PERFORMANCE OF WIRELESS COMMUNICATION SYSTEMS IN ULTRA-WIDEBAND INTERFERENCE AND NON-GAUSSIAN NOISE by AMIR MASOUD NASRI-NASRABADI M.Sc., Tehran University, 2004 B.Sc., Sharif University of Technology, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2008 © Amir Masoud Nasri—Nasrabadi, 2008 Abstract The growing demand for high speed wireless applications and the scarcity of the spectral resources has necessitated the development of new concepts that enable more efficient uti lization of the frequency spectrum. Ultra—wideband (UWB) is an emerging technology that is capable of utilizing the spectral resources more effectively by sharing the spectrum with other applications. This spectrum sharing however can potentially result in harm ful interference from UWB systems to devices co—existing in the same frequency band. Therefore, in order to guarantee peaceful co—existence, the effects of UWB interference on co—existing systems have to be carefully analyzed. Towards this goal, in this thesis, we first study the effects of multi—band (MB) orthog onal frequency—division multiplexing (OFDM) UWB interference on a generic uncoded narrowband (NB) system. For this purpose, we develop analytical expressions for the am plitude probability distribution (APD) and the bit error rate (BER) performance of the NB system in the presence of MB—OFDM interference. We use the obtained results to assess the accuracy of the Gaussian approximation (GA) for MB—OFDM UWB interference. We show that for most channel models and signal bandwidths the GA is unable to accurately predict the NB system performance and the exact BER analysis has to be used to obtain meaningful results. We also analyze the effects of UWB interference in a more general framework that allows us to study the impact of general types of non—Gaussian noise and interference on generic uncoded victim systems. Specifically, we present a unified asymptotic symbol error 11 Abstract rate (SER) analysis of linearly modulated systems impaired by fading and generic non— Gaussian noise and interference. Our analysis also encompasses diversity reception with equal gain and selection combining and is extended to binary orthogonal modulation. The obtained asymptotic results show that for high signal—to—noise ratios (SNRs) the SER of the victim system depends on the moments of the non—Gaussian noise and interference. Furthermore, we study the impact of different types of UWB interference on victim sys tems that employ a combination of bit—interleaved coded modulation (BICM) and OFDM. For the UWB interferer we consider MB—OFDM, direct—sequence UWB (DS—UWB), and impulse radio UWB (IR—UWB) interference formats following recent standards or standard proposals developed by the IEEE or the European Computer Manufacturers Association (ECMA). Besides the exact analysis we calculate the BER of the BICM—OFDM system for the case when the UWB interference is modeled as additional Gaussian noise. Our results show that while the GA is very accurate for DS—UWB interference, it may severely over- or underestimate the true BER for MB—OFDM and IR—UWB interference. Finally, we analyze the performance of BICM—based systems in the presence of generic non—Gaussian noise and interference. In particular, we present an asymptotically tight upper bound for the BER and a closed—form expression for the asymptotic performance of receive diversity BICM single—carrier and BICM—OFDM victim systems.. Our analysis shows that, if the popular Euclidean distance metric is used for Viterbi decoding, BICM— based systems optimized for additive white Gaussian noise (AWGN) are also optimum for any other type of noise and interference with finite moments. 111 Table of Contents Abstract ii Table of Contents iv List of Tables x List of Figures xii List of Abbreviations xviii Notation xxii Acknowledgments xxiv 1 Introduction 1 1.1 UWB Technology 3 1.1.1 Historical Development of UWB 4 1.1.2 Advantages of UWB 5 1.2 UWB Standards 6 1.3 UWB Co—existence 7 1.3.1 UWB Interference Analysis 9 1.3.2 Impact of a Generic Non—Gaussian Impairment 9 1.4 Motivation and Background 10 iv Table of Contents 1.5 Contributions and Organization 12 2 Performance of Generic Uncoded NB Systems in MB—OFDM Interfer ence 16 2.1 Introduction 16 2.2 System Model 18 2.2.1 Signal and Channel Model 18 2.2.2 MB—OFDM UWB Signal Model 20 2.2.3 NB Receiver Processing 22 2.3 MB—OFDM UWB Interference Analysis 23 2.3.1 Exact Moment Generating Function (MGF) of i[k] 24 2.3.2 Approximate Models for MB—OFDM UWB Interference 28 2.3.3 Amplitude Probability Distribution (APD) 30 2.4 BER of BPSK NB Signals 31 2.4.1 Preliminaries 32 2.4.2 BER in Non—Fading Channel 33 2.4.3 BER in General Fading Channels 34 2.4.4 BER in Rayleigh Fading 36 2.5 Results and Discussions 37 2.5.1 APD Plot 38 2.5.2 BER in Non—Fading Channel 38 2.5.3 BER in Fading Channel 43 2.5.4 Impact of Different UWB Channel Models 44 2.6 Conclusions 45 3 Performance of Generic Uncoded Systems in Non—Gaussian Noise . . 47 3.1 Introduction 47 v Table of Contents 3.2 Preliminaries 49 3.2.1 Some Definitions and Notations 49 3.2.2 Signal Model 50 3.2.3 Admissible Types of Noise and Examples 51 3.2.4 Mellin Transform of Composite Noise 55 3.3 Single—Branch Reception 56 3.3.1 Basic Error Probability Result 57 3.3.2 Exact SER Expression for M—PAM 59 3.3.3 Asymptotic SER of Linear Modulations 59 3.3.4 Asymptotic SER of BOM 62 3.4 Diversity Combining 63 3.4.1 Equal Gain Combining (EGC) 64 3.4.2 Selection Combining (SLC) 66 3.4.3 Comparison of EGC and SLC 67 3.5 Numerical Results and Discussions 68 3.5.1 Single-Branch Reception 69 3.5.2 Diversity Combining 71 3.6 Conclusions 73 4 Performance of Generic BICM—OFDM Systems in UWB Interference 75 4.1 Introduction 75 4.2 System Model 77 4.2.1 BICM-OFDM System 77 4.2.2 Channel Models 79 4.2.3 UWB Signal Model 81 4.3 MGF of UWB Interference Signal 85 vi Table of Contents 4.3.1 MGF of MB—OFDM Interference 85 4.3.2 MGF of DS—UWB Interference 86 4.3.3 MGF of IR—UWB Interference 87 4.4 BER of BICM-OFDM Systems 88 4.4.1 Conditional MGF of Metric Difference 88 4.4.2 Union Bound for BER 89 4.4.3 Gaussian Approximation 92 4.5 Results and Discussion 94 4.5.1 System Parameters 94 4.5.2 Verification of Theoretical Results 95 4.5.3 Effect of System Parameters on Performance 96 4.5.4 Influence of Distance Between Interferer and Victim Receiver . 104 4.6 Conclusions 107 5 Performance of Generic BICM—Based Systems in Non—Gaussian Noise 108 5.1 Introduction 108 5.2 System Model 110 5.2.1 System Model 110 5.2.2 Fading and Noise Model 112 5.3 Upper Bound for BER 113 5.3.1 MGF of Metric Difference 114 5.3.2 Upper Bound 115 5.4 Asymptotic Analysis 116 5.4.1 Laurent Series Expansion of F(sln) 117 5.4.2 Asymptotic BER 118 5.4.3 Diversity Gain, Coding Gain, and Design Guidelines 120 vii Bibliography Appendices Table of Contents 145 A Periodicity of ‘1’iITk(s, Tk) B Asymptotic BER in Rayleigh Fading 152 153 C The Mellin Transform C.l Definition and Existence C.2 Basic Properties C.3 Product of Two Independent RVs . C.4 Sum of Two Independent RVs 155 155 155 156 157 D Spatially Correlated Fading Channels . 159 5.4.4 Uncoded Transmission 5.5 Calculation of the Noise Moments and MGFs 5.5.1 Spatially I.I.D. Noise 5.5.2 Spatially Dependent Noise 5.5.3 Monte—Carlo Method 5.6 Numerical and Simulation Results 5.7 Conclusions 121 122 122 125 129 129 138 6 Conclusions and Future Work 6.1 Research Contributions . 6.2 General Conclusions 6.3 Suggestions for Future Work 139 139 141 142 viii Table of Contents E Related Publications 161 ix List of Tables 2J MB—OFDM parameters[1]. 23 3.1 Pdf pa(a) of fading amplitude a 0 and corresponding series parameters Pk, , and S for the series expansion pa(a) = Pk(a2/7)’. y = 4f’{a2} 51 3.2 Pdf Pn(fl) and Mellin transform Ms(s) = M{p(n)} for different types of noise. Noise variance o = 1 in all cases. Generalized Gaussian noise: ri(C) = [F(1/C)/F(3/C)]c’/2BPSK interference with fixed channel phase (CP): S is the set of all 2’ possible sums of the ±dk, 1 k < ]• 8 contains all positive elements of S 53 3.3 Parameters ,8 and d, for M—ary modulation schemes X E {PAM, PSK, QAM,BOM} 58 4.1 Parameters of considered UWB formats. These values were taken from the respective standards/standard proposals [1, 2, 3]. We note that for IR—UWB there are also optional channels with B > 500 MHz 82 4.2 Conditional MGFs [k](sIg[k],r) for considered UWB formats. For MB— OFDM n1, n, tc1, and are chosen such that /3[frt, k] 0 for n [ni, flu] or/and [‘‘ is]. For DS—UWB and IR—UWB the limits Icj, are chosen to ensure 3j [it, k] 0 and /3 [it, k, [it]J 0 for ic [it1, ‘cJ, respectively. The integrals required for calculation of /3 [ic, kl and i3 [it, k, ‘O[ic]] for DS—UWB and IR—UWB can be easily evaluated numerically, respectively 86 x List of Tables 5.1 Pdfpa(a) of fading amplitude a for popular fading models and corresponding values for a. and ad. We have omitted the subscripts k and 1 for convenience. The parameters for Rayleigh (Chh), Ricean (Jih, Chh), and Nakagami—m (m, Caa) fading are defined in Appendix D. The parameters for Nakagami—q (q, b) and Weibull (c) fading are defined as in [4] 113 5.2 MGF n(s) and scalar moments m(i) of types of noise considered in Section 5.5. All variables in this table are defined in Section 5.5. (SC) and (OFDM) means that the type of noise is relevant for BICM—SC and BICM—OFDM, respectively 123 5.3 Vector moments m(i) of types of noise considered in Section 5.5. All vari ables in this table are defined in Section 5.5. (SC) and (OFDM) means that the type of noise is relevant for BICM—SC and BICM—OFDM, respectively. 126 xi List of Figures 1.1 FCC spectral mask for indoor operation of UWB systems 8 2.1 NB signal and MB—OFDM signal in a) frequency domain and b) time domain for NB 3. We note that for the time domain representation of the MB— OFDM signal only the contribution of the relevant first frequency band is depicted 22 2.2 APD plot for different normalized NB signal bandwidths B8//f. o = 1, fo = 10f, and NB 3. Exact APD [Eq. (2.28)], GA [Eq. (2.25)], and IGA [Eq. (2.26)] 37 2.3 BER vs. B8/f for different NB pulse shapes and different SIRs. 10 1og0( SNR) = 10 dB, fo = 10f, and NB = 3 39 2.4 BER vs. fo/f for different SIRs. Interference limited case (SNR —÷ cc), B8 = and NB 3. Markers: Simulation results. Solid lines: Numerical results [Eq. (2.35)] 40 2.5 BER vs. B3/Lf for different fo/f and different SIRs. 101og0(SNR) = 10 dB and NB = 3. Exact BER [Eq. (2.35)], GA [Eq. (2.33)], and IGA [Eq. (2.34)] 41 2.6 BER vs. B8/Zf for different NB. 101og0(SNR) = 10 dB, lOlog10(SIR) = 15 dB, and fo 10zf. Exact BER [Eq. (2.35)], GA [Eq. (2.33)], and IGA [Eq. (2.34)] 42 xii List of Figures 2.7 BER vs. B3/f for different fading parameters m. 10 1og0(SNR) 10 dB, 101og0(SIR) = 10 dB, fo 10f, and NB = 3. Exact BER [Eq. (2.39)], GA [Eq. (2.36)], and IGA [Eq. (2.37)] 43 2.8 BER vs. B3/L\f for different interference channels. 10 1og0(SNR) = 10 dB, 10 1og0(SIR) = 15 dB, and fo = 10Lf3. Exact BER for flat channel [Eq. (2.35)] and simulation results for one realization of CM1 and CM4 UWB channel models 45 3.1 SER vs. SNR for 8—PSK over a Nakagami—m fading channel with s—mixture noise (e 0.25, i’t = 10). Markers: Simulated SER. Solid lines: Asymptotic SER [Eq. (3.18)] 69 3.2 SER vs. SNR for BPSK over a Nakagami—m fading channel with m = 2 and different types of noise discussed in Sections 3.2.3, 3.2.4. s—mixture noise: Example El) in Section 3.2.3. M—PSK interference with random CP: Exam ple E3) in Section 3.2.3. Rayleigh faded BPSK interference: Example E5) in Section 3.2.4. BPSK interference with fixed CP: Example E2) in Section 3.2.3. Markers: Simulated SEa. Solid lines: Asymptotic SER [Eq. (3.18)]. 70 3.3 SER vs. SNR for 16—QAM with bandwidth B8 over a Nakagami—m fading channel with m = 2 and UWB interference. Markers: Simulated SER. Solid lines: Asymptotic SER for MB—OFDM interference [Eq. (3.18)]. Dashed lines: Asymptotic SER for DS—UWB interference [Eq. (3.18)] 71 xiii List of Figures 3.4 SER vs. SNR per branch for BPSK over a Ricean fading channel with Ricean factor K 2, L = 2 diversity branches, and Ricean faded M—PSK interfer ence. The interference channel has Ricean factors of K1 = 0 and K1 = 10, respectively. All diversity paths have the same average SNR. EGC and SLC are considered. Markers: Simulated SER. Solid lines: Asymptotic SER for EGC [Eq. (3.29)]. Dashed lines: Asymptotic SER for SLC [Eq. (3.33)]. . 72 3.5 SER vs. SNR per branch for 8—PSK over a Rayleigh fading channel with EGC. All diversity paths have the same average SNR. Markers: Simulated SER. Solid lines: Asymptotic SER for Rayleigh faded BPSK interference (two interferers) and Gaussian noise [Eq. (3.29)]. Dashed lines: Asymptotic SER for Gaussian noise [Eq. (3.29)] 73 4.1 Block diagram of considered system model including the BICM—OFDM transmitter (Tx), the BICM—OFDM receiver (Rx), and I UWB interfer ers 77 4.2 BER vs. SIR for IR-UWB (Nb = 32, L = 16), BPSK DS-UWB (L = 24), and MB—OFDM. Case C3 for channel, I = 1, SNR — co, 4—PSK, and R = 1/2 95 4.3 BER vs. for IR—UWB with different Nb and L. Case C3 for channel, I = 1, SNR = 15 dB, SIR = 10 dB, 4—PSK, and R = 1/2 97 4.4 BER vs. Z.f8 for BPSK DS-UWB (L = 24), 4-BOK DS-UWB (L = 24), and MB—OFDM. Case C3 for channel, I = 1, SNR = 15 dB, SIR = 10 dB, 4—PSK, and R = 1/2 98 xiv List of Figures 4.5 BER vs. LS.f3 for IR-UWB, 4-BOK DS-UWB, and MB-OFDM. Code rates of R = 1/2, 2/3, 3/4, 5/6, 7/8, and 1 (uncoded) are considered. Case C4 for channel, IEEE 802.16 puncturing patterns, I = 1, SNR = 10 dB, SIR = 8 dB, and 4—PSK 99 4.6 BER vs. for IR-UWB, 4-BOK DS-UWB, and MB-OFDM. Code rates of R = 1/2 (SIR 8.6 dB), R = 2/3 (SIR = 11.2 dB), R = 3/4 (SIR = 13.7 dB), and R = 5/6 (SIR = 16.8 dB) are considered. Case C3 for channel, IEEE 802.16 puncturing patterns, I = 1, SNR —* oc, and 4—PSK 100 4.7 BER vs. Zf3 for IR-UWB (Nb = 32, L = 16), 4-BOK DS-UWB (L = 24), and MB—OFDM. Results for Cl and C3 are compared. I = 1, SNR 15 dB, SIR = 10 dB, 4—PSK, and R = 1/2 101 4.8 BER vs. /f3 for IR-UWB, 4-BOK DS-UWB, and MB-OFDM. I = 1, 2, 5, 10, 100 i.i.d. interferers are considered. Case C3 for channel, SNR —* oc, SIR = 8 dB, 4—PSK, and R = 1/2. The markers (“V”) indicate simulation results for MB—OFDM and I = 2 102 4.9 BERvs. dfora) = 312.5kHzandb) f8 = 4.125MHz. IR—UWB (Nb = 32, L = 16), 4—BOK DS—UWB (L 24), and MB—OFDM interference. Case C3 for channel, NLOS UWB path loss model from [5], I = 1, SNR = 10 dB, 4—PSK, and R = 1/2 105 4.10 Data rate of IEEE 802.lla BICM—OFDM sysem vs. d for IR—UWB, BPSK DS—UWB, and MB—OFDM. Case C3 for channel, LOS UWB path loss model from [5], IEEE 802.lla puncturing patterns, I = 1, f3 = 312.5 kHz, and SNR = 25 dB 106 xv List of Figures 5.1 BER of BICM—SC and BICM—OFDM impaired by GMN (6—mixture noise, 6 0.1, ic = 100) and NBI, respectively, vs. SNR y. R = 3/4, Rayleigh fad ing, 4—PSK, and NR = 1. BICM—SC: Flat time—selective fading, N = 972, and BfT = 0.007. BICM—OFDM (N = 64): Frequency—selective Rayleigh fading with L = 10 and B equal power, sub—carrier centered NBI signals with I, 1, 1 i B, t = 7. BICM—OFDM (N = 128): Frequency— selective Rayleigh fading with L = 20 and B equal power, sub—carrier centered NBI signals with I, = 1, 1 j-t B, ,c = 2. Solid lines with mark ers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)] 131 5.2 BER of 131CM—SC and BICM—OFDM impaired by various types of noise vs. SNR 7. Rayleigh fading, R = 3/4, 4—PSK, and NR = 1. BICM—SC: N = 972 and BfT = 0.007. BICM-OFDM: N 64 and L = 10. GMN I: 6—mixture noise, 6 0.01, ic 100. GMN II: e—mixture noise, 6 = 0.1, it = 100. GMN III: 6—mixture noise, 6 = 0.1, Ic = 10. Asynchronous CCI: Two asynchronous equal power 4—PSK CCI signals, I,, = 1, u E {1, 2}, = 0, 3 t < 10, raised cosine pulses gl,,L(t), t e {1, 2}, with roll—off factor 0.3, r1, = 0.3T, jt E {1, 2}, ic = 2. NBI I: One sub—carrier—centered NBI signal, I = 1,12 = 13 = 14=15 0, ,c= 9. NBIII: 2 equal power, sub—carrier—centered NBI signals, I = 12 = 1, 13 = 14 = 15 = 0, ,c = 14. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)] 132 xvi List of Figures 5.3 BER of BICM—OFDM impaired by NBI (3 equal power, sub—carrier—centered NBI signals, I = 12 = 13 = 1, ic = 10) vs. SNR 7. Rayleigh fading, L = 20, 4—PSK, N = 128, B = 3, and NR = 1. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)] 133 5.4 BER of BICM—SC impaired by GMN (e—mixture noise, = 0.25, ic = 10) and AWGN, respectively, vs. SNR ‘y. Ideal i.i.d. fading, R = 7/8, 16—QAM, and NR = 1 134 5.5 BER of uncoded SC transmission impaired by SD- and SI—GMN (E—mixture noise, € = 0.1, ,c = 10), respectively, vs. SNR 7. NR = 2, Nakagami—m fading spatial correlation Pa = 0.9, and 4—PSK 135 5.6 BER of BICM—SC impaired by AWGN/ACGN vs. SNR 7. Spatially i.i.d./spatially correlated, temporally i.i.d. Rayleigh fading, R = 7/8, 4—PSK, and NR = 2. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)] 136 5.7 BER of BICM—OFDM system with sub—carrier spacing L\f3 impaired by IR— UWB [3] (Nb = 8 bursts per symbol and L chips per burst) and MB—OFDM UWB [1], respectively, vs. SNR . Ideal i.i.d. Rayleigh fading, R = 5/6, 4—PSK, and NR = 1. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. For comparison the bound and the asymptotic BER for AWGN are also shown 137 xvii List of Abbreviations 4—BOK 4—ary Bi—Orthogonal Keying ACGN Additive Correlated Gaussian Noise APD Amplitude Probability Distribution AWGN Additive White Gaussian Noise BER Bit Error Rate BICM Bit—Interleaved Coded Modulation BOM Binary Orthogonal Modulation BPPM Binary Pulse Position Modulation BPSK Binary Phase Shift Keying CC Convolutional Code CCI Co—Channel Interference CDMA Code—Division Multiple Access CM Channel Model CP Channel Phase DAB Digital Audio Broadcasting DFT Discrete Fourier Transform DS Direct—Sequence DS—UWB Direct—Sequence Ultra—Wideband DTV Digital Television DVB Digital Video Broadcasting xviii List of Abbreviations ECMA European Computer Manufacturers Association ED Euclidean EDGE Enhanced Data Rates for GSM Evolution EGC Equal Gain Combining EIRP Effective Isotropic Radiated Power FCC Federal Communications Commission FEC Forward Error Correction FF Frequency—Flat FWA Fixed Wireless Access GA Gaussian Approximation GMN Gaussian Mixture Noise GPS Global Positioning System GSM Global System for Mobile Communication IEEE Institute of Electrical and Electronic Engineers IFFT Inverse Fast Fourier Transform IGA Impulsive Gaussian Approximation i.i.d. Independent, Identically Distributed i.n .d. Independent, Non—Identically Distributed JR Impulse Radio ISM Industrial, Scientific and Medical LHS Left Hand Side LOS Line—of—Sight MB Multi—Band MGF Moment Generating Function ML Maximum—Likelihood xix List of Abbreviations M—PAM M—ary Pulse Amplitude Modulation M—PSK M—ary Phase Shift Keying M—QAM M—ary Quadrature Amplitude Modulation MRC Maximum—Ratio Combining NB Narrowband NBI Narrowband Interference NLOS Non—Line—of—Sight NTIA National Telecommunications and Information Administration OFDM Orthogonal Frequency Division Multiplexing PAM Pulse Amplitude Modulation pdf Probability Density Function PEP Pairwise Error Probability PSD Power Spectral Density QAM Quadrature Amplitude Modulation QPSK Quaternary Phase Shift Keying REC Rectangular RHS Right Hand Side RV Random Variable SC Single—Carrier SD Spatially Dependent SER Symbol Error Rate SFS Severely Frequency—Selective SI Spatially Independent SIR Signal—to—Interference Ratio SLC Selection Combining xx List of Abbreviations SNR Signal—to--Noise Ratio SRN Square—Root Nyquist SRRC Square—Root Raised Cosine TFC Time—Frequency Code TG Task Group UMTS Universal Mobile Telecommunications System UNIT Unlicensed National Information Infrastructure UWB Ultra—Wideband WB Wideband WIMAX Worldwide Interoperability for Microwave Access WLAN Wireless Local Area Network WMAN Wireless Metropolitan Area Network WPAN Wireless Personal Area Network xxi Notation Throughout this thesis, bold upper case and lower case letters denote matrices and vectors, respectively. The remaining notation and operators used in this thesis are listed as follows: (.)* Complex conjugation [.]T Transposition [.]H Hermitian transposition det(.) Matrix determinant I Absolute value of a complex number I II L2—norm of a vector Re{ } Real part of a complex number Im{.} Imaginary part of a complex number Statistical expectation with respect to x Pr{.} Probability of an event I(.) Zeroth order modified Bessel function of the first kind 00 —tx—iF(.) Gamma function, F(x) = e t dt Q(.) Gaussian Q—function, Q(x) = f°e2/2 dt Laplace transform of a function, 1{p(x)} j°p(x)e8x dx o(.) Order of a function, f(x) is o(g(x)) if limx_o f(x)/g(x) 0 diag(x) A matrix with the elements of vector x on the main diagonal Convolution operator X x X identity matrix xxii Notation All—zero column vector of length X Pf(1 u2) Gaussian RV with mean [I and variance a2 u v Asymptotic equivalence of u and v xxiii Acknowledgments My greatest debt of gratitude is to Professor Robert Schober for his invaluable guidance, advice and technical insight from the very early stages of this research. He has provided me with continued encouragement and support in various ways during the course of my Ph.D. I am especially grateful that he was always accessible and willing to help his students with their research. In addition, his perpetual energy, drive, and determination have been a great source of motivation for all his students, including me. I will continue to be inspired by his enthusiasm, clarity of thought, and professional integrity. I am also especially thankful to Professor Lutz Lampe for his helpful and constructive comments he provided me on some of the work we have done together. I also gratefully acknowledge the encouragement he has always offered me. I would also like to thank the other members of my doctoral committee, the University examiners, and the External examiner Professor Marco Chiani for their time and effort and their valuable feedback and suggestions. I am also thankful to all my colleagues at the Department of Electrical and Computer Engineering at the University of British Colombia, for creating a congenial and stimulating environment for research. Lastly, and most importantly, I wish to thank my family, my parents and my brothers for their endless love, support, and encouragement over all these years. To them I dedicate this thesis. xxiv Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Bell Canada, and the University of British Columbia Graduate Fel lowship. xxv Chapter 1 Introduction The demand for high speed wireless communications has dramatically increased in recent years due to the rapid growth of new applications such as wireless access to the Internet and multi—media services. The total frequency spectrum that is made available for wireless communications, however, cannot keep pace with these increasing demands. As a result, the spectrum has become increasingly scarce, facing the wireless industry with a major challenge to provide sufficient spectral resources for emerging applications. On the other hand, the primary method of spectrum allocation in the United States and most other countries has been based on granting licenses that offer exclusive access to the frequency spectrum. This inflexible spectrum allocation is very inefficient as it results in idle portions of the spectrum that cannot be reused by other applications. In fact, recent studies by the Spectrum Policy Task Force of the US Federal Communications Commission (FCC) [6] have shown vast temporal and geographic variations in spectrum utilization ranging from 15% to 85% in frequency bands below 3 GHz. Furthermore, measurements performed in [7] have indicated that in the frequency range above 3 GHz the bands are even more poorly utilized. The limited spectrum together with inefficient spectrum usage, necessitate the devel opment of new solutions that exploit the scarce spectrum more effectively. These new solutions require a paradigm shift from fixed and licensed spectrum allocation to unli censed spectrum allocation methods that enable more efficient spectrum utilization by allowing different users to share the spectral resources. This unlicensed spectrum allo 1 Chapter 1. Introduction cation has been successfully used in a number of previous FCC regulations such as the 2.4 GHz Industrial, Scientific and Medical (ISM) band, the 5—6 GHz Unlicensed National Information Infrastructure (UNII) band, and the 57—64 GHz microwave band. Motivated by the success of previous unlicensed spectrum allocations, the wireless in dustry has actively petitioned the FCC to grant more unlicensed bands. These efforts have resulted in a landmark rule—making by the FCC in 2002, in which the FCC released new regulations allowing the unlicensed operation of wireless devices over an enormous band width of 3.1—10.6 GHz. Since then, a growing interest has arisen in academia and industry in developing new techniques that facilitate spectrum sharing among different applications in the newly allocated unlicensed band. In addition to efficient utilization of the spectrum, the developed techniques must guar antee the peaceful co—existence among the unlicensed spectrum users. Therefore these techniques have to ensure that the interference caused as a result of spectrum sharing remains limited. Two basic co—existence approaches proposed in this regard are ultra— wideband (UWB) technology [8] and cognitive radio (CR) [9]. UWB technology is an underlay approach that requires the spectrum users to operate over very large bandwidths but at very low transmit power levels. On the other hand, CR is an overlay approach that relies on spectrum sensing and adaptive spectrum allocation to avoid interference to higher priority users. More information about the CR approach can be found in [10]. This thesis mainly focuses on the UWB approach and in particular, on the co—existence of UWB and other wireless communication systems. Specifically, in this thesis, we provide a comprehensive analysis of the effects of UWB interference on the performance of co—existing wireless systems. We explain the UWB co—existence issue and the approach pursued in this thesis towards UWB interference analysis in Section 1.3 in some detail. However, before proceeding, to set the stage for the rest of the thesis we discuss the UWB technology 2 Chapter 1. Introduction as well as some previous and recent developments in the UWB field. More explicitly, in Section 1.1, we introduce FCC’s definition of UWB, give a brief overview of the historical development of UWB, and explain the main advantages of UWB techonology. We elaborate on the current status of wireless standards developed based on UWB technology in Section 1.2. After discussing the UWB co—existence issue in Section 1.3, we review relevant work previously done in the literature and highlight the contributions of this thesis in Sections 1.4 and 1.5, respectively. 1.1 UWB Technology In FCC’s UWB regulations released in 2002, UWE is defined as any wireless transmission system that occupies a bandwidth in excess of 500 MHz or has a fractional bandwidth of more than 209o [11]. The fractional bandwidth is defined as Bf = fH— fL . 100% (1.1) where fL and fH denote the upper and lower —10 dB emission point, respectively, and f is the center frequency given as f = (fH + fL)/2. In contrast, wideband (WB) signals are defined as signal with fractional bandwidth between 1% and 20%, while narrowband (NB) signals have fractional bandwidths of less than 1% [12]. An example of a wireless standard using NB technology is the IEEE 802.llb Wireless Local Area Network (WLAN) [13] which employs 22 MHz of bandwidth and a center frequency in the 2.4 GHz ISM band resulting in a fractional bandwidth of Bf 0.9%. FCC’s 2002 regulation marked the emergence of modern UWB technology. However, UWB is not a new technology and has a history as long as wireless communications itself. The concept and advantages of modern UWB communications are best understood by 3 Chapter 1. Introduction reviewing the historical development of UWB. 1.1.1 Historical Development of UWB The origins of UWB date back to the late 1800’s when Marconi used spark—gap signals with enormous bandwidths for sending and receiving Morse codes. The spark—gap signalling relied on transmitting noisy signals which were then detected by simple amplitude detector receivers [14]. The spark—gap transceivers were in fact quite inefficient at the time as they were unable to take advantage of the wide signal bandwidth to effectively capture the received energy. This inefficiency in signal reception resulted in low signal—to—noise ratios (SNRs) at the receiver that were typically compensated for by increasing the transmit signal power. The high transmit power levels employed in these early wireless devices together with their large operational bandwidths raised many concerns regarding the spectrum sharing and the interference inflicted upon other communication systems. These issues gradually drove wireless communications to narrower signal bandwidths. In fact, in 1912 the Radio Act was released in the United States to resolve the interference issues by requiring the wireless signals to occupy bandwidths as narrow as possible [14]. This act marked the start of a movement towards NB wireless systems. Although the march towards NB wireless systems lasted about a century, the significant benefits of WB communications were acknowledged in several instances. In 1948 Claude Shannon provided revolutionary ideas that changed the traditional perspectives in wireless communications [151. In particular, he showed that if proper transmission and detection techniques are employed, the transmission rate can be drastically increased by spreading the signal energy over a large bandwidth. Assuming a fixed transmit power, increasing the signal bandwidth also reduces the power spectral density (PSD), which in turn reduces the amount of interference caused to other communication systems. Shannon’s ideas led to 4 Chapter 1. Introduction the development of WB spread—spectrum techniques such as direct—sequence (DS) code— division multiple access (CDMA) which were later used in a variety of wireless systems. Another turning point in UWB history came in the 1960s when the study of impulse measurements showed the possibility of using short time—domain pulses with huge band width occupancy for radar and positioning applications. Ross and Robbin published several patents during the years 1973—1987 which pioneered the use of UWB signals in a number of application areas including communications and radar systems [16]. These patents along with several other patents and papers, set up the foundation for modern UWB communi cations. Although the attractive features of UWB were acknowledged by many researchers, the application of UWB technology was mainly limited to radar systems before 2001 [11]. With scarce bandwidth resources, UWB bandwidth requirements seemed to be too high to be fulfilled. But things changed drastically in 2002 as a result of FCC’s new regulations allowing the unlicensed operation of UWB radios over a huge bandwidth of more than 7 GHz. This move renewed the interest in UWB in industry and academia and initiated a productive activity to explore new aspects of this technology, resulting in numerous papers, patents, and standards on UWB related subjects. 