 0 12288 306.0 6120 16 8 1/3 1 0 18432 613.2 12264 16 8 1/3 1 0 36864 Tab le 4.13 Frame structure of RC5 of the IS-2000 reverse link [14] thogonal, and are assigned as follows (+1 is abbreviated by + and -1 by -): - P C H : W$2 = (+ + + + + + + + + + + + + + ++) - E A C H : W$ = (+ + - - + + ) - R C C C H : W | = (+ + - - + + ) - D C C H : W£6 = (+ + + + + + + + ) - F C H : = (+ + + + + + + + ) - SCH1: W2 = (+- ) or = (+ + — ) - SCH2: W2A = (+ + ) or W£ = (+ + ++) 262 Block Size N m J 384 6 6 768 6 12 1536 6 24 3072 6 48 6144 7 48 12288 7 96 576 5 18 2304 6 36 4608 7 36 9216 7 72 18432 8 72 36864 8 144 Table 4.14 Parameters of the IS-2000 reverse link interleaver [14]. Modulation, Complex Spreading and Pulse Shaping The data channels are mapped onto either the / or the Q branches, but not both: this corresponds to B P S K modulation, as opposed to Q P S K modulation in the case of the forward link. The P C H , D C C H and SCH2 are mapped onto the /-branch, while the F C H , S C H l , E A C H and R C C C H are mapped onto the Q-branch. The resulting I/Q channel is spread by a complex spreading sequence. The real part of the latter is the product of a short P N sequence PNi with a long P N sequence PNi. The imaginary part is obtained by multiplying the 1-chip delayed version of PNt with a short P N sequence PNq, decimating by a factor 2 the resulting sequence, and multiplying the decimated sequence by PNi and W\ ~ (-1—). For S R I , PNi, PNq and PNi are the same as those specified for the reverse link of IS-95B: PNi = P i V 7 , PNq = PNQ, PNt = PNL. For SR3, PNi and PNq are the first 3 x 2 1 5 chips of the offset versions (by 19 and 19 + 2 1 9 chips, respectively) of the P N sequence with generator polynomial given by (4.12). PNt is obtained by multiplexing the following 3 sequences: PNr, the product of PNi and its 1-chip delayed version, and the product of PNt and its 2-chips delayed version. The PN-spread sequences on each branch are then shaped using a baseband F I R filter, modulated by / and Q carriers, combined and transmitted. 263 4.3 Smoothing for V B R Video over Multicode and Multirate D S / C D M A 4.3.1 Introduction As discussed in Chapter 3 and in the previous section, 2.5G and 3G D S / C D M A mobile networks have the ability to change the transmission rate either on a connection or frame-by-frame basis [191]. The adaptation method used depends on the type of system. IS-95B systems use multicode transmission in order to provide higher transmission rates [191]: several codes are assigned to a high-rate service requested by a user, and the data stream is demultiplexed among these codes. IS-2000 systems use multirate transmission instead [14]: the data rate of a traffic channel can be increased (with the chip rate remaining the same) to support a higher source rate. Wideband C D M A ( W - C D M A , [257], [258]) uses a combination of multirate and multicode transmission. Various methods have been proposed to support V B R video over D S / C D M A networks. If the source rate is assumed constant (e.g. due to perfect source rate control), or if the peak source rate is within the maximum transmission rate, then a simple approach consists in transmitting at a constant rate. This wil l however result in a reduced video quality (in the source rate control case), or a high bandwidth requirement (in the peak rate transmission case). However, for simplicity and benchmarking purposes, such an approach was used in [259] for the forward link of an IS-95 system and in [190] for the reverse link of a multicode IS-95B system. Alternatively, the transmission rate can be varied adaptively in order to follow the variations of the video source. This was done for example in [75] for the multicode case. If each code retains the same power as in the single-code case, this approach wil l lead to higher interference to other users, and thus a reduction in the cell capacity. The multirate (variable spreading gain) case was treated for example in [260] for IS-2000 systems, where a decoding delay was introduced in order to be able to better support the rate variations. However, an increase in the transmission data rate wil l lower the processing gain, which wil l result in a higher B E R if the transmission power is kept constant. One way to minimize the drawbacks of the previous approaches is to perform real-time smoothing of the video stream. The ideal goal of smoothing is to permit the transmission 264 at a nearly constant rate of a V B R source. To allow this, a certain amount of buffering must be introduced either at the transmitter or the receiver, resulting in a certain delay in the decoding. If the video frames are all available beforehand to the transmitter (as in stored video), the latter can determine the optimal sequence of transmission rates (or schedule [49]) before it initiates transmission, and then keep it frozen. For this of-fline case, in [49], the authors present an algorithm which achieves the "greatest possible reduction in rate variability" for stored video. They derive a formal proof of the opti-mality of the algorithm, and propose a practical 0(n) implementation for it based on a shortest-path approach. A review and comparison of various offline smoothing algorithms (including the previous) are offered in [68]. If the video frames are available progressively to the transmitter (as in real-time interactive or streaming video), the latter has to deter-mine the best schedule in a progressive way, either on a frame-by-frame basis, or based on a sliding window of frames. A n online version of the optimal algorithm of [49] is proposed in [51]. Further work on online smoothing is presented for example in [66], where the proposed algorithm tries to send as many frames as possible at the same rate with a given look-ahead interval, or in [69], which also proposes online versions of the algorithm of [49]. A n optimal algorithm for both online and offline video was proposed in [67] using dynamic programming. While there have been many studies on source rate control for wireless video (e.g. [55], [54], [261], [56]), most previous published work on smoothing was carried out in the framework of wireline communications, such as the Internet or asynchronous transfer mode (ATM) networks, which are relatively free of errors due to noise or interference. In these systems, a rate increase doesn't necessarily translate into a B E R increase or a capacity reduction, as is the case for current D S / C D M A cellular systems. Hence most previous work on smoothing was thus concerned mainly with optimizing certain characteristics of the transmission rate (such as peak and variance), but without looking at the effect on the P S N R of the decoded video sequence (which is influenced by the B E R of the communication link). In D S / C D M A cellular systems, the choice of the transmission rates wil l have an effect on the processing gain and/or transmitted power, and hence on the resulting B E R (and hence on the P S N R ) or cell capacity. Hence the 265 goal of this section is to illustrate such an interplay between rate, delay, and P S N R , through the application of smoothing concepts to D S / C D M A cellular systems. To this end, we describe a real-time smoothing scheme whose goal is to maximize the S N R over the highest number of frames, and hence obtain a minimum B E R for a minimum transmitted power. The presented algorithm relies on some principles intro-duced in [51] for the design of the Sliding Window algorithm (SLWIN), but is designed for best operation in a D S / C D M A cellular system. It wi l l be denoted as SLWIN2. It attempts to minimize the peak rate and the rate variability of the transmitted signal (as does SLWIN) , but subject to the additional constraint of maintaining the minimum average transmission bandwidth at all times. This last constraint is introduced in order to maximize the processing gain (for multirate systems) or minimize the number of codes transmitted in parallel (for multicode systems), which wil l lead to a lower B E R , if the total power assigned to the video user has to remain constant, or a higher cell capacity, if the total power assigned to the video user can be varied. No source rate control is applied (i.e. varying the quantizer step size or skipping