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Channel aware strategies in wireless communications Elkashlan, Maged 2006

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Channel Aware •Strategies in Wireless Communications by Maged Elk'ashlan •B'.Sc, Arab Academy for-Science.and'Technology, 1997 M.Sc , Arab Academy for- Science and Technology,. 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS: F O R T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES (Electrical and Computer Engineering) T H E UNIVERSITY OF BRITISH C O L U M B I A August 2006 © Maged Elkashlan, 2006 11 Abstract Channel-aware resource allocation is widely considered to be crucial for realizing high data rates in wireless networks. This thesis considers the resource allocation problem in a multiuser wireless network using link adaptation. Link Adaptation (LA), which loosely refers to changing transmission parameters over a link, such as modulation, coding rate, power, etc., in response to changing channel conditions is considered to be a powerful means of achieving higher efficiency or throughput in wireless networks. The adaptation of the transmission parameters is performed according to the predicted future quality of the channel, also referred to as the channel state (CS). The objectives of this thesis are threefold: devise new methods for the improvement of multicarrier code division multiple access ( M C - C D M A ) as an effective signaling scheme in correlated fading channels, the development of new channel aware algorithms with adaptive subchannel allocation, and the investigation of the statistics of general order selection in correlated Nakagami fading channels. A novel adaptive subcarrier allocation algorithm is developed for M C - C D M A to im-prove the overall bit error rate (BER) performance. The proposed method is suitable for use in correlated fading channels. This algorithm assigns users to subcarrier groups that provide favorable fading characteristics, while simultaneously reducing the amount Abstract 111 of interference caused to other users. This method examines the effect of equalizing the interference i n each subcarrier group while maintaining a reduced correlation among the subcarrier fading processes for a group. Consequently, subcarriers are separated into non-contiguous groups to maximize frequency diversity and minimize multiple access in-terference ( M A I ) . Previously proposed adaptive subcarrier allocation algorithms, are considered to be greedy algorithms. They are greedy i n the sense that they consider every reservation request individually, and make the choice that looks best at the moment. Adapt ive allocation using the simple greedy method gives an optimal solution for a single user system. Adapt ive bit allocation using the greedy method cannot give an optimal solution for multiuser cases. It is possible that the sub-channels w i t h the largest channel gain for one user are also the largest for another user. We present i n this thesis a new class of dynamic resource allocation schemes that are based on al l the users' subband gains. Due to the time-varying nature of the wireless channel, dynamic resource allocation makes full use of multiuser diversity to achieve higher performance. In this thesis, we formulate the multiuser subcarrier allocation problem and propose an iterative algorithm to perform the subcarrier allocation. This method involves the ordering of subband gains. Thus, the analytical foundation is the theory of order statistics. The users are recursively assigned to that subcarrier which provides the highest possible signal-to-noise ratio ( S N R ) . W h i l e ensuring that no more than one user is assigned to the same subband. Our objective is to find low-complexity schemes which can improve system capacity and throughput, and simultaneously minimiz ing the M A I . Abstract iv Finally, the cumulative distribution function (cdf) (and hence outage probability) of the r-th order branch SNR in correlated Nakagami—m fading is studied. The accuracy of a simple exchangeable approximation to reduce the computational load is examined. The effects of correlation on the error performance of channel aware systems that involve the ordering of channel gains in a correlated Nakagami fading environment is investigated. A useful analytical formula for the B E R of the r-th order statistic of a set of arbitrarily correlated and hot necessarily exchangeable diversity branch gains is described and shown to be applicable in the performance analysis of various diversity systems. V Contents Abstract ii Contents v List of Figures ix List of Abbreviations