1.1.2 Advantages of UWB The large bandwidth used in UWB systems results in several interesting advantages for these systems. In particular, in multi—path channels with high time—dispersions, large system bandwidths enable the receiver to resolve the multi—path arrivals. This reduces the possibility of a destructive combination of waveform amplitudes and therefore leads to less severe fading in the resolved paths and more robustness against fading [17]. Furthermore, increasing the bandwidth enables the design of communication systems 5 Chapter 1. Introduction in the “power—limited regime” where the Shannon capacity increases linearly with the transmit power. This is in contrast to the commonly pursued “bandwidth—limited regime” method in wireless communication where the capacity increases only logarithmically with the transmit power. Therefore, high data rates can be achieved by WB systems without resorting to increased transmit power or exploiting higher order modulations and advanced error control coding. The large bandwidths used by UWB systems translates to very narrow pulses in the time—domain. As a result, UWB radios are potentially able to offer timing precision much better than the global positioning system (GPS) and other radio systems. High timing precision along with good material penetration properties, makes UWB systems excellent candidates for short range radar applications [17, 18]. Finally, the large bandwidth and the low transmit power employed in UWB systems imply that UWB signals have a small PSD and therefore, at least in theory, can appear as low power noise to other communication systems with overlapping spectrum. Thus, UWB systems can underlay other existing licensed and unlicensed systems, resulting in more efficient spectrum utilization. As mentioned before, this potential for spectrum sharing and efficient utilization of the spectrum has been a major motivation for the introduction of modern UWB technology. 1.2 UWB Standards Following FCC’s regulations allowing the commercialization of UWB systems and mo tivated by the highly desirable characteristics of this technology, an effort was initiated by the IEEE towards the standardization of UWB systems. Due to the low power and therefore short range nature of UWB technology, the IEEE 802.15 wireless personal area networks (WPANs) working group was chosen as the main body for developing UWB— 6 Chapter 1. Introduction based standards. To address the demands for both high rate and low rate applications, two task groups (TGs), namely TG3a and TG4a, were formed under IEEE 802.15, which were responsible for developing standards for short—range, high data—rate applications and low—rate, positioning and localization devices, respectively. Both task groups TG3a and TG4a received several proposals from academia as well as industry. A down—selection procedure was then used to choose the proposal most suitable for the target applications considered by each group. In TG3a, two proposals, namely multi—band (MB) orthogonal frequency—division multiplexing (OFDM) UWB [19] and DS—UWB [2], were selected in the down—selection process. These proposals are based on OFDM and DS—CDMA transmission technologies which have both been traditionally employed in NB and WB wireless communication systems. The battle between the two final contender proposals however led to an impasse as neither proposal could achieve the majority vote in the down—selection procedure. As a result the task group abandoned the standardization process, leaving both options available for future UWB systems. The MB—OFDM UWB proposal was later adopted by the European Computer Manufacturers Association (ECMA) [1]. In task group TG4a, however, a single proposal was chosen as the final standard in the down—selection procedure. This proposal is based on impulse ra dio (IR) technology [3] and has characteristics similar to traditional IR—UWB techniques [20, 21]. 1.3 UWB Co—existence While UWB technology has generated a great deal of interest, it has also created a great deal of controversy. As mentioned earlier, UWB systems are designed to share the spectral resources with other services that operate in the same frequency band. Although this sharing of the spectrum provides for more efficient use of the spectrum, it also results in 7 Chapter 1. Introduction —4O —45 N : : : -50 ___3.l . . 1:99 : : : . . . > . . . . . . C . . . o -60 . 0 : : : : : 0 . . . E : : : : -65 w . cri . : : : : : -70 : D . . . .. :096 1.61 : : : : : —80 I 100 101 Frequency in GHz Figure 1.1: FCC spectral mask for indoor operation of UWB systems [8]. a potential for UWB devices to cause harmful interference to co—existing communication systems. To limit the interference caused by UWB systems to nearby devices, the FCC has specified some limitations on the maximum effective isotropic radiated power (EIRP) at different frequencies, by regulating a spectral mask as a part of its rule making in 2002. This spectrum mask in shown in Fig. 1.1 for the indoor operation of UWB communication systems. Europe and Asia have introduced similar but more conservative spectral masks that allow lower emission levels for UWB devices. The FCC’s restrictions on the UWB transmit power provide some level of protection against UWB interference. However, since the interference power grows exponentially with decreasing distance between the UWB interferer and the victim receiver, depending on the location of the UWB interferer these limitations may not be sufficient to avoid noticeable 8 Chapter 1. Introduction performance degradations in co—existing wireless systems. This issue has raised concerns among many wireless service providers and spectrum license holders, and has left them opposed to the introduction of UWB. Therefore given the controversy surrounding the co—existence of UWB systems and licensed systems, the assessment of interference caused by UWB devices is of great importance to guarantee peaceful co—existence and to gain worldwide acceptance for UWB technology. 1.3.1 UWB Interference Analysis Motivated by the importance of the UWB co—existence issue, in this thesis, we investi gate the effects of UWB interference on co—existing communication systems. As will be explained in more detail in Section 1.4, most existing results on UWB co—existence have been obtained based on the Gaussian approximation (GA), i.e., under the assumption that the UWB interference signal at the victim receiver is Gaussian distributed. Some recent studies [22, 23, 24, 25], however, have shown that this assumption can be severely violated in practical scenarios and therefore can lead to highly inaccurate predictions for the system performance. Since the accurate evaluation of the UWB interference effects is of crucial importance for co—existence analysis, in this thesis, we conduct a complete co—existence analysis by carefully taking into account the non—Gaussian characteristics of UWB interfer ence. In particular, we provide an analytical framework that allows us to accurately study the interference effects of different types of UWB on practically relevant victim systems. 1.3.2 Impact of a Generic Non—Gaussian Impairment We also analyze the effect of UWB interference in a more general framework that allows for the consideration of more general types of non—Gaussian noise and interference. Based on this framework, we investigate the performance degradation of a victim receiver in 9 Chapter 1. Introduction the presence of a generic non—Gaussian impairment. For the non—Gaussian impairment we consider practically relevant examples such as co—channel and adjacent channel interference [26, 27], and impulsive noise [28]. The obtained results are applicable to different types of practically important victim receivers, most commonly used fading models, and practically relevant types of noise. 1.4 Motivation and Background The co—existence analysis of UWB and other communication systems has received con siderable attention recently. As a result, the impact of interference from different types of UWB systems have been investigated in a number of studies using experiments, sim ulations, and analytical methods. In particular, the effect of MB—OFDM interference on an IEEE 802.lla WLAN receiver was evaluated experimentally in [29]. The impact of MB—OFDM interference on a C—band digital television (DTV) receiver was studied via experiments and simulations in [22] and [30], respectively. Extensive simulations were used in [31] to study the interference effects of IR—UWB on the Global System for Mo bile Communication (GSM), the Universal Mobile Telecommunications System (UMTS)/ Wideband CDMA (WCDMA), and GPS receivers. Unfortunately, in the aforementioned studies the influence of the FCC spectral mask has not been taken into account. Further more, while these experimental and simulative studies provide useful results on the system performance, their applicability is limited to a particular victim system. Thus, it is difficult to deduce from these results either qualitative or quantitative performance predictions for other existing or future communication systems. Most analytical studies on the effects of UWB interference in the literature have been conducted based on the GA. The GA implies that the aggregate impairment at the victim receiver, which consists of the UWB interference and the Gaussian distributed background 10 Chapter 1. Introduction noise, is also Gaussian distributed. Therefore the analytical results already available in the literature for Gaussian noise can be employed to analyze the system performance. This assumption has been used in [5, 321 to evaluate the performance of IEEE 802.lla WLAN systems in the presence of UWB interference. An analysis of the performance of a NB system in the presence of a Rayleigh faded UWB interferer has been performed in [33] by modeling the UWB interference signal as a white Gaussian process. The GA has been used in [34] to analyze the performance degradation of UMTS, GPS, and fixed wireless access (FWA) systems in IR—UWB interference. The accuracy of the GA for the UWB interference, however, has been questioned by a number of studies recently which have shown the inaccuracy of this assumption in practical scenarios for different types of UWB technologies. In [22], experimental tests were used to demonstrate that the distribution of the interference caused by MB—OFDM UWB systems can substantially deviate from Gaussian. In [23, 24], it was shown that the interference from MB—OFDM UWB devices can be modeled as impulsive noise if the bandwidth of the victim receiver contains several MB—OFDM sub—carriers. A semi—analytical approach was employed in [25] to evaluate the performance of a NB system in the presence of IR—UWB interference and the inaccuracy of the GA was again pointed out for various NB system parameters. This lack of reliable performance predictions for victim receivers in UWB interference motivates the precise UWB interference analysis performed in Chapters 2 and 4. The performance degradation suffered by victim receivers in the presence of other types of non—Gaussian impairment has also been considered in literature. In particular, the performance of binary phase shift keying (BPSK) and quadrature amplitude modulation (QAM) modulated systems in the presence of impulsive noise has been studied in [35] and [36], respectively. The effects of co—channel interference on the performance of different 11 Chapter 1. Introduction communication systems have been analyzed in e.g. [28, 36, 37]. The methods used for performance analysis in these studies, however, are only applicable to particular noise types and thus cannot be generalized to other types of non—Gaussian noise. This provides enough impetus for evaluating the performance of victim systems impaired by generic non—Gaussian noise in Chapters 3 and 5. Furthermore, it is well—known that asymptotic analysis for high SNRs is a useful tool for studying the performance of communication systems, as it results in closed—form expression for system performance that are easy to evaluate and are therefore suitable for system design. Asymptotic results for system performance are available in the literature for various modulation schemes, types of fading, and diversity combining techniques, cf. e.g. [38]—[42]. All these results, however, have been obtained for AWGN. In Chapters 3 and 5, we show that an asymptotic analysis can also be employed to obtain closed—form expressions for the performance of victim systems impaired by generic non—Gaussian noise. 1.5 Contributions and Organization As explained in Section 1.3, the main goal of this thesis is to investigate the perfor mance degradation of victim systems impaired by UWB interference and/or a generic non—Gaussian impairment. Specifically, this thesis aims at studying the victim system’s performance in the following four scenarios: • A generic uncoded NB systems impaired by MB—OFDM interference (Chapter 2). • A generic uncoded system impaired by generic non—Gaussian noise (Chapter 3). • A generic coded system exploiting the popular combination of bit—interleaved coded modulation (BICM) and OFDM impaired by UWB interference (Chapter 4). • A generic BICM—based system impaired by generic non—Gaussian noise (Chapter 5). 12 Chapter 1. Introduction More specifically, in Chapter 2, we investigate the effect of MB—OFDM UWB interference on a generic uncoded NB system. For this purpose, we first derive the exact moment generating function (MGF) of MB—OFDM UWB interference. Based on this result, we develop analytical expressions for the amplitude probability distribution (APD) and the bit error rate (BER) performance of the NB system. These expressions can be efficiently numerically evaluated and the presented analysis is general enough to encompass non— fading and various fading channels. We show that for NB systems with, respectively, much smaller and much larger bandwidths than the MB—OFDM sub—carrier spacing a GA and an impulsive GA (IGA) of the MB—OFDM UWB interference lead to accurate performance predictions. However, for most channel models and signal bandwidths, the exact BER analysis has to be used to obtain meaningful results. An exception is the Rayleigh fading NB channel where both GA and IGA yield tight approximations of the exact BER regardless of the NB system bandwidth. Having establishing the non—Gaussian behavior of the UWB interference in Chapter 2, in Chapter 3, we turn our attention to more general types of non—Gaussian noise and inter ference and analyze the effects of these impairments on a generic uncoded victim receiver. Specifically, we present a unified asymptotic symbol error rate (SER) analysis of linearly modulated uncoded systems impaired by fading and generic non—Gaussian noise1. The derived asymptotic closed—form results are valid for different types of fading channels and all noise processes with finite moments. Our analysis also encompasses diversity reception with equal gain and selection combining and is extended to binary orthogonal modulation. We show that for high SNRs the SER of the victim system depends on the moments of the non—Gaussian noise. Since the noise moments can be readily obtained for all commonly encountered noise probability density functions (pdfs), the provided SER expressions are ‘Unless stated otherwise, in the rest of this thesis, by “noise” we refer to any additive impairment of the received signal, i.e., our definition of noise also includes what is commonly referred to as “interference”. 13 Chapter 1. Introduction easy and fast to evaluate. Furthermore, we show that the diversity gain only depends on the fading statistic and the number of diversity branches, whereas the combining gain depends on the modulation format, the type of fading, the number of diversity branches, and the type of noise. An exception are systems with a diversity gain of one, since their combining gain and asymptotic SER are independent of the type of noise. However, in general, in a log—log scale for high SNR the SER curves for different types of noise are parallel but not identical and their relative shift depends on the noise moments. We pursue the study of UWB interference effects further in Chapter 4, where we analyze the impact of different types of UWB interference on BICM—OFDM systems. For this purpose, we develop an analytical framework for performance analysis of generic BICM— OFDM systems impaired by UWB interference. MB—OFDM, DS—UWB, and IR—UWB interference formats are considered for the UWB interferer following recent IEEE/ECMA standards or standard proposals. Besides the exact analysis we calculate the BER for the case when the UWB interference is modeled as additional Gaussian noise. Our results show that in general the BER of the BICM—OFDM system strongly depends on the UWB format and the OFDM sub—carrier spacing. While the GA is very accurate for DS—UWB, it may severely over- or underestimate the true BER for MB—OFDM and IR—UWB interference. Our analysis is applicable to victim systems such as IEEE 802.11 WLANs, IEEE 802.16 Worldwide Interoperability for Microwave Access (WiMAX), and 4th generation mobile communication systems. Furthermore, since the ECMA MB—OFDM standard is also based on the BICM—OFDM concept, our analysis can also be used to evaluate the impact of other UWB signals on ECMA MB-OFDM UWB systems. In Chapter 5, we present a framework for performance analysis of BICM—based sys tems in the presence of non—Gaussian noise. The considered BICM—based systems employ BICM—OFDM or a combination of BICM and single—carrier (SC) transmission technique 14 Chapter 1. Introduction (BICM—SC) combined with multiple receive antennas. The proposed analysis is very gen eral and is applicable to BICM systems impaired by general types of fading, and general types of noise and interference with finite moments such as additive white Gaussian noise (AWGN), additive correlated Gaussian noise, Gaussian mixture noise, co—channel inter ference, NB interference, and ultra—wideband interference. We present an asymptotically tight upper bound for the BER and a closed—form expression for the asymptotic BER at high SNRs. We show that the diversity gain of the BICM—based systems only depends on the free distance of the code, the type of fading, and the number of receive antennas but not on the type of noise. In contrast their coding gain strongly depends on the type of noise via the noise moments. Our analysis shows that, if the popular Euclidean distance metric is used for Viterbi decoding, BICM—based systems optimized for AWGN are also optimum for any other type of noise and interference with finite moments. Finally, in Chapter 6, we conclude this thesis by providing some perspective on the results obtained in this dissertation and identifying areas of future research. 15 Chapter 2 Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.1 Introduction In this chapter, we study the performance of generic uncoded NB2 systems in the presence of MB—OFDM UWB interference. The case of an uncoded victim system is considered as an starting point for interference analysis as it allows us to concentrate on the effects of UWB interference without being concerned about the complicated structure of the victim receiver. A detailed treatment of the performance of coded systems in the presence of UWB interference is provided in Chapter 4. Moreover, the motivation for considering MB—OFDM technology for the UWB system arises from the lack of results on the effects of MB—OFDM UWB interference in the literature. In fact, most existing UWB interference studies have considered either DS—UWB or IR—UWB, cf. e.g. [25, 33, 34, 43]. This is surprising as MB—OFDM had been a strong candidate for standardization by the IEEE for high—rate WPANs [44] and has been recently adopted as a standard by the ECMA [1]. The performance of an IEEE 802.lla WLAN receiver in the presence of MB—OFDM in terference was evaluated experimentally in [29]. Simulations and experiments were used in [30] and [22], respectively, to study the the performance degradation of a C—band DTV re 2j this chapter, by NB system we refer to any system whose bandwidth is relatively small compared to the bandwidth of the considered UWB system. Therefore, our definition differs from the conventional definition of NB system presented in Chapter 1. 16 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference ceiver impaired by MB—OFDM interference. In [23, 24], it was shown that the MB—OFDM UWB interference can be modeled as impulsive noise provide that the bandwidth of the victim receiver is sufficiently large to contain several MB—OFDM sub—carriers. However, a general and exact analysis of MB—OFDM interference is not available in the literature. Therefore in this chapter, we provide a comprehensive analysis of MB—OFDM inter ference effects. For this purpose, we first derive the exact MGF and the exact APD [22] of MB—OFDM interference. Based on the MGF, we obtain analytical expressions for the exact BER of an uncoded NB system impaired by MB—OFDM UWB interference and AWGN. These expressions allow for an efficient numerical evaluation and are applicable to non—fading channels as well as all commonly encountered fading channel models including Rayleigh, Ricean, Nakagami—m, and Nakagami—q fading. The obtained APD and BER results show that when the NB signal bandwidth is much smaller and much larger than the MB—OFDM sub—carrier spacing, the MB—OFDM interference model can be simplified using the GA and the IGA, respectively. We also provide the MGF for the GA and the IGA, respectively, and derive the corresponding APD and BER expressions for the NB receiver. Using APD and BER results we further show that the accuracy of the GA and the IGA does not only depend on the NB signal bandwidth but also on the properties of the NB channel. We demonstrate that for Rayleigh fading and sufficiently high signal—to— interference ratios the BERs for GA and IGA are identical and closely approximate the exact BER regardless of the NB signal bandwidth. For other fading models and non— fading channels GA and IGA generally become accurate if B3 < O.O5f and B3 > respectively, where B3 and /f denote the NB signal bandwidth and the MB—OFDM sub— carrier spacing, respectively. For O.O5Lf B3 5f the provided exact BER expressions have to be used to obtain meaningful results. For these NB signal bandwidths the BER 17 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference of the NB receiver strongly depends on the carrier frequency offset between NB signal and MB—OFDM signal, the NB signal bandwidth, the number of MB—OFDM frequency bands, and the NB pulse shape. The rest of this chapter is organized as follows. In Section 2.2, the considered system and transmission model is introduced. In Section 2.3, the exact MGF of MB—OFDM in terference and two simple approximations thereof are derived, and exact and approximate expressions for the APD of MB—OFDM interference are provided. Exact and approxi mate BER expressions for a generic uncoded system impaired by MB—OFDM interference, AWGN, and (possibly) fading are developed in Section 2.4. Numerical results are presented and discussed in Section 2.5, and conclusions are drawn in Section 2.6. 2.2 System Model In this section, the considered signal and channel model is introduced. In addition, the MB—OFDM UWB signal and the NB receiver processing are described in detail. 2.2.1 Signal and Channel Model We consider the scenario where a MB—OFDM UWB transmitter is in close proximity to a coherent NB receiver. In order to capture the main effects of MB—OFDM interference on a NB signal while maintaining mathematical tractability, we adopt the same simple channel model3 as e.g. [24, 25, 29j. In particular, the received signal in equivalent complex baseband representation is modeled as r(t) = a(t) s(t) + g e9 i(t — r) + n(t), (2.1) 3We note that for large NB signal bandwidths a more elaborate, frequency—selective channel model may be more appropriate. However, for such a model an analytical interference study may not be feasible. In addition, our results in Fig. 2.8 suggest that frequency selectivity of the interference channel does not have a large impact on the error rate of the NB system. 18 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference where s(t), 1(t), and n(t) denote the transmitted NB signal, the transmitted MB—OFDM UWB (interference) signal, and AWGN with one—sided PSD N0, respectively. We assume that the NB channel is approximately time invariant during one NB symbol duration T3. Therefore, the NB channel gain can be modeled as a constant ac(t) = a a where a Ia and 8a denote the magnitude and the phase of a, respectively. For non—fading channels a = 1, whereas for fading channels a is a positive random variable with pdf pa(a) and variance u E{a2} = 1. Due to the proximity of the MB—OFDM transmitter and the NB receiver and the lack of synchronization between the two devices, the associated channel is modeled as non—fading with constant magnitude g and uniformly distributed phase O e [—it, it). The relative delay r of the MB—OFDM signal compared to the NB signal is uniformly distributed in the interval [0, T8). The NB signal s(t) is given by s(t) = x[k] p(t - kT3), (2.2) where E3, x[k], and p(t) denote the symbol energy, the kth transmit symbol, and the NB pulse shape, respectively. In the following we assume BPSK modulation for the NB system, i.e., x[k] e {±1}. This assumption simplifies the exposition but all fundamental and qualitative results obtained in this chapter also apply to higher order modulations. In particular, the general approach4 presented in [45] can be used to extend our analysis to any QAM and pulse amplitude modulation (PAM) format. We further assume that p(t) has a square—root Nyquist (SRN) frequency response [26], which is true for example for band—limited square—root raised cosine (SRRC) and time—limited rectangular (REC) pulse shapes. 41n [451, it has been shown that the I3ER of PAM can be expressed as a sum of terms wjPe(bja) where w and b are constellation dependent constants and e (a) is the BER for BPSK for channel amplitude a. A similar statement is true for the QAM case, cf. [45]. 19 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.2.2 MB—OFDM UWB Signal Model The adopted model for the MB—OFDM UWB transmit signal i(t) closely follows the ECMA standard [1]. More precisely, i(t) is modeled as [1] ( DO N/2 i(t) = — a,[ic] w(t ei2’tfot (2.3) V ‘i k=—OO n=—N/2 \ nO where N, and T denote the number of non—null sub—carriers, the sub—carrier spacing, and the OFDM symbol duration, respectively, and where we took into account that the zeroth sub—carrier is a null sub—carrier. In the following, we discuss the remaining variables in Eq. (2.3) more in detail. MB—OFDM data symbols a[kj: The MB—OFDM data symbols a[k] are taken from a 4—QAM alphabet, i.e., a[k] E {±1 ± j}. The ECMA standard [1] specifies seven different data rate modes for MB—OFDM UWB systems. The statistical properties of the data symbols a[k] are a function of the MB—OFDM data rate. Specifically, for MB—OFDM data rates of 320 and 480 Mb/s the data symbols a[kj can be assumed independent, identically distributed (i.i.d.) with respect to both time index k and sub—carrier index n. In contrast, for data rates below 320 Mb/s each OFDM symbol is repeated once in time, i.e., a[2k] = a[2k + 1], Vk, and for the two lowest data rates (55 and 80 Mb/s) the MB—OFDM data symbols have also a conjugate symmetry with respect to sub—carrier index n, i.e., a[k] = a[k], 1 ii N/2. We note that all data rate modes also involve channel coding but this does not have a noticeable effect on the interference statistics since the coded bits are interleaved before they are mapped to an[kl. 20 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference OFDM pulse shape w(t): The OFDM pulse shape w(t) is rectangular and given by [1] 11 Tpt<Tj—TG w(t) = (2.4) 0 otherwise where T and TG T — (Tp + 1/f) denote the prefix duration and the guard interval, respectively. Frequency offsets fo and fMB[k]: fMB[k] and fo are a time—dependent, periodic and a constant frequency offset, respectively. fMB[k] e {(n — 1)Nf b e {1, 2, ..., NB}} is used to switch between the NB MB—OFDM frequency bands [1], and fo represents the offset of the NB signal with respect to the zeroth sub—carrier of the first MB—OFDM band (fMB [k] = 0). Each MB—OFDM frequency band is active only 100/NB % of the time, and the order in which the different bands are used is determined by a so—called time—frequency code. In the following, we assume that the time—frequency code fMB[k] = (kmodNB)Nfj, (2.5) is applied and that the NB signal lies fully in the first MB—OFDM frequency band. We note that our analysis can be easily adapted to the case where the NB signal is affected by two MB—OFDM frequency bands and to other time—frequency codes. In general, different time—frequency codes will lead to different NB system error rates. The frequency hopping of the MB—OFDM signal and its effect on the NB signal are schematically shown in frequency and time domain in Figs. 2.la) and 2.lb), respectively. In Fig. 2.1, NB = 3 is valid and for the time domain representation of the MB—OFDM signal only the contribution of the relevant first frequency band is depicted. For all numerical examples presented in Section 2.5 the MB—OFDM parameters are 21 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference a) f NB Signal j’BTi i/sf b) 2T — t S MB-OFDM Signal Figure 2.1: NB signal and MB—OFDM signal in a) frequency domain and b) time domain for NB = 3. We note that for the time domain representation of the MB—OFDM signal only the contribution of the relevant first frequency band is depicted. chosen according to the ECMA standard [1] unless stated otherwise, cf. Table 2.1. 2.2.3 NB Receiver Processing The NB receiver performs matched filtering and sampling of the continuous—time received signal r(t). Taking into account the SRN property of the NB pulse shape p(t) and the normalization f° p(t) 2 dt = the discrete—time received signal can be expressed as r[k] = r(t) ®p*(_t)tkTS = ax[k] +i[k] +n[k], (2.6) where i [k] g e i (t — r) 0 p (—t) ItkTs and n [k] denote the complex MB—OFDM UWB interference and the complex AWGN, respectively. The resulting decision variable for NB Signal fo 22 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference BPSK transmission is Table 2J: MB—OFDM parameters [1]. r[k] Re{eSarc[k]} = ax[k] + i[k] + n[kl, (2.7) with real AWGN n[k] Re{eiOan[k]} and i[k] Re{e_Saic[k]} = Re{ge8i(t— r) ®p*(_t)t_kT}, (2.8) where phase 0 — Qa is uniformly distributed in [—it, it). The estimated data symbol i[k] is obtained as [kj = 1 if r[k] 0 and b[k] = —1 if r[k] <0. For future reference, we define the SNR and the signal—to—interference ratio (SIR) as SNRA E{(ax[k])2}— 1 — E8 — ‘{2(ri[k]) ——jc, SIR ‘{(ax[k])2}— 1 ‘{2(i[k]) — respectively. Here, o ‘{(ri[k])2}and cr ‘{(i[k])2}denote the variance of ri[k] and the variance of i [k], respectively. 2.3 MB—OFDM UWB Interference Analysis In this section, we first derive an expression for the exact MGF I,E(s) of the MB—OFDM UWB interference i[k]. Subsequently two approximate models for MB—OFDM UWB in- Number of non—null sub—carriers N 122 Sub—carrier spacing = 4.125 MHz OFDM symbol duration T 312.5 ns Prefix duration T = 60.61 ns Guard interval T0 = 9.47 ns Frequency bands NB = 3 and (2.9) 23 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference terference are provided and the associated MGFs are given. In addition, exact and ap proximate expressions for the APD are developed. 