frames) in order to maintain a constant encoded video quality. The algorithm must also take into account the fact that in standardized communication systems, the set of allowable transmission rates is limited, and the transmission frames' periods usually differ from the video frames' pe-riods; in previous descriptions of smoothing algorithms for wireline systems, with few exceptions, no constraints were imposed on the allowable set of transmission rates, and the transmission and video frame sizes were often assumed equal. While the difference in frame sizes only affects the practical implementation, it wi l l be shown through examples that the limited set of transmission rates can lead to different rate schedules and P S N R performances for S L W I N and SLWIN2. We perform two case studies using both SLWIN and SLWIN2: we apply them to real-time H.263 video communication over the reverse and forward links of both multicode IS-95B and multirate IS-2000 systems, using detailed simulation models, in the case of a circuit-switched H.324 connection [245]. We compare the results obtained with both schemes in terms of peak, mean and variance of rate, and P S N R . The smoothing algorithms can be applied equally well to the reverse or forward links of such systems, and to packet-switched connections (c.f. [49], Section III-C). Indeed, our work considers 266 "non-collaborative" smoothing schemes [262], which can be used on both forward and reverse links. In contrast, a recent paper [262] (which also considered smoothing over wireless D S / C D M A systems) focused on "collaborative" schemes, which are useful on the forward link of cellular systems; however it presented numerical results for stored video only, using a simplified channel model. The plan of this section is outlined as follows. After describing the motivations behind using smoothing in Section 4.3.2, Section 4.3.3 presents a smoothing framework for the transmission of V B R video, which is used to derive the S L W I N and SLWIN2 algorithms described in Section 4.3.4. Section 4.3.5 presents and discusses representative numerical results obtained with the algorithms and simulation model of the previous sections. 4.3.2 Motivation for Smoothing The traffic generated by video sequences can be very bursty, especially when the interval between I-frames is small. Let Nr be the interval (in number of video frames) between two intra-coded (I) frames. Examples of a video trace, corresponding to the H.263-encoded Miss America video sequence, are given in Figs. 4.21 and 4.22 for A / = 132 and A / = 20, respectively. The spikes correspond to I-frames, while the rest of low-rate traffic is made out of P-frames 1. The Miss America video sequence features the head and shoulders of a woman talking to the camera and making occasional head movements, and is of low motion; it is typical of a videoconferencing application. Other examples of a video trace, corresponding to the H.263-encoded Foreman video sequence, are given in Figs. 4.23 and 4.24 for Nj = 132 and A 7 / = 20, respectively. This sequence features a foreman talking in an expressive way to the camera, and includes more motion than the Miss America sequence, as can be evidenced by comparing the rate variability of the traces; it is typical of a videophone application.. In order to support the frequent source rate variations, IS-95B and IS-2000 systems could be made to adapt their transmission rate, preferably on a frame-by-frame basis. 1 In practical very-low bit-rate live video communication systems, only segments of frames (e.g. mac-roblocks) instead of whole frames are intra-coded, in order to minimize the bit rate. This does not require any changes to the smoothing algorithms discussed in this chapter. 267 In the case of multicode IS-95B systems, this could be done by varying the number of codes assigned to the video service. However, if the power of each code is equal to that of a single-code user, the total transmitted power of the video user wi l l vary frequently and exhibit a high peak-to-average ratio (PAR) . Conversely, if the total power assigned to a multicode user needs to remain constant regardless of the number of codes, the S N R per code wil l decrease resulting in a higher B E R . In the case of multirate IS-2000 systems, if the transmission rate of the video user is increased by a certain factor F to accommodate a higher source rate, the processing gain PG of the receiver decreases by the same factor F. This results in a reduction of robustness to multiple-access and multipath interference, and thus a higher B E R for the high-rate user. To compensate for the lower PG in order to maintain a constant B E R , the transmitted power of the video user needs to be increased to F times the power needed when the processing gain is F times higher. If the transmitted power (in addition to the transmission rate) is adapted to the source rate, this wi l l also lead to frequent power variations and high P A R ' s for the transmitted power. These rapid power variations and high P A R ' s are not desirable due to the following: • The increased power of the video user wil l create extra multiple-access interference to the other users of the system, thus reducing the cell capacity. The high-power video user can further swarm a voice user which operates close to him; • High P A R ' s reduce the power amplifier efficiency; • Frequent rate and power changes increase the signaling overhead. Moreover, the burstiness of the video stream may lead to peak source rates which are higher than the maximum rate for transmission. It would thus be favorable to minimize the peak, average and variance of the transmis-sion rate, which wil l lead to lower and more constant power requirements for a variable-power system, or a lower B E R for a fixed-power system. In the next section we examine the constraints which govern the selection of the transmission rates; these constraints are then taken into account in the design of a smoothing algorithm, whose goal is to minimize the peak and variance of the transmission rate, while maintaining the minimum average bandwidth at all times. 268 Video frame number Figure 4.21 Video trace of Miss America, with 1 out of 132 I frames. 800 ,1 , , , 1 , 0 20 40 60 80 100 120 Video frame number Figure 4.22 Video trace of Miss America, with 1 out of 20 I frames. 269 2500 0 20 40 60 80 100 120 Video frame number Figure 4.23 Video trace of Foreman, wi th 1 out of 132 I frames. 2000 ,1 i i i i i 1 0 20 40 60 80 100 120 Video irame number Figure 4.24 Video trace of Foreman, wi th 1 out of 20 I frames. 270 4.3.3 V B R Video Constraints We first present the system parameters and notations used to describe the rate adap-tation framework, then lay down the rate constraints which must be observed by the system. 4.3.3.1 Video Source The source outputs video frames of equal duration TF seconds. Let s(i) denote the source rate of video frame i, in bits/second. Let S(i) = s(i)TF be the number of bits carried by video frame i, and Sr(z) = Yl)=i t>e the total number of bits in video frames 1 through i. 4.3.3.2 Transmitter Let Tc be the duration (in seconds) of an IS-95B or IS-2000 frame, which is not necessarily equal to TF. Similarly to before, r(i) denotes the data rate for C D M A frame i, R(i) = r(i)Tc is the number of data bits in C D M A frame i, and Rr(i) = Yl)=i ^s the total number of bits in C D M A frames 1 through i. 4.3.3.3 Video Decoder Let Tmax be the maximum end-to-end delay that can be tolerated between video users. The actual delay is given by Td = Tp + Tb, where: • Tp = NpdTF, where Npd is a real number, is the sum of the propagation delay and processing delays (due to video and channel coding/decoding, interleaving, etc.). • Th = NbdTF, where Nbd is a real number (which is chosen to be an integer to clarify expositions), is a buffering delay introduced in order to smooth the video stream. We can then write Td = NdTF, where Nd = Npd + Nbd is the total number of video frames by which the decoding is delayed, from the start of the transmission. Hence, the video decoder starts processing the received bitstream Td seconds after the transmission began. In the following, without loss of generality, we wil l take Tp = 0 and thus Nd = Nbd in order to simplify the exposition. 271. The video encoder and decoder are synchronized, thus the rate of decoded video frame i, d(i) is equal to the rate of encoded frame i, s(i). However, the decoded video frames lag their corresponding encoded video frames by Td, as illustrated in Fig. 4.25. Hence, d(i) = s(i - Nd) if i > Nd, and 0 if i < Nd. Also, define D(i) = d(i)Tc, and DT(i) = J2)=i If the decoder buffer has a finite buffer capacity of b bits, then the maximum number of bits that can be received by the decoder by frame i is BT(i) = DT(i) + b. tbO) ts(0. Video frames (encoder end) C D M A frames Video frames (decoder end) te(i) ts(i+D te(i+l) tb(i+D 1 1 i i+1 i+2 i+N d i+N d+l j j+l J+2 J+3 J+4 J+5 J+6 J+7 i+1 Figure 4.25 Rate adaptation framework. 4.3.3.4 Rate Constraints Both video encoder and decoder are equipped with a buffer used to smooth out rate variations. Let Be(i) be the number of bits which need to be buffered at the video encoder at the end of encoded video frame i. Similarly, let Bd(i) be the number of bits which need to be buffered at the video decoder at the end of encoded video frame i. Equivalently, Bd(i + Nd) is the number of bits which need to be buffered at the video decoder at the end of decoded video frame i (c.f. F ig . 4.25). Then the constraints which must be observed by the video communication system, for each video frame i, are given by [263]: 0 < Be{%) < B\ e maxi (4.14) 0 < B\i + Nd) < Bdmt (4.15) 272 where B^ax and B!^ax a r e the encoder and decoder buffer sizes, respectively (in bits). In this work, we make the assumption that the video encoder buffer is of sufficient size so as to avoid any overflow, i.e. Be(i) < B^ax for all i. Moreover, encoder underflow, which occurs when the encoder is not delivering enough bits to the transmitter, can be prevented by reducing the transmission rate or suspending transmission of video data altogether (it is unlikely for the case at study, due to the high source bit rate), hence 0 < Be(i) for all i. Then, the constraints of Eq. (4.14) need not be dealt with. Our main concerns are thus the possibilities of decoder buffer underflow and overflow [263], i.e. the violation of the constraints of Eq. (4.15). Decoder buffer underflow, i.e. when Bd(i + Nd) < 0, occurs when the receiver does not deliver enough bits on time to the video decoder, which leads to a loss of one or several video frames by the decoder. Decoder buffer overflow, i.e. when Bd(i + Nd) > 5 ^ a x , occurs when too many bits need to be stored in the decoder buffer before the beginning of video decoding, which leads to the discarding by the buffer of one or several received video frames. Let BJp^ + Nd) (different from Rr{i + Nd)) be the total number of bits transmitted by the end of video frame i + Nd, hence Bd(i + Nd) = Rlj.{i + Nd) - ST{i)- Then Eq. (4.15) can be written as: 0<B!r(i + Nd)-ST^)<Bdmax. (4.16) In the case of our system with unequal video and C D M A frame sizes, Eq. (4.16) becomes: J i 0 < $ > ( j ) l b + r(l + l)(hTF - ITC) -J2s(j)TF < B*ax (4.17) 3 = 1 J"=l where h = i + Nd — 1, and I = [hTF/Tc\, which means that in the time spanned by h video frames there are I full C D M A frames (the first term on the left-hand side of Eq. (4.17)), and possibly an additional fraction of a C D M A frame (the second term on the left-hand side of (4.17)) i f hTF/Tc is not an integer. Note that for Eq. (4.17) to be equivalent to the decoder buffer underflow constraint given in [263], we should have h = i + Nd. However, this wil l allow the constraint Eq. (4.17) to hold throughout decoded video frame i only if there were no buffer underflow at the beginning of this 273 frame, and if the transmission rate was constant during the whole duration of this video frame (as in [263]). This doesn't apply to the general case of unequal video and C D M A frame sizes, since a transmission rate change during decoded video frame i can cause the decoder buffer to underflow even if the constraint Eq. (4.17) is satisfied at the beginning of the video frame. However, by choosing h = i + Nd — 1, all of decoded video frame i w i l l be received on time if there is no buffer underflow at the beginning of it. Hence constraint Eq . (4.17) is more conservative than the one given in [263]. In order for the decoder buffer overflow constraint to still be respected, as indicated in Eq. (4.17), B!^ax has been replaced with B^ax = B^ax - rmaxTf, where rmaxTf is the maximum number of bits received over the interval of one video frame. In the special case where TF = Tc, one can choose h = i + Nd, and Eq. (4.17) reduces to 0 < RT(l) - ST(i) < B^ax, i.e. the term r(l + l)(hTF - ITC) disappears from the inequalities. If TF ^ Tc, the term r(l + l)(hTF — ITC) can lead to an non-integer number of bits in the evaluation of (4.17), and in all cases makes the latter less easy. We've therefore used the following simplified form in our algorithms: /' i 0 < J2r(j)Tc - 5>( j )T F < Bdmax (4.18) 3=1 3=1 where I' = \hTF/Tc~\, and \x~\ denotes the smallest integer larger than x. In order to make Eq. (4.18) as conservative as (or more than) Eq . (4.17), we can simply replace A ^ with Nd — 1 in the algorithms. 4.3.4 Smoothing Algorithms for V B R Video over D S / C D M A Two smoothing algorithms are described in the next sections. The first one, termed SLWIN2, is a more general version of that initially presented in [264], [265] (where it was denoted as Method 2), by allowing a sliding window length A ^ > 1 (Nw = 1 in [264], [265]), and imposing constraints on the decoder buffer capacity ( B ^ a i —> oo in [264], [265]). The second algorithm is equivalent to that presented earlier in [51] and denoted as SLWIN, and wil l serve as a benchmark. It has been modified in order to be able to deal 274 with unequal source and transmission frame lengths. It should be noted that the case of different transmission granularities has also been treated in [266], for an IP network. Both algorithms are based on a sliding window mechanism in which the transmission schedule is updated for every new video frame made available (larger updating intervals can be considered [51], in order to decrease computational complexity). For a sliding window of length Nw, in order to incorporate video frame i into the schedule computation, the algorithms need to have knowledge of the following Nw — 1 video frames. W i t h this mechanism, the total end-to-end delay wil l now be Td = Tb + Tw = NdTf, where Tw = (Nw — 1)TF and thus A ^ = Nbd + Nw — 1. A n alternative in order to avoid this extra delay would be to try to predict the following Nw — 1 frames (c.f. [66], [51]). A t the start of video frame i, thus at time ts[i] = (i — l)TF, the algorithms can modify the rates of the C D M A frames whose transmission begins before time td[i] = ts[i] + Td. There are Nj = NdTF/Tc or NdTF/Tc+l such frames, the first case occurring if there are no incomplete frames within the Td duration. A t the end of video frame i, thus at time te[i] = ts[i] + TF = iTF, the C D M A frames whose transmission started before te[i] now have their rate determined; thus if a C D M A frame is shared by video frames i and i + 1 (such as C D M A frame j + 1 in Fig . 