2.3.1 Exact Moment Generating Function (MGF) of i[k] Before we calculate the exact MGF, we first reformulate and simplify the analytical ex pression for i [kj. Reformulation of i[k] Based on Eqs. (2.3) and (2.8) the MB—OFDM interference signal can be expressed as oo N/2 i[kj=Re , (2.10) I ,=—oo where the coefficients c(kT3 — icT — -r) are defined as — icT — r) ‘p*(t) ® ei2 i(t_T_’Ti)w.(t — r — kT) ej2(fMB[k1+jo)(tT) t=kT3 (2.11) Adopting the time—frequency code in Eq. (2.5) and assuming that the NB signal lies fully in the first MB—OFDM frequency band, Eq. (2.10) can be simplified to lao N/2 i[kl = Re g e8ej20ITj!3fl(rk — NnicT) afl[NBIc] , (2.12) icOO n=—N1/2 nO 24 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference where Tk kT3 — r and 3(t) is given by oo T—T i3(t) = ei2i+fp*(u — t) w(u) du = = f e i+fo)uP*( — t) du. (2.13) Alternatively, using Parseval’s theorem [26], /3(t) may be obtained from (t) = ei2j+t ej2t_/(j)+TP*(f +nf+ fo) sin(f/f) df, (2.14) where P(f) denotes the Fourier transform of p(t). Since any NB pulse shape p(t) is essentially limited in time and frequency, the coefficients ,8, (rig — NBIcT) are approximately zero if, respectively, i and n exceed certain upper and lower limits, cf. Eqs. (2.13) and (2.14). We note that for other time—frequency codes specified in [1] similar simplified expres sions for i[k] can be found. Conditional Moment Generating Function Now, we are ready to derive the MGF of i[k] conditioned on Tk and phase offset e I)iI8,Tk(s) E{ejO,Tk}. (2.15) For evaluation of Eq. (2.15), the statistical properties of the MB—OFDM data symbols a[k] are important. We first note that because of the frequency hopping dictated by the time—frequency code in Eq. (2.5), the NB signal is affected only by every NBth MB—OFDM symbol. Therefore, assuming NB 2 the repetition of every MB—OFDM symbol in time 5Although Eq. (2.15) differs from the usual definition of the MGF by the negative sign in the exponential, we still refer to ‘iIe,rk(s) as MGF. Strictly speaking jIo,rk(s) as defined in Eq. (2.15) is the Laplace transform of the conditional pdf of i[kj. 25 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference for data rates below 320 Mb/s has no impact on the interference statistics. However, the dependence introduced by the conjugate symmetry for MB—OFDM data rates of 55 and 80 Mb/s affects the interference statistics seen by the NB signal if the NB signal spectrum is non—zero for both positive and negative sub—carrier frequencies. Therefore, we consider in the following two different cases. Cl) No conjugate symmetry (MB—OFDM data rates of more than 80 Mb/s): In this case, the MB—OFDM data symbols can be modeled as i.i.d. random variables and using Eq. (2.12) we can rewrite Eq. (2.15) as :: N/2 iI8,Tk(’) = fJ Jlj é {exp (—sRe {geei2 0I\JB1Ti/3fl(rk — NBFcT)afl[NBl])} ,rk}. ,c=—oo n=—N/2 nO (2.16) If we furthermore take into account that the MB—OFDM data symbols are equi—probable 4—QAM symbols, we arrive at the conditional MGF N/2 = fJ JJ cosh (s Re{ge°e2 0NBkTu/3(rk — NBIcT)}) k=—OO n=—N/2 nØO cosh (s — NBicT)}) . (2.17) Finally, for numerical evaluation of 4j1rk,e(s) we replace Eq. (2.17) by kkj fl cosh (s Re{ge i20ITi/3fl(rk — NBIcT)}) no cosh (s — NBIcTj)}) , (2.18) where tj, n1, and n are suitably chosen upper and lower limits which ensure /3(rk — NBIT)} 0 if i [—‘ci, ‘c} or n [—nj, n}, cf. Section 2.3.1. Depending on the system parameters, typically 20 to 100 terms are required in Eq. (2.18) to achieve an accurate 26 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference approximation of Eq. (2.17). 02) Conjugate symmetry (MB—OFDM data rates of 55 and 88 Mb/s): Using similar steps as above but taking into account a[k1 = a[k1, 1 n N/2, the conditional MGF is obtained as N/2 il8,Tk (s) = fJ fJ cosh (s — NBkT)}) cosh (s Irn{ge e320T6fl(rk — NBkT)}) , (2.19) where 7(t) 3(t) + ,8_(t) and S(t) /3(t) — ,8_(t). For numerical evaluation of Eq. (2.19) the products can be truncated in a similar way as in Eq. (2.18). In the remainder of this chapter, we will assume Case Cl) and i.i.d. MB—OFDM data symbols. However, all analytical results presented are also valid for Case C2) and conjugate symmetric data symbols if the conditional MGF in Eq. (2.19) is used instead of the one in Eq. (2.17). Exact Moment Generating Function Since we assume that both the phase offset 8 and the delay r of the MB—OFDM interferer are uniformly distributed random variables, cf. Section 2.2.1, we have to average jI9rk(s) with respect to °g and r or equivalently with respect to 0 and rk, respectively. After averaging over 8, we can express the MGF of i[k] conditioned on -rk as jITk(8, Tk) {iIe,rk()} ± f ii8,rk(8) dO. (2.20)2r Since it is shown in Appendix A that jIrk(s, Tk) is periodic in Tk with period the unconditional MGF can be obtained by averaging tjITk (s, Tk) over any interval of length NBT. Since -r is assumed to be uniformly distributed, Tk is also uniformly distributed and 27 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference the exact MGF of i[kj can be computed as ,E(S) = 1 f fI9,Tk(s)dOdm. (2.21)27rNBT NBT 27r E (s) can be efficiently numerically evaluated as Eq. (2.21) only involves two integrals with finite limits and jIe,T(s) may be replaced by the approximation in Eq. (2.18). 2.3.2 Approximate Models for MB—OFDM UWB Interference In this subsection, we discuss two simplified MB—OFDM UWB interference models which are good approximations of the exact model if the NB signal bandwidth B3 1/T3 is much smaller and much larger than the MB—OFDM sub—carrier spacing respectively.6 Gaussian Approximation (GA) We first consider the case where the NB signal has a very small bandwidth B3 compared to the sub—carrier spacing frequency i.e., B3 << /f which also implies T3 >> T,. In this case, Eqs. (2.13) and (2.14) show, respectively, that/3(rk — NBIc) is non—zero over a large interval of ic and a small interval of n since p(t) is a broad pulse in the time domain whereas P(f) is a narrow pulse in the frequency domain. Therefore, in Eq. (2.12) for certain values of n many independent random variables with non—zero (and similar) variances are added in the summation over ic. This suggests that the central limit theorem can be invoked and i[k] may be approximated as a real Gaussian random variable with MGF i,GA(S) = e82J2. (2.22) 6We note that the exact amount of spectrum occupied by the NB signal heavily depends on the adopted pulse shape p(t), of course. In order to facilitate a unified treatment of different pulse shapes, we refer to B3 = 1/T3 as the NB signal bandwidth here. 28 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference In this case, the multi—band nature of the MB—OFDM UWB interference does not affect the MGF. We note however that for a given B3 the number of non—zero terms added in Eq. (2.12) decreases with increasing NB implying that the GA becomes less accurate. Impulsive Gaussian Approximation (IGA) Now, we assume that the bandwidth B3 of the NB signal7 is large compared to the sub— carrier spacing frequency i.e., B3 >> and T3 << T. Thus, if fMB[k] = 0 the MB—OFDM signal impairs the NB signal for several NB symbol durations T3. After the MB—OFDM signal has hopped to a different center frequency fMB[k] > 0 it will not affect the NB signal for a duration of NBT — 1/f, where we took into account that i[k] is also zero during the prefix and the guard interval. These time—domain considerations show that only approximately a fraction of p 1/(NBfTI) NB symbols are affected by the MB-OFDM signal. IfaNB symbol is impaired by the MB—OFDM signal (i.e., if fMB[kl = 0),/3(Tk—NBIc) will have non—zero values only for a single value of ic = ic0. However, for ic0, /3fl(Tk — NBFo) will be non—zero for a large number of sub—carriers n since P(f) is a broad pulse, cf. Eq. (2.14). Therefore, i[k] in Eq. (2.12) involves a sum of many independent random variables with similar variances and the central limit theorem applies. To summarize, for B3 >> lOOp % of the NB symbols are impaired by approximately Gaussian interference, whereas 100 (1 — p) % of the NB symbols do not experience any interference. Therefore, in this case, the MGF of i[k] can be approximated as Ii,IGA(S) = 1 p + pe82(20). (2.23) 7We again note that in this chapter, “narrowband” implies that the bandwidth of the signal is small compared to the total MB—OFDM UWB bandwidth, i.e., B5 <<NBNLSf. 29 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference We note that similar IGA models for MB—OFDM have also been reported in [23, 24]. 2.3.3 Amplitude Probability Distribution (APD) The National Telecommunications and Information Administration (NTIA) recommends APDs for assessment of UWB waveforms [22, 23]. The APD is the complementary cumu lative distribution function of the amplitude of the interference signal i[k] APD(y) Pr{i[kfi > y}. (2.24) We note that the APD as defined in Eq. (2.24) deviates slightly from the one in [22, 23]. In [22, 23] the APD of the complex UWB interference i[k] is considered, whereas we consider the APD of the real interference i[k], cf. Eq. (2.8), since only i[k] affects the decision variable, cf. Eq. (2.7). For the GA and the IGA it is straightforward to show that the corresponding APDs are given by APDGA(y) = 2Q () (2.25) and APDIGA(y) 2pQ () (2.26) respectively, where Q(x) 4= j°e2/2 dt denotes the Gaussian Q—function [26]. For the exact MB—OFDM interference model a closed—form expression for the APD cannot be found but APDE(y) can be evaluated numerically. For this purpose it is useful to note that the exact pdf pj,E(X) of i[k] is an even function of x, since its Fourier transform j,E(JW) is a real and even function of w [46]. Therefore, exploiting basic properties of the Laplace 30 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference transform [47], we can express the exact APD as -y c+j00 APDE(y) = 2 f Pi,E(X) dx = -- f ,E(s) e8 (2.27) —00 0—300 where c is a positive constant lying in the region of convergence of the integral. An efficient numerical evaluation of Eq. (2.27) is possible by applying a Gauss—Chebyshev quadrature rule, cf. e.g. [47]. This results in 2 K/2 APDE(y) (Re{,a(cs) e} + kIm{,E(csk) e’}) , (2.28) where Sk l+jk, k tan([2k—1]ir/[2K]), and Kis a sufficiently large integer. According to [47] the best choice for c is that real—valued s which minimizes j E(S)e8. Since the minimum of Fi,(s) e’9 is difficult to compute, we use c = 8mjfl = y/o instead, where 5min minimizes F, GA(S) e_Ys. We found that this choice guarantees fast convergence of the sum in Eq. (2.28) and K = 200 is sufficient to obtain accurate results. We note that although the evaluation of Eq. (2.28) requires the numerical integration of three integrals [Eqs. (2.13), (2.21)], the computation of APDE(y) is quite fast, since all integrals have finite limits. 2.4 BER of BPSK NB Signals In this section, the BER of BPSK NB signals impaired by AWGN and MB—OFDM UWB interference is derived for transmission over non—fading and fading channels, respectively. 31 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.4.1 Preliminaries In this subsection, we establish some basic results that are helpful for BER calculation. First, it is convenient to introduce the MGF +n,x(s), X e {E, GA, IGA}, of the sum of MB—OFDM UWB interference and AWGN for the different interference models discussed in Section 2.3. Exploiting the fact that i[k] and m[k] are statistically independent i+n,x(s) can be obtained as = I,x(s) Ia(s), X E {E, GA, IGA}, (2.29) where Ia(s) e82/2 denotes the MGF of n[k]. Using Eqs. (2.22), (2.23), and (2.29) it is not difficult to show that for a given NB channel gain a and BPSK signaling the GA and the IGA result in BERs _______ Pe,GA(a) = Q () (2.30) and _________ Pe,IGA(a) = (1- p) Q () + pQ( (2.31) respectively. On the other hand, for any of the considered interference models and a given NB channel amplitude a, we may express the BER of the BPSK NB receiver as —a c+jOO P6,x(a) Pi+n,x(X) dx = +n,x(s) e_as , x e {E, GA, IGA}, (2.32) where we have used as a similar approach as for derivation of Eq. (2.27). In the following subsections, we use Eqs. (2.30)—(2.32) for calculation of the BER for non—fading and fading channels. 32 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.4.2 BER in Non—Fading Channel In this case, we may set a = 1 in Eqs. (2.30)—(2.32) and replace Pe,x(a) by Pe,. Using the definitions of the SNR and the SIR in Eq. (2.9), the GA and the IGA result in a BER of GA = Q (VSNR-’ + SIR_i) (2.33) and _________________ Pe,IGA = (1 — p) Q (VNR) + P (-‘ +2(PSIR)_1), (2.34) respectively. For SIR — oc, Eqs. (2.33) and (2.34) simplify to the well—known BER expression for BPSK transmission over a pure AWGN channel as expected, cf. [26, Eq. (5.2- 5)]. We note that Eq. (2.34) can also be found in [23, 24] and is reported here only for completeness. For the exact MB—OFDM UWB interference model a closed—form solution for Fe, E cannot be derived and the exact BER has to be evaluated numerically. This can be done efficiently by applying the Gauss—Chebyshev quadrature rule already used for derivation of Eq. (2.28). This yields K/2 Pe,E (Re{+fl,E(csk) ecsk} + kIm{+fl,E(csk) e_c3k}) , (2.35) where sk, k, K, and c are defined after Eq. (2.28). A good choice for c guaranteeing fast convergence of the sum in Eq. (2.35) is C Smjn = 1/(o + of), where 8mjfl minimizes i+n, GA(S) c_s. 33 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.4.3 BER in General Fading Channels For fading channels the NB amplitude a is a random variable with pdfpa(a). Furthermore, we introduce the pdfp7(7) of the squared amplitude 7 = a2. The average BER in fading channels can be obtained by averaging the expressions for Pe,x(a), X e {E, GA, IGA}, given in Eqs. (2.30)—(2.32) over either pa(a) or For example, exploiting the alternative representation of the Q—function Q(x) = 1 ,-/2 _______ ;fo [48] when averaging Eqs. (2.30) and (2.31) over p(7) yields 7r/2 Fe, GA = / (s-’ + SIR) 2 ) d (2.36) and ir/2 ir/2 i—p 1 /SNR\ p 1 / 1 Pe,IGA = 2) d+ J (SNR’ + (pSIR)_i)sifl2) d, (2.37) respectively. Here, t7(s) E{e8}denotes the MGF (as defined here) of ‘y. For the most important fading distributions (Rayleigh, Ricean, Nakagami—m, Nakagami—q) closed—form expressions for I(s) can be easily obtained, cf. e.g. [48, Table 3]. For example, for Rayleigh and Nakagami—m fading 7(s) = 1/(1 + s) and Is(s) = 1/(i + s/m)m result, respectively. Using Eqs. (2.36) and (2.37) the BER for the GA and the IGA can be efficiently computed by numerically evaluating one—dimensional integrals with finite limits. We note that for Rayleigh fading Eqs. (2.36) and (2.37) can be further simplified and closed—form expressions for Fe, GA and Fe, IGA can be obtained, cf. Section 2.4.4. This is also possible for Nakagami— m fading by exploiting results in [49]. On the other hand, averaging Eq. (2.32) over pa(a), we obtain for both the exact and 34 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference the approximate interference models c+joo Pe,X = E{Pe,x(a)} = f i+n,x(s) a(S) , X E {E, GA, IGA}, (2.38) c-joo where 1a(s) E{e_as} denotes the MGF of a. Applying again a Gauss—Chebyshev quadra ture rule, we obtain 1 K/2 Pe,X (Re{+ x(csk) Fa(csk)} + kIm{(I+fl,x(csk)1a(CSk)}), (2.39) where X e {E, GA, IGA} and Sk, k, K, and c have been defined before. The amplitude MGF a(S) is readily available in the literature for all relevant fading distributions. For example, the MGFs I(s) for Rayleigh, Ricean, Nakagami—q, and Nakagami—m fading can be found in [50, Table 1j8. Consequently, Eq. (2.39) can be used for efficient numerical computation of both the exact and the approximate BERs for any relevant fading distri bution. For fading channels it is more difficult to find good values for c than for non—fading channels since 1a(8) typically involves confluent hypergeometric series, cf. [50, Table I]. A suitable value is c = Re{s’}/2, where s’ denotes that zero of 1a(s) which has minimum real part. s’ can be found using standard numerical procedures [51]. Note that for the GA and the IGA Eq. (2.39) is an alternative to Eqs. (2.36) and (2.37), respectively, where both options entail approximately the same complexity. 8Strictly speaking the characteristic function a(—jW) is given in [50, Table II. However, the MGF can be easily obtained by applying the substitution w = js. 35 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.4.4 BER in Rayleigh Fading For Rayleigh fading Eqs. (2.36) and (2.37) can be further simplified to Pe,GA (1 — 1 (2.40)2 \ /i + SNR’ + SIR’J and Pe,IGA = 2 (i — +SNR’ — i + SNR’ + pSIR-’’ (2.41) respectively. For SIR — oo both Eq. (2.40) and Eq. (2.41) simplify to the well—known BER expression for BPSK transmission over a Rayleigh fading channel impaired by AWGN, cf. [26, Eq. (14.3.7)]. Furthermore, it can be shown that for SNR>> 1 and SIR>> 1 the GA, the IGA, and the exact MB—OFDM UWB interference model all lead to the same BER Pe,X (SNR-’ + SIR’), X E {E, GA, IGA}. (2.42) For the GA and the IGA Eq. (2.42) can be proved by applying 1//1 + x 1 — x/2, x <<1, in Eqs. (2.40) and (2.41), whereas the exact case is discussed in Appendix B. Eq. (2.42) shows that for high SNRs and high SIRs the exact BER Pe,E in Rayleigh fading only depends on the SNR and the SIR but is independent of other system parameters such as B3, and Jo. In the next chapter, we generalize this result to other types of non—Gaussian noise and interference. In particular, we show that for sufficiently high SNRs and SIRs, the performance of an uncoded system impaired by Rayleigh fading and generic non—Gaussian noise and interference is independent of the noise and interference statistics. 36 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference Figure 2.2: APD plot for different normalized NB signal bandwidths B3//.Sj. o = 1, fo = 1Of, and NB = 3. Exact APD [Eq. (2.28)], GA [Eq. (2.25)], and IGA [Eq. (2.26)]. 2.5 Results and Discussions In this section, we use our analytical methods presented in the previous sections to study the performance of BPSK NB receivers impaired by MB—OFDM UWB interference, AWGN, and possibly fading. The MB—OFDM UWB parameters given in Table 2.1, an SRRC NB pulse with roll—off r = 0.35, and the channel model introduced in Section 2.2 are em ployed unless stated otherwise. In addition, to avoid distortion of the results by the null sub—carriers, we set fo = 1OLf unless specified otherwise. 0.1 100.Pr{Ii{k]I >y}% 37 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 2.5.1 APD Plot Fig. 2.2 shows APD plots for MB—OFDM UWB interference with o 1, fo 10f, and NB = 3. The curves in Fig. 2.2 were generated using the results in Section 2.3.3 and adopting the format recommended by the NTIA in [22, Appendix D], i.e., the x—axis shows the percentage of time for which i[k]I exceeds the ordinate. The GA yields approximately a straight line and is approached by the exact APD for small (normalized) NB signal bandwidths B3/f, whereas the IGA is approached for large B8/f, cf. Section 2.3.2. The more often a given ordinate is exceeded on average (i.e., the larger the corresponding x—axis value), the higher the corresponding bit error rate will be. For communications applications the x—axis interval below 1% is most relevant as it reflects the behavior of the tails of the corresponding pdf. Therefore, Fig. 2.2 shows that for certain NB signal bandwidths MB—OFDM UWB interference is more favorable than Gaussian noise with equal variance, whereas impulsive noise has a more detrimental effect in the interval of interest. For example, i[kj exceeds 201og10(y) = 6 dB for B8/Lf = 0.15 only i0 % of the time, while Gaussian noise and impulsive noise exceed this level 0.5% and 4% of the time, respectively. The APDs in Fig. 2.2 suggest that the impact of MB—OFDM UWB interference on a NB receiver strongly depends on the bandwidth of the NB signal. 2.5.2 BER in Non—Fading Channel In Fig. 2.3, we investigate the dependence of the exact BER on the NB pulse shape as a function of B3//f. SRRC impulses with various roll—off factors r are compared with a REC pulse of duration T8 = 1/B8 for lOlog10(SNR) = 10 dB, fo = 10f, NB = 3, and various SIRs. As expected, all exact BERs approach the respective GAs and the IGAs for very small and very large NB signal bandwidths, respectively. The GA is faster approached for smaller r than for larger r since in the former case the corresponding time domain pulse 38 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference iSRJ 102 1.0_i io° 0l 102 B2/zf Figure 2.3: BER vs. B3/if for different NB pulse shapes and different SIRs. 101og0(SNR) 10 dB, fo = 10f, and NB = 3. Exact BER [Eq. (2.35)], GA [Eq. (2.33)], and IGA [Eq. (2.34)]. Markers: Simulation results. shape decays more slowly. For medium NB bandwidths (0.05 B3//.f 5) the exact BER depends significantly on the NB pulse shape, especially for low SIRs. The simulation results for r = 0.2 and r = 0.8 confirm the exact BER analysis. The effect of the frequency offset fo between the NB signal and the MB—OFDM signal on the exact BER is studied in Figs. 2.4 and 2.5. Fig. 2.4 shows the exact BER vs. fo/’f for the interference limited case (i.e., SNR —+ cc) assuming B8 = and NB 3. Numerical results (solid lines) and simulation results (markers) are in excellent agreement. For 10 1og0(SIR) > 6 dB the BER strongly depends on fo, where values of fo close to the sub—carrier center frequency of 10f result in a lower BER. 39 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference ! I .... () LI LI LI *:::::[[ 0-g- - ôôj”b0 p--O0 O--C-OO I I 101og0(STFt)=O, 1, ,11 dB 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4 10.5 fo/f Figure 2.4: BER vs. fo/f for different SIRs. Interference limited case (SNR —* cc), B3 = f,, and NB = 3. Markers: Simulation results. Solid lines: Numerical results [Eq. (2.35)]. We note that as long as border effects can be neglected the BER is periodic in fo with period Fig. 2.5 depicts BER vs. B3/Lf for 101og0(SNR) = 10 dB, NB = 3, various fo/f1 and various SIRs. The exact BER approaches the GA faster if the NB carrier frequency is between two MB—OFDM sub—carriers (e.g. fo/f = 10.5), and it becomes independent of fo if the NB signal bandwidth exceeds twice the MB—OFDM sub—carrier spacing (B3 > 2f). In Fig. 2.6, we investigate the impact of NB on the exact BER for 10 log10(SNR) = 10 dB, 10 1og0(SIR) = 15 dB, and fo = While the exact BER for large B3 and the IGA depend on NB, the exact BER for small B3 and the GA are independent of NB. 10_i 1 10_s 1 06 10_i 1 08 9.5 40 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 10 2 —exact BER - -;-::1P gjo(). 16dB : : —101og1(SIR) = 21dB - . . E 102 1.0_i 10° 101 102 B5/f Figure 2.5: BER vs. B8//f for different fo/f and different SIRs. 10 1og0(SNR) = 10 dB and NB = 3. Exact BER [Eq. (2.35)], GA [Eq. (2.33)], and IGA [Eq. (2.34)]. However, the exact BER approaches the GA faster for small NB as expected, cf. Section 2.3.2. The difference between GA and IGA increases with increasing NB, as in the latter case the interference becomes more impulsive for larger NB. We note that even for NB = 1 (i.e., conventional OFDM with only one band) GA and IGA are not identical because of the non—zero prefix duration T and the non—zero guard interval TG, which are responsible for p 1/(fT) < 1 in this case. Figs. 2.3, 2.5, and 2.6 show that for a given SIR NB signals with small bandwidths B8 are less affected by MB—OFDM UWB interference than NB signals with large B8. In addition, for non—fading channels with the considered parameters the IGA upper—bounds the exact BER, whereas the GA is approached from below by the exact BER for small NB 1’ 10 41 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 10_2 • :::: :: — exact BER — GA :•:•:: :;:: --IGA i0 — :: : : 1,2,. . :; 1& 10 ;;,,,I I 10_2 10_i 100 10’ 102 B3/.f Figure 2.6: BER vs. B3/zf for different NB. 101og0(SNR) = 10 dB, 101og0(SIR) = 15 dB, and fo = 10LSj. Exact BER [Eq. (2.35)], GA [Eq. (2.33)], and IGA [Eq. (2.34)]. signal bandwidths. In fact, from the comparison of the GA and the exact BER in Figs. 2.3, 2.5, and 2.6 we can also conclude that in non—fading channels for low—to—medium values of B3 the pdf of MB—OFDM UWB interference has a more favorable impact on error performance than the Gaussian pdf, cf. discussion in Section 2.5.1. Figs. 2.3, 2.5, and 2.6 are also in agreement with the discussion presented in Subsection 2.3.2, as they show that the GA and the IGA become accurate for B3 << /f and B3 >> respectively. For the GA in particular, these figures shows that the GA leads to a close approximation for the performance when Pr T/T3 << 1, and further illustrate that the GA looses its accuracy as Pr increases. 42 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference —exact5ER 1 :. :: —GA 10 ._____.__.:._ .L.r io_2 ,._._: _•. t io3 .. t. . .. ,.. io4 10_2 10_I 100 101 102 B3/tXf Figure 2.7: BER vs. B3/Af for different fading parameters in. 101og0(SNR) = 10 dB, 101og0(STR) = 10 dB, fo = 10f, and NB = 3. Exact BER [Eq. (2.39)], GA [Eq. (2.36)], and IGA [Eq. (2.37)]. 2.5.3 BER in Fading Channel Fig. 2.7 illustrates the dependence of the exact BER on the NB signal bandwidth B3 for Nakagami—m fading with 10 log10(SNR) = 10 dB, 10 1og0(SIR) 10 dB, fo = 10zSf, and NB = 3. We note that in = 1 and in —f oo correspond to Rayleigh fading and no fading, respectively. It is interesting to observe that the performance difference between GA and IGA increases with increasing in. Similarly, while the exact BER is almost independent of B3 for in = 0.5, 1, 2 (cf. also Section 2.4.4), a strong dependence of the exact BEE on B5 can be observed for in 5. This clearly shows that MB—OFDM UWB interference should not be analyzed in isolation (as is done with the APD plots advocated by the NTIA), but only in connection with the underlying NB channel. Furthermore, similar observations 43 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference as in the previous subsection are in order regarding the GA and IGA. From Fig. 2.7 we clearly observe that the GA and the IGA become accurate for B3 << /f and B3 >> respectively. The GA becomes inaccurate as Pr T/T3 increases especially for higher values of m. Finally, Fig. 2.7 shows that the performance degradation of the NB system caused by MB—OFDM interference also strongly depends on the fading statistics. In the next chapter we generalize this result and show that this is generally true for the performance of an uncoded system in the presence of generic non—Gaussian noise and interference. 2.5.4 Impact of Different UWB Channel Models So far we have assumed a flat channel model. While this is an appropriate model for small NB signal bandwidths B3, a frequency—selective channel model may be more appropriate for large B3. Therefore, we compare in Fig. 2.8 the analytical BER obtained for the channel model described in Section 2.2 with simulation results for one (typical) realization of the CM1 and CM4 UWB channel models defined in [52]. We have assumed a flat and non—fading NB channel for all BER curves shown in Fig. 2.8, i.e., we assume that only the MB—OFDM interference channel is frequency selective. For CM1 and CM4 the impulse response is multiplied by a log—normal shadowing term, cf. [52]. To separate the effects of frequency—selectivity and log—normal shadowing, we show in Fig. 2.8 simulation results obtained with and without the log—normal term. Fig. 2.8 shows that the exact BER obtained for a flat interference channel can serve as a good estimate for the BER for CM1 and CM4 UWB interference channels. The log—normal shadowing causes the simulated BER to deviate from the exact BER also for small NB signal bandwidths, since the averaging of individual interference terms which is essential for application of the central limit theorem does not affect the log—normal shadowing. For large NB signal 44 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference 10 — flatchannel,exactBER — — CM1 UWB channel, simulation . ... •.:: . • —. CM4 UWB channel, simulation . v without shadowing : 0 with shadowing ;. . 1.: . 102 10_I 100 101 102 B3/zf Figure 2.8: BER vs. B3//..f for different interference channels. 10 1og0(SNR) = 10 dB, 10 1og0(SIR) = 15 dB, and fo = 10f. Exact BER for flat channel [Eq. (2.35)] and simulation results for one realization of CM1 and CM4 UWB channel models. bandwidths and no shadowing the deviation from the exact BER is larger for the more severely frequency—selective CM4 channel than for the CM1 channel. However, for CM4 and large B3 the effects of shadowing and frequency—selectivity seem to balance each other and the simulated BER is very close to the exact BER. 2.6 Conclusions In this chapter, we have provided a comprehensive analysis of MB—OFDM UWB inter ference and its impact on BPSK NB receivers. Based on the MGF of MB—OFDM UWB interference, efficient methods for numerical evaluation of the exact interference APD and 45 Chapter 2. Performance of Generic Uncoded NB Systems in MB—OFDM Interference the exact BER of a BPSK NB receiver have been developed. Furthermore, we have in troduced a GA and an IGA for MB—OFDM UWB interference which result in simpler APD and BER expressions than the exact interference model. Our numerical results sug gest that, in general, the GA and the IGA give accurate results for B8 < O.O5Lf and B8 > respectively. For the special case of a Rayleigh fading NB channel both ap proximations are tight for sufficiently high SNR and SIR regardless of B8. However, for O.O5Lf B5 in general, the BER of the NB receiver strongly depends on the carrier frequency offset fo between the NB signal and the MB—OFDM signal, the NB signal bandwidth, the number of MB—OFDM frequency bands, and the NB pulse shape. For the GA in particular, we showed that the accuracy is only guaranteed for Pr = T/T8 << 1, and further demonstrated that the approximation error incurred by using the GA increases with increasing Pr 46 Chapter 3 Asymptotic Performance of Generic Uncoded Systems in Non—Gaussian Noise and Interference 3.1 Introduction In this chapter, we extend the results obtained in the previous chapter for MB—OFDM UWB interference to generic non—Gaussian noise and interference. In particular, we pro vide a unified asymptotic analysis of the SER performance of generic linearly modulated uncoded systems impaired by generic non—Gaussian noise9 assuming asymptotically high SNRs. While the analytical expressions for the SER are often quite involved, the asymp totic analysis allows us to obtain simple yet accurate performance approximations that are desirable for system design [261. Most asymptotic results available in the literature have been obtained for impairment by AWGN [38]—[42]. While AWGN may often be the dominant noise source, as explained in Chapter 1, there are many practical applications where non—Gaussian noise impairs the received signal. Examples include co—channel and adjacent channel interference in mobile cellular systems [26, 27], impulsive noise in wireless and powerline communications [37]. 9As mentioned in Chapter 2, here “noise” refers to any additive impairment of the received signal and therefore also includes what is commonly referred to as “interference”. 