4.25), its rate wil l be determined by the former only. 4.3.4.1 SLWIN2 The goal of SLWIN2 is to minimize the peak and variability of the transmission schedule, subject to the constraint of maintaining at all times the minimum necessary transmission bandwidth required to respect constraints Eq. (4.18). In other words, for each new available video frame, SLWIN2 chooses among all schedules with the minimum necessary bandwidth the one which provides the smallest peak rate and variance. In order to verify the previous constraint, it is allowed to change the transmission rates on a frame-by-frame basis. The way in which the transmission rates are chosen is described below, in response to both buffer underflow and overflow conditions. When a higher bandwidth is necessary in order to avoid buffer underflow, the al-gorithm progressively increases the rates of the C D M A frames with the lowest rates, starting with the earliest frames, until Eq. (4.18) is verified. More specifically, when a 275 rate increase is needed, the algorithm identifies the earliest frame with the lowest rate, and determines whether it can assign to it a higher rate without overflowing the decoder buffer. If it's possible, it increases its rate to the next higher one, and starts over the same procedure. If it's not possible, it identifies the next earliest frame with the same rate, or if there isn't any, the earliest frame with the next higher rate. When the transmission bandwidth must be decreased in order to avoid buffer overflow, the algorithm progressively decreases the rates of the C D M A frames with the highest rates, starting with the latest frames, until Eq. (4.18) is verified. This procedure is simply the reverse of that detailed above. The pseudocode for an implementation of this algorithm is given below, along with explanatory comments. The following indices and quantities are used in the pseudocode: - if ( ic) is the indice of the video ( C D M A ) frame currently under consideration, and tv (tc) is its ending time; - iw ( iw-c) is the indice of the earliest video ( C D M A ) frame of the sliding window , and tw (tWjC) is its starting time; - iSp ( hp.c) is the indice of the earliest video ( C D M A ) frame to which the algorithm can backtrack (i.e. all previous frames have their rates fixed), given that isp > iw (isp_c > iw.c), and tsp (£j,p_c) is its starting time; - U ( id-c) is the indice of the last video ( C D M A ) frame at which a buffer underflow condition was detected, and ib ( ib_c) is the indice of the last video ( C D M A ) frame at which a buffer overflow condition was detected; - Rates\\ is an array containing all the permissible transmission rates, and iT[i] is the indice of the rate assigned to C D M A frame i, with minimum (maximum) value ir.min i^r -max) • Initialization if = 0,ic = 0,tv = 0,tc = 0 iw — 1, iyj_c ~~ 1, tyj — 0, tw_c = 0 isp = .1, isp-c = 1, tSp — 0, tsp_c — 0 id = 1) id.c = 1, h — 1, ib.c = 1 / / Assign the minimum rate to a l l C D M A frames 276 ir[i] = 1, r[i] = Rates[iT[i]] for all i Loop if — if + I, tv = tv + Tf, ic = ic + 1, tc = tc + Tc, RT[0] = 0, continue = 1 while (continue —= 1) { // Update RT RT[ic] = RT[ic-l}+r[ic]Tc while (tc < tv) {ic = ic + l,tc = tc + Tc, RT[ic] = RT[ic - 1] + r[ic]Tc } 1/ Buffer underflow is detected if (RT[ic] < DT[if}) { II Find the earliest CDMA frame with the min. rate imin = min arg min{r[«] : i £ [max(?sp_c, iw_c),... , ic}} 11 Assign to it the next higher rate, if possible if (ir[imin] + 1 > V_max) { if {imin = = «c) { Signal buffer underflow } } else { v[*min] = ir \imiri\ ~t~ 1 } ^[^min] = flfltes[!r[irajn]] // The underflow position is stored, the algorithm backtracks to the earliest possible frame id = if, if = max(isp,iw), tv — m a x ( i s p , £ w ) + Tf id-c ~ ici ic = max(i Sp_ c , iw_c), tc = max(iSp c , twc) + Tc } / / Buffer overflow is detected else if (RT[ic}> BT[if}) { 11 Find the latest CDMA frame with the max. rate imax — max arg max{r[i] : i £ [max(i S p_ c ,w), . . . ,ic}} II Assign to it the next lower rate, if possible if {ir[imax] - 1 < ir.min) { if {imax == max(isp_c,iw_c)) { Signal buffer overflow } } else { ir \imax\ = ir \imax\ ~ 1 } // The overflow position is stored, the algorithm backtracks to the earliest possible frame, updates iSp/isp-c ib — if, if = max(iSj,, iw), tv = max(iSp, tw) + Tf, isp = i„ + 1, tsp = (isp — i.)Tf ib.c — ici ic = rnax(iSp_c, iw_c), tc — max(i S j U ; ) t f f l . c )+T c , isp.c — ib.c+1) tsp-c = {isp-c~ 1)TC 277 /•/ The constraints are respected else { // Process the next video frame, unless the last one has already been reached if (if == Nd + Nf) { continue = 0 } else { if = if + 1, tv = tv + Tf, ic = ic + 1, tc = tc + Tc } } // Update the starting position of the sliding window if (if >iw + Nd + Nw) { iw = iw + 1, tw = tw + Tf while (tyj_c <C tyj) •[ iw-c ~ ^w_c ~f 1, U^J_C ~ ^W_C } } } In [260], we used another algorithm to calculate the schedule. When a higher band-width is necessary in order to avoid buffer underflow, the algorithm progressively increases the rates of the earliest C D M A frames, until Eq. (4.18) is verified (instead of the C D M A frames with the lowest rates). When the transmission bandwidth must be decreased in order to avoid buffer overflow, the algorithm progressively decreases the rates of the lat-est C D M A frames (instead of the C D M A frames with the highest rates). This algorithm wil l be denoted as SLWIN2' (Method 1 of [264]). Unlike SLWIN2, it does not attempt to make the transmission schedule as smooth as possible: it simply chooses the rates so as to avoid buffer underflow and overflow. The advantage of using smoothing algorithms over such a scheme wil l be quantified in Section 4.3.5. 4.3.4.2 S L W I N The goal of SLWIN is to make the transmission schedule as smooth as possible, i.e. with minimum peak rate, rate variance and effective bandwidth [51]. To this end, it creates a series of constant-rate runs, by attempting to minimize the rate and maximize the length of each run. Hence, unlike SLWIN2, the transmission rate is not changed on a frame-by-frame basis, but on a run-by-run basis, as further detailed below. When a rate change is needed, a new run at a higher/lower rate is started at the earliest (leftmost) point in the schedule. Hence, when a higher bandwidth is necessary in order to avoid buffer underflow, the algorithm increases the rates of all the C D M A frames comprised between the start of the run (ra.ax(ispjC,iwj:)) and the occurence of 278 the underflow (id_c), and repeats it until Eq. (4.18) is verified. When the transmission bandwidth must be decreased in order to avoid buffer overflow, the algorithm decreases the rates of all the C D M A frames comprised between the start of the run (max(i s p_ c , iWJC)) and the occurence of the overflow (ib_c), and repeats it until Eq . (4.18) is verified. The pseudocode for SLWIN, tailored to deal with unequal video and transmission frame sizes, is obtained by making the following additions and modifications to the pseu-docode for SLWIN2: Initialization Loop while (continue == 1) { // Update RT II Buffer underflow is detected if (RT{ic] < DT[if]) { II Assign the same rate to the following CDMA frames which begin within the same video frame % — lc, t — tc while (t < Tv) { i = i + 1, t = t + Tc if (i > imin) { ir[i] = ir[i - 1], r[i] = Rates[ir[i]] } } } I/ Buffer overflow is detected else if (RT[ic\ > BT[if]) { I/ Find the earliest CDMA frame with the max. rate imax = min arg max{r[i] : i G [max(i s p_ c, iw_c),... , ic]} II The overflow position is stored, the algorithm backtracks to the earliest possible frame, updates iSp/iSp_c if (id > if) { ib = if, if = max(isp,iw), tv = max(tsp,tw) + Tf, isp = ib + l, tsp = (isp - l ) T f 279 } else { i ib — ifi isp = rnax ( i^ + 1,iw), tSp = (iSp l)Tf, if = iSp, tv = tSp -j- Tf ib-c = ia isp-c = rnax ( i^_ c + l , i « ; _ c ) ) tsp_c — (isp-c l)-^ci ic ~ isp-a tc = tspjc } // Assign the same rate to the following CDMA frames which begin within the same video frame % = %c, t = tc while (t < Tv) { i = i + 1, t = t + Tc if (i > imin) { ir[i) = ir[i - 1], r[i] = Rates[iT[i]] } } } 11 The constraints are respected else { iS(if==Nd + Nf){ if (id — — Nd + Nf) { continue = 0 } else { // Tries to reduce the rate of the last run, if possible isp = rnax(z^ -f-1, iw), tsp — (i$p 1)Tf, if = isp, tv — tSp -j- Tf isp-c ~ rnax ( i^_ c -j- l,iw-c) •> tsp_c — (isp-c 1)TC, ic = iSp_ci tc = t§p_c ~\~ Tc if (iT[ic] — 1 < ir-min) { continue = 0 } else { ir[ic] = ir[ic] — 1> r[ic] = Rates[iT[ic]] } I/ Assign the same rate to the following CDMA frames-which begin within the same video frame % %Q, t t^ while (t < Tv) { i = i + 1, t = t + Tc if (i > i m i n) { ir[i] = iT[i ~ 1], r[i] = Rates[iT[i\] } } } } else { if = if + 1, tv = tv + Tf, ic = ic + 1, tc = tc + Tc 11 Holds the same rate throughout the current run if (if == id + 1 OR if = ib + l OR ic < imin OR ic < imax) { r[ic] = Rates[ir[ic]] } else { ir[ic] — v [ i c ~~ 1]> r[*c] — -Ra£es[ir[?c]] } Z = Z Q , t — tc while (t<Tv) {i = i + l,t = t + Tc if ( i ; = = id + 1 OR i / = i 6 + 1 OR i < i T O j n OR i < i m Q X ) { r[i] = Rates[ir[i]] } 280 else { ir[i] — ir[i — 1], r[i] = Rates[ir[i]] } } } } // Update the starting position of the sliding window } 4.3.4.3 Comparison between S L W I N and SLWIN2 The main difference between SLWIN and SLWIN2 lies in their adaptation of the transmission rate on a run-by-run and frame-by-frame basis, respectively. Given that S L W I N tries to maintain long constant-rate runs between "critical points", at the end of the run it may have consumed more bandwidth than actually needed by the data. This can lead to a higher mean rate than the SLWIN2 algorithm, which is more flexible in its assignment of rates in order to consume the minimum necessary bandwidth (c.f. F ig . 4.31 and associated discussion in Section 4.3.5.2). One possibility could be to use the "gating" capability of cdma2000 systems, i.e. the transmitter would stop transmitting (i.e. the rate is 0 kbps) when there is no more useful data to send. However, this would still result in a higher mean rate for S L W I N than for SLWIN2, since the same amount of useful data would be transmitted in a shorter time interval. Nevertheless, changing the transmission rate on a frame-by-frame basis presents some drawbacks. The number of rate changes can be larger for SLWIN2 than for SLWIN: frequent switching between rates can be undesirable (c.f. [267]). SLWIN2 has a higher computational complexity than SLWIN, with the gap increasing with the size of the smoothing window. If there is no restriction on the values of the rates which can be used, i.e. in the case of a fluid model (used for example in the theoretical discussions of [49], [267]), then S L W I N and SLWIN2 wil l give identical performances. Section 4.3.5 presents sample numerical results illustrating differences between SLWIN and SLWIN2 in terms of peak, mean and variance of rate. 281 4.3.5 Performance Evaluation Results and Discussion 4.3.5.1 Simulation Setup Fig . 4.26 is a general diagram of the simulated IS-95B or IS-2000-based video com-munication system. Below we describe the various components which make up the video transmission link: the source, the transceiver and the multiple-access channel. In this thesis we do not tackle issues related to the network layer, and assume throughout that the circuit-switched video user is granted the required (maximum) transmission band-width at the start of the connection: this allows us to focus on the mechanisms affecting the lower layers, as stated in the introduction. QCIF video H.263 encoder H.223 mux Convolutional encoder + interleaver Spreading + modulation IS-95B/ H.324 Video Subsystem c d m a 2 0 0 0 Transceiver Fading multipath channel Interference_ + A W G N Video display H.263 decoder H.223 demux Viterbi decoder + deinterleaver Demodulation + despreading Figure 4.26 I S - 9 5 B or IS-2000-based system simulation model. Video Source The source coder is a H.263-compliant video compressor, which encodes Q C I F frames at a rate of 30 frames/second. The optional modes are disabled. The video bitstream is then packetized according to the level 2 protocol of H.223 Annex B [251]. Each picture corresponds to one video packet (VP) , which is then broken down into one or more multiplex packets (MP) , such that the payload of each M P does not exceed 254 bytes. A synchronization flag (SF) of 2 bytes and a basic header (BH) of 3 bytes are prepended to 282 each M P , according to the syntax of [251]. Moreover, an optional header (OH) of 1 byte follows the B H . Its purpose is to allow the recovery of the previous M P , in the case the latter is lost due to an error in its SF or B H . A C R C of 2 bytes is appended to each V P . To evaluate the average P S N R , we use Np = 120 frames of the Miss America or Foreman H.263 video sequences, playing at 30 frames/s. Results are averaged over 20 independent simulation runs, each run using the same Np video frames. The average P S N R for each simulation run, PSNR,., is calculated as the average of the P S N R ' s PSNRf(i) of each decoded frame i = 1, 2 , . . . , NF, i.e.: NF PSNRr = — ^PSNRjii). (4.19) F i=i The P S N R of each frame is calculated as in [259] as the weighted sum of the PSNR's for the weighted luminance and chrominance components of that frame: PSNRf(i) = PSNRfiY{i) + 0-3PSNRftCh(i) + 0.3PSNRfjCT(i) (4.20) where: / 255 2 \ PSNRfx(i) = 10 log — = „ . (4.21) V i v f e E £ 1 E £ i M * > 0 - r , ( i f c , 0 ) V In Eq. (4.21), Nt denotes the number of lines in the picture (=144 for Q C I F ) , Np denotes the number of pixels per line (=176 for QCIF) , ox(k, I) and rx(k, I) are the values of the original and reconstructed 8-bit pixel values, respectively, for component x € {Y, Cb, Cr}. No error concealment is used in the simulations. If the decoder crashes during a run, the PSNR' s for the individual unprocessed frames are taken to be zero. If the decoder freezes during a run, the P S N R for this run is taken to be equal to the average of the PSNR's of the other runs. Reverse Link IS-2000 Transceiver The IS-2000 uplink transceiver is as described in Section 4.2.7. In our system, the video data is transported by the SCH1, while the F C H is used to carry voice and/or 283 control information. The D C C H 2 and SCH2 channels are omitted for our purposes. The spreading rate is Rc = 1.2288 x iV Mcps, where N = 3 (Spreading rate 3), and the chip period Tc = l/Rc. Data is processed by a rate 1/4 convolutional encoder, and interleaved 3 according to the pattern specified in Section 4.2.7. At the BS receiver, the channel multipaths are collected by a Rake receiver with Lr fingers and a resolution of T c , and are maximal-ratio combined. The reverse despreading and coherent demodulation operations are then performed, followed by deinterleaving and hard-decision Viterbi decoding. The analytical uncoded B E R performance of such a system was derived in Chapter 3. In our simulations, the F C H has a fixed data rate of Rp = 9600 bps. The S C H l can take one of the following rates: 9600, 19200, 38400, 76800 or 153600 bps. Note that lower and higher rates are possible, but wil l not be considered here due to the high source rate and the high mobility scenario, respectively. We assume that the powers of the P C H and the F C H are equal, which is a worst-case assumption since the power of the P C H is likely to be chosen a few factors smaller than that of the F C H . The power of the S C H l is also taken to be equal to that of the F C H , resulting in lower SNR's and higher B E R ' s for higher data rates. This allows us to observe the effect of smoothing on the video user, in terms of P S N R . If the power was increased proportionally to the rate on a frame-by-frame basis (as we did in [260]), the P S N R of the video user would remain constant, but the capacity of the cell would be decreased due to the extra interference contributed by the video user. While numerically obtaining the cell capacity as a function of the smoothing algorithms or parameters would be interesting, it would prove to be a formidable task in terms of simulation time, given the fact that each user's physical layer is simulated. Moreover, as mentioned in Section 4.3.2, varying the power on a frame-by-frame basis would increase the signaling complexity. These practical facts motivate our decision to keep the power of the video user constant, despite our knowledge that the degradations 2 For all the systems under consideration in this section, the signaling information associated with the rate changes is assumed to be carried by the control channels (e.g. D C C H for the IS-2000 uplink) and handled by higher-layer protocols. Since the multiple-access interference contributed by this extra signaling isn't large enough to have a noticeable effect on the simulation results, the control channels haven't been included in the simulation models in order to limit their complexity. 3 In [264], [265], the interleaver was switched off, leading lo higher BER's . 