47 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise Furthermore, future wireless systems are expected to operate in the presence of UWB in terference which, as shown in Chapter 2, can exhibit highly non-Gaussian characteristics. Although analytical expressions for the SER are available for some types of non—Gaussian noise and interference, a general asymptotic result giving insight into how system perfor mance is affected by the type of noise (in addition to the type of fading) is missing in the literature. In this chapter, we derive simple and elegant asymptotically tight expressions for the SER of linear modulation schemes impaired by fading and (possibly) non—Gaussian noise. Thereby, we assume that the receiver does not know which type of noise is present and applies the detection rule which is optimum for Gaussian noise. The main restriction that we impose on the noise is that the Mellin transform Ma(s) [53] of its pdf exists for s 1, or equivalently, all the noise moments are finite. Most practically relevant types of noise meet this condition. We also extend our asymptotic SER results to binary orthogonal modulation (BOM), equal gain combining (EGC), and selection combining (SLC). Furthermore, we show that for high SNR the SER depends on the Mellin transform Ma(s) of the pdf of the noise process, or equivalently, on the noise moments’0. Interestingly, we find that the diversity gain of the communication system only depends on the type of fading and the number of diversity branches, whereas the combining gain” is also affected by the type of noise. Therefore, in a log—log scale for high SNR the SER curves for different types of noise are parallel. For the special case of a system with a diversity gain of one for high SNR the SER becomes independent of the type of noise. The remainder of this chapter is organized as follows. In Section 3.2, the considered ‘°As shown in Appendix C (cf. Section C.4), the Mellin transform of the noise pdf Ma(s) is closely related to the noise moments. Therefore, the dependence of SER on the Mellin transform Ma(s) also implies its dependence on the noise moments. combining gain is also sometimes referred to as “coding gain” in the literature, e.g. [39]. 48 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise signal and noise models are introduced. In Section 3.3, the asymptotic SERs of linear modulation formats and BOM are derived. These results are extended to systems with EGC and SLC in Section 3.4. The derived analytical results are verified by simulations for some representative and relevant special cases in Section 3.5, and conclusions are drawn in Section 3.6. 3.2 Preliminaries In this section, we present the considered signal, channel, and noise models. However, first we introduce some definitions and notations. 3.2.1 Some Definitions and Notations Mellin transform and noise moments: The Mellin transform M(s) = M{p(x)} f0°°p(x)x’ dx of a function p(x) will play an important role in derivation of the asymptotic results in this chapter. A detailed discussion of the Mellin transform and its properties can be found in Appendix C. Furthermore, as shown in Section C.4 of Appendix C, the Mellin transform of the noise pdf Ma(s) is closely related to the noise moments. As a result, the Mellin transform Ma(s) and the noise moments may be used interchangeably. Diversity and combining gain: It is well—known that for transmission over flat fading chan nels impaired by Gaussian noise the SER at high SNR can be expressed as [26, 39] FE (G7)d (3.1) where -y denotes the average SNR, and G and Gd are referred to as the combining gain and the diversity gain, respectively. We will show that Eq. (3.1) is also valid for general types of non—Gaussian noise meeting the mild assumptions outlined in Section 3.2.3. 49 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 3.2.2 Signal Model For clarity of presentation, we restrict our attention for the moment to the single—receive— branch case. The extension to diversity reception is provided in Section 3.4. Assuming a frequency—nonselective channel and perfect phase and timing synchronization for the desired signal, the received signal in complex baseband representation can be modeled as r=ax+n (3.2) where a, x, and n, denote the real—valued fading gain, the complex transmitted symbol, and the complex noise, respectively. We assume that a, x, and n are mutually independent random variables (RVs). The results derived in this chapter are applicable to all fading processes whose amplitude pdf Pa (a) can be expanded into a power series around a = 0 for high SNR, cf. Section 3.3. In particular, we will consider Rayleigh, Ricean, Nakagami—m, Nakagami—q, and Weibull fading and the corresponding pdfs pa(a) are given in Table 3.1. We emphasize that in general the noise n may include both channel noise and interference. Unless stated otherwise12,in our analysis, we assume that the SER can be obtained by only considering rRe{r}=ax-i--n (3.3) where x Re{x} and ii Re{n}. The validity of this assumption is obvious for one— dimensional modulation schemes such as BPSK and M—ary PAM (M—PAM). The same is true for M—ary QAM (M—QAM) if the real and imaginary parts of n are i.i.d. RVs [26j. For general M—ary phase shift keying (M—PSK) the above assumption always involves an approximation. For convenience, we adopt the normalization a2 ‘{n2} = 1, y {a2}, and u 12n Section 3.2.4, for BOM it is necessary to also consider the imaginary part of r. 50 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise Table 3.1: Pdf pa(a) of fading amplitude a 0 and corresponding series parameters pk, , and 6 for the series expansion pa(a) = Z°=pk(a2/7)’’. = Fading Model Pdf of Fading Amplitude a and Series Parameters a2’\Rayleigh Pa(a) = exp (—-;;;-} Pk — 2(_i)k_1 (k—i)! 6=0 Ricean pa(a) 2(K±1)a exp (_K — (1+K)a2) ‘o (2a/+’)) K 0 Pk 2(K + 1)ke_K k-i (-i)’K£_dk=O (k—i—ic)! (ic’)2 =1, 6=0 a2 j ( ba2Nakagami—q Pa(a) = yTE2 exp ( (i_b2)y) ° (1_b2)y) Lk/2J (_i)b/2)’b-i+q2 Pk = (i_b2)k+h/2 (ic!)2 (k—2ic)! 0q<1 =1, 6=0 TU Nakagami—m pa(a) = () a2m exp (_) m 1/2 Pk = 2(_i)lc_lmk+m_l r(m)(k—1)! =1, ö=m—1 c/2 c/2l Weibull pa(a) = c (12) a exp [— (r(i + 2/c)) j C > 0 Pk = (_l)k_1 (r(1!.2/c))ck/2 (k—i)! = c/2,6 = 0 e{x2}= 2, i.e., 7 is the average SNR per symbol. 3.2.3 Admissible Types of Noise and Examples For the presented asymptotic performance analysis method to be applicable, the RVs n and n have to fulfill the following assumptions. AS1) The pdf p(n) of ii is an even function, i.e., pn(fl) = pn(—n). 51 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise AS2) The fundamental strip of the Mellin transform Ma(s) M{p(n)} of Pn(fl) includes the interval [1, oc), i.e., Ma(s) exists for 1 Re{s} < oo. This implies that all noise moments are finite. AS3) For two—dimensional linear modulation schemes and for BOM we assume that n has a rotationally symmetric pdf p(.), i.e., n and have the same pdf for all real AS4) The Mellin transform Ma(s) of pn(fl) and the coefficients Pk of the series expansion of pa(a) given in Table 3.1 fulfill the condition PkMn(2* + 26 + 1)p, Pk+lMn(2(k+1)+26+1) > ‘ where > 0 and S 0 are constants that depend on the fading pdf and are also specified in Table 3.1. We note that AS1) is mainly made for convenience as it simplifies our exposition and holds for most types of noise of practical interest. Similar results as in Sections 3.3 and 3.4 could also be derived for non—even pn(n). AS2) is necessary and holds for most practically relevant types of noise. We note, however, that AS2) does not hold for alpha—stable processes with c < 2 which have been occasionally used in the past to model impulsive noise, cf. e.g. [54j. For alpha—stable processes, Ma(s) does not exist for s> O + 1. AS3) is not necessary for one—dimensional linear modulation schemes but considerably simplifies the asymptotic analyses of two—dimensional modulation formats and BOM, respectively. AS4) is necessary and holds for most practically relevant combinations of noise and fading. For all considered fading distributions Ipk/Pk+1 increases at least with k for k — oo. This means A54) is fulfilled as long as M(2k + 26 + 1)/M(2(k + 1) + 26 + 1)1 does not decrease faster than 1/k for k —* oc. 52 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise Table 3.2: Pdf pn(fl) and Mellin transform Ma(s) = M{p(n)} for different types of noise. Noise variance o = 1 in all cases. Generalized Gaussian noise: ij(C) = [I(l/C)/1(3/C)]c/2.BPSK interference with fixed channel phase (CP): S is the set of all 2’ possible sums of the ±dk, 1 k I. S contains all positive elements of S. Noise Model pn(ri), I(s), and Ma(s) Gaussian Noise Pn(fl) = —i--- exp ( 2T) Mn(s)-)F() 2 Generalized Gaussian Pn(fl) = 2P(1/C)(,7(C))h/c’ exp (17(0)) Noise, C> 0 Ma(s) = 2P(i/c) () (i7C)) V”1 ck exp / 2Gaussian Mixture p,,(m) = k=1 flk) r(f)2 s—i>Z= Ck = 1 Ma(s) 2/ k=1 Ck >D CkOk = 1 BPSK Interference Pn(fl) = r 6(n — d) with Fixed CP Ma(s) = Z€s+ d ldkI2 = 1 M—PSK Interference pn(n) — 2 In < Idil, 1=17r.J’d1—n Id1’—’P() i = 1with Random CP Ma(s) = 2/F(±ih’2) IdkI2 = 2 Gaussian noise obviously fulfills AS1)—AS3). AS4) is also fulfilled for Rayleigh, Ricean, Nakagami—q, Nakagami—m, and Weibull fading with 0 < c 2. For Weibull fading IM(2k + 26 + 1)/M(2(k + 1) + 26 + 1) decreases as 1/kc’2 and p(, 6) = 0 follows for c > 2, i.e., AS4) is not met in this case. In the following, we will briefly discuss three relevant non—Gaussian types of noise which also fulfill at least AS1), AS2), and AS4). The pdfs p(n) and Mellin transforms Ma(s) for these noises as well as those for Gaussian noise and generalized Gaussian noise are summarized in Table 3.2. El) Gaussian mixture noise: Gaussian mixture noise is often used to model the combined 53 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise effect of Gaussian background noise and man—made or impulsive noise, cf. e.g. [37, 55, 56]. In this case, the pdf of n is given by Pnc(Thc)2C exp(_2 (3.5) where ck > 0 and > 0 are parameters. Two popular special cases of Gaussian mixture noise are Middleton’s Class—A noise [55] and c—mixture noise. For c—mixture noise I = 2, = 1 — c, c2 = c, = o, and °2 = ito, where c is the fraction of time when the impulsive noise is present, ,c is the ratio of the variances of the Gaussian background noise and the impulsive noise, and = 1/(1 — e + ice). It is easy to verify that AS1)—AS3) are valid for Gaussian mixture noise. AS4) is also fulfilled for all types of fading considered in this chapter except for Weibull fading with c> 2. E2) BPSK interference with fixed channel phase: The complex and real interference (noise) from I independent, symbol synchronous’3BPSK signals ik E {±1}, 1 k I, can be modeled as nc—dc,kik and n=dkik (3.6) respectively, where dc,k j’Pdc and dk dc,k cos(pdCk) denote the complex and the real gain of the kth interference channel, respectively. The interference channel phases c°d,k are assumed to be constant. We note that BPSK interference with fixed channel phase only fulfills Al), A2) and AS4), i.e., the validity of the presented asymptotic analysis is restricted to one—dimensional linear modulation schemes in this case. Note also that in this case AS4) is also fulfilled for Weibull fading with c> 2. 13We note that even if the BPSK interference signals are not symbol synchronous with the desired signal, for n, and n a similar model as in Eq. (3.6) results, cf. e.g. [27], and the mathematical tools developed in this chapter are still applicable. However, because of space limitations, we only consider the symbol synchronous case here. 54 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise ES) M—PSK interference with random channel phase: In this case, n and n are also given by Eq. (3.6) but the phases c°dck, 1 < k < I, are mutually independent RVs uniformly distributed in the interval (—n, ir] and k e {ei2m/Mlm e {O, 1, ..., M — 1}}. The randomness of the phases çod may be due to e.g. the lack of phase synchronization between the interferers and the desired signal. Because of the uniformly distributed phases, AS3) holds in addition to AS1), AS2), and AS4). Again, AS4) holds for all types of fading considered. 3.2.4 Mellin Transform of Composite Noise In general, the noise n may be the sum or the product of different RVs nk, 1 k I. In this case, the framework developed in this chapter is applicable as long as AS1)—AS4) hold and we explain in Appendix C how the Mellin transform Ma(s) of p(n) can be obtained from the Mellin transforms of the pdfs of k, 1 k I. To illustrate the application of the results in Appendix C, we briefly discuss two practically relevant examples. E4) Ricean faded M—PSK interferer: A Ricean faded M—PSK interferer can be modeled as n = n1 + n2, where n1 and n2 represent the direct and the specular (Rayleigh) component, respectively. n1 can be modeled by Example E3) and n2 is a real Gaussian RV. The Mellin transform Ma(s) can be obtained by applying Eq. (C.9)14, where M1(s) and M2(s) are given in Table 3.2. E5) Rayleigh faded multiple BPSK interferers: If multiple synchronous BPSK interference signals k, 1 < k I, originate from the same transmitter (e.g. base station), they arrive over the same channel at the receiver (e.g. moblile station or base station of another cell) for the desired signal. Therefore, if the interference channel is Rayleigh faded, this type of ‘4We note that n1 and 2 are statistically independent although they involve the same M—PSK inter ference signal. 55 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise interference can be modeled as ri = n,n2, where n1 and n2 represent the fading gain (real Gaussian RV) and the interference signal (modeled as in Example E2)), respectively.’5 In this case, Ma(s) can be obtained from Eq. (C.5), where M1(s) and M2(s) are again given in Table 3.2. This interference model applies for example to synchronous CDMA systems after despreading where the coefficients d,k in Eq. (3.6) denote the correlation of the desired user’s signature sequence with the signature sequences of the users in an neighboring cell [26]. For more complicated types of noise Eqs. (C.5) and (C.9) may have to be applied repeatedly. Alternatively, in cases where a closed—form expression for Ma(s) cannot be found or if only measurements of ri are available, the Mellin transform Ma(s) may also be estimated using Monte Carlo integration of Eq. (C.1). Exploiting that pn(ri) is an even function, an estimate of Ma(s) is given by 1 N I[(s) = (3.7) k=1 where n[k], 1 k N, are realizations of the RV n. For a sufficiently large number N of samples the estimate J[(s) will approach Ma(s). Of course, the validity of AS2) and AS4) has to be verified. However, this can often be accomplished without knowing p(ri) or Ma(s) explicitly. 3.3 Single—Branch Reception In this section, we develop exact and asymptotic expressions for the SER of linear mod ulation schemes such as M—PAM, M—PSK, and M—QAM with a single diversity branch. In addition, we also consider the asymptotic SER of BOM. ‘51t is interesting to note that a similar interference model as in E5) also holds for an asynchronous Rayleigh faded BPSK interferer, cf. [27]. 56 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 3.3.1 Basic Error Probability Result The calculation of the SER of linear modulation schemes involves the evaluation of the probability Pe(d) that the received signal r = ax + n is larger than a certain threshold ay, where d y — x > 0. Conditioned on n we obtain Pe(dn) = Pr{ax + n > ayn} = Fa (s), (3.8) where Fa(a) j’pa(U) du is the cumulative distribution function (cdf) of a. Averaging Pe(dln) with respect to n yields Pe(d) = e{Pe(dn)} f pn(fl)Fa () dn, (3.9) where we have exploited Fa(a) = 0 for a < 0. For most practically relevant types of noise and fading the integral in Eq. (3.9) cannot be solved in closed form. However, as long as closed—form expressions for both p(ri) and Fa(a) are available, numerical integration methods can be applied for computation of Pe(d) based on Eq. (3.9). Unfortunately, this numerical approach does not reveal how Pe(d) depends on the SNR. Therefore, to simplify Eq. (3.9), we assume that for high SNR ‘y the pdf pa(a) can be expanded into a series 1 °° 2 tk+ö pa(a) = - Pk (s-) , a 0, (3.10) where > 0 and 5 > 0 are real—valued constants and Pk are real—valued coefficients, cf. Table 3.1. With Eq. (3.10) the cdf Fa(a) can be expressed as 1 2 k+ Fa(a) = () . (3.11) 57 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise Table 3.3: Parameters ,8 and d for M—ary modulation schemes X e {PAM, PSK, QAM, BOM}. Modulation Scheme 4 M-PAM 1 - 17’ BPSK(M=2) 1 M-PSK (M 4) 1 /sin () M-QAM 2(1_s) 1BOM(M=2) 1 Applying Eq. (3.11) in Eq. (3.9) and assuming that AS2) in Section 3.2.3 holds, we can express the error probability as Pe(d) = Pk + 1) (d276. (3.12) The infinite series in Eq. (3.12) converges if AS4) is fulfilled. To see this we may use the so—called quotient criterion [53] to show that the series converges if > ° d2(p(,S))h/t (3.13) Therefore, as long as p(, 5) > 0 the series will start to converge for some finite SNR value o. The exact value of ‘Yo depends on both the type of fading and the type of noise, of course. For example, for Nakagami—m fading and Gaussian noise, we obtain p(l, m — 1) 1/(2m) and = 2m/d. This shows that for larger m (i.e., less severe fading) higher SNRs are required for convergence. 58 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 3.3.2 Exact SER Expression for M—PAM Although the main emphasis of this chapter is on SER approximations offering insight into the system behavior at high SNR, it is worth noting that for one—dimensional modulation schemes such as BPSK and M—PAM, Pe(d) in Eq. (3.12) can be used to derive an expression for the exact SER. In particular, using similar steps as for the Gaussian case in [26, Ch. 5] and assuming that AS4) is fulfilled and the SNR is high enough for Eq. (3.12) to converge, we obtain pAM =2 3M e (dM) PM Pk M(2(k± ö) +1) ((dM27(6 (3.14) where /3M and dM are given in Table 3.3. The SER of BPSK pSK can be obtained from pSK = p2PAM To verify the result in Eq. (3.14), we can consider the special case of M = 2, Nakagami—m fading, and Gaussian noise. Using , ö, and Pk from Table 3.1 and Ma(s) from Table 3.2, it is straightforward to show that under these conditions Eq. (3.14) can be simplified to [39, Eq. (9)]. We emphasize that the technique in [39] is limited to Gaussian noise, whereas Eq. (3.14) is valid for any type of noise fulfilling AS1), AS2), and AS4) in Section 3.2.3. We note that PjM can be evaluated numerically also by combining Eqs. (3.9) and (3.14). This numerical approach also succeeds at low SNRs, where Eq. (3.12) does not converge, but does not reveal the connection between the Mellin transform of p(n) and the SER. 3.3.3 Asymptotic SER of Linear Modulations As has been shown in Section 3.3.1 the series in Eq. (3.12) will converge for high enough SNR if AS4) is fulfilled. In that case we can also approximate Pe(d) by the first term of 59 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise the sum in Eq. (3.12). Based on this observation the asymptotic SER of linear modulation schemes can be obtained pX X Pi M(2( + 6) + 1) dx 2 —(t+6) MM 6 ((M)7) (. where X stands for PAM, PSK, and QAM, respectively. The respective values of and d are summarized in Table 3.3, cf. also [26]. We note that for M—PSK with M 4 and M—QAM AS3) in Section 3.2.3 is necessary to ensure that for high enough SNR and d are independent from the pdf of i-i. Eq. (3.15) shows that Eq. (3.1) not only holds for Gaussian noise but also for the more general class of noises considered in this chapter. In particular, comparing Eq. (3.1) and Eq. (3.15), the diversity gain is Gd = + S and the combining gain is given by 7 \1/Gd G_dx21 ‘-‘d 316c M /3plM(2Gd+1) Therefore, for high SNR in a log—log scale the slope of the SER curves (—Gd) only depends on the fading statistic but is independent of the noise statistic. On the other hand, the relative shift of the SER curves (Ge) depends on both the fading and the noise statistics. The pdf p(rI) of the noise n influences the combining gain via its Mellin transform Ma(s) for s = 2Gd + 1. Since Gd = + 6 depends on the fading process, the fading statistic also determines in part what effect the type of noise has on G and on the asymptotic SER. To further highlight this point, we specialize Eq. (3.15) in the following. 1) Rayleigh, Ricean, and Nakagami—q fading: In this case, = 1 and S = 0, cf. Table 3.1. Since M(3) = o/2 = 1/2 holds always, we obtain vx 6iiP1 M) 7 60 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise i.e., surprisingly the asymptotic SER for Rayleigh, Ricean, and Nakagami—q fading is independent of the noise statistic. If we further specialize Eq. (3.17) to BPSK and Rayleigh fading, we obtain pSK = 1/(47), which is a famous result for Gaussian noise [26J. However, our analysis here shows that this result is also valid for a much larger class of noises. ) Nakagami—m fading: Using = 1, 8 m — 1, and Pi = 2mm/P(m) in Eq. (3.15) yields 2/3mm_lM(2m+1) F(m) (dM)2m7m i.e., the SER depends on M(2m + 1). Therefore, in a log—log scale non—Gaussian noise will result in a horizontal shift of the SER curve by GN(m) log10 dB (3.19) compared to Gaussian noise. If GN(m) is negative, the SER caused by the non—Gaussian noise is lower than that caused by Gaussian noise. The opposite is true if GN(m) is positive. For the special case of c—mixture noise, Eq. (3.19) simplifies to GN(m) = 10 log10 ( — + ) dB. (3.20) For example, for c = 0.01 and ic = 100, we obtain GN(0.5) = —2.24 dB, GN(1) = 0 dB, GN(2) = 7.03 dB, and GN(3) = 10.34 dB, which clearly shows that given the same noise statistic, different fading statistics may cause significantly different combining gains. 3) Weibull fading: Adopting = c/2, 6 0, and P1 = (F(1 + 2/c))d/, we obtain pX 2 /3 (F(1 + 2/c))d/ M(c + 1) 3 21 c(d)c7/ ( . ) i.e., the asymptotic SER depends on M(c + 1). Similar to the Nakagami—m fading case, 61 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise non—Gaussian noise causes a horizontal shift of the SER curve. A comparison of Eqs. (3.18) and (3.21) shows that for Weibull fading with parameter c this shift is simply given by GN (c/2), i.e., the SER curves of Nakagami—m fading with parameter m and those of Weibull fading with parameter c 2m are shifted by the same amount if the noise is non— Gaussian instead of Gaussian. Note however that in case of Weibull fading the restriction c 2 is necessary since AS4) is not met for Gaussian noise and c> 2. 3.3.4 Asymptotic SER of BOM With the definitions r Re{r} and Im{rc}, in BOM, the output of the two correlators is {r /a + n, = n} and {r = n, 1 ‘/a + n} if bit “1” and “0” are transmitted, respectively, where n Re{n} and ñ Im{n} [26j. Assuming coherent detection, one defines the decision variable as (3.22) where x E {+v’} and ñ n — ñ. Comparing Eq. (3.22) with Eq. (3.3) it is obvious that the framework developed in Sections 3.3.2 and 3.3.3 is also applicable to BOM. For calculation of the Mellin transform Mfl(s) of ñ we note that ñ can be expressed as ñ = nI [cos(e) — sin(e)} = InI cosQ + -/4) (3.23) where e denotes the phase of n. Since according to AS3) in Section 3.2.3 e is uniformly distributed n = nj cos(e) and n cos(e + ir/4) have the same pdf. Therefore, from Eq. (C.4) we obtain Mn(s) = (\/)‘ Ma(s). Using this resuft, we obtain Eq. (3.15) with 130M = 1/2 and dM = 1, cf. Table 3.3. A comparison of BOM and BPSK yields /320M = /3’51 and dM = d’</\/ which shows 62 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise that BOM suffers from an SNR loss of 3 dB compared to BPSK. While this 3 dB loss is a well—known fact for Gaussian noise [26], our analysis here shows that it also holds for a much larger class of noises independent of the fading statistic. For completeness we note that the exact SER of BOM can be obtained by replacing Ma(s) by Mn(s) in Eq. (3.14). 3.4 Diversity Combining In this section, we extend the framework introduced in Section 3.3 to EGC and SLC. We assume that the signal model in Eq. (3.3) is valid for L diversity branches, i.e., rj=atx+ni, (3.24) where rj, al, and i denote the received signal, the fading amplitude, and the noise in the lth branch. Furthermore, we assume = 1, 1 <1 <L, and the SNR of the lth branch is S{a}, i.e., different branches may have different SNRs. For convenience, we assume that the fading gains in different branches are statistically independent and follow the same distribution (e.g. all branches are Nakagami—m distributed with the same m). The latter restriction is only made to arrive at simple and insightful results and the extension to the case where different branches follow different distributions is straightforward. We also assume that the noise RVs i, 1 1 <L, have the same pdf p(n) and fulfill AS1)—AS4) in Section 3.2.3. 63 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 3.4.1 Equal Gain Combining (EGC) In coherent EGG the complex received signals of all branches are co—phased and combined. The resulting decision variable is given by =r1ãx+ñ (3.25) - L - L where a = al and ii n. For the framework developed in Section 3.3 to be applicable to EGC, we require the series expansion of the pdf pa(a) of a and the Mellin transform Mfl(s) of the pdf of ñ. For general s and mutually dependent ri, 1 1 L, the calculation of Mn(s) may be quite involved. However, for the most important special case where the m1, 1 1 L, are mutually statistically independent and s is an integer, Mn(s) can be easily obtained by applying Eq. (C.9). For the series expansion of pa(a) we first note that the Laplace transform of pa(a) can be expressed as [50] a(s) £{p(a)} = flai(S) (3.26) where a1(S) 1{pa(a)} and paj(a) denotes the pdf of a1. Considering Eq. (3.10) 1aj(5) can be expressed as at(5) P(2(k + 6))(5271)_(k+3) (3.27) By combining Eqs. (3.26) and (3.27) we can obtain a series expansion for 4a(s) which then can be used to obtain the desired series expansion for pa(a). The first term of this 64 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise expansion is given by 1 P2 6 L L 2 Lp(a) = r(2L(+6)) () + ° ([I (a2/71))]. (3.28) With Eq. (3.28) and a similar approach as in Section 3.3 the asymptotic SER of linear modulation schemes with EGC can be written as pX zf [P(2( + 6))]L Mn(2L( +6) + 1) r X 2 M — L ( +6) P(2L( + 6)) ((dy) ) . (3.29) We note that Eq. (3.29) is only a valid asymptotic approximation of the true SER if the corresponding series expansion converges, cf. Eq. (3.12). To analytically verify this conver gence, we would have to establish a criterion similar to A54) by replacing Pk and Ma(s) in Eq. (3.4) with the expansion coefficients of p(a) and Mn(s), respectively. Unfortunately, the full series expansion of pa(a) is quite involved and offers little insight. However, ex tensive comparisons of Eq. (3.29) with simulation results suggest that Eq. (3.29) is a valid asymptotic approximation for the SER for Rayleigh, Ricean, Nakagami—q, and Nakagami— m fading with arbitrary number of diversity branches and all types of noise considered in this chapter. Specializing this result for BPSK to Ricean fading and Gaussian noise leads to pIC (1 + K)Le_LL/[2lL! jE which is in perfect agreement with [38, Eq. (5)1, [4, Eq. (9.38)1. We assume for simplicity equal branch SNRs j = y, 1 1 L, the diversity gain follows as Gd = L( + 6), whereas the combining gain is given by G — dx 2 1 Gd P(2Gd) 1/Gd 30 -( M) pf[P(2Gd/L)]LMn(2Gd+1)) . (. ) 65 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise From Eqs. (3.29) and (3.30) we observe that while the slope of the SER curves is not influ enced by the type of noise, the SER curve for non—Gaussian noise experiences a horizontal shift compared to the SER curve for Gaussian noise. This horizontal shift corresponds to a difference in the combining gain and depends on Mn(2Gd + 1). For Rayleigh, Ricean, and Nakagami—q fading Gd = L and the Mellin transform M(2L + 1) (and therefore also the SER and G) depends on the type of noise if L> 1. 3.4.2 Selection Combining (SLC) In SLC, only the path with the largest fading amplitude is considered for detection. There fore, the decision variable can be modeled as r=axH-n (3.31) where a max{ai, a2, ..., aL} and n has the same pdfp(n) as ru, 1 1 L. Since the cdf of a is given by Fa(a) = flf=i Fa) [4], where Fat(a) is the cdf of aj, we can obtain the expansion Fa(a) = 2+ ö)L (a2)+6 + o (ñ (a2/7l)6) (3.32) cf. Eq. (3.11). Therefore, exploiting Eq. (3.32) and using similar steps as in Section 3.3.1, we can express the asymptotic SER of linear modulations with SLC as pX P1 M (2L(±S)+ 1) j ((d)2 -(+ö) (3.33) Similar to the EGC case, we have verified the validity of Eq. (3.33) by comparing it with simulation results for Rayleigh, Ricean, Nakagami—q, and Nakagami—m fading and all types 66 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise of noise considered in this chapter. For BPSK transmission over a Rayleigh faded channel with Gaussian noise Eq. (3.33) simplifies to pSK = (2L)!/ [224L! J71], which can be shown to be identical to [4, Eq. (9.268)] for L = 2 and high SNRJ6 If we assume again equal branch SNRs 71 = ‘y, 1 1 <L, from Eq. (3.33) we obtain a diversity gain of Gd = L( + ö) and a combining gain of / r>L—lrL \ 1/Gd G_dx21 ‘ d 334c M kj3pfLLMfl(2Gd+1) Similar to the EGC case for L > 1 the SER and the combining gain depend on the type of noise also for Rayleigh, Ricean, and Nakagami—q fading. 3.4.3 Comparison of EGC and SLC It is interesting to compare the combining gains achievable with EGC and SLC. For this purpose, we define the relative gain G5(L) as the ratio of Eqs. (3.30) and (3.34) GE/s(L) = ( LLF(2Gd) Mfl(2Gd+1)\1/Gd \..(2Gd)L_1[r(2Gd/L)]L Mn(2Gd+ 1)) i.e., for high enough SNRs EGC achieves a gain of 10 log10(G5)dB over SLC. The first term on the right hand side of Eq. (3.35) is only influenced by the type of fading, whereas the second term is affected by both the type of noise and the type of fading. If we assume Rayleigh, Ricean, or Nakagami—q fading, Gd = L and Eq. (3.35) simplifies to GE/s(L) — ((2L)! M(2L + 1) 1/L (336c \, 2LMn(2L+1) ) 16More precisely, the variables p and g, which are defined in [4], have to be set to p = 0 and g = 1 in [4, Eq. (9.268)1 to obtain p2PSK 3/[8-y172j for high SNR. 67 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise If we furthermore assume Gaussian noise, we obtain G5(L) = [(2L)!/(2LL)]l > 1, i.e., EGC always outperforms SLC. For dual diversity and i.i.d. noise RVs ri, and n2 Eq. (3.36) can be simplified to __________ = / 6M(5) (3 37c \ / V2Mn5+3 where Eq. (C.9) and 114(3) = 1/2 have been exploited. As an example, we may consider the case of an interference limited system where n1 and n2 are due to a Ricean faded M—PSK interferer with Ricean factor K1 and uniformly distributed channel phase’7. The Mellin transform M(5) for this case can be calculated as explained in Example E4) in Section 3.2.4. The resulting G5(2) is GE/s(2) = /6+ 12K + 3K (3 38)C V 4+8K+3 which is a monotonic decreasing function in K1. For example, from Eq. (3.38) we obtain that the performance gain of EGC compared to SLC is 0.88 dB, 0.23 dB, and 0 dB for K1 = 0, K1 = 10, and K1 —b co, respectively. 3.5 Numerical Results and Discussions In this section, we verify the derived analytical expressions for the asymptotic SER for different practically relevant cases. First, the case of a single diversity branch is considered, then results for diversity combining are presented. ‘TWe note that since the two interference processes are Lid. and the pdf of the Ricean fading channel gain with uniformly distributed channel phase is rotationally symmetric, n1 and n2 are statistically independent although they involve the same M—PSK interference signal. 68 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 100 .::: .:::::::::::::::::::::::::::::::::::::::::: v simulatedSER V — asymptotic SER (theory) 101 V iO_2 :.: : , :0.5 :::::::::::::::::2:::..::::::::::.::::::::::::::::: .::::: : : : : : :: . : . : : : . rn: 4.1::.:: 10 10_6 : : ::: : :: :: .: 1o • : : Vfl :4:::::::: 10 I I I I I 0 5 10 15 20 25 30 35 40 45 50 10 1og0(SNR) [dB] Figure 3.1: SER vs. SNR for 8—PSK over a Nakagami—rn fading channel with E—mixture noise (€ 0.25, ,c = 10). Markers: Simulated SER. Solid lines: Asymptotic SER [Eq. (3.18)]. 3.5.1 Single—Branch Reception Fig. 3.1 shows the SER of 8—PSK modulation in Nakagami—m fading and c—mixture noise ( = 0.25, ,c = 10). As expected, for high enough SNR the simulation curves (markers) closely approach the asymptotic results obtained from our analysis (solid lines). In Fig. 3.2, we show the SER (which is identical to the bit error rate in this case) for BPSK modulation in Nakagami—m fading (m = 2) for some types of noise discussed in Sections 3.2.3 and 3.2.4. Fig. 3.2 clearly illustrates that for a given SNR the SER caused by non—Gaussian noise and interference may be considerably lower or higher than that caused by Gaussian noise. Fig. 3.3 shows the SER of a NB signal having bandwidth B3 and employing 16—QAM in 69 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 100 1:::;.::::::: .:I::::::::::J:::. Gaiissiarr,oise :..:::::::..::c::::::::;::::::::::::::: V o—mixturenoise(o=0.01,K=100) -t N., N 0 E—mixture noise (€ = 0.1 K = 1000)10 X M—PSK interference with random CP (1=1) N. N 0 Rayleigh faded BPSK interference (1=2) N. BPSK interference with fixed CP (1=2) - V . 10 2 : . .:::: .:.: .::::::::::::::::::::::::::::::::: 10 1o . 10 ::.::::: ::.(/) :.::.: :.:: ::.:::::::::::: . :.: 10 HH io7 : 1 0 5 10 15 20 25 30 35 40 10 1og0(SNR) [dB] Figure 3.2: SER vs. SNR for BPSK over a Nakagami—m fading channel with m = 2 and different types of noise discussed in Sections 3.2.3, 3.2.4. c—mixture noise: Example El) in Section 3.2.3. M—PSK interference with random CP: Example E3) in Section 3.2.3. Rayleigh faded BPSK interference: Example E5) in Section 3.2.4. BPSK interference with fixed CP: Example E2) in Section 3.2.3. Markers: Simulated SER. Solid lines: Asymptotic SER [Eq. (3.18)]. Nakagami—m fading (m = 2) with a MB—OFDM and a DS—UWB interferer, respectively. The NB pulse shape is a square—root raised cosine filter with roll—off factor 0.35 and the receiver employs the corresponding matched filter. The MB—OFDM and DS—UWB inter ferers were generated according to the corresponding IEEE 802.15.3a standard proposals [44, 57]. Since a closed—form calculation of the related Mellin transforms is too involved, we estimated M(s = 5) using Eq. (3.7) and then calculated the asymptotic SER using Eq. (3.18). This semi—analytical approach is much faster than directly simulating the SER. Interestingly, Fig. 3.3 shows that while for B = 1 MHz the MB—OFDM interferer yields a 70 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise 100 “ v MB—OFDM B = 1 MHz (simulation) :: DS—UWB, B= 1 FV1Hz(sirnulation) ‘“cS.’s..’ MB-OFDM,B5=2OMHz(simulation) io’ D vI—fz(sirriulation) — IIB—OFDM(theory) . — - DS-UWB (theory) “ ‘S ‘S ....‘‘S5 ‘.‘.N. -2 .5... :: .5::.:.: ::::. : ..:::: :.: ::.;::J>ç:..’. .....:.:::. ‘S. ‘S D.’: •• “ ‘5 10 iO_6 I I 10 15 20 25 30 35 40 10 log10(SNR) [dB] Figure 3.3: SER vs. SNR for 16—QAM with bandwidth B3 over a Nakagami—m fading channel with m 2 and UWB interference. Markers: Simulated SER. Solid lines: Asymp totic SER for MB—OFDM interference [Eq. (3.18)]. Dashed lines: Asymptotic SER for DS—UWB interference [Eq. (3.18)]. lower SER than the DS—UWB interferer, the opposite is true for B8 = 20 MHz. 3.5.2 Diversity Combining Fig. 3.4 shows the SER of BPSK with EGC and SLC, respectively, in Ricean fading (K = 2). We assume L = 2 and identical SNRs for both diversity branches. The BPSK signal is impaired by a Ricean faded M—PSK interferer with Ricean factor K1 and uniformly distributed phase, cf. Sections 3.2.4, 3.4.3. The relative performance loss of SLC compared to EGC is smaller for K1 = 10 than for K1 = 0 as predicted by Eq. (3.38). It is also interesting to note that both EGC and SLC achieve a better performance for the larger K1. 71 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise N’. I _____________________________:: ::::: : : + EGG, l = 0 (simulation) .: v EGG, l = 10 (simulation) , SLG, K = 0 (simulation) 0 SLG, K1 = 10 (simulation) “<) — EGG (theory) — SLG (theory) N cçN-’ •‘“. -N I N’.NN’ N A : :::::::::::::::::::::::::::::: ::.ç:::.::::::: T N : N’>’±•• . -“.. - . - N - . . N N N N’. s.c N - N ‘ N N - —6 : N10 .::- ::::::::::::::::::::::::: ::.:::*:_222426 10 log10(SNR) [dB] Figure 3.4: SER vs. SNR per branch for BPSK over a Ricean fading channel with Ricean factor K = 2, L 2 diversity branches, and Ricean faded M—PSK interference. The interference channel has Ricean factors of K1 = 0 and K1 = 10, respectively. All diversity paths have the same average SNR. EGC and SLC are considered. Markers: Simulated SER. Solid lines: Asymptotic SER for EGC [Eq. (3.29)]. Dashed lines: Asymptotic SER for SLC [Eq. (3.33)1. In Fig. 3.5, we consider the SER of 8—PSK in Rayleigh fading with EGC. We assume that the composite noise impairing the received signal is the sum of two Rayleigh faded BPSK interferers [cf. Example E5) in Section 3.2.41 and Gaussian noise, where the inter ference power is 10 dB higher than the Gaussian noise power. For comparison we also consider the case of purely Gaussian noise. As expected from Eq. (3.17) for L = 1 both types of noise yield the same asymptotic SER. In contrast, assuming identical SNRs for L = 2 and L = 3 purely Gaussian noise is less favorable than the composite noise. 72 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise H .: ,.: v Rayleigh faded BPSK interferers + Gaussian noise (simulation)a... C Gaussian noise (simulation) -l — Rayleigh faded BPSK interferers + Gaussian noise (theory)10 ——Gaussiannoise(theory) 10 1og0(SNR) [dB] Figure 3.5: SER vs. SNR per branch for 8—PSK over a Rayleigh fading channel with EGC. All diversity paths have the same average SNR. Markers: Simulated SER. Solid lines: Asymptotic SER for Rayleigh faded BPSK interference (two interferers) and Gaussian noise [Eq. (3.29)]. Dashed lines: Asymptotic SER for Gaussian noise [Eq. (3.29)]. 3.6 Conclusions In this chapter, we have presented a powerful novel approach to the asymptotic SER anal ysis of linearly modulated signals impaired by fading and (possibly) non—Gaussian noise. Thereby, the only major assumption on the considered noise is that the Mellin transform Ma(s) of its pdf exists for s 1, or equivalently, all the noise moments remain finite, which is true for most practically relevant types of noise. Based on this assumption we have provided general and simple—to—evaluate SER approximations for linear modulation formats with single—branch reception, EGC, and SLC and for BOM, which become tight for high SNR. Our analysis has shown that the diversity gain is independent of the noise statistic and only depends on the fading statistic and the number of diversity branches. In 73 Chapter 3. Performance of Generic Uncoded Systems in Non—Gaussian Noise contrast, the combining gain depends on both the type of fading and the type of noise via the Mellin transform of the noise pdfs or the noise moments. Therefore, in a log—log scale for high SNR the SER curves for different types of noise are all parallel and their relative shift depends on the noise moments. 74 Chapter 4 Performance of Generic BICM-OFDM Systems in UWB Interference 4.1 Introduction In Chapter 2, we studied the performance of generic uncoded victim systems in MB—OFDM interference. In this chapter, we extend the results obtained in Chapter 2 to coded systems and in particular, investigate the impact of UWB interference on generic BICM—OFDM systems. BICM—OFDM is considered for the victim system since this scheme has recently become the most important air interface in wireless communications. In particular, BICM— OFDM has been used e.g. in IEEE 802.11 WLAN [13] and IEEE 802.16 WiMAX [58] and is also likely to be adopted by most future wireless communication standards. For the UWB system we consider the three major technologies that will be used in commercial products in the near future. Specifically, we concentrate on MB—OFDM UWB, DS—UWB, and IR—UWB which have been proposed in the ECMA standard [1], the IEEE 802.15.3a standard proposal [2], and the IEEE 802.15.4a standard [3], respectively. The effects of UWB interference on the performance of specific types of victim systems that employ BICM—OFDM have been studies in [5, 30, 32, 33]. In particular, The perfor 75 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference mance degradation suffered by IEEE 802.lla WLANs as a result of generic IR—UWB and DS—UWB interference was investigated by computer simulations in [5, 32] and [33], respec tively. The impact of MB—OFDM UWB and DS—UWB interference on the BER of C—band DTV, which employs OFDM and convolutional coding without bit interleaving, was stud ied via simulations in [30]. Although the simulative studies in [5, 30, 32, 33] are quite comprehensive, their applicability is limited to a particular victim BICM—OFDM system. Therefore it is difficult to generalize these results to other existing or future BICM—OFDM systems that use e.g. different OFDM sub—carrier spacings and/or different code rates. In this chapter, we provide an analytical framework that allows us to accurately cal culate the BER of a generic BICM—OFDM system impaired by UWB interference and AWGN. The presented framework is general enough to include various different channel models for the BICM—OFDM system and the UWB interference. Furthermore, we derive a simple GA for the exact BER which is easy to compute. We note that our results are not only applicable to WB BICM—OFDM victim systems such as IEEE 802.lla WLAN and IEEE 802.16 WiMAX but also to UWB BICM-OFDM victim systems such as ECMA MB OFDM UWB. We show that the impact of UWB interference on BICM—OFDM strongly depends on the UWB format and on the OFDM sub—carrier spacing /f5. For example, while the GA is very accurate for DS—UWB and all sub—carrier spacings of practical in terest (e.g. L\f8 < 10 MHz), for IR—UWB and MB—OFDM UWB the GA is only accurate for Lf8 < 100 kHz. In fact, for IR—UWB the GA may lead to a substantial underestima tion of the effect of UWB interference and overly optimistic performance predictions. For MB—OFDM UWB the GA overestimates the effect of the interference for /f3 < 2 MHz and underestimates it for /f3 > 2 MHz. On the other hand, while the performance of the BICM—OFDM system is practically constant over the entire bandwidth of the IR—UWB and MB—OFDM UWB systems, it may be frequency dependent for DS—UWB due to the 76 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference Figure 4.1: Block diagram of considered system model including the BICM—OFDM trans mitter (Tx), the BICM—OFDM receiver (Rx), and I UWB interferers. possibly non—flat PSD of DS—UWB. The remainder of this chapter is organized as follows. In Section 4.2, the system model is presented, and in Section 4.3 the MGFs of the considered UWB formats are provided. The performance of BICM—OFDM in UWB interference is analyzed in Section 4.4. In Section 4.5, this performance analysis is used to study the impact of the considered UWB formats on BICM—OFDM, and conclusions are drawn in Section 4.6. 4.2 System Model The considered system model comprises one BICM—OFDM transmitter, one BICM—OFDM (victim) receiver, and I UWB interferers, cf. Fig. 4.1. In this section, we present the mathematical models for all components of the system. We note that for convenience all signals and systems are represented by their complex baseband equivalents. 4.2.1 BICM-OFDM System We assume that the victim system employs the popular BICM—OFDM concept, e.g. [59, 60]. Therefore, coding is performed along the frequency axis over the N sub—carriers of a single OFDM symbol using the concatenation of a convolutional encoder of rate R, an interleaver, n(t) 77 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference and a memoryless mapper. In particular, the elements of the codeword c [ci, c2,. . , CmN] are interleaved, and the interleaved bits are broken up into sub—sequences of m bits each, which are subsequently mapped to symbols x[k] from a constellation X of size Xj M = 2 to form the transmit sequence x [x[—N/2j, x[—N/2 + 1],... , x[N/2 — 1]j (N is even). Subsequently, the transmit symbols x[k] are modulated onto the N OFDM sub—carriers resulting in the baseband transmit signal s(t) = x[k] k(t), (4.1) where k(t) ei2 fktw(t) is the kth OFDM sub—carrier waveform with sub—carrier spacing /f3. Here, w8(t) is a rectangular pulse with w3(t) = \/j, —T < t < 1/f8, and w3(t) = 0 otherwise, where denotes the length of the cyclic prefix. The duration of one OFDM symbol is T8 = + 1/f3. s(t) is transmitted over a multipath channel with an impulse response h(t) which is zero outside the interval [0, The received signal r(t) is impaired by AWGN n(t) and I UWB interferers i,1(t), 1 ii I. Consequently, the received signal can be modeled as r(t) = s(t) ® h(t) + n(t) + g(t) 0 ei2 t_m)j(t — ru), (4.2) where g(t), r, and f,, are the (causal) channel impulse response of length Tg, the delay, and the frequency offset of the vth UWB signal, respectively. Both r and f are defined relative to the victim signal. The typical OFDM receiver processing involving low—pass filtering, sampling, and discrete—time Fourier transform (DFT) can be equivalently represented by filtering of the received signal with a bank of matched filters bk(t) = (—t), —1/f3 t 0, —N/2 k N/2 — 1, and bk(t) = 0 otherwise, and subsequent sampling [61]. Assuming coherent reception, the sampled output of the kth matched filter is obtained as 78 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference r[kJ e01 r(t) ® k(t)t=O = a[k]x[k] + n[k] + i[k], — <k — 1, (4.3) where i[kj is the (effective) UWB interference, H[kJ h(t)e_i2f8ktdt = c[k1eieHUd1 is the gain of the kth sub—carrier with magnitude c[k] and phase eH[k], and n[k] is AWGN. The UWB interference is given by i[k] e1k(t) ® g(t) 0ei2Mt_T v(t — r)Io gv[k]iv[k] (4.4) with gv[k} = ejeH[Iej2TG,1(kf3), (4.5) i,[k] = ?/k(t) ®ei2vtiv(t — Tv)tO, (4.6) where we have assumed that the frequency response of the UWB channel G(f) f1;rg g (t) e_j2ftdt is approximately constant over the range of frequencies where ‘k (t) has significant energy. Throughout this chapter we assume that the i [k], g,, [k], and r,1 of different UWB interferers v are independent RVs. The statistical properties of the BICM—OFDM victim and the UWB interference channels will be discussed in the next subsection. 4.2.2 Channel Models The performance of the BICM—OFDM system depends on the statistical properties of c[k] and 9v [k], ii I. To arrive at mathematically tractable results, we assume that the BICM—OFDM victim and the UWB interference channels are either severely frequency— 79 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference selective (SFS) or frequency—flat (FF) over the bandwidth B3 of the BICM—OFDM system. In the following we characterize c[kJ and g[kJ for the resulting four special cases. Cl) SFS Victim and Interference Channels: If all involved channels fluctuate sig nificantly within the bandwidth B3 of the BICM—OFDM system, assuming ideal bit— interleaving we may model H[k] as i.i.d. .f(0, 1) and g[k] as i.i.d. .A/(0, o11) RVs (Rayleigh fading), where o,1 .f { Ig1,[k] 2 } We note that different UWB interferers may have dif ferent variances due to e.g. different distances from the victim receiver. The c[kj are i.i.d. Rayleigh distributed. The assumption of i.i.d. Rayleigh fading along the frequency axis is popular in the BICM—OFDM literature, e.g. [60]. C2) FF Victim and SFS Interference Channels: If the victim channel frequency re sponse is constant, c[k] 1, Vk, is valid. The interference channel gain g[k] is still modeled as i.i.d. f(0, o,,) distributed RV. C3) SFS Victim and FF Interference Channels: In this case, the a[k] are i.i.d. Rayleigh distributed and the phase of the i.i.d. Gaussian H[kl renders the g[k] i.i.d. RVs that can be modeled as g[k] = th cf. Eq. (4.5). Thereby, th, g,,[k]j is constant and e[k] is an i.i.d. RV which is uniformly distributed in [—it, it). Similar to Cl, different UWB interferers may have different gains g,, due to e.g. different distances from the BICM—OFDM receiver. Assuming a frequency flat interference channel is popular in the literature, e.g. [62, 63], as it allows an unobscured view of the degradation caused by the interference signal. This model may also be practically relevant if the BICM—OFDM bandwidth B3 is small and the interference source is in close proximity to the victim receiver. C) FF Victim and Interference Channels: Now, a[k] = 1, Vk, is valid and g[k] is modeled as g[k] = , e’, where ,. is still constant and e. is identical for all sub—carriers and uniformly distributed in [—it, it). This channel model eliminates all fading effects and thus, allows us to separate the performance degradation caused by the interference from 80 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference the degradation caused by signal fading. We note that practical channels will be in between the four considered extreme cases. However, as will be shown in Section 4.5, many qualitative results regarding the impact of UWB on BICM—OFDM systems are practically independent of the adopted channel models. 4.2.3 UWB Signal Model In this chapter, we consider the following UWB signal models: MB—OFDM [1], DS—UWB [2], and IR—UWB [3]. We note that the three underlying standards/standard proposals [1, 2, 31 also include optional forward error correction (FEC) coding. However, since the applied FEC coding does not change the relevant statistical properties of the UWB transmit symbols, it can be ignored for the purpose of interference analysis. For simplicity of notation, we drop the subscript v in the UWB signal i(t) in the following if no confusion arises. Various system parameters for the considered UWB formats can be found in Table 4.1. MB-OFDM Signal Model The adopted MB—OFDM model closely follows the ECMA standard [1]. In particular, the MB—OFDM signal is modeled as N/2 i(t) = a[i] ej2 (t—kT) w(t — icT) 2Mjt, (4 7) tç=—co n=—N/2 nØO where N, T, and an[it1 e {±1 ±j} denote the number of sub—carriers, the sub—carrier spacing, the symbol duration, and the data symbols of the MB—OFDM signal, respectively. The MB—OFDM pulse shape w(t) is given by w(t) = 1, T t T — TG, and w(t) = 0 81 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference Table 4.1: Parameters of considered UWB formats. These values were taken from the respective standards/standard proposals [1, 2, 3]. We note that for IR—UWB there are also optional channels with B8 > 500 MHz. r MB-OFDM [1] Number of sub—carriers N1 = 128 Sub—carrier spacing = 4.125 MHz Symbol duration T1 = 312.5 ns Prefix duration T 60.61 ns Guard interval TG = 9.47 ns Frequency bands NB = 3 DS—UWB [2] Chip duration T = 0.762 ns Bandwidth B1 = 1.3 GHz Chip pulse shape p(t) Square—root Nyquist pulse Roll—off factor 0.3 IR-UWB [3] Chip duration T = 2.02 ns Bandwidth B1 = 499.2 MHz Chip pulse shape p(t) Square—root Nyquist pulse Roll—off factor 0.6 otherwise, where T and TG T1 — (Tp + 1/f) denote the zero—prefix duration and the guard interval, respectively. Moreover, in Eq. (4.7), fMB[It] e {(ri — 1)Nl/.fI rib E {1, 2, ..., NB}} is a periodic, time—dependent frequency offset used to switch between the NB MB—OFDM frequency bands. In order to simplify the exposition, in the following we assume that the victim OFDM signal lies fully in the first MB—OFDM frequency band and is affected only by every NBth MB—OFDM symbol. We note that as mentioned in Chapter 2 (cf. Subsection 2.2.2), the statistical properties of the data symbols a[ic] depend on the MB—OFDM data rate. For the two highest data rate options (320 and 480 Mb/s) the a[ic] are i.i.d., whereas for data rates below 320 Mb/s each OFDM symbol is repeated once in time, i.e., a[2ic] = a[2ic + 1], and for the two lowest data rates (55 and 88 82 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference Mb/s) conjugate symmetry with respect to the sub—carrier index is also enforced, i.e., a[ic] = a[ic], 1 n N/2. DS—UWB Signal Model For DS—UWB we closely follow the IEEE 802. 15.3a standard proposal [2], where both BPSK and 4—ary bi—orthogonal keying (4—BOK) are considered for modulation. In case of BPSK modulation, the DS—UWB signal is given by a[k]p(t-kT), (4.8) where a[ic] e {±1} and T denote the data symbols and the symbol duration, respectively. Furthermore, the DS—UWB pulse shape is p(t) cip(t — lT), where c1, p(t), and T denote the spreading sequence of length L’8, the chip waveform, and the chip duration, respectively. For 4—BOK signaling the UWB interference signal is given by i(t) = a[k]pb[k](t — iT), (4.9) where a[ic] E {±1} and b[ic] e {0, 1} represent the 4—ary data symbol. The pulse shape is defined as pb(t) Z’ cj,bp(t — lTd) with orthogonal spreading sequences c1,0 and c1,. For T, T, L, and p(t) the same definitions as in the BPSK case apply. Both a[t] and b[ic] are i.i.d. sequences. The standard proposal [2] envisions two frequency bands of operation: a lower band from 3.1 0Hz to 4.85 0Hz and a higher band from 6.2 GHz to 9.7 GHz. The chip durations T and the bandwidths B are different in both bands. The values given in Table 4.1 are 18We note that in Chapter 3, we used L to denote a different quantity, i.e., the number of diversity branches at the receiver. 83 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference valid for the lower operating band. The data rate of DS—UWB is adjusted via the spreading sequence length which varies between L = 1 and L = 24, cf. [2, Tables 3—6j. We note that the PSD of DS—UWB is flat for L < 12 since in that case the spreading sequences have only a single non—zero chip. However, for L = 12 and L = 24 the PSD is not flat since there are multiple non—zero chips. IR-UWB Signal Model For IR—UWB we adopt the signal model proposed by the IEEE 802.15.4a standardization committee [3] which employs a combination of BPSK and binary pulse position modulation (BPPM). An IR—UWB symbol has duration T and consist of Nb burst of duration Tb Ti/Nb. Similarly, a burst consist of L chips of duration T Tb/L. The IR—UWB signal can be modeled as co L—1 i(t) = a[itj p[i, 1] p(t — iT — h[!]Tb — IT0 — b[ic] TppM), (4.10) where a[ic] e {±1} and b[tc] E {0, 1} denote the i.i.d. BPSK and BPPM data symbols, respectively. Furthermore, p0(t) is the chip waveform and TppM is the delay for BPPM. p[ic, 1] e {±1} and h[k] e {0, 1, ..., Nh — 1} are the scrambling sequence and hopping sequence, respectively, where Nh denotes the number of hopping positions. Both p[ic, 1] and h[ic] are pseudo—random sequences and therefore they are assumed to be (approximately) i.i.d. Similar to DS—UWB, IR—UWB has a lower operating band from 3.2 GHz to 4.7 GHz and a higher operating band from 5.9 GHz to 10.3 GHz. The only mandatory data rate is 0.85 Mb/s in the lower operating band, but in both bands there are several optional data rates ranging from 0.1 Mb/s to 27.24 Mb/s. Different data rates are realized by selecting appropriate combinations of Nb E {8, 32, 128} and L = 2’, i e {0, 1,..., 9}, cf. [3, Table 84 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference 39a] for details. 4.3 MGF of UWB Interference Signal For the error rate analysis presented in Section 4.4 the MGF of Re{g,jk]i[k]} conditioned on g[k] and -r defined as [k](sIg[k], r) Pc] iv[k]} jg[k], r} plays a major role. Therefore, in this section, we calculate I(sg[k}, T) for the considered UWB formats. For this purpose, it is useful to note that Eq. (4.6) can be simplified to i[k] = /757 / 2 v_3ti(t — r) dt. (4.11) For simplicity of notation we will drop in the remainder of this section the subscript ii, 1 < v < I, in f,,, rb,, g,,[k], and i(t), respectively. For convenience, the results of this section are summarized in Table 4.2. 4.3.1 MGF of MB—OFDM Interference The MGF of MB—OFDM interference was derived in Chapter 2. As explained in Chapter 2, because of the hopping between different frequency bands, the repetition in time of the MB—OFDM data symbols a[ic} for data rates of less than 320 Mb/s does not affect the MGF. It was further shown that the conjugate symmetry of the MB—OFDM data symbols for data rates of 55 and 80 Mb/s does have an impact on the MGF but the impact on the (uncoded) BER was found to be minor. Therefore, here we concentrate on MB—OFDM data rates of more than 80 Mb/s, but note that all our results can be easily extended to lower data rates, (cf. Chapter 2). Adjusting the derivations in Chapter 2 to the problem at hand yields the conditional MGF given in Table 4.2. 85 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference Table 4.2: Conditional MGFs [k](sg[k], r) for considered UWB formats. For MB—OFDM n, ri, ‘ti, and Ic are chosen such that /3[ic, k] 0 for n [n1, flu] or/and , 0 [icj, ‘ca]. For DS—UWB and IR—UWB the limits Icj, ‘u are chosen to ensure /3j [ic, k] 0 and /3j[Ic, k,t9[ic]] 0 for ic 0 is1, respectively. The integrals required for calculation of 3j[ic, kj and i3i[’c, k,t9[ic]] for DS—UWB and IR—UWB can be easily evaluated numerically, respectively. MB-OFDM t[k](sg[k], r) cosh (s Re{g[kj/3[,c, k]}) cosh (s Im{g[k]/3[ic, k]}) /3[ic, k] = (em[Idl —et) = 2ir(f — /.f8k+ flLS.f) l=min{1/zf—r—kNBT,T—TG},1l=max{—r—KNBT,Tp} = E(TG — T — T)/(NBTi)1, u = L(’/f — T — r)/(NBT)J DS-UWB ‘[k](sIg[Jcj,r) cosh(sRe{g[k] >DZc118[ic, k]}) (BPSK) 4)[](sg[k}, r) cosh(s Re{g[k] ‘ cj,/3j[ic, k]}) (4—BOK) i3 [ic, k] = fluei2(f_fsc)tp(t) dt lu1/fsTTi1Tc,liTiTilTc IR-UWB [k](sIg[k], r) * fJZ’ cosh(s Re{g[k]3j[ic, k, hTb + bTppM]}) /3l[P, k,9[’1] = fluei2(f_)tpc(t) dt lu =1/fs_r_,cTj_lT-9[tc],ljt=_r_,cTj_lT--t9[,c] 4.3.2 MGF of DS—UWB Interference Exploiting Eqs. (4.8) and (4.11) we first rewrite i[k] for BPSK DS—UWB as i[k] =a[k]c1j[, , (4.12) where /3j[k, k] is defined in Table 4.2. Based on Eq. (4.12) and the fact that the a[tt] E {±1} are i.i.d., [k](sg[k],r) can be obtained as given in Table 4.2. 86 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference Similarly, for 4—BOK the interference signal can be expressed as 00 L—1 i[k] = a[ic]cj,6i3i[ic,k} (4.13) where [i, kJ is identical to that of the BPSK case. Averaging the MOF first over a[ic] we obtain 00 L—1 [kJ(sg[k], r,b[ic]) = fl cosh ( Re {[k1 Cj,b[]j[, k] }). (4.14) The final result for ‘I[k](sjg[k1, r) is given in Table 4.2 and is obtained by averaging r, b[ic]) in Eq. (4.14) with respect to b[k] and noting that b[ic] e {0, 1} is an i.i.d. RV. 4.3.3 MGF of IR—UWB Interference For calculation of the MGF of IR—UWB interference it is convenient to rewrite Eq. (4.10) as i(t) = â[,l]p(t — - lT - [ic]), (4.15) where a[it, 1] a[ic]p[ic, 1] and 9[ic] h[lc]Tb + b[/c]TppM. We note that ã[ic, 1] E {+1} is i.i.d. and z9[ic] is i.i.d. and has pdf Nh—i 1 p,9) = — hTb — bTppM). (4.16) h=O b=O From Eq. (4.11) we obtain for the effective discrete—time interference i[k] = (4.17) 87 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference with /3i[’c, k, [ic]] as defined in Table 4.2. Using now i[k] from Eq. (4.17) and averaging over ã[ic, 1], which is i.i.d. with respect to both ic and 1, we obtain ik1(8Ig[k], r, = fifi cosh (s Re{g[k]1[,k,9[k}}}). (4.18) The final result for 4[](sg[k], r) given in Table 4.2 is obtained by exploiting that t9[ij is an i.i.d. RV when averaging [k](sg[kJ,T,z9Ic]) with respect to 9[k]. 4.4 BER of BICM-OFDM Systems In this section, we derive a tight upper bound on the exact BER of a BICM—OFDM victim receiver impaired by UWB interference and AWGN. In addition, we also provide a simple and easy—to—evaluate GA for this upper bound. However, first we briefly review the bit metric used for Viterbi decoding in the BICM—OFDM receiver and calculate the conditional MGF of the metric difference. 4.4.1 Conditional MGF of Metric Difference As customary, we assume that the BICM—OFDM system employs Viterbi decoding with branch metric [59] )[k] mm {r[k]—a[k]x[k]I2} (4.19) x[k}EX for the ith bit of the kth sub—carrier. Here, X denotes the subset of all symbols in constellation X whose label has value b E {O, 1} in position i. For BER calculation the 88 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference MGF of the metric difference (x[k],z[k]) Ir[k] — c[k]x[k]I2— Ir[k] — a[k]z[k]12 = —c2[k]d[kJ + 2o [k] dxz[kl Re{e ‘1(i[k] + rt[k])} (4.20) is of interest, where x[k] and z[k] x[k] denote the transmitted symbol and another symbol in X, respectively. Furthermore, we used in Eq. (4.20) the definition x[k] — z[k] dxz[k1e, where d[k] and 9d[k] denote the magnitude and phase, respectively. Since both i[kJ and n[k] are rotationally symmetric RVs, cf. Section 4.2, Cases C1—C4, we can express the conditional MGF of (x[kj, z[k]) as [k,Z[k(sIg[k], r,c[k]) {e_3klz[} =e2z[k1(1+8)s fl (2a[k]d [k}sg[k] , ru), (4.21) where we used the definitions g[k] [gi[k] ... gi[k]] and r [Ti ... Ti], the fact that n[k] is .Ar(0,0-) distributed with MGF 4(s) = ‘{e_811ll} = e2°, and the assumption that the UWB interferers are mutually independent. 4.4.2 Union Bound for BER The union bound for the BER of a convolutional code of rate R = k/ri (k and n are integers) is given by Pb - w(d) P(c —* ê), (4.22) ddmin where c and ê are two distinct code sequences with Hamming distance d that differ only in 1 1 consecutive trellis states. Furthermore, w(d) and dmjn denote the total input weight of error events at Hamming distance d and the minimum Hamming distance of the 89 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference code, respectively. P(c — ê) is the pairwise error probability (PEP), i.e., the probability that the decoder erroneously chooses the code sequence ê when the code sequence c is transmitted (assuming that c and ê are the only possible code sequences). It is convenient to express the PEP in terms of an MGF as [59] c+joo P(c -* è) = -- f e{(sI)} (4.23) c—joo where c is a small positive constant that lies in the region of convergence of the integral. The MGF I’(sIr) is conditioned on -r since the delay -r is constant for one OFDM symbol, and thus the bit de—interleaving does not result in an averaging with respect to T. W(slr) depends on the BICM—OFDM victim and the UWB interference channel models. In particular, for Cases C1—C3 specified in Section 4.2, the bit de—interleaving renders the g[k] involved in the considered error event c —f ê statistically independent. Thus, following the same steps as in [59] for the AWGN channel and exploiting Eq. (4.21), we obtain ‘I’(slr) = (1(sIr))d with N1 m2N {ea2z (1+su)s II r1,)} },j1 ii=1 (4.24) where z [k] represents the nearest neighbor of x [k] in with b being the bit complement of b, i.e., we invoke the BICM expurgated bound from [59]. We note that for C2 we may set = 1 and omit the averaging over c in Eq. (4.24), whereas for Cl and C3 o is Rayleigh distributed. Similarly, for Cl and C2 g, are .IV(O, RVs, whereas for C3 g1 = je with uniformly distributed phase e. For C4 g[k] = g [gi g ... gi] is constant over all sub—carriers, and thus is not affected by the bit de—interleaving. Since a = 1 for C4, we obtain ‘IJ(slr) = ‘g{(I’(sIg, r))d} with 90 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference m 1 I J(sg,T) = m2N e k](1+su [k1(2dXZ[k]sgVrL,) (4.25) k=— i1 b=O x[k]EX where g, = ,e9v with i.i.d. uniformly distributed e. An approximate upper bound for the BER of a BICM—OFDM victim receiver impaired by UWB interference and AWGN can be obtained by truncating the union bound in Eq. (4.22) and using Eq. (4.23) in combination with Eqs. (4.21) and (4.24) or (4.25). Thereby, the integral in Eq. (4.23) can be efficiently evaluated using a Gauss—Chebyshev quadrature rule, cf. [47] for details. The conditional MGFs ‘i[k] (2c[kjd[k]sIg[kj, r) in Eq. (4.21) can be found in Table 4.2 and we note that the proposed method is general enough to include heterogeneous interference scenarios involving different types of UWB interference (e.g. one MB—OFDM and one IR—UWB interferer). Furthermore, our BER results are applicable to any UWB format as long as the corresponding conditional MGF can be obtained. The averaging over r in Eq. (4.24) involves an I—dimensional integration which is potentially problematic for large I. However, numerical evidence shows that ‘I’(sr) is almost independent of r and assuming that r, 1 I, is uniformly distributed in the BICM—OFDM symbol interval, it suffices to average r over few (e.g. 5) values in [0, T8) to get accurate BER results. The averaging over c in Eq. (4.24) is not problematic as it only involves a one—dimensional integral. The same is true for the averaging over g in Eq. (4.24) since only I independent one—dimensional integrals have to be evaluated which can be done efficiently numerically. The evaluation of ‘I’(sr) based on Eq. (4.25) is more challenging since here averaging over g requires an I—dimensional integral. Therefore, in this case, the proposed analytical BER evaluation method is most suitable for scenarios 91 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference with few UW13 interferers (e.g. I 3). Although the analytical method proposed in this subsection enables the efficient eval uation of the performance of BICM—OFDM systems, it requires numerical computation of up to three integrals and the conditional MGFs in Table 4.2 also involve sums and prod ucts. In the next subsection, we provide an easy—to—evaluate approximation for the BER of BICM—OFDM based on a GA of the UWB interference signal. Similar GAs have been proposed for uncoded single—carrier transmission in [25, 64, 65]. 