284 incurred by a variable B E R might not be acceptable to the end user. Forward Link IS-2000 Transceiver The IS-2000 downlink transceiver is as described in Section 4.2.6. As in the reverse link, the video data is transported by the SCH1, while the F C H is used to carry voice and control information. The other channels are omitted. We use the parameters of Radio Configuration 3 [14]: the spreading rate is Rc = 1.2288 Mcps, the code rate is 1/4. The transmitter uses turbo encoding; the turbo encoder and interleaver are as specified in Section 4.2.6. At the MS receiver, the channel multipaths are collected by a two-finger Rake re-ceiver with a resolution of Tc and are maximal-ratio combined (alternatively, transmitter diversity could be used). The reverse despreading and coherent demodulation operations are then performed, followed by channel deinterleaving and turbo decoding. The turbo decoder uses the M A P (i.e. B C J R ) algorithm (c.f. Appendix F) . The analytical uncoded B E R performance of such a system was derived in Chapter 3. In our simulations, the SCH's can take one of the following rates: 9600, 19200, 38400, 76800 or 153600 bps. The power of the S C H is made to remain constant for all rates, resulting in lower SNR's for higher data rates, but thus avoiding a reduction in the cell capacity, as discussed in the previous section on the IS-2000 reverse link. Reverse Link IS-95B Transceiver The IS-95B uplink transceiver is as described in Section 4.2.5. The coded and interleaved symbol stream carrying the video information is demultiplexed into a number of sub-streams. Each symbol sub-stream c of user k is assigned to a different traffic channel. A t the BS receiver, the channel multipaths collected by the Rake receiver are equal-gain combined with a resolution of Tc. The reverse despreading and noncoherent de-modulation operations are then performed, followed by deinterleaving and hard-decision Viterbi decoding. The analytical uncoded B E R performance of such a system was derived in Chapter 3. Each channel has a maximum data bit rate of 9.6 kbps. A user can then transmit at a total aggregated data bit rate of 9.6 x Ncodes kbps, where the number of codes Ncodes 285 varies from 1 to 8. W i t h a coded bit rate Rb = 3 x 9.6 = 28.8 kbps and a chip rate Rc = 1.228800 Mbps, this results in a processing gain of N = (\og2M)Rc/Rb = 256 for each channel. In our simulations, a system with Ncodes codes wil l be limited to transmit the same total power as a system with only one code: the power of each traffic channel of the multicode video user wi l l be equal to l/Ncodes times the power of the traffic channel of a single-code user. This preserves system capacity at the expense of a higher B E R for the multicode video user, and hence a degradation in the received video quality. We wil l show how the smoothing algorithms manage to limit this degradation. Forward Link IS-95B Transceiver The IS-95B downlink transceiver is as described in Section 4.2.4. The management of the multiple codes, the set of allowed rates and the peak power specification are similar to those described in the previous section on the IS-95B uplink. The MS uses a Rake receiver with a resolution of T c , which performs M R C on the collected multipaths. The analytical uncoded B E R performance of such a system was derived in Chapter 3. Cellular Environment As in the previous chapter, the frequency-selective channel is modeled as a tapped delay line filter with an impulse response given by h(t, r) = Ylt^Q1 cti(t)ej61^5(r — n), where Lc is the number of multipath components, ati(t) and 9i(t) are the time-variant amplitude and phase, respectively, of the complex short-term fading coefficient of the Ith path, and T\ is the delay of the latter. It is assumed that there is no ISI. Samples of each ai(t) are Rayleigh-distributed. A single-cell system is considered in order to simplify the simulations, however numerical results could be obtained the same way for a multiple-cell environment as we have done in in [190], [260]. We assume that all the users in the system, except for the desired video user, transmit voice only (using a single code for IS-95B, or the F C H at 9.6 kbps for IS-2000). Moreover they are uniformly distributed over the cell, which is a common assumption. The activity factor (AF) of the voice is 0.375, while that of the video source is 1. The mobile speed is v = 100 km/h , and there are Lr = 2 Rake fingers. Perfect power control, synchronization and channel estimation are also assumed for simplicity.. 286 4.3.5.2 Comparison of Transmission Rates for S L W I N and SLWIN2 This section compares the transmission schedules produced by S L W I N and SLWIN2, in terms of peak, variance and mean of the rates, for different systems and parameters. The effects of these quantities on the decoded P S N R wil l be assessed in Section 4.3.5.3. IS-2000 Reverse Link Fig. 4.27 plots the peak and mean, and Fig. 4.28 the variance of the transmission rates for the SLWIN and SLWIN2 algorithms, as the delay Nd is varied. The sliding window length is A ^ = 10 and the decoder buffer size = 64k. For this set of parameters, it can be seen that while the peak rate is similar for both algorithms, the mean and variance of SLWIN2 is lower than that of SLWIN. It can be explained by the better ability of SLWIN2 to adjust to source rate variations: indeed, S L W I N smooths the video stream by producing piecewise constant-rate transmission segments, while SLWIN2 changes the transmission rate on a frame-by-frame basis. However, as mentioned previously, this can result in many more rate changes for SLWIN2, and the computational complexity of SLWIN2 is higher than SLWIN. The peak rate decreases with Nd for both SLWIN and SLWIN2. The variance and mean of the transmission rates decrease in quasi-monotonous and monotonous fashions, respectively, for SLWIN2, while they vary in an irregular (but overall decreasing) fashion for SLWIN. Fig . 4.29 plots the peak and mean, and Fig. 4.30 the variance of the transmission rates for the SLWIN and SLWIN2 algorithms, as the sliding window length Nw is varied. The delay is Nd = 9 and the decoder buffer size 5 ^ a x = 64k. The peak and variance decrease with Nw for both SLWIN and SLWIN2, but for Nw larger than 10 there is not much improvement. The average rate is constant for SLWIN2 but fluctuates lightly for SLWIN. Figs. 4.31 and 4.32 plot the cumulative schedule RT with A ^ = 1 and Nw = 10, respectively, for both SLWIN and SLWIN2, with Nd = 9 and B^ax = 64k, along with the cumulative decoded data DT. It is noticed that the schedule of SLWIN2 follows DT more closely than SLWIN does, i.e. it exhibits a lower variance and consumes a lower average bandwith, as evidenced previously in Figs. 4.27-4.30. In the case of SLWIN, the high variability of the rate is due to the fact that the schedule is computed on a 287 run-by-run basis. For example, in Fig. 4.31 a new constant-rate run is started at video frame 21, and extends until video frame 30 (an underflow critical point): since the rate stays constant between these two points, at the end of the run the cumulative rate Rp is much bigger than the minimum needed (Dp)- In contrast, in the case of SLWIN2, the rate is changed on a frame-by-frame basis between these two locations, and thus at the end of the variable-rate run the cumulative rate RT is very close to DT. For larger A V s , e.g. in Fig 4.32, the schedules become smoother for both S L W I N and SLWIN2, due to the latters' knowledge of future frames. F ig . 4.33 plots the same quantities but for Nd = 120, Nw = 1 and Bmax = 8k, and also along the cumulative buffer capacity Bp. In this case the schedules are very close alike. Fig. 4.34 plots the transmissions rates of the Nc = \(NF + Nd- 1)7>] = 214 C D M A frames used to transport the Np = 120 frames of the Miss America video sequence, with Nd = 9. It can be seen that, for this particular case, SLWIN2' has a peak transmission rate of 153600 bps, versus 76800 bps for SLWIN2. Moreover, it can be seen that more rate variations are observed with SLWIN2' . Fig. 4.35 plots the transmissions rates of the Nc = \(NF + Nd - 1)TF] = 399 C D M A frames used to transport the same Np = 120 frames, but now with Nd = 120. In this case SLWIN2' still has a peak transmission rate of 153600 bps, while that of SLWIN2 is merely 19200 bps. The transmission schedule for SLWIN2 is much smoother than in the previous case. The transmission schedule for SLWIN2' has significant variations over a certain number of frames, and then exhibits the minimum rate for the rest of the sequence: indeed, the algorithm of SLWIN2' tries to meet the rate constraint as early as possible. IS-2000 Forward Link Figs. 4.36, 4.37 and 4.38 plot the peak, variance and mean of the transmission rates, respectively, for the SLWIN and SLWIN2 algorithms, as the delay Nd is varied. The sliding window length is Nw = 10 and the decoder buffer size Bmax = 64k. For this set of parameters, it can be seen that the peak rate is similar for both algorithms, and decreases with Nd for both SLWIN and SLWIN2. However, the mean and variance of 288 SLWIN2 are lower than that of SLWIN. As mentioned before, it can be explained by the better ability of SLWIN2 to adjust to source rate variations. Figs. 4.39 and 4.40 plot the variance and mean, respectively, of the transmission rates for the SLWIN and SLWIN2 algorithms, as the sliding window length Nw is varied. The delay is Nd = 12 and the decoder buffer size B ^ n 3 . = 64k. The variance decreases with Nw for both S L W I N and SLWIN2, but for Nw larger than 10 there is not much improvement. The average rate is constant for SLWIN2 but fluctuates lightly for SLWIN. Fig. 4.41 plots the cumulative schedule RT for both S L W I N and SLWIN2, with Nd = 12, Nw = 10 and Bmax = 64k, along with the cumulative decoded data DT. It is noticed that the schedule of SLWIN2 follows DT more closely than S L W I N does, i.e. it consumes a lower average bandwith, as evidenced previously in Fig . 