4.4.3 Gaussian Approximation If the UWB interference signals i[k] are modeled as .A[(O, u[k]) distributed with o-[k] e{I[kfi2}, the variance of n[kj + i[k] conditioned on g[k] is given by a2[k] = o + g,[k] o11[k]. For concreteness, we consider in the following again Cases Cl—C4. In particular, we assume that the g[k] are either independent RVs or that they are constant with g,[k] = th, &, Vk, v. We note that, unlike for the exact analysis in Section 4.4.2, in the latter case expectation over g[k] is not necessary since the variance u2[k] only depends on g[k]2 = , which is not a RV. Therefore, for both cases the PEP in Eq. (4.23) can be simplified to c+jao P(c -* ê) = - f ((s))d (4.26) c-joo m 1 (s) = m2N (sd[k]), (4.27) k=— i1 b0 x[k]EXt where (sd[kj) {e21+8J8 fi Eg {e2I9 2J,182} } (4.28) 92 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference We note that the considered MB—OFDM, DS—UWB with L < 12, and IR—UWB formats have a constant PSD, i.e., o[k] = o- is constant over all BICM—OFDM sub—carriers. Consequently, in these cases, the averaging over the sub—carriers in Eq. (4.27) can be omitted. However, for DS—UWB with L = 12 and L = 24 the PSD is not flat and the averaging over the sub—carriers is necessary. In the following, we will further simplify Eq. (4.28) for the four interference channel models C1—C4 introduced in Section 4.2. Cl) If H[k] and g[k] are i.i.d. .A1(0, 1) and ..Af(0, o) RVs, respectively, we obtain I e2z[k1[1918 (4.29) fl(1 + J C2) If a[k] = 1, Vk, and g[k] is i.i.d. f(0,o), Eq. (4.29) is still valid if we omit the expectation and set a = 1. C3) If g,{kfi is constant and a[k] an i.i.d. .Af(0, 1) RV, we can simplify Eq. (4.28) to ‘‘(sId[k]) = 1 . (4.30) 1 + sd[k][1 + s(o + o[kj)] C4) If both a[k] and Ig[kj are constant, we can simply omit the expectations in Eq. (4.28) and set a = 1 and = , respectively. An approximation for the BER of a BICM—OFDM receiver can be obtained by combining Eqs. (4.22) and (4.26)—(4.30). The integral in Eq. (4.26) can again be efficiently evaluated using a Gauss—Chebyshev quadrature rule. We note that computation of the GA for Cl— C4 is unproblematic in all cases since at most two one—dimensional integrals have to be evaluated. We note, however, that our results in the next section will show that the accuracy of the GA strongly depends on the UWB format and on the sub—carrier spacing /f8 of the 93 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference BICM-OFDM system. 4.5 Results and Discussion In this section, after verifying the validity of our analytical results from Section 4.4 with simulation results in Subsection 4.5.2, we use the analytical results to highlight the effects of the considered UWB formats on a BICM—OFDM system in Subsection 4.5.3 and to investigate the influence of the distance between the UWB transmitter and the BICM— OFDM receiver in Subsection 4.5.4. However, first we specify the system parameters used for the simulations and numerical evaluations. 4.5.1 System Parameters Throughout this section we consider a BICM—OFDM system with N = 64 sub—carriers. We found however that the performance is almost independent of N for N 16, which will be true for most practical systems. For BICM we assumed ideal interleaving and we adopted the rate 1/2 standard convolutional code with generator polynomials [133, 171] (octal representation), which is used e.g. in both IEEE 802.lla WLANs [13] and IEEE 802.16 WiMAX [58], and higher code rates were obtained via puncturing. In the following, SNR and SIR refer to the SNR and SIR per information bit, respec tively. Unless stated otherwise, ft, = 2f = 8.25 MHz19, 1 v I, 4—PSK modulation with R = 1/2 is used, Case C3 for the channel model is assumed20,and only I 1 inter ferer is present. The various parameters for the UWB signals were taken from Table 4.1 19We note that, when plotted as a function of the SIR, the performance of BICM—OFDM is practically independent of the frequency offset as long as the BICM—OFDM signal lies fully in the UWB frequency band. 20We concentrate on frequency flat interference channels in order to be able to separate the effects of the UWB channel from those of the UWB signal itself, as it is customary in the literature [62, 631 However, we also show results for Case Cl which assumes a frequency—selective UWB channel. 94 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference 10_i :i::::: :::I:::: :1::::::: :1 — Theory -- Simulation . - - GA 102 . :. i0 .::::::::L::::: :. .: :::. :::::IRUWAf:MHz): 1104 i05 .::: . f .::: :. ::. MB-OFPM(f= 4125 MHz).. 106 BPSK DS-UWB (sf, =4 125 MHz) :::::::::::::::::I4—OD:IvI:CA .:1:MH:)L:: .:::: I I I I I I 3 4 5 6 7 8 9 10 11 12 SIR [dB] Figure 4.2: BER vs. SIR for IR—UWB (Nb = 32, L = 16), BPSK DS—UWB (L = 24), and MB—OFDM. Case C3 for channel, I = 1, SNR —* oc, 4—PSK, and R = 1/2. and [1, 2, 3]. 4.5.2 Verification of Theoretical Results In Fig. 4.2 we show the BER vs. SIR for MB—OFDM, BPSK DS—UWB (L = 24), and IR—UWB (Nb = 32, L = 16) interference for different BICM—OFDM sub—carrier spacings sf4. An interference limited scenario is considered (i.e., SNR —* oo). Fig. 4.2 includes simulation results, theoretical results obtained by evaluating the analytical expressions in Section 4.4, and the GA. For the theoretical results the union bound in Eq. (4.22) was truncated after the first 8 terms. As can be observed from Fig. 4.2, for relevant BERs (e.g. BER < 1O), where the union bound becomes tight, the theoretical results are in 95 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference perfect agreement with the simulations for all considered UWB formats and sub—carrier spacings. We note that although the analytical BER expressions involve integrals, which have to be evaluated numerically, it took only minutes to compute the theoretical BER curves in Fig. 4.2. In contrast, the simulations for Fig. 4.2 took several days to finish21. We observe from Fig. 4.2 that the BER strongly depends on the UWB interference format and the sub—carrier spacing. The GA is only a good approximation for DS—UWB. However, it is interesting to note that all BER curves have the same asymptotic slope as the GA. Having confirmed the accuracy of the derived analytical results, we will use these results in the following subsections to investigate the influence of various system parameters. 4.5.3 Effect of System Parameters on Performance Since the effects of all BICM—OFDM and UWB parameters on the BER crucially depend on the BICM—OFDM sub—carrier spacing f3, we show in this subsection the BER as a function of /.f3 In this context, it might be helpful to note that the sub—carrier spacings for IEEE 802.lla WLANs [13] and IEEE 802.16 W1MAX [58] are /f3 312.5 kHz and Zf8 89.3 kHz, respectively. However, our results are not limited to narrowband and wideband BICM—OFDM systems and can also be used to evaluate the impact of UWB interference on UWB BICM—OFDM systems. An example for the latter case is the ECMA MB—OFDM system [1] which also uses the BICM—OFDM concept and has a sub—carrier spacing of /.f8 = 4.125 MHz. UWB Format: In Figs. 4.3 and 4.4 we investigate the impact of the UWB format on the BER and the validity of the GA for SNR = 15 dB and SIR = 10 dB. Fig. 4.3 shows 21Both the simulations and numerical evaluations were performed on the same computer (with two Intel Xeon 3.6 GHz processors). The simulation program was written in C, whereas MATLAB was used for the numerical evaluations. 96 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference I,, — IR—LJWB (Nb 8):;::: ::: : ::: : :; :: : : : : : : S : ::: : : : : : : : : : — - IR-UWB (Nb = 32) / L = 4 1 IR_UWB(Nb=128) • -, GA : I I—. -4 : : : :: : :110 - L=512 / L=128 / J=32 4f L=16 . s::: : ::.‘‘.::.: s.:: . S.. L=8 10 6 — — ‘- .———— . • 106 .fs [Hz] Figure 4.3: BER vs. f3 for IR—UWB with different Nb and L. Case C3 for channel, I = 1, SNR = 15 dB, SIR = 10 dB, 4—PSK, and R = 1/2. that for IR—UWB the validity of the GA strongly depends on f3 and on the data rate of the IR—UWB system. For the lowest data rate of 0.1 Mb/s (NbL = 4096) IR—UWB is highly impulsive and the GA is not accurate in the considered f3 range. Similarly, for the mandatory data rate of 0.85 Mb/s (NbL = 512) the GA leads to overly optimistic performance predictions for /f3 100 kHz. For example, for the sub—carrier spacing used in IEEE 802.lla WLANs and Nb = 32 the GA suggests that the BER is approximately by a factor of 1.5 lower than it actually is. For very high IR—UWB data rates (L = 2) the GA is fairly accurate in the considered f3 range since the IR—UWB data signal is less impulsive in this case. Similarly, Fig. 4.4 shows that the GA is very accurate for BPSK and 4—BOK DS—UWB. We note that we only show results for the lower operating band 97 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference — DS-UWB (BPSK) — - DS-UWB (4-BOK) MB-OFDM : • — GA . : : ::. . . : io I 10 108 io7 /f5 [Hz] Figure 4.4: BER vs. /f8 for BPSK DS-UWB (L = 24), 4-BOK DS-UWB (L = 24), and MB—OFDM. Case C3 for channel, I = 1, SNR 15 dB, SIR = 10 dB, 4—PSK, and = 1/2. and L = 24 (corresponding to the lowest DS—UWB data rates). However, the results for smaller L (corresponding to higher data rates) and the higher operating band are very similar and are also in excellent agreement with the GA. For MB—OFDM interference the GA may over- or underestimate the performance depending on /f3 and is accurate only for Lf8 100 kHz. For the sub—carrier spacing of IEEE 802.lla WLANs the GA leads to slightly too pessimistic performance predictions. On the other hand, if one MB—OFDM system is impaired by another, interfering MB—OFDM system, the GA underestimates the true BER by a factor of 4. Finally, we note that for the relatively small sub—carrier spacings used in IEEE 802.16 WiMAX the GA is very accurate for MB—OFDM, DS—UWB, and IR—UWB (for data rates of 0.85 Mb/s or more). 98 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference :: ________________ R — 3 4 — — ..4 / 1 — —7 _____ : : : : 1: :10 . .;::::::::::::::::::.::::::::::.::;:.:::(::.::: — — - — — —— 4 —1_ .— — — — — — — — —————— I .%. . I . I I :•. : : : 108 —. — — — I- __ 4• __ — — 4 — — ———. — — — % # ::.:.::.:: .::::::: io9 io5 106 /f8 [Hz] Figure 4.5: BER vs. /f3 for IR—UWB, 4—BOK DS—UWB, and MB—OFDM. Code rates of R 1/2, 2/3, 3/4, 5/6, 7/8, and 1 (uncoded) are considered. Case C4 for channel, IEEE 802.16 puncturing patterns, I = 1, SNR = 10 dB, SIR = 8 dB, and 4—PSK. Puncturing: In Fig. 4.5 we show the BER for different code rates and SNR = 10 dB and SIR = 8 dB. For puncturing the puncturing patters from the IEEE 802.16 standard [58] were adopted. Since we assume that both the BICM—OFDM channel and the interference channel are frequency flat (Case C4), Fig. 4.5 provides an unobscured view of how the impact of UWB interference is affected by puncturing. Fig. 4.5 shows that the general shape of the BER curves for all UWB interference formats is not affected by the puncturing. To get a better understanding of the influence of puncturing on the shape of the BER curves, we chose in Fig. 4.6 the SIR for different code rates such that the GA yields the same BER for all considered code rates. In this case, we assume Case C3 for the channel and SNR —* oo. Fig. 4.6 shows that for C3 the impact of the non—Gaussianity of the UWB 99 106 io7 f8 [Hz] Figure 4.6: BER vs. zf3 for IR—UWB, 4—BOK DS—UWB, and MB—OFDM. Code rates of R = 1/2 (SIR = 8.6 dB), R = 2/3 (SIR = 11.2 dB), R = 3/4 (SIR = 13.7 dB), and = 5/6 (SIR = 16.8 dB) are considered. Case C3 for channel, IEEE 802.16 puncturing patterns, I = 1, SNR —p cc, and 4—PSK. interference is less pronounced for higher code rates. We note that higher code rates correspond to a code with a smaller minimum distance dmin and thus, a smaller diversity order [59]. Therefore, our results are in agreement with the findings in [65] where it was shown that (for an uncoded single—carrier system) the impact of the non—Gaussianity of the interference on the BER is larger for channels with higher diversity order. Interference Channel: In Fig. 4.7 we study the impact of the interference channel on the performance for SNR = 15 dB and SIR = 10 dB. In particular, we compare Cl (severely frequency—selective UWB channel) with C3 (frequency—fiat UWB channel). Remarkably, the general shape of the BER curves is not affected by the interference channel. However, 100 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference DS-UWB(4-BOK, L = 24) : : :::: :::: E F :::: : : A :: : : : : ::: 1o5 ,.,‘ _.. .. —,..,‘.— .••--:•:- 106 - _r_ - .::::: . . .1 .:;;i Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference 1 ., .. —IR——UWB(analysis) DS—UWB(analysis) :::::::::::.::::::::::::::;::::::: — — MB-OFDM (analysis) : • — GA (analysis) :.: v MB—OFDM (Cl I d victim channel simulation) MB—OFDM (Cl, B = 10, simulation) : — .4 0 IR—UWB(C3,B=20,simulation) 4/10 * IR—UWB (C3 B=7 simulation) MB—OFDM (C3 ii d victim channel simulation) x MB—OFDM (C3, B = {5, 9, 20, 40}, simulation) . , :-“%: %...:::::::.f . . . S C3 * Bi b — -,.. S .-.— 10 io6 zXf3 [Hz] Figure 4.7: BER vs. f8 for IR—TJWB (Nb = 32, L = 16), 4—BOK DS—UWB (L = 24), and MB—OFDM. Results for Cl and C3 are compared. I = 1, SNR = 15 dB, SIR = 10 dB, 4—PSK, and R = 1/2. a frequency—selective interference channel leads to a performance loss for all UWB formats and the non—Gaussian behavior of the BER curves is also slightly more pronounced in this case. Besides the numerical results which assume that the frequency domain gains c[k] of the victim channel are i.i.d. Rayleigh RVs, we also show in Fig. 4.7 simulation results (markers) for both i.i.d. and non—i.i.d. victim channels. For the non—i.i.d. case we assume that the victim channel impulse response h(t) has 10 i.i.d. taps and coding and interleaving is performed over B OFDM symbols. The channel gains in different OFDM symbols are assumed to be statistically independent due to e.g. frequency hopping inside the bandwidth of the UWB signal. Thus, for large B the frequency domain fading gains are approximately i.i.d. as far as the dominant error events of the code are concerned. 101 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference I — R—UVVB (Nb = 32, L = 128) : : ::: : ::: ::: .:: : : : :::::. : : : : : : : : : : ::.. : : DS-UWB (4-50K, L = 24) :: : . : ::: ::: : : : : : : : . . ::::::::::: :: :: ::: : :: - - MB-OFDM . . . :. . . .—.. 1105 I s _\- - - - - - ‘ : 106 io5 106 /f3 [Hz] Figure 4.8: BER vs. ‘f for IR-UWB, 4-BOK DS-UWB, and MB-OFDM. I = 1, 2, 5, 10, 100 i.i.d. interferers are considered. Case C3 for channel, SNR —÷ cc, SIR = 8 dB, 4—PSK, and R = 1/2. The markers (“V”) indicate simulation results for MB—OFDM and I=t2. In contrast, for small B some correlation will be left and the i.i.d. assumption may lead to overly optimistic performance predictions. However, the results in Fig. 4.7 show that the simulation results agree well with the numerical results even for relatively small B. Multiple Interferers: In Fig. 4.8 we investigate the effect of multiple interferers on the BER. To facilitate this comparison SIR = 8 dB (SNR —f cc) was assumed independent of the number of interferers I. Furthermore, Case C3 was adopted for the channel and both the interference channels and the interferers were i.i.d. As can be observed from Fig. 4.8, for MB—OFDM and IR—UWB the total interference becomes more Gaussian as the number of interferers grow. This behavior can be explained by the central limit theorem. However, while for MB—OFDM the GA is accurate for f8 < 2 MHz for I 5 interferers, for the 102 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference considered IR—UWB format even 100 interferers are not sufficient to make the GA tight in the considered range of sub—carrier spacings. For DS—UWB increasing the number of interferers has almost no effect since, in this case, the interference signal is practically Gaussian already for I = 1. To confirm the numerical BER results, we have also included simulation points for MB—OFDM with I = 2. Here, the frequency domain channel gains a[kl were simulated as i.i.d. Rayleigh RVs. General Conclusions Concerning GA: For system—level simulations a simple interference model is desirable and it is convenient if UWB interference can be modeled as additional Gaussian noise. Figs. 4.3—4.8 allow us to draw some general conclusions for BICM—OFDM systems in this regard. These conclusions are practically independent of the modulation scheme, the code rate, the channel of the BICM—OFDM system, and the interference channel. First, we note that the GA is very accurate for DS—UWB in the entire range of relevant sub—carrier spacings (e.g. /.f8 10 MHz). On the other hand, for IR—UWB operating with the mandatory data rate of 0.85 Mb/s the GA underestimates the BER of the BICM—OFDM system and becomes tight only for f3 100 kHz. For MB—OFDM the GA overestimates the BER for /f < 2 MHz but underestimates it for Zf3 > 2 MHz. The GA becomes tight again for L.f3 100 kHz. Furthermore, taking into account that /f3 1/T8, and noting that if 1/T, is valid for MB—OFDM UWB systems, based on Figs. 4.3—4.8 we conclude that in general, the accuracy of the GA is a function of Pr T/T3, i.e., the ratio between the symbol durations of the UWB system and the victim system. Specifically, the error incurred when using the GA increases with increasing Pr This observation is quite general and invariably applies to all the three considered UWB formats. 103 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference 4.5.4 Influence of Distance Between Interferer and Victim Receiver In this subsection, we investigate the influence of the distance between the UWB inter ferer and the victim receiver on the performance of the BICM—OFDM system. For this purpose, we adopt the line—of—sight (LOS) and the non—LOS (NLOS) UWB path loss models from [5] and assume that the maximum of the PSD of the UWB transmit signal is -41.3 dBm/MHz (FCC spectral mask). We note that the performance for MB—OFDM and IR—UWB interference is practically independent from the frequency offset as long as the BICM—OFDM signal lies entirely in the UWB bandwidth and the unmodulated zeroth sub—carrier of MB—OFDM is avoided. In contrast, the BICM—OFDM performance is fre quency dependent for DS—UWB with L = 12 and L = 24 because of the non—flat DS—UWB PSD. This frequency—dependence is not visible if the BER is parameterized by the SIR, cf. Figs. 4.3—4.8, since the interference statistics are practically frequency independent and the frequency—dependent interference power is “hidden” in the SIR. Impact on BER: In Fig. 4.9 we investigate the BER degradation caused by a UWB interferer with distance d from the victim receiver. The transmit power, the transmitter— receiver distance, and the noise figure of the BICM—OFDM system are such that SNR = 10 dB is achieved. In Fig. 4.9 we compare the performance for the three considered UWB interference formats assuming an NLOS UWB path loss model [5] and the sub—carrier spacings used in a) IEEE 802.11aWLANs (312.5 kHz) and b) MB—OFDM (4.125 MHz). As mentioned before, for DS—UWB (L = 24) the performance depends on the frequency offset and we show the best—case and the worst—case performances along with the corresponding GAs. These GAs are in perfect agreement with the performance for DS—UWB. The GA for the worst case is also valid for IR—UWB (Nb = 32, L = 16) and MB—OFDM since in that case the maximum PSD of -41.3 dBm/MHz applies. However, while for = 312.5 104 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference -610 Figure 4.9: BER vs. d for a) f3 = 312.5 kHz and b) /f3 = 4.125 MHz. IR—UWB (Nb = 32, L = 16), 4—BOK DS—UWB (L = 24), and MB—OFDM interference. Case C3 for channel, NLOS UWB path loss model from [5], I = 1, SNR = 10 dB, 4—PSK, and R = 1/2. kHz this GA is also fairly accurate for IR—UWB and MB—OFDM, for Zf3 = 4.125 MHz it is accurate only for MB—OFDM. The difference between the best case and the worst case for DS—UWB is larger for the smaller sub—carrier spacing, since the larger sub—carrier spacing results in an averaging over a larger range of frequencies. Impact on Data Rate: Now, we adopt for the BICM—OFDM system the IEEE 802.lla parameters (including puncturing patters), i.e., there are 8 possible data rates from 6 Mb/s to 54 Mb/s which are realized by different combinations of modulation schemes (BPSK, 4—PSK, 16—QAM, 64—QAM) and coding rates (R 1/2, 2/3, 3/4). In Fig. 4.10, we show the BICM—OFDM data rate as a function of the interferer—victim 105 -e-IR-UWB • • • •. -4-- MB-OFDM -V- worst—case DS—UWB • • -- best—case DS—UWB 10 -610 a) 2 4 6d[m] b) 2 4 6 8d[m] 10 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference 6G I I I _____I II I _____ I 20 10 —e— IR_UWB(Nb=32, L= 128) MB-OFDM V- worst-case DS-UWB (BPSK, L = 24) best-case DS—UWB (BPSK, L = 24) —x-GA C I I I 0 4 6 8 10 12 14 d[m] Figure 4.10: Data rate of IEEE 802.lla BICM—OFDM sysem vs. d for IR—UWB, BPSK DS—UWB, and MB—OFDM. Case C3 for channel, LOS UWB path loss model from [5], IEEE 802J1a puncturing patterns, I = 1, zf = 312.5 kHz, and SNR = 25 dB. receiver distance d for MB—OFDM, BPSK DS—UWB (L = 24), and IR—UWB (N = 32, L = 128), an LOS UWB path loss model [5], and SNR = 25 dB. We assume that the BICM—OFDM system switches to a lower data rate if the BER exceeds 10—6. We note that only 5 out of the possible 8 data rates are actually chosen since in some cases higher data rate modes (using a larger signal constellation but a lower code rate) have a better performance than lower data rate modes (with a smaller signal constellation but a higher code rate) because of the higher diversity order of the code they use. Fig. 4.10 clearly shows that the data rate of an IEEE 802.lla WLAN system suffers significantly if a UWB transmitter is in its vicinity. For DS—UWB interference we show again the best and worst case performances in Fig. 4.10 along with the respective GAs. The 106 Chapter 4. Performance of Generic BICM—OFDM Systems in UWB Interference GA is very accurate for DS—UWB in both cases and a significant performance improvement can be achieved if the BICM—OFDM system operates at a frequency where the DS—UWB PSD is lower than -41.3 dBm/MHz. Note, however, that DS—UWB with L < 12 has a fiat PSD and these “spectrum holes” do not exist. We note the GA is also accurate for MB—OFDM but overly optimistic for IR—UWB. 4.6 Conclusions In this chapter, we have provided an analytical performance evaluation framework for generic BICM—OFDM systems impaired by UWB interference. The considered UWB for mats closely follow IEEE/ECMA standards or standard proposals. Besides the accurate analysis we have also proposed a simple GA where the UWB interference is treated as additional Gaussian noise. Our results show that while for DS—UWB the GA is applicable for all practically relevant BICM—OFDM sub—carrier spacings (e.g. f3 < 10 MHz), for MB—OFDM and IR—UWB the GA becomes tight only for Z.f3 < 100 kHz. For larger sub—carrier spacings the GA may severely underestimate the BER for IR—UWB interfer ence employing the mandatory data rate of 0.85 Mb/s. In contrast, for MB—OFDM the GA overestimates the BER for f3 < 2 MHz but underestimates it for Lf3 > 2 MHz. For example, for IEEE 802.lla WLANs the GA overestimates the BER for MB—OFDM interference but underestimates it for IR—UWB interference. In contrast, if a MB—OFDM system is interfered by other UWB systems, the GA underestimates the BER for both IR— UWB and MB—OFDM interference. Finally, our numeral results confirm that in general, the accuracy of the GA for different UWB interference formats is governed by Pr = T/T3, i.e., the ratio between the symbol durations of the UWB system and the victim system. In summary, the presented results give important insights into the impact of various UWB interference formats on the performance of BICM—OFDM systems and shed new light on the validity of the popular GA. 107 Chapter 5 Performance of Generic BICM—Based Systems in Non—Gaussian Noise and Interference 5.1 Introduction In Chapter 4, we studied the performance of BICM—OFDM victim systems in UWB inter ference. In this chapter, we extend the results obtained in the previous chapter to generic non—Gaussian noise and interference. In contrast to Chapter 4, we consider BICM—based victim systems that are equipped with multiple receive antennas, and employ either BICM— SC or BICM—OFDM. We emphasize that BICM can provide superior diversity gains in fading channel compared to other coded—modulation techniques. As a result, BICM—based schemes have been adopted by a number of recent standards and are also expected to play a major role in future wireless system [661. The highly desirable performance gains achieved by BICM in fading channels can be mainly attributed to its ability in efficiently extracting high orders of time and frequency diversity when combined with SC modulation [59] and OFDM, respectively. More robustness against fading can be achieved in these systems by employing multiple antennas at the receiver and thereby exploiting the available space diversity. 108 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise Similar to other communication systems, BICM—based systems are usually optimized for AWGN. In practice however, as discussed in the previous chapter, these systems are likely to operate in the presence of non—Gaussian interference caused by co—existing UWB underlay systems. Furthermore, they can also be subject to other non—Gaussian impair ments such as narrowband interference (NBI), co—channel interference (CCI), correlated Gaussian interference, and man—made impulsive noise. Therefore, it is of both theoretical and practical interest to investigate how the performance of BICM—SC and BICM—OFDM systems designed for AWGN environments is affected by non—Gaussian noise.22 We note that almost all existing performance studies of BICM are limited to AWGN. For example, union bounds for the BER of BICM—SC were provided in [59, 67] and similar expressions for BICM—OFDM can be found in [60]. The combination of BICM—OFDM and spatial diversity techniques was analyzed in [60, 68, 69]. In contrast, only few analytical results are available for non—AWGN types of noise. Namely, the performance of BICM—SC in Mid dleton’s Class A impulsive noise and of BICM—OFDM in UWB interference was analyzed in [701 and [711, respectively. Motivated by the lack of general performance results, in this chapter, we provide a math ematical framework for performance analysis of BICM—SC and BICM—OFDM systems with multiple receive antennas in fully interleaved fading and non—AWGN environments. This framework is very general and applicable to arbitrary linear modulation formats, all com monly used fading models, and all practically relevant types of noise with finite moments. We first develop a general easy—to—compute upper bound on the BER of BICM systems. We also derive closed—form asymptotic BER expressions for BICM—SC and BICM—OFDM systems which provide significant insight into the impact of system parameters such as the modulation format, the free distance of the code, the type of fading, and the type of noise 22We note that similar to Chapter 3, in the this chapter the term “noise” refers to any additive impair ment of the received signal, and also includes what is commonly referred to as “interference”. 109 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise on performance. In particular, we show that while the diversity gain of BICM systems is not affected by the type of noise, the coding gain depends on certain noise moments. We note that in Chapter 3 we have derived analytical results for the asymptotic performance of generic uncoded systems in non—AWGN channels. However, both the analysis techniques and the results in Chapter 3 are not applicable to BICM. The rest of this chapter is organized as follows. In Section 5.2, the considered BICM— SC and BICM—OFDM system models are introduced. The proposed upper bound and asymptotic approximation for the BER are presented in Sections 5.3 and 5.4, respectively. Various practically relevant noise models are discussed in Section 5.5. The presented analysis is verified via computer simulations in Section 5.6, and conclusions are drawn in Section 5.7. 5.2 System Model We consider BICM—SC and BICM—OFDM systems with NR receive antennas. For conve nience, all signals and systems are represented by their complex baseband equivalents. 5.2.1 System Model The BICM transmitter consists of a convolutional encoder of rate R, an interleaver, and a memoryless mapper [59]. Specifically, the codeword c [c1,c2,.. . , cmcKc] of length mK is generated by a convolutional encoder and interleaved. The interleaved bits are broken up into blocks of m bits each, which are subsequently mapped to symbols xk from a constellation X of size X M = 2m to form the transmit sequence x [xi, x2,. . . , XKc] of length K. Assuming perfect synchronization and demodulation, for both BICM—SC 110 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise and BICM—OFDM the signal observed at the NR receive antennas can be modeled as rk=\/hkxk+nk, 1kK, (5.1) where hk [hk,1 ... hk,NR]T with E{WhkI2} = NR and k [flk,1 •.. k,NR]T with = NR contain the fading gains hk,l and the noise variables k,j, 1 1 NR, respectively, and ‘y denotes the SNR per receive antenna. As customary in the literature, cf. e.g. [59, 67, 68], for our performance analysis we assume perfect interleaving, which means that hk and k can be modeled as i.i.d. random vectors and only their first order pdfs are relevant. In the following, we will discuss the assumptions necessary for the validity of this model more in detail, and we will provide simulation results for the case when these assumptions are violated in Section 5.6. BICM—SC: For BICM—SC we assume transmission over a fiat fading channel and coding over B frames of N data symbols, i.e., K = NB. The channel is time—variant within one frame and changes independently from frame to frame (e.g. due to frequency hopping). For sufficiently large N and/or B assuming that the time—domain fading vectors hk are i.i.d. is justified [59]. BICM—OFDM: We consider a BICM—OFDM system with N sub—carriers where one codeword spans B OFDM symbols, i.e., K = BN. We assume that the length of the OFDM cyclic prefix exceeds the length of the channel impulse response and that the channel changes independently from OFDM symbol to OFDM symbol. Thus, modeling the frequency—domain channel gains hk as i.i.d. vectors implies that the channel is severely frequency selective and/or B is sufficiently large. Practical BICM—SC and BICM—OFDM systems that employ interleaving and coding over B > 1 frequency—hopped frames include the GSM and the Enhanced Data Rates for GSM Evolution (EDGE) mobile communication systems (N = 116/N = 348, B = 8), and 111 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise the ECMA MB—OFDM UWB system (N = 128, B = 3; future versions of the standard may use up to B = 15) [1], respectively. 