4.38. x 10 6 h I Peak rate: SLWIN Peak rate: SLWIN2 Mean rate: SLWIN Mean rate: SLWIN2 0.4 N ,/N_ 0.6 Figure 4.27 IS-2000 peak and mean transmission rates against Nd/Np for SLWIN and SLWIN2 with NT = 20, Nw = 10, Ban 64k, for the Miss America bitstream. 289 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N d / N p Figure 4.28 IS-2000 variance of the transmission rates against Nd/NF for S L W I N and SLWIN2 wi th Nj = 20, Nw = 10, -B^az = 64k, for the Miss America bitstream. - Peak rate: SLWIN • Peak rate: SLWIN2 Mean rate: SLWIN - - Mean rate: SLWIN2 7 - | 6.5 - ' 6 " ; 0 20 40 60 80 100 120 N w (video frames) Figure 4.29 IS-2000 peak and mean transmission rates against Nw for S L W I N and SLWIN2 wi th Ni = 20, Nd = 9, B ^ a I = 64k, for the Miss America bitstream. 290 •S 2.5 SLWIN SLWIN2 60 N (video frames) 80 Figure 4.30 IS-2000 variance of the transmission rates against Nw for S L W I N and SLWIN2 wi th Ni = 20, Nd — 9, B m a x = 64k, for the Miss America bitstream. DT - - RT SLWIN — R r SLWIN2 20 40 60 80 Video frame number Figure 4.31 IS-2000 transmission rate schedule for S L W I N and SLWIN2 wi th TV/ Nd = 9, Nw = 1, B m a x = 64k, for the Miss America bitstream. = 20, 291 Video frame number Figure 4.32 IS-2000 transmission rate schedules for S L W I N and S L W I N 2 wi th Nj Nd = 9, Nw = 10, B^ax = 64k, for the Miss America bitstream. Video frame number Figure 4.33 IS-2000 transmission rate schedules for S L W I N and S L W I N 2 wi th N{ Nd = 120, Nw = 1, B m a x = 8k, for the Miss America bitstream. 292 ,'l I Ml I Method 1 Method 2 60 80 100 120 140 CDMA frame number 160 Figure 4 . 3 4 IS-2000 transmission rates for SLWIN2 (i.e. Method 2) and S L W I N 2 ' (i.e. Method 1) wi th Nj = 20, Nd - 9, Nw- = 1, B ^ = 64k, for the Miss America bitstream. Figure 4 . 3 5 IS-2000 transmission rates for S L W I N 2 (i.e. Method 2) and S L W I N 2 ' (i.e. Method 1) wi th NT = 20, Nd = 120, Nw = 1, B ^ a x = 64k, for the Miss Amer ica bitstream. 293 x 10 1.3h SLWIN SLWIN2 0.7 20 60 80 N (video frames) 100 120 Figure 4.36 IS-2000 peak transmission rates against Nd for S L W I N and S L W I N 2 wi th TV/ = 20, Nw = 10, B m a x — 64k, for the Foreman bitstream. SLWIN SLWIN2 140 N d (video frames) Figure 4.37 IS-2000 variance of the transmission rates against Nd for S L W I N and SLWIN2 wi th TV/ = 20, Nw = 10, NF = 120 and B m a x = 64k, for the Foreman bitstream. 294 SLWIN SLWIN2 140 N r f (video frames) Figure 4.38 IS-2000 mean transmission rates against Nd for S L W I N and SLWIN2 with TV/ = 20, Nw = 10, NF = 120 and B^ax = 64k, for the Foreman bitstream. SLWIN SLWIN2 120 N w (video frames) Figure 4.39 IS-2000 variance of the transmission rates against Nw for S L W I N and SLWIN2 wi th TV/ = 20, Nd = 12, B m a x = 64k, for the Foreman bitstream. 295 SLWIN SLWIN2 O.g I 1 1 1 1 1 0 20 40 60 80 100 120 N w (video frames) Figure 4.40 IS-2000 mean transmission rates against Nw for S L W I N and SLWIN2 wi th TV/ = 20, Nd = 12, B m a x = 64k, for the Foreman bitstream. Video frame number Figure 4.41 IS-2000 transmission rate schedules for S L W I N and SLWIN2 wi th TV/ = 20, Nd = 12, Nw = 10, B^ax = 64/c, for the Foreman bitstream. 296 IS-95B Reverse Link Fig. 4.42 plots the peak and mean, and Fig . 4.43 the variance of the transmission rates for the SLWIN and SLWIN2 algorithms, as the delay Nd is varied. The sliding window-length is Nw = 10 and the decoder buffer size 5 ^ a x = 64k. In comparison to the results of the previous section, these quantities are closer together for S L W I N and SLWIN2. Indeed, in this case there is a finer granularity of the rates as compared with the IS-2000 case: SLWIN then offers a performance closer to its ideal one, which occurs when a fluid model for the rates is assumed. The same can be said from an examination of plots of these quantities versus Nw. Figs. 4.44 and 4.45 plot the transmissions rates of the C D M A frames used to trans-port the NF = 120 frames of the Miss America video sequence, for Nd = 9 and Nd = 120 respectively. In both cases, SLWIN2' has a peak transmission rate of 76.8 kbps, as com-pared to 67.2 kbps (for Nd = 9) and 28.8 kbps (for Nd = 120) with SLWIN2. Also, more rate variations are observed with SLWIN2' . In the case Nd = 9, the small buffering delay limits the smoothing ability of SLWIN2, while in the case Nd = 120, the transmission schedule for SLWIN2 is much smoother. In contrast, for Nd = 120, the transmission schedule for SLWIN2' behaves as described for the IS-2000 system: it has significant variations over a certain number of frames, and then exhibits the minimum rate for the rest of the sequence. IS-95B Forward Link Fig. 4.46 plots the cumulative schedule RT for both S L W I N and SLWIN2, with Nd = 60, Nw = 10 and - B ^ a a . = 64k, along with the cumulative decoded data DT. The schedules are seen to be very close together, and in fact mostly overlap: as explained in the previous section on the IS-95B uplink, the finer granularity of the transmission rates makes in sort that S L W I N and SLWIN2 produce similar outputs. The effect of A ^ on the schedule produced by SLWIN2 is illustrated in Fig . 4.47, with the same parameters as before: larger A V s are seen to produce smoother schedules. 297 3.5 2.5 - Peak rate: SLWIN • Peak rate: SLWIN2 Mean rate: SLWIN - - Mean rate: SLWIN2 0.6 Figure 4.42 IS-95B peak transmission rates against Nd/NF for S L W I N and SLWIN2 wi th Ni = 20, Nw = 10, B m a x — 64k, for the Miss America bitstream. x 10 ~i 1 SLWIN - - SLWIN2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.43 IS-95B variance of the transmission rates against Nd/NF for S L W I N and SLWIN2 wi th Nj = 20, Nw — 10, B m a x = 64k, for the Miss America bitstream. 298 x 10 III itll I'll' i I " |IH| J l " I 'HI I ill 11. l'» I M II Method 1 Method 2 80 100 120 140 C D M A frame number F i g u r e 4 . 4 4 IS-95B transmission rates for SLWIN2 (i.e. Method 2) and S L W I N 2 ' (i.e. Method 1), wi th Ni = 20, Nd — 9, Nw = 1, i ? ^ a x = 64k, for the Miss America bitstream. Method 1 Method 2 150 200 250 C D M A frame number 400 F i g u r e 4 . 4 5 IS-95B transmission rates for SLWIN2 (i.e. Method 2) and S L W I N 2 ' (i.e. Method 1), wi th Ni = 20, Nd = 120, Nw = 1, B^ax = 64k, for the Miss America bitstream. 299 Video frame number Figure 4.46 IS-95B transmission rate schedules for S L W I N and S L W I N 2 wi th JVj = Nd = 60, Nw = 10, B m a x = 64fc, for the Foreman bitstream. 300 4.3.5.3 Simulation Results: P S N R This section presents the performance evaluation and comparison of the S L W I N and SLWIN2 algorithms in terms of P S N R . The results are obtained through computer sim-ulations, and illustrate the effects of smoothing and transceiver parameters. IS-2000 Reverse Link Fig. 4.48 plots the average P S N R as a function of the number of users K, for Nj = 20, Nd = 9, for the SLWIN, SLWIN2 and SLWIN2' algorithms with Nw = 1 and Nw = 10. In the case Nw = 1, SLWIN2 performs better than SLWIN, especially for a larger number of users. When Nw = 10, the two algorithms show a similar performance. This stems from the observations made in Section 4.3.5.2: when a small sliding window length is used, e.g. Nw = 1, with SLWIN the variance and mean of the rate are higher than with SLWIN2. Hence more C D M A frames are transmitted at high rates, which leads to a lower average processing gain and thus a higher B E R (with the assumption made of no power adaptation). Furthermore, many of these high-rate h igh-BER frames are used to transport the I-frames from the video sequence (since these I-frames are responsible for the high-rate), which wil l have a greater impact on the decoded video quality since they are used for predicting the following P-frames. While a higher Nw improves the performance for SLWIN, it also introduces a larger end-to-end delay, which may not be tolerable. F ig . 4.49 plots the results for JVj = 132. Similar observations can be made, however for a large number of users SLWIN2 performs better than S L W I N even for a higher Nw. In both figures, it can be seen that SLWIN and SLWIN2 allow a dramatic gain in the P S N R over SLWIN2' : this shows the benefit of using smoothing algorithms. Fig. 4.50 illustrates the effect of the buffering delay Nd on the average P S N R , using SLWIN2. W i t h a delay Nd = 3, the video quality degrades rapidly for K > 15. However, if all of the video frames are buffered prior to decoding (Nd = 120), the P S N R remains almost constant, at the expense of a longer delay. The effect of Nr on the P S N R is shown in F ig . 4.51, with Nd = 9. A higher Nr means a higher-rate for the video sequence, and thus a higher B E R . It also means that the video sequence is refreshed more often, and thus error propagation can be stopped earlier. However, in this case it is seen that the higher B E R offsets the gain in P S N R made possible by more frequent refreshing. 301 302 35 30 25 5 I Nd=3 -e- Nd=9 „ Nd=120 jl I I 1 I I 1 0 10 20 30 40 50 60 Number of users K F i g u r e 4 . 5 0 A v e r a g e P S N R for JV> = 20, Nw = 1, B ^ a x = 64k a n d va r ious A d ' s , for S L W I N 2 i n the IS-2000 u p l i n k case, for the M i s s A m e r i c a b i t s t r e a m . F i g u r e 4 . 5 1 A v e r a g e P S N R for Nd = 9, Nw = 1, = 64k a n d var ious A / ' s , for S L W I N 2 i n the IS-2000 u p l i n k case, for the M i s s A m e r i c a b i t s t r e a m . 303 IS-2000 Forward Link Let Niter denote the number of turbo decoding iterations. Tmax = NmaxTF denotes the maximum total end-to-end delay allowed. Then, assuming a pipeline architecture for the turbo decoder (which leads to a worst-case turbo decoding delay), Niter and Nbd must satisfy NiterTc + NbdTF < Tmax, where Tp = NiterTc and Tb = NbdTF. F ig . 4.52 plots the average P S N R as a function of the number of users K, with Nr = 20, Nmax = 13, Nw = 1, for the S L W I N and SLWIN2 algorithms and different sets of {Niter, Nbd} which satisfy the previous constraint. It can also be seen that increasing the number of iterations at the expense of Nbd is beneficial for up to about 6-8 iterations: after this the returns of the turbo decoder diminish, and thus it is preferable to rely on a larger Nbd to improve performance. Surely if the turbo decoding delay is made much smaller than NiterTc, the number of iterations wil l have a negligible effect on the delay budget, and hence a larger Nbd wil l be affordable. Fig . 4.53 shows the effect of Nbd for NiteT fixed and equal to 1, with JV/ = 20, Nw = 1, for the SLWIN2 algorithm. It can be seen that in this case the P S N R is not greatly influenced by varying over the previous range of 6-9. Hence this confirms that the number of turbo decoding iterations was the determining factor in improving the P S N R in Fig. 4.52. The effect of the number of diversity branches Lr on the P S N R is examined in Fig. 4.54, with NT = 20, Nbd = 9, Nw = 1, Bmax = 64k, Niter = 6, and SLWIN2. A higher Lr results in a degradation of the P S N R , given a certain number of users. This corroborates the observations made in Section 3.4.3, where it was explained that the use of diversity could actually increase the B E R due to increased multicode interference. Note however that these results are for a single-cell system with low background noise: for a multi-cell system, diversity wil l be useful in combating interference (which isn't synchronous anymore) contributed from other cells. Fig . 4.55 indicates that as the speed of the mobile increases, so does the P S N R . Indeed, the errors induced by the fading process are less correlated because of the higher Doppler shift, enabling the turbo decoder to correct more errors. However, if C L P C and the effect of estimation errors are taken into account, the reverse is most likely to be 304 true: the gain due to a more accurate tracking of the channel variations can offset the B E R penalty due to correlated errors. Fig. 4.56 plots the P S N R for code rates = 1/3 and R = 1/4, and the same pa-rameters as before. It is seen that the P S N R is considerably improved for K > 15 when R = 1/4 is used instead of R = 1/3. 30 • • " ^ — — \ . s x v \ \ N V \ \ -\ \ \ ^ \ \ V \\ \ \ \ \ \ \ \ \ \ \ V \ \ \ \ \ \ . . \ . . \ i \ ^ \ \ \ \ \ \ \ \ \ A _f_ SLWIN2, 1^=12, N.t =1 - t - S L W I N ' \ r 1 2 - O _ ^ S L W I N 2 , 1^=11, N i | a=2 _<>_ SLWIN, N B D =11 ,N i t a =2 _Q_ SLWIN2, 1^=10, N. L E=4 _0_ SLWIN, l^D=10, N, T E=4 ,. SLWIN2, N = 9 , N, =6 —*— bd iter v SLWIN, N d , N., =6 — X - Dd iter \ \ \ \ *> 5 10 15 20 25 30 35 Number of users K F i g u r e 4.52 Average PSNR for Nj = 20, Nw = 1, B^ax = 64k, for SLWIN and SLWIN2, in the IS-2000 downlink case, for the Foreman bitstream. IS-95B Reverse Link Fig . 4.57 plots the average P S N R as a function of the number of users K, for A j = 132, Nw = 1 and B^^ = 64k and different Nd's, for the two algorithms. The difference in performance between SLWIN and SLWIN2 is now smaller than for the previous case, as was explained in Section 4.3.5.2, and also due to the fact that less smoothing needs to be done since the source rate is less bursty (only one I-frame is present). It is seen again that higher delays cause the P S N R to degrade more gracefully, but at the expense of a latency penalty. As illustrated in Figs. 4.58 and 4.59, for S L W I N and SLWIN2 respectively, higher A w ' s weren't found to improve much the P S N R . 305 Figure 4.53 Average P S N R for A// = 20, Nw = 1, B „ a i = 64k, 1 turbo iteration, and various Nbd's, for SLWIN2 i n the IS-2000 downlink case, for the Foreman bitstream. Figure 4.54 Average P S N R for JVj = 20, NM = 9, NW = 1, B^AX = 64k, 6 turbo iterations, and various numbers of diversity branches L R , for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream. 306 5 10 15 20 25 30 35 Number of users K Figure 4.55 Average P S N R for Nr = 20, Nbd = 9, Nw = 1, B^ax = 64k, 6 turbo iterations, and various values of the mobile speed v, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream. 25 10 5 1 i 1 - t - R=1/4 - * - R=1/3 l i t — " \ ~~ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N S _ l I I I I I 10 15 20 25 30 35 Number of users K Figure 4.56 Average P S N R for JV 7 = 20, Nbd = 9, Nw = 1, B m a x = 64k, 6 turbo iterations, and different code rates R, for SLWIN2 in the IS-2000 downlink case, for the Foreman bitstream. 307 Fig. 4.60 plots the average P S N R for a higher number of I-frames, i.e. A 7 / = 20, and Nw = 10. A much higher Nd is seen to be needed to maintain an acceptable decoded video quality at high interference levels. The effect of the number of diversity branches Lr on the P S N R is illustrated in Fig. 4.61, with Nr = 132, Nbd = 9, Nw = 1, Bdmax = 64k, and SLWIN2. As Lr is increased from 1 to 2, a clear improvement of the P S N R is observed. However, the improvements are much smaller for higher Lr's. This is in line with the constatations made in Section 3.2.3 (c.f. in particular Figs. 3.10, 3.11), i.e. for multicode D S / C D M A systems using noncoherent M-ary modulation and E G C , having more than two diversity branches doesn't bring a much bigger improvement due to the combining loss. Fig . 4.62 shows the effect of the mobile speed on the P S N R : as in F ig . 4.55, for all other parameters being constant, a lower Doppler shift due to a smaller v leads to burstier error patterns and a lower average P S N R . 25 5 NX V \ \ \\ \\ \ \ \ \ \ \ V V \\ \ \ A \ \ \ * \ SLWIN2, N=6 0 SLWIN2, N=9 D SLWIN2, N=120 SLWIN, Nd=6 SLWIN, N=9 „ SLWIN, N =120 V \ \ \ \ •-"to 0 5 10 15 20 25 30 35 Number of users K F i g u r e 4.57 Avera