5.2.2 Fading and Noise Model Fading Model: The fading gains can be expressed as ak,jeiek,l, where ak,j and are mutually independent RVs. Specifically, ek,1 is uniformly distributed in [—ir, it) and ak,j is a positive real RV characterized by its distribution pa,1(ak,l) or equivalently by its MGF 1a,j(5) {e’d}. Correlated fading can be modeled via the joint pdf Pa(ak) or the joint MGF a(S) e{e_E8takI}, s [s1 ... SNR]T, of the elements of ak [ak,1 ... ak,NR]T cf. e.g. [4, 72, 73]. For the asymptotic analysis in Section 5.4, we require the fading channel to be asymptotically spatially i.i.d., i.e., for ak —f °NR the joint pdf can be expressed as NR pa(ak) fJpa(ak,l), (5.2) where Pa@) = 2aca + o(a2’) (5.3) with fading distribution dependent constants o and 0d Eq. (5.2) is obvious for i.i.d. and independent, non—identically distributed (i.n.d.) fading [38, 39, 74], and we prove its validity for the most popular correlated fading models (Rayleigh, Ricean, and Nakagami— m) in Appendix D. For these correlated fading models and for independent Nakagami—q and Weibull fading, the fading pdfpa,j(ak,l) and parameters c and d are specified in Table 5.1. We note that a similar table was provided in Chapter 3 (cf. Table 3.1) for independent fading models. The results provided in Table 3.1 are therefore not applicable to correlated fading models. Noise Model: The proposed analysis is very general and applicable to all types of noise 112 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise Table 5.1: Pdf pa(a) of fading amplitude a for popular fading models and corresponding values for ci and We have omitted the subscripts k and 1 for convenience. The parameters for Rayleigh (Chh), Ricean (jIh, Chh), and Nakagami—m (m, Caa) fading are defined in Appendix D. The parameters for Nakagami—q (q, b) and Weibull (c) fading are defined as in [41. for which all joint moments of the elements of k exist. This is a mild condition which is met by most practically relevant types of noise and interference, see Section 5.5 for several examples. An exception is a—stable noise, which is sometimes used to model impulsive noise [54], as the higher order moments of cr—stable noise do not exist. Note that our analysis is applicable to other types of impulsive noise such as Middleton’s Class—A model and c—mixture noise. 5.3 Upper Bound for BER In this section, we present an upper bound for the BER of BICM systems operating in non— AWGN environments. Compared to the upper bound obtained in Chapter 4 (cf. Eqs. (4.22), (4.23), (4.24), and (4.25)), the upper bound provided in this section is more general, as 113 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise it applies to BICM—SC as well as BICM—OFDM victim systems and also allows for the consideration of multiple receive antennas. 5.3.1 MGF of Metric Difference We assume standard Viterbi decoding with Euclidean (ED) branch metric [59] A[k] mm {Hrk — /5hkxkW2} (5.4) Xk eX for bit i, 1 i m, of symbol xk. Here, X denotes the subset of all symbols in constellation X whose label has value b E {O, 1} in position i. Although the ED metric is not optimum for non—Gaussian noise, it is employed in most practical systems since the pdf of the noise, which is necessary for optimum maximum—likelihood (ML) decoding, is usually not known at the receiver. For derivation of the proposed upper bound it is convenient to first calculate the MOF of the metric difference II’k —4/jhkzkjI2— Irk — = d[k]7IIhkII2— (5.5) where xk denotes the transmitted symbol and zk is the nearest neighbor of xk in with b being the bit complement of b, and Xk — zk d[k]eied[I1 with ED d[kj > 0. Since we assume the phases 0k,1 of hk,1 to be uniformly distributed, in Eq. (5.5), we have absorbed ejOd[c] in hk without loss of generality. Based on Eq. (5.5) the MGF I(xk,zk)(s) of(xk, zk) can be expressed as = ‘flk{e flk} = (—2 d[k],,/as)} (5.6) 114 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise where 11k [ei1nk,1 ... e_3O1.Rnk,NR1T and k{e2T}} is the MGF of ñk If the phases of the noise components k,1, 1 1 NR, are mutually independent and uniformly distributed in [—it, it), cI(s) = I’,(s) Enk{e_8T1{1k}} is valid and I(s) in Eq. (5.6) can be replaced by Ia(s). Further simplifications are possible if both the phases and the amplitudes of k,j, 1 < 1 < NR, are mutually independent. In this case, we can express I(s) as NR Fn(s) = fJ1n1(slRe{nk,1}), (5.7) where only the scalar MGFs 41(s) nk,l{e_8{,1}} of the elements k,j e8’”nj,j of ñk are required. If the phases of the k,j, 1 1 NR, are uniformly distributed in —sRe1n[—it, it), In1(s) = T(s) = Eflk{e ki} is valid, i.e., only the scalar MGFs of the noise components are required. The scalar MGFs cI (s) of several practically relevant types of noise are collected in Table 5.2, cf. Section 5.5. If I(s) cannot be calculated in closed form, it can be computed by numerical integration. However, even if closed—form expressions for the MGF are available, calculation of k,zk() in closed form is usually not possible, and evaluation of Eq. (5.6) entails NR numerical integrals. 5.3.2 Upper Bound Assuming a convolutional code of rate R = k/m (k and n are integers) the union bound for the BER of BICM is given by [91 Pb- w(d)P(c—*ê), (5.8) d=df where c and ê are two distinct code sequences with Hamming distance d that differ only in 1 1 consecutive trellis states, w(d) denotes the total input weight of error events at 115 Chapter 5. Performance of Generic BICM—Ba.sed Systems in Non—Gaussian Noise Hamming distance d, and df is the free distance of the code. P(c —# ê) is the PEP, i.e., the probability that the decoder chooses code sequence ê when code sequence c ê is transmitted. Invoking the expurgated bound from [59], the PEP can be expressed as c+jcx/ 1 \d P(c I (mcmc k,Zk)(S)) (5.9)i=1 b=O XkEX, where c is a small positive constant that lies in the region of convergence of the integrand. The integral in Eq. (5.9) can be efficiently evaluated numerically using a Gauss—Chebyshev quadrature rule, cf. [47]. As will be shown in Section 5.6, Eqs. (5.8) and (5.9) constitute an asymptotically tight upper bound on the true BER. This bound is a generalization of similar bounds in [59, 68] for AWGN to arbitrary types of noise (and interference). Unfortunately, the integrals in Eqs. (5.6) and (5.9) obscure the impact of system and channel parameters on performance. This motivates the asymptotic analysis in the next section, which leads to closed—form results. 5.4 Asymptotic Analysis In this section, we analyze the asymptotic behavior of the upper bound in Eq. (5.8) for high SNR, i.e., -y —* oo. For this purpose, it is convenient to consider the conditional PEP c+joo 1 1 dsP(c—*ctn)=--_ j (sIn)—, (5.10) c—joG where n [nf ... I/ mci \ (sjn) = (mc2mc xk,zk)(SIflk)) (5.11) i1 b=O XkEX / 116 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise and (xk,zk)(snk) = ak,ek{e kZk)} with channel phase vector 0k defined as [ek,l .. ek,NRJT. The conditional PEP in Eq. (5.10) is given by the sum of the residues of 1(s)n)/s at poles lying in the left hand side (LHS) of the complex s—plain (including the imaginary axis) [47]. In order to investigate the singularities of 1(sln)/s, we derive the Laurent series representation of ‘I(sn) around s = 0 for the asymptotic case of 7 —p oo in the following subsection. 5.4.1 Laurent Series Expansion of (sn) Exploiting the fact that the elements of ak and ek are asymptotically i.i.d., cf. Section 5.2.2, for —* oo we can rewrite I)(xk,zk)(snk) as NR xk,zk) (Sflk) JJ Lk,zk)(5k,1), (5.12) where (Xk ,zk) (s Ink,1) 8ak,1,ek,t { e [k] ak t12 e2 /57d [k] ak,tRe{k,I} s}. Using the Taylor series expansion ex = Z the integral f0°° x1e_PX2dx = p’/2P(j/2) [53, 3.462], and Eq. (5.3), k,Zk)(sIrik,1) can be expressed as (zk,zk)(8Iflk,t) = ak,i,ek,t {e Iak,d2 (2 d[k] ak,lRe{nk,1} s)/i! } =(7d[k]s)ad 2P(ad + i/2)ee,1{Re{nk,l}}s’ + 0(7 d) (5.13) Using ‘ekl{Re{ñk,j}} = i even, and ‘ekl{Re{ñk,1}} = 0, i odd, in Eq. (5.13) leads to xk,zk)(8tflk,1) =(7d[k]s)ad iInk,1I25z + 0(7_ad), (5.14) 117 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise where ,8 is defined as 2P(c + i)F(i + 1/2) — P(ad + i) 5 15(2i)!P(i+1) — (i!)2 The asymptotic Laurent series expansion of (sn) is obtained from Eqs. (5.11), (5.12), and (5.14) as NR d (sIn) = X(, NR, d) (7S)_adNRd (n Zkl(S)) +0(7-adNftd) (5.16) with zk,l(S) Z- !3iInk,li2sand modulation dependent constant d mc 1 X(ad,NR,d) (mcmc (dz[k])adNR) . (5.17) In the next subsection, we will use Eq. (5.16) to calculate a closed—form expression for the asymptotic BER. 5.4.2 Asymptotic BER As mentioned before, the conditional PEP in Eq. (5.10) is given by the sum of the residues of I(sn)/s in the LHS of the complex s—plain. Using d’Alembert’s convergence test [53, 0.222] it is easy to show that zk,1(S) is convergent for all s. Thus, (fl zk,l(s))° is also convergent for all s. Consequently, the first term on the right hand side (RHS) of Eq. (5.16), which dominates for high SNR, is convergent for s 0, i.e., for high SNR the only singularity of 1’(s{n)/s is at s = 0. Thus, the asymptotic conditional PEP is given by the residue of 1(sln)/s at s = 0 or equivalently by the coefficient associated with s0 in the series expansion of the first term on the RHS of Eq. (5.16). Assuming cdNRd is an 118 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise integer this leads to d NR P(c —* ê n) = X(ad, NR, d) Rd7dNREl fi fJ /31In,z2 tj++id=QdNRd k=lj1+..+jN=ik 1=1 +0 (5.18) Based on Eqs. (5.8) and (5.18) a closed—form expression for the asymptotic unconditional BER Pb Wc(df) E{P(c —* ê rz)} can be obtained as Pb _ (f) X(ad, NR, df) M(ad, NR, df)7f, (5.19) where M(ad,NR,d)= jl.jNRmfl(j1,...,jNR), (5.20) i1+”+iddNRd k=1 j1++JNRZk with the joint noise moments m(ji,. . . ,iNR) {lnk,12’. . . flk,NR2N1}. (5.21) In arriving at Eqs. (5.19)—(5.21) we have used the assumptions that (a) the first term in the summation in Eq. (5.8) is asymptotically dominant, (b) the union bound is asymptotically tight, (c) the noise vectors k are i.i.d., and (d) all joint moments of the elements of k exist. Assumption (d) is necessary since the terms absorbed in0(7dNR) in Eq. (5.18), contain sums of products of elements of k, cf. Eq. (5.13), which have been neglected in (5.19). Eq. (5.19) is a generalization of similar asymptotic expressions for AWGN in [59, 68] to non—AWGN channels. Depending on the properties of the noise, evaluation of mn(ji, . . . ,jNR) may be cumbersome. However, for two important special cases significant 119 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise simplifications are possible. Case 1 (spatially i.i.d. noise): If the components of k are independent, Eq. (5.20) simplifies to M(ad,NR,d) = /31m(j). . /3Ndmfl(jNftd) (5.22) jl++jNRd=adNRd with scalar noise moments m(j) {nk,l2} which are independent of k and 1. Case 2 (cd = 1): If 0d = 1, which is true for example for (possibly spatially correlated) Rayleigh, Ricean, and Nakagami—q fading, Eq. (5.20) simplifies to M(1, NR, d) (NRd)! il+..+id=NRd (j1,jd) m(ii). . . m(i) (5.23) with vector noise moments m(i) Closed—form expressions for the moments mn(j) and m(i) of several important types of noise are provided in Tables 5.2 and 5.3, respectively, cf. Section 5.5. 5.4.3 Diversity Gain, Coding Gain, and Design Guidelines To get more insight, it is convenient to express the asymptotic BER as Pb (G7) [39], where Gd and G denote the diversity gain (i.e., the asymptotic slope of the BER curve on a double logarithmic scale) and the coding gain (i.e., a relative horizontal shift of the BER curve), respectively. Considering Eq. (5.19), we obtain Gd = adNRdf (5.24) 10 10 (wc(df)X(ad, NR, df)G [dB] ——log10 — — log10 — log10 M(cd, NR, df). (5.25) 120 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise From Eq. (5.24) we observe that the diversity gain of BICM is independent of the type of noise. The coding gain in Eq. (5.25) consists of three terms, where the first, the second, and the third term depend on the fading channel, the modulation scheme and the code, and the type of noise, respectively. The primary goal of BICM design is to maximize df for a given decoding complexity in order to maximize Gd (and to minimize the asymptotic BER). Gray labelings (yielding smaller X(ad, NR, df) than non—Gray labelings) and codes with small w(df) are advantageous for maximizing the second, modulation and coding dependent term in Eq. (5.25). Once df is fixed, the last term in Eq. (5.25) cannot be further influenced through system design making the BICM design guidelines effectively indepenent of the type of noise in the system. Thus, our results show that BICM systems optimized based on the guidelines provided in [591 for systems operating in fading and AWGN are also optimum for non—AWGN environments as long as the ED metric is used for Viterbi decoding. 5.4.4 Uncoded Transmission While BICM is the main focus of this chapter, based on Eq. (5.19) it is also possible to compute the asymptotic BER of uncoded transmission with maximum—ratio combining (MRC) at the receiver. In this case, df = 1, k = 1, and w(1) = 1 are valid. Furthermore, assuming a regular signal constellation such as M—QAM or M—PSK, it is easy to see that X(d, NR, 1) = Nmin/(mcd), where Nmin and dmin are the average number of minimum distance neighbors and the minimum distance of X, respectively. Therefore, the asymptotic BER of uncoded transmission with MRC can be expressed as Pb Nmjn M(d, NR, 1)7-QdNR (5.26) mcdmjn 121 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise where M(ad, NR, 1) = Zj1+...+jNR=dNR !-i /3jm(ji,. . . , iNs), which can be further simplified for ad = 1 and spatially i.i.d. noise, cf. Section 5.4.2. In particular, for ad = 1 we obtain M(1, NR, 1) = mfl(NR)/NR!, cf. Eq. (5.23), and it can be shown that for Rayleigh and Ricean fading (for both of which ad = 1 holds) Eq. (5.26) is identical to [75, Eqs. (12), (16)]. However, Eq. (5.26) is more general than the results in [75] since it is not limited to Rayleigh and Ricean fading and is also applicable to e.g. Nakagami—m, Nakagami—q, and Weibull fading. 5.5 Calculation of the Noise Moments and MGFs In this section, we discuss several practically relevant types of noise and compute the corresponding MGFs 1n(s) and moments mn(ji,.. . ,jjr) required for evaluation of the upper bound in Section 5.3 and the asymptotic BER in Section 5.4, respectively. In the following, we add superscripts “t” and “f” to the noise variables k and riki to distinguish between time—domain and frequency—domain noise, respectively. 5.5.1 Spatially I.I.D. Noise For spatially i.i.d. noise only the scalar MGFs en(s) and the scalar moments m(i) have to be computed for evaluation of the upper bound and the asymptotic BER, respectively, cf. Eqs. (5.7) and (522), and Table 5.2. In the following, we will consider two relevant examples for spatially i.i.d. noise. AWGN Although the main focus of this chapter is non—AWGN, the presented results are also valid for AWGN. Since the DFT does not affect the statistical properties of AWGN, the results in this subsection are valid for both BICM—SC and BICM—OFDM. We note that although 122 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise Table 5.2: MGF n(s) and scalar moments m(i) of types of noise considered in Section 5.5. All variables in this table are defined in Section 5.5. (SC) and (OFDM) means that the type of noise is relevant for BICM—SC and BICM—OFDM, respectively. [ Noise type Noise MGF I(s) Scalar moment m(i) AWGN exp(s2/4) i! GMN (SC) c exp(so/4) i! Ck GMN (OFDM) Zki+...+kj=N Ck1 ,...,k1 i! Zki+...+kj=N ck1 k1 k1 x_exp(s2o1 k1/) CCI (SC) exp(s2o/4) i! NBI (OFDM) p1 Zj= EkeJ’..rp.j (Z1=E=1ZkEJ, COO,k c0 exp(s2a,,k/4) + ci exp(s2o/4) +cia) the AWGN case was covered extensively in the literature, e.g. [59, 67, 68], our results are still more general than existing results as they allow for spatially correlated fading and more general fading models. For example, for Ricean fading (ad = 1) we obtain from Eq. (5.22) with the help of Eq. (5.15) and Table 5.2 M(1, NR, d) = (2c1). Thus, with Eq. (5.19) and Table 5.1 we get Pb (2NRdf_ 1) (wc(df) exp (_c h)) X(1,NR,df)7Rdf, (5.27) which is a new result. For NR = 1, we may rewrite Eq. (5.27) as Pb (2d_1) wc(df) [(1 + K)e]’fX(1, 1, df)ydf with Ricean factor K ILh2/, where uh and o denote the mean and the variance of hk,1. In contrast, for Ricean fading with NR = 1 the Chernoff bound was used in [59] and [68] to investigate the asymptotic behavior of BICM—SC and BICM—OFDM, respectively, since “a closed—form expression for the PEP for arbitrary K is missing” [59]. Comparing our result with the asymptotic Chernoff bound [59, Eq. (62)] shows that the Chernoff bound is by a factor of4df/(2df1) > 1 larger than the asymptotic BER, i.e., for df = 3 and df = 6 the Chernoff bound is horizontally shifted by 2.7 dB and 123 Chapter 5. Performance of Generic BICM—Ba.sed Systems in Non—Gaussian Noise 1.6 dB compared to the asymptotic BER, respectively. Spatially Independent Gaussian—Mixture Noise (SI—GMN) GMN is often used to model the combined effect of Gaussian background noise and man made or impulsive noise, cf. e.g. [37, 55, 70]. If the phenomenon causing the impulsive behavior affects the antennas independently, the GMN is spatially i.i.d. [28]. In the follow ing, we will discuss the impact of SI—GMN on BICM—SC and BICM—OFDM separately. BICM—SC: The GMN model is a time—domain model, i.e., the time domain noise is distributed according to pn(fl,j) = — exp (_2), 1 i NR, (5.28) where c > 0 and ? > 0 are parameters, and cj = 1. Two popular special cases of Gaussian mixture noise are Middleton’s Class—A noise [55] and f—mixture noise. For f—mixture noise I 2, c1 = 1 — €, c2 = , o o, and o = ico, where E is the fraction of time when the impulsive noise is present, K is the ratio of the variances of the Gaussian background noise and the impulsive noise, and = 1/(1 — e + ice) 1. BICM—OFDM: Taking into account the fact that GMN is rotationally symmetric, it can be shown that if the pdf of n1 follows Eq. (5.28), the pdf of n1 is given by Pn(fl,1)= exp (m12), 1 NR, (5.29) k1+--•+k=N k1 ,..,k1 k1 ,.,k1 which is again an SI—GMN model with parameters ck1 ,...,k1 (kl,,kI)c’ . c’ and o ..,k1 (k1 + .. . + kjo)/N. We note that the spectral i.i.d. asumption for n1 is justified only if the interleaver spans several OFDM symbols, i.e., B >> 1, since the noise after DFT in one OFDM symbol will be spectrally dependent. 124 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 5.5.2 Spatially Dependent Noise For most practically relevant types of spatially dependent noise the phases of the elements of k are not independent. In these cases it is difficult to find closed—form expressions for the joint MGF Ia(s) and the joint moments mn(ji,.. . , jM). Therefore, unless stated otherwise, we will concentrate in the following examples on the important special cases ad = 1 (with arbitrary NR) and NR = 1 (with arbitrary ad), where only the vector moments m(i) and the scalar moments m(i) of the noise are required, respectively. Additive Correlated Gaussian Noise (ACGN) in BICM—SC Systems In BICM—SC systems, correlated Gaussian noise n may be caused by narrowly spaced receive antennas [76]. Correlated Gaussian interference n = hkbk + ñ is caused by a synchronous co—channel interferer transmitting i.i.d. PSK symbols bk over a spatially correlated Rayleigh fading channel with gains hk and AWGN ñ. In both cases n is fully characterized by its covariance matrix C, E{n(n)H}, and the corresponding vector moments m(i) are given in Table 5.3, where ?q, 1 1 NR, denotes the eigenvalues of cnn. Asynchronous Co—Channel Interference (CCI) in BICM—SC Systems Another common type of non—AWGN impairment in BICM—SC systems is asynchronous CCI. We consider coding over B different hopping frequencies and assume that at hop ping frequency u, 1 B, in addition to AWGN ñ, there are I,, Rayleigh faded asynchronous CCI signals leading to time—domain noise k = IlkqA[’t] g,[11b,,.L[k — 1] + ñ, 1 k N, (5.30) i=1 1=k 125 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise Table 5.3: Vector moments m(i) of types of noise considered in Section 5.5. All variables in this table are defined in Section 5.5. (SC) and (OFDM) means that the type of noise is relevant for BICM—SC and BICM—OFDM, respectively. Noise type ] Vector moment m(i) Ak1 kNRACGN (SC) ,‘ 1 NR k1 AkNRCCI (SC) Zkj+...+kNR=i 1,S•• A’’NBI (OFDM) i! c0 Zkl+..+kNR=i 1,’,k” NR,p,v,k (i+NR—1)! +C1 (NR—1)! (i+NR—1)! ‘cç-IGMN (SC) (NR—i)! L.dk=1 Ck (i+NR—1)! k+•••+k=NCk1 k1GMN (OFDM) (NR-1)’ where hk,{i] and b,,[l] e ( Mi,,.,: M—ary symbol alphabet) denote the temporally i.i.d. zero—mean Gaussian random channel vector and the i.i.d. symbols of the ith interferer at the th hopping frequency, respectively. Furthermore, gj,[l] g,11(1T + where g,(t), T, and r are the effective pulse shape, the symbol duration, and the time offset of the ith interferer at the th hopping frequency, respectively. We assume that g,(1T + 0 for i < k1 and i > k, denote the set of all possible values Zk1 g,[1]b,,[l] by S, and define S,, x.. . x If I, = 0, we formally set = {0} with ISI = 1. With these definitions, the pdf of can be expressed as Pn(fl) = det(Cs) exp (_(fl)Hcifl), (5.31) where 1/(SB) and C3 Zi + aINR (ai: variance of elements of Eq. (5.31) shows that CCI in BICM—SC systems can be interpreted as correlated Gaussian mixture noise. For future reference we denote the ratio of the total CCI variance and the total AWGN variance by ic, cf. Section 5.6. The scalar moments m(i) (valid for NR = 1) and vector moments m(i) of asynchronous CCI are given in 126 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise Tables 5.2 and 5.3, respectively, where we have replaced C5 by u for NR = 1 in Table 5.2, and in Table 5.3, 1 1 NR, are the eigenvalues of C8. Narrowband Interference (NBI) in BICM—OFDM Systems We consider a BICM—OFDM system with coding over B different hopping frequencies. At hopping frequency p, 1 t B, the received frequency—domain signal is impaired by AWGN and I Rayleigh faded PSK NBI signals. The corresponding frequency—domain noise model is fl 9k,pjZ]btL[i]hkjL[2] + ñ,1, 1 k N, (5.32) where b[ij is the PSK symbol of the ith interferer at the tth hopping frequency affecting the set of sub—carriers via gk,1[i] exp[—jir(N — 1)(k + f,/if8)/N+ j] sinir(k + f,i/f8)]/sin[7r(k +f1,/.f3) N] [77]. Here, f,,,j and q,j denote the frequency and phase of the ith interferer at hopping frequency t relative to the user, respectively, and f8 is the OFDM sub—carrier spacing. Since we consider NBI, the same interference fading vector hk, [i] (modeled as spatially correlated zero—mean Gaussian random vector) affects all sub—carriers in JV,j. For f1,j = vLf, the NBI affects only sub—carrier v, i.e., J\f,j while, in theory, for f,j v/if the NBI affects all sub—carriers. However, gk,[i] decays quickly and we limit J\f,j such that gk,,.L[i] 0 for k 0 .A4j. Finally, we assume that no sub—carrier is affected by two narrowband interferers at a given hopping frequency, i.e., fl .Af,j2 = 0, i1 i2. The pdf for this general interference scenario is given by B I pn(fl) = 1.1=1 i=1 kE.A4L,j ±N2NR exp (_2), (5.33) 127 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise where o denotes the variance of the elements of the AWGN ñ, c0 1/(BN), c1 Id,/(BN), C,1,,k and C, Eq. (5.33) shows that, similar to CCI in BICM—SC systems, NBI in BICM—OFDM systems can be interpreted as correlated Gaussian mixture noise. We denote the ratio of the total NBI variance and the AWGN variance by ic, cf. Section 5.6. The corresponding moments m(i) and m(i) are provided in Tables 5.2 and 5.3, respectively, where we have replaced for NR = 1 in Table 5.2, and in Table 5.3, 1 I < NR, are the eigenvalues of Spatially Dependent (SD) GMN SD—GMN is an appropriate model for impulsive noise if all antennas are affected simulta neously by the phenomenon causing the impulsive behavior [28j. BICM-SC: The joint pdf for SD—GMN 4 is given by I t2 N2’R exp (— ‘ ), (5.34) where c and u are defined similarly as for SI—GMN is Section 5.5.1. Since the phases of the elements of n are independent random variables, the joint MGF n(s) can be calculated to I(s) = cj exp(o s/4). Furthermore, in this particular case, even a closed—form expression for the joint moment mn(ji,... , .JN), cf. Eq. (5.21), can be found as I ‘ç’ 2(jl+”.+jNR) mfl(J1,...,JNR) =J1!”3NR!ciu . (5.35) i=1 BICM—OFDM: The DFT operation at the receiver transforms the noise pdf in Eq. (5.34) into p(m) = exp (_HmiI2) (5.36) k1+•-i-k=N Ici ,...,k1 128 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise where the same definition is used for ck1 k1 and k as for SI—GMN, cf. Eq. (5.29). In this case, the joint MGF can be obtained as = cJ1 k1 exp(o k1 s?/4). (5.37) kj+••+k1=N 1=1 The corresponding joint moment is given by 2(j1+”+jNR) mn(31,.. . ,JN) = JN! Ck1 k1 (5.38) 5.5.3 Monte—Carlo Method For complicated types of noise such as UWB interference, it may be difficult to calculate the moments m(i), m(i), and mn(ji,. . . , jpj) in closed form. In such cases, a similar Monte—Carlo simulation method as described in Chapter 3 (cf. Subsection 3.2.4) can be used to obtain the noise moments. More specifically, the noise moments may be obtained by Monte—Carlo simulation of Eqs. (5.21), (5.22), or (5.23) and subsequently be used in Eq. (5.19) for calculation of the asymptotic BER. We note that this semi—analytical approach is much faster than a full simulation since the moments are independent from the SNR ‘y and have to be computed only once. 5.6 Numerical and Simulation Results In this section, we verify our derivations in Sections 5.3—5.5 with computer simulations and employ the presented theoretical framework to study the performance of BICM in non—AWGN environments. For the simulations, we consider both idealized channels with temporally i.i.d. channel and noise vectors, and non—ideal channels generated based on the models presented in Sections 5.2.1 and 5.5. In the non—ideal case, for BICM—SC 129 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise we assume a frame size of N = 972 and a normalized fading bandwidth BfT of 0.007, which are typical values for the DAMPS mobile communication system [78]. For BICM— OFDM we consider systems with N = 64 and N = 128 sub—carriers transmitting over channels with L = 10 and L = 20 i.i.d. impulse response coefficients. For all simulations shown, a pseudo—random interleaver was employed. Throughout this section we adopt the standard convolutional code with rate R = 1/2 and generator polynomials [133, 171] (octal representation). Higher code rates are obtained via puncturing and, unless specified otherwise, 4—PSK modulation and NR = 1 receive antennas are used. The parameters of the adopted noise models are specified in the respective captions of Figs. 5.1—5.7. In Fig. 5.1, we show simulation results for BICM—SC and BICM—OFDM impaired by GMN and NBI, respectively. In both cases, coding (R = 3/4) and interleaving is performed over different numbers of frames B. Besides the simulation results we also show the upper bound and the asymptotic BER derived in Sections 5.3 and 5.4, respectively. For high enough SNR and BICM—OFDM with N = 128 and the severely frequency— selective channel with L = 20 the analytical results are accurate even for B = 3. In contrast, for BICM—SC and BICM—OFDM with N = 64 and L = 10 the interleaver is not able to generate i.i.d. channels for small B which leads to performance degradation and the corresponding simulated BER exceeds the upper bound (which was derived assuming i.i.d. channels). However, as B increases, the simulation results approach the upper bound and the asymptotic BER also in these cases for high SNR. Note that for non—delay critical applications, such as data transmission, large B can be afforded. In Fig. 5.2, we show the BER of BICM—SC and BICM—OFDM (N = 64) for Rayleigh fading and various different noise and interference scenarios. Fig. 5.2 shows that the simu lated BERs (solid lines with markers), which were generated with non—ideal channels and for different B, approach the upper bound (solid lines without markers) and the asymp 130 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 100 \\ ICM-OFDM(N128) — Upperbound(theory) — — Asymptotic BER (theory) I :: II: -e— B = 3 (simulation) : t (soiiuon) BKM OFDM (N 64) B = 15 (simulation ) Figure 5.1: BER of BICM—SC and BICM—OFDM impaired by GMN (c—mixture noise, = 0.1, ic = 100) and NBI, respectively, vs. SNR -y. R 3/4, Rayleigh fading, 4—PSK, and NR = 1. BICM—SC: Flat time—selective fading, N = 972, and BfT = 0.007. BICM— OFDM (N = 64): Frequency—selective Rayleigh fading with L 10 and B equal power, sub—carrier centered NBI signals with I,, = 1, 1 B, ic = 7. BICM—OFDM (N = 128): Frequency—selective Rayleigh fading with L = 20 and B equal power, sub—carrier centered NBI signals with I, = 1, 1 < j.t < B, ic = 2. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. totic BER (dashed lines) for high SNR. In particular, for the BER region of BER < i0, which is difficult to simulate, the proposed analytical results are accurate approximations for the true BER. The upper bound is again not a true upper bound for the simulated BER because of the non—ideal channel. In accordance with our findings in Section 5.4.3, Fig. 5.2 shows that for high SNR all BER curves are parallel, i.e., all considered types of noise lead to the same diversity gain of Gd = df = 5. Nevertheless, there are large 10_I 1 02 1 io_5 0 5 10 15 SNR [dB] 131 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 100 0 5 10 15 20 25 30 SNR [dB] Figure 5.2: BER of BICM—SC and BICM—OFDM impaired by various types of noise vs. SNR 7. Rayleigh fading, R = 3/4, 4—PSK, and NR = 1. BICM—SC: N = 972 and BfT = 0.007. BICM—OFDM: N 64 and L 10. GMN I: €—mixture noise, = 0.01, ic = 100. GMN II: e—mixture noise, = 0.1, Ic 100. GMN III: f—mixture noise, = 0.1, Ic = 10. Asynchronous CCI: Two asynchronous equal power 4—PSK CCI signals, I, 1, t e {1, 2}, I, = 0, 3 10, raised cosine pulsesg1,(t), e {1, 2}, with roll—off factor 0.3, r1, = 0.3T, E {1, 2}, ic = 2. NBI I: One sub—carrier—centered NBI signal, I = 1, ‘2 = 13 = 14 = 15 = 0, ,c = 9. NBI II: 2 equal power, sub—carrier—centered NBI signals, I = 1 = 1, 13 = 14 = I = 0, Ic = 14. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. performance differences between different types of noise because of the different coding gains G. Fig. 5.2 confirms that OFDM is far more robust to GMN than SC if BICM is used in both cases. For GMN II BICM—OFDM outperforms BICM—SC by 5 dB at high SNR and approaches the performance in AWGN. This is an interesting result, since a pre vious comparison in [70] had shown that BICM—SC is more robust to GMN than uncoded 10_i 1 02 1 o 1 10 iO_8 1 1010 ‘S\ —a-— BICM—OFDM:NBII(B=5) :HiuHH. —-— BICM—OFDM:NBIII(B=5) ::.::: —AWGN I I I 132 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise SNR[dB] Figure 5.3: BER of BICM—OFDM impaired by NBI (3 equal power, sub—carrier—centered NBI signals, I = I = 13 = 1, i = 10) vs. SNR 7. Rayleigh fading, L = 20, 4—PSK, N = 128, B = 3, and NR 1. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. OFDM. Note, however, that for BICM—OFDM a relative large B is necessary to make the GMN approximately spectrally independent, whereas for BICM—SC GMN is temporally independent even for B = 1, cf. Section 5.5.1. In Fig. 5.3, we investigate the effect of the code rate R on the performance of BICM— OFDM (N = 128) in NBI for a non—ideal Rayleigh fading channel with L = 20 and B = 3. Fig. 5.3 shows that as the code rate increases, the diversity gain increases since the free distance of the code increases, cf. Eq. (5.24). While the upper bound (solid line without markers) approaches the simulation results (solid lines with markers) for BER < 10 in all cases, the convergence of the upper bound to the asymptotic BER (dashed lines) is slower 100 1 0_2 f 10 —e-- R0=1/2 —v.- R0 = 213 —e— R0 = 3/4 . R = 7’8 5 10 133 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 10° 1 02 I 1 0° b_b 1012 0 50 Figure 5.4: BER of BICM—SC impaired by GMN (E—mixture noise, = 0.25, tt = 10) and AWGN, respectively, vs. SNR -y. Ideal i.i.d. fading, R 7/8, 16—QAM, and NR = 1. for small (R = 1/2) and large (R = 7/8) code rates. For R = 1/2, df is large making the asymptotic BER curve very steep, which leads to an over—estimation of the BER at low SNRs. For R = 7/8, the slow convergence can be explained by the large relative weight of terms neglected in asymptotic BER expressions (e.g. w(df + 1)/w(df) = 56). For comparison, R = 3/4 shows a much faster convergence since w(df + 1)/w(df) = 5. In Fig. 5.4, we consider the impact of the type of fading on BER for GMN and AWGN. Note that idealized channels with i.i.d. coefficients have been used to obtain the simulations shown in Fig. 5.4 and, in contrast to the other figures in this chapter, 16—QAM was employed instead of 4—PSK. Since the type of fading affects the diversity gain Gd = cddf, the asymptotic slopes of the BER curves for Nakagami—m (cd = m = 2) and Weibull (cd = c/2 = 2/3) fading differ from the asymptotic slopes of the BER curves for Rayleigh, 5 10 15 20 25 30 35 40 45 SNR [dB] ‘ 134 Figure 5.5: BER of uncoded SC transmission impaired by SD- and SI—GMN (c—mixture noise, c 0.1, ic = 10), respectively, vs. SNR y. NR = 2, Nakagami—m fading spatial correlation Pa = 0.9, and 4—PSK. Ricean, and Nakagami—q fading, since for the latter three ad = 1 holds. It can also be observed that the performance loss caused by GMN compared to AWGN decreases with decreasing diversity order. In Fig. 5.5, we show the BERs of uncoded SC transmission over correlated Nakagami—m channels with NR = 2 receive antennas and impairment by SD- and SI—GMN (both cases: c—mixture noise with c = 0.1, ,c = 10). The spatial fading correlation coefficient is Pa = 0.9. Note that for uncoded transmission the temporal i.i.d. asumption for fading and noise is not required. Fig. 5.5 shows that for uncoded transmission the derived upper bound is very tight even at low SNR and approaches the asymptotic BER at high SNR. Thereby, the asymptotic BER converges faster to the upper bound for channels with smaller diversity 135 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 10 1 08 EE:EE:EEEEES:SHSSH 4.!!’ 5 10 15 2025 SNR [dB] 10-Il 0 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 100 ., I I I I —e— lid. fading, AWGN lid. fading, ACGN = 0.9) .. correlated fading h = 0.9), AWGN 10 g(pO9) ACGN(p=09) b_b :‘.\ 10’ I I I 0 5 10 15 20 25 30 SNR [dB] Figure 5.6: BER of BICM—SC impaired by AWGN/ACGN vs. SNR 7. Spatially i.i.d./spatially correlated, temporally i.i.d. Rayleigh fading, R = 7/8, 4—PSK, and NR = 2. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. gain. Furthermore, Fig. 5.5 confirms that spatial noise dependencies lead to significant performance degradations. In Fig. 5.6, we consider the BER of BICM—SC impaired by temporally i.i.d., spatially uncorrelated/correlated (fading correlation Ph = 0.9) Rayleigh fading and AWGN/ACGN (noise correlation p = 0.9) for NR = 2. Fig. 5.6 shows that, while noise correlation has also adverse effects on performance, fading correlation is more harmful. Furthermore, the convergence of the asymptotic BER to the union bound is negatively affected by the spatial fading correlation. Finally, in Fig. 5.7, we consider the BER of BICM—OFDM impaired by UWB inter 136 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise 10 _________________________________ -e-- IR_UWB(L=16,t=4M ::::: IR-UWB (L= 128, = 4MHZ) 10_i “ ....N\........................ __ MB_OFDMUWB(f=4MHz) 2 :::::ç::::::;:.::::::::::::: BOAUWBf—i) 10 H :: \ V V 10 .: ,,: I 110 02’5 30 SNR [dBj Figure 5.7: BER of BICM—OFDM system with sub—carrier spacing Lf3 impaired by IR— UWB [3] (Nb = 8 bursts per symbol and L chips per burst) and MB—OFDM UWB [1], respectively, vs. SNR ‘y. Ideal i.i.d. Rayleigh fading, R = 5/6, 4—PSK, and NR = 1. Solid lines with markers: Simulated BER. Solid lines without markers: BER bound [Eq. (5.8)]. Dashed lines: Asymptotic BER [Eq. (5.19)]. For comparison the bound and the asymptotic BER for AWGN are also shown. ference and temporally i.i.d. Rayleigh fading. We consider MB—OFDM and IR—UWB interference following the EMCA [1] and the IEEE 802.15.4a [3] standards, respectively. Specifically, for IR—UWB we assume Nb = 32 bursts per symbol and L chips per burst [3]. The MGF required for the upper bound in Eq. (5.8) was obtained using the methods pro posed in [71]. Since, due to the complicated nature of the interference signal, closed—form expressions for the moments are difficult to obtain, we used the Monte—Carlo approach discussed in Section 5.5.3 for calculation of the moments for evaluation of the asymptotic BER given in Eq. (5.19). Fig. 5.7 nicely illustrates that the coding gain in UWB interfer 137 Chapter 5. Performance of Generic BICM—Based Systems in Non—Gaussian Noise ence strongly depends on the sub—carrier spacing of the victim BICM—OFDM system and the format of the UWB interference. 5.7 Conclusions In this chapter, we have presented a framework for performance analysis of BICM—SC and BICM—OFDM systems impaired by fading and non—Gaussian noise and interference. The proposed analysis is very general and applicable to all popular fading models (including Rayleigh, Ricean, Nakagami—m, Nakagami—q, and Weibull fading) and all types of noise with finite moments (including AWGN, ACGN, GMN, CCI, NBI, and UWB interference). In particular, we have derived an asymptotically tight upper bound for the BER which allows for efficient numerical evaluation and a simple closed—form expression for the asymp totic BER. Our analysis reveals that while the coding gain is strongly noise dependent, the diversity gain of the overall system is not affected by the type of noise. This result is important from a practical point of view since it shows that at high SNRs the BER curves of BICM systems optimized for AWGN will only suffer from a parallel shift if the impairment in a real—world environment is non—Gaussian. 138 Chapter 6 Conclusions and Future Work In this chapter, we conclude the thesis by summarizing our results and highlighting the contributions of this dissertation. We also suggest several topics for further research. 6.1 Research Contributions In this thesis, we studied the effects of UWB interference on co—existing victim systems in Chapters 2 and 4, and showed that the UWB interference can have highly non—Gaussian characteristics in different practical scenarios. We generalized the results obtained in Chap ters 2 and 4, to more general types of non—Gaussian noise and interference in Chapters 3 and 5, respectively. In particular, in Chapter 2, we provided a comprehensive analysis of the performance of generic uncoded NB systems in MB—OFDM UWB interference. Assuming BPSK mod ulation for the NB system, we employed an MGF—based approach to obtain the exact interference APD and the exact BER of an uncoded NB receiver impaired by MB—OFDM interference. Based on the BER and APD results we showed that for NB system band widths of B3 < O.O5/.f and B8 > the NB system performance can be accurately predicted using the GA and the IGA, respectively. For O.O5Lf B3 however, the BER of the NB system strongly depends on various NB system parameters such as carrier frequency offset between NB signal and MB—OFDM signal, the NB signal bandwidth, the number of MB—OFDM frequency bands, and the NB pulse shape. We further showed 139 Chapter 6. Conclusions and Future Work that in this case, both GA and IGA are inaccurate in modeling the MB—OFDM UWB interference and therefore the proposed exact analysis has to be used to obtain reliable performance results. We studied the asymptotic SER of linearly modulated uncoded systems impaired by fading and generic non—Gaussian noise in Chapter 3. The only major assumption on the considered noise was that its moments exist, which is true for most practically relevant types of noise. Based on this assumption we provided asymptotically tight SER approxi mations for linear modulation formats with single—branch reception, EG C, and SC and for BOM. The obtained approximations were closed—form and therefore fast and easy to eval uate. Our asymptotic analysis showed that the diversity gain is independent of the noise statistic and only depends on the fading statistic and the number of diversity branches. In contrast, the combining gain was shown to be dependent on both the type of fading and the type of noise through the noise moments. In Chapter 4, we provided a union bound on the BER performance of generic BICM— OFDM systems impaired by MB—OFDM UWB, DS—UWB and IR—UWB interference. Via simulations, we demonstrated that this bound is tight for BER values of practical interest. Using the obtained BER results we showed that the GA provides a close approximation for DS—UWB interference for all practically relevant BICM—OFDM sub—carrier spacings e.g. f8 < 10 MHz. We further showed that for MB—OFDM and IR—UWB interference the GA is generally unable to accurately predict the system performance and it only becomes valid for < 100 kHz. For MB—OFDM interference, the GA overestimates the BER for f3 < 2 MHz but underestimates it for f8 > 2 MHz. The GA always underestimates the BER in the presence of IR—UWB interference and becomes more inaccurate with increasing /f3. We also used our numerical results to assess the accuracy of the GA for the performance of practically important victim receivers in UWB interference. In 140 Chapter 6. Conclusions and Future Work particular, we could show that for IEEE 802.lla WLANs the GA overestimates the BER for MB—OFDM interference but underestimates it for IR—UWB interference. In contrast, the GA underestimates the BER for both IR—UWB and MB—OFDM interference when it is used to evaluate the performance of a MB—OFDM victim receiver interfered by other UWB systems. We analyzed the performance of receive diversity BICM—SC and BICM—OFDM systems impaired by fading and generic non—Gaussian noise and interference in Chapter 5. The proposed analysis was very general and applicable to all popular fading models and all types of noise with finite moments. In particular, we derived an asymptotically tight upper bound for the BER which allows for efficient numerical evaluation and a simple closed— form expression for the asymptotic BER. Using the obtained BER results we showed that the diversity gain is independent of the noise type but depends on the type of fading, the number of receiver antennas, and the free distance of the convolutional code. In contrast, the coding gain was shown to be dependent on the types of fading, the employed modulation and coding schemes, the number of receiver antennas, and the type of noise via the noise moments. Our analysis showed that BICM systems optimized for AWGN are also optimum for any other type of noise and interference with finite moments provided that the popular ED metric is used for Viterbi decoding. 6.2 General Conclusions Our findings in Chapters 2 and 4 show that in general, the accuracy of the widely used GA for the UWB interference is only guaranteed if Pr T/T8 << 1, where T and T3 denote the symbol durations of the UWB system and the victim system, respectively. However, in most practical scenarios Pr 15 not small enough to guarantee the tightness of the GA. For larger values of Pr, the accuracy of the GA is generally a function of various 141 Chapter 6. Conclusions and Future Work system parameters such as the assumed UWB signal format, the type of underlying fading channel, and the transmission and reception schemes employed in the victim system. In such cases, the provided exact/accurate analytical results should be used to obtain reliable performance predictions. Furthermore, the results obtained in Chapters 3 and 5 help us draw some general con clusions regarding the asymptotic performance of generic victim systems in non—Gaussian noise. More explicitly, we conclude that while the diversity gain is always independent of the noise type, it is in general a function of the fading type and the transmission and re ception schemes employed in the victim system. On the other hand, the combining/coding gain not only depends on the type of fading and the transmission and reception schemes employed in the victim system but also on the type of noise through the noise moments. These results are important from a practical point of view as they indicate that at high SNRs, the BER curves of victim systems optimized for AWGN will only suffer from a parallel shift if the impairment in a real—world environment is non—Gaussian. 6.3 Suggestions for Future Work 1. In Chapters 3 and 5, we assumed coherent detection for the considered BICM—based victim receivers. As a natural extension of the results developed in these chapters, differentially coherent and non-coherent BICM—based schemes can be considered for the victim receiver. 2. Although BICM—based schemes provide high performance gains in fading channels compared to other coded modulation schemes, their performance can be further im proved by using iterative decoding and demodulation at the receiver. The resulting technique is called BICM iterative decoding (BICM—ID) [79] and has received some 142 Chapter 6. Conclusions and Future Work attention recently. It is of interest to investigate the performance of BICM—ID victim receivers in the presence of UWB interference and other types of non—Gaussian noise. 3. In Chapters 3 and 5, the considered BICM schemes were based on convolutional coding, i.e., they employed convolutional binary encoders and Viterbi decoders at the transmitter and receiver, respectively. Current BICM—based IEEE standards such as the IEEE 802.16 and the IEEE 802.lla specify convolutional coding as the mandatory coding scheme but also allow an optional use of turbo coding. The op tional turbo coding scheme employs a convolutional or a block turbo encoder at the transmitter in conjunction with an iterative decoder at the receiver, to improve the system performance at the cost of increased overall system complexity. It is desirable to extent the framework developed in Chapters 3 and 5 to turbo coding. 4. In this thesis, we have considered victim receivers that are designed to exhibit optimal performance for Gaussian distributed noise at the receiver. When non—Gaussian noise is present at the receiver, the considered receivers will therefore have sub— optimal performance. The knowledge of noise statistics can be exploited at the receiver to design new techniques with optimal performance in the presence of non— Gaussian noise with known statistics. While in practice this information is usually not available at the receiver, the design and performance analysis of such receivers are still of interest, as they can be used to reveal the best achievable performance in the presence of non—Gaussian noise. 5. After studying coded and uncoded systems, it is interesting to see how close these systems perform to the channel capacity, which serves as an upper bound for the transmission data rate. In this regard, the channel capacity can be obtained in the presence of UWB interference and other types of non—Gaussian noise and then 143 Chapter 6. Conclusions and Future Work be compared to the actual data rate of the considered system. While conventional expressions are available for channel capacity in AWGN, the literature on channel capacity in the presence of UWB interference and other non—Gaussian impairments is sparse. 144 Bibliography [1] ECMA, “Standard ECMA-368: High Rate Ultra Wideband PHY and MAC Stan dard,” [Online] http://www. ecma-international. org/publications/standards/Ecma 368.htm, Dec. 2005. [2] IEEE P802.15, “DS-UWB Physical Layer Submission to IEEE 802.15 Task Group 3a (Doe. Number P802. 15-04/0137r4) ,“ Jan. 2005. [3] I. P802.15.4a, “Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LR-WPANs): Amend ment to Add Alternate PRY,” Jan. 2007. [4] M. Simon and M.-S. Alouini, Digital Communication over Fading Channels. Hoboken, New Jersey: Wiley, 2005. [5] J. Bellorado, S. Ghassemzadeh, L. Greenstein, T. Sveinsson, and V. 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Tung, “Mellin Transform Applied to Uncertainty Analysis in Hydrol ogy/Hydraulics,” Journal of Hydraulic Engineering, vol. 116, pp. 659—674, May 1990. 151 Appendix A Periodicity of T?jTk(s, Tk) The periodicity of jIrk(s, -rk) can be proved by replacing Tk by rL Tk+NB)s.Tj in Eq. (2.20) 1 N1/2 iTk(s,Tk) = i— /‘ J1[ II J2ir n=—N/2 nO cosh (s Im{gejOei0lTu/3n(rk + NB)T — NBIcT)}) d8, cc N/21 j j-j fJ cosh(s — NBIc’T)})— 2ir 27r n=—N/2 nO cosh (s — NBIc’T)}) d9 cc N/2 = _ f JJ fJ cosh(s — NBlc’T)})2ir c’oQ n=—N/2 no cosh (s Jm{ge29Ii0 ‘T,3fl(rk — NBIc’Ti)}) d8’ = jITk(8, Tk), (A.1) where we have used the substitutions ic’ = — ) and 0’ = 0+ 2IrfONBATi, respectively. 152 Appendix B Asymptotic BER in Rayleigh Fading Based on Eq. (2.6) we can express the exact BER as Pe,E = Pr{ab[k] + i[k] + n[kj — ab[kfi2 > ab[k] + i[k} + n[k] + ab[k]j2} Pr{a + a(i[k] + n[k])*/2 + a(i[k] + n[k])/2 < O}. (B.1) For Rayleigh fading a is a zero—mean Gaussian random variable. Using the results of [26, Appendix B] we can express the exact BER conditioned on i[k] as 1 ( Io(y2)e_Y2 PeIi,E = — /i + 2o)’ (B.2) where y2 Iic[kH2/(2\/1 + 2o) and Io() denotes the zeroth order Bessel function of the first kind. Assuming Ii[kH2 << 1 and o << 1 and using the relations 1/’1 + x 1 — x/2, Io(x) 1 —x2/4, and e 1 — x, which are all valid for x << 1, we obtain from Eq. (B.2) P€IeE = o/2 + i[k]I2/4. (B.3) Averaging Eq. (B.3) over the (unknown) pdf of the interference yields the (unconditional) BER Pe,E = o/2 + u?/2, (B.4) 153 Appendix B. Asymptotic BER in Rayleigh Fading where we have used {i[kJI2}= 2o which follows from the fact that i[k] is rotationally symmetric since °g is uniformly distributed in [—ir, 7r). Now, Eq. (2.42) is easily obtained by combining Eq. (B.4) with Eq. (2.9). 154 Appendix C The Mellin Transform In this appendix, we discuss the Mellin transform and its properties in some detail. For further reading we recommend [80, 81]. C.1 Definition and Existence The Mellin transform Ma(s) M{p(n)} of p(n) is defined as Ma(s) f ns_lpm(n) dn (c.1) where both Pn(fl) and s may be complex in general. The Mellin transform exists if f°° Re{s}—1 p(n) dn is finite. The interval cj < Re{s} ci for which Ma(s) exists is referred to as the fundamental strip. Tables of Mellin transforms can be found in [80]. In the remainder of this appendix, p(n), p,1 (n), and p2 (n) denote the pdfs of n, n1, and n2, respectively. Furthermore, we use the notations Ma(s) M{p(n)}, M1(s) M{p1(n)}, and M2(s) M{p2(n)}. C.2 Basic Properties The Mellin transform has many useful properties. A detailed discussion of these properties in the context of RVs and pdfs can be found in [81]. Here, we only state most important 155 Appendix C. The Mellin Transform properties of the Mellin transform without proof. 1) Scaling: M{p(cn)} = c Ma(s), c > 0. (C.2) 2) Linearity: M{cxjp1(n) +c2pn(n)} = cM1(s) + c2M2(s). (C.3) 3) Scaling of the RV: Let n1 = cin, c > 0, then M1(s) = Ma(s). (C.4) C.3 Product of Two Independent RVs The Mellin transform in Eq. (C.1) is only defined for positive n, whereas pn(n) is an even function of n, cf. AS1) in Section 3.2.3. Usually this discrepancy is not a problem and we can just ignore pn(n) for n < 0 when calculating the Mellin transform. However, care must be taken when calculating the Mellin transform of the product of two RVs. Let n = n12, where n1 and n2 are two independent RVs with even pdfs, respectively. The pdf of n can be expressed as pn(n) = f°pn1(n)2n/ dni/niI. Using the def inition in Eq. (C. 1) and exploiting Pn1 (n) = p1 (—n) andp2(n) = pn2(n), it is easy to show that the Mellin transform of pn(n) can be expressed as Ma(s) = 2M1(s)Ms) (C.5) which differs from [81, Eq. (15)] by a factor of two since in [81] non—negative RVs n1, n2 were assumed. 156 Appendix C. The Mellin Transform C.4 Sum of Two Independent RVs Let n = ri1 + n2, where n1 and m2 are independent RVs. For general s it is difficult to find a simple relation between Ma(s) and M1 (s), M2(s). In the following, we show however that such a relation exists if s is an integer. First, we establish a relation between Ma(s) and the moments ma(s) of ri. If s is an even integer, we obtain 2M(s + 1) = ma(s) Th8Pn(fl) dn. (C.6) Similarly, if s is odd, we can express 2M(s + 1) as 2M(s + 1) = thu(s) n8(n) dn (C.7) where (n) = p(n), n 0, and (n) = —p(n), n <0. Note that ma(s) = 0 and thu(s) = 0 for odd and even s, respectively. Recall that the Laplace transform of the pdf of n is given by ‘Tfl, (p) = I (p) 12(p), where 71(p) £{pn1(fl)} and I2(p) £{p2(m)}.3 For even s, the moments ma(s) of ri can be calculated from ma(s) = = (_1) () Skl(1) = (_1)8 () m(s - k)m2(k) (C.8) where m1(s) and m2(s) denote the sth moment of n1 and n2, respectively. It is easy 23Following the literature, in general we use “s” as transformation variable for both Mellin and Laplace transform. In this appendix, however, we deviate from this practice to avoid confusion and use “p” as transformation variable for the Laplace transform. 157 Appendix C. The Mellin Transform to show that Eq. (C.8) also holds for odd s if ni0 is replaced by m. Therefore, taking Eqs. (C.6) and (C.7) into account, the Mellin transforms of pnQn) for integer s can be calculated from Ma(s) = 2 (s_i) M1(s — 2k)M(2k + 1), S odd , 1. (C.9) 2 (21) M1(s — 2k)M(2k + 2), s even 158 Appendix D Spatially Correlated Fading Channels In this appendix, we prove Eq. (5.2) for correlated Rayleigh, Ricean, and Nakagami—m fading. For simplicity of notation, we drop subscript k in this appendix. Ricean Fading: For Ricean fading the pdf of the channel vector h is given by ph(h) = NR det(Chh) exp [—(h — h) — (D.1) where ILh {h} and Chh — M’h) (h — I-”h)} are the channel mean and channel covariance matrix, respectively. For h —> we can rewrite Eq. (D.1) as ph(h) = exp( LhChhiIh) + o(l). (D.2) Based on Eq. (D.2) and the relation h12 = a it can be shown that Eqs. (5.2) and (5.3) hold for correlated Rayleigh (th = ONR) and Ricean (p ONR) fading with c and 0d as specified in Table 5.1. Nakagami—m Fading: For Nakagami—m fading the joint MGF of a?, 1 1 NR, is given by [4] a2(8) E {ex (_ a? SI) } = det(INR + SCaa/m)m, (D.3) 159 Appendix D. Spatially Correlated Fading Channels where S diag{s}, and Caa and m denote the channel correlation matrix and the fading parameter, respectively. The behavior of the joint pdf pa2(a, ..., aNR) of a?, 1 1 NR, for a —* °NR can be deduced from the behavior of Ia2(S) for Sj —+ oc, 1 1 NR, which is given by NR NH a2(S) = mNRmdet(Caa)_mflsim + (flsIm). (D.4) Consequently, we obtain NH 2(m—1) NH Pa2(a, ..., a) =mmdet(Caa)_mfl( +0 (fla?(m_1)), (D.5) which clearly shows that the a1, 1 <1 NR, are asymptotically i.i.d., i.e., Eqs. (5.2) and (5.3) are valid. The corresponding parameters c and 0d are provided in Table 5.1 and can be obtained by exploiting the relation between pa2(a, ..., a) and pa(a). 160 Appendix E Related Publications The following is a list of publications that are based on the research conducted for this thesis. Journal Papers 1. A. Nasri and R. Schober. Performance of BICM—SC and BICM—OFDM Systems with Diversity Reception in Non—Gaussian Noise and Interference. Submitted to the IEEE Transactions on Communications. 2. A. Nasri, R. Schober, and L. Lampe. Performance of BICM—OFDM Systems in UWB Interference. Accepted for Publication in the IEEE Transactions on Wireless Communications, 2008. 3. A. Nasri, R. Schober, and Y. Ma. Unified Asymptotic Analysis of Linearly Modulated Signals in Fading, Non—Gaussian Noise, and Interference. Accepted for Publication in the IEEE Transactions on Communications, 2007. 4. A. Nasri, R. Schober, and L. Lampe. Analysis of Narrowband Communication Sys tems Impaired by MB—OFDM UWB Interference. IEEE Transactions on Wireless Communications, 6:4090—4100, November 2007. Conference Papers 5. A. Nasri and R. Schober. Performance of BICM—OFDM Systems in Non—Gaussian 161 Appendix E. Related Publications Noise and Interference. Submitted to the IEEE Global Telecommunications Confer ence (Globecom), March 2008. 6. A. Nasri, R. Schober, and L. Lampe. Performance Evaluation of BICM—OFDM Systems Impaired by UWB Interference. Accepted for presentation at the IEEE International Communications Conference (ICC), Beijing, China, May 2008. 7. A. Nasri, R. Schober, and M. Yao. Asymptotic SER Analysis of EGC and SC in Fading and Non—Gaussian Noise and Interference. In Proc. IEEE International Communications Conference (ICC), Glasgow, June 2007. 8. L. Lampe, A. Nasri, and R. Schober. Interference from MB—OFDM UWB Systems: Exact, Approximate, and Asymptotic Analysis. In Proc. IEEE International Confer ence on Ultra—Wideband (ICUWB), Invited Paper, Best Paper Award, Boston, September 2006. 9. A. Nasri, R. Schober, and Y. Ma. Unified Asymptotic Analysis of Linearly Modulated Signals in Fading and Noise. In Proc. IEEE Global Telecommunications Conference (Globecom), Finalist for Best Student Paper Award, San Francisco, December 2006. 10. A. Nasri, R. Schober, and L. Lampe. Comparison of MB—OFDM and DS—CDMA Interference. In Proc. IEEE International Conference on Ultra— Wideband (ICUWB), Boston, September 2006. 11. A. Nasri, R. Schober, and L. Lampe. Performance of a BPSK NB Receiver in MB— OFDM UWB Interference. In Proc. IEEE International Conference on Communica tions (ICC), Istanbul, June 2006. 162 Appendix E. Related Publications 12. A. Nasri, R. Schober, and L. Lampe. Analysis of NB BPSK in MB—OFDM UWB Interference and Fading. In Proc. IEEE Vehicular Technology Conference (VTC), Melbourne, May 2006. 163
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Performance of wireless communication systems in ultra-wideband interference and non-Gaussian noise Nasri-Nasrabadi, Amir Masoud 2008
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Title | Performance of wireless communication systems in ultra-wideband interference and non-Gaussian noise |
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Nasri-Nasrabadi, Amir Masoud |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | The growing demand for high speed wireless applications and the scarcity of the spectral resources has necessitated the development of new concepts that enable more efficient utilization of the frequency spectrum. Ultra-wideband (UWB) is an emerging technology that is capable of utilizing the spectral resources more effectively by sharing the spectrum with other applications. This spectrum sharing however can potentially result in harmful interference from UWB systems to devices co-existing in the same frequency band. Therefore, in order to guarantee peaceful co-existence, the effects of UWB interference on co-existing systems have to be carefully analyzed. Towards this goal, in this thesis, we first study the effects of multi-band (MB) orthogonal frequency-division multiplexing (OFDM) UWB interference on a generic uncoded narrowband (NB) system. For this purpose, we develop analytical expressions for the amplitude probability distribution (APD) and the bit error rate (BER) performance of the NB system in the presence of MB-OFDM interference. We use the obtained results to assess the accuracy of the Gaussian approximation (GA) for MB-OFDM UWB interference. We show that for most channel models and signal bandwidths the GA is unable to accurately predict the NB system performance and the exact BER analysis has to be used to obtain meaningful results. We also analyze the effects of UWB interference in a more general framework that allows us to study the impact of general types of non-Gaussian noise and interference on generic uncoded victim systems. Specifically, we present a unified asymptotic symbol error rate (SER) analysis of linearly modulated systems impaired by fading and generic non- Gaussian noise and interference. Our analysis also encompasses diversity reception with equal gain and selection combining and is extended to binary orthogonal modulation. The obtained asymptotic results show that for high signal-to-noise ratios (SNRs) the SER of the victim system depends on the moments of the non-Gaussian noise and interference. Furthermore, we study the impact of different types of UWB interference on victim systems that employ a combination of bit-interleaved coded modulation (BICM) and OFDM. For the UWB interferer we consider MB-OFDM, direct-sequence UWB (DS-UWB), and impulse radio UWB (IR-UWB) interference formats following recent standards or standard proposals developed by the IEEE or the European Computer Manufacturers Association (ECMA). Besides the exact analysis we calculate the BER of the BICM-OFDM system for the case when the UWB interference is modeled as additional Gaussian noise. Our results show that while the GA is very accurate for DS-UWB interference, it may severely over or underestimate the true BER for MB-OFDM and IR-UWB interference. Finally, we analyze the performance of BICM-based systems in the presence of generic non-Gaussian noise and interference. In particular, we present an asymptotically tight upper bound for the BER and a closed-form expression for the asymptotic performance of receive diversity BICM single-carrier and BICM-OFDM victim systems.. Our analysis shows that, if the popular Euclidean distance metric is used for Viterbi decoding, BICM- based systems optimized for additive white Gaussian noise (AWGN) are also optimum for any other type of noise and interference with finite moments. |
Extent | 6366060 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-03-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0067048 |
URI | http://hdl.handle.net/2429/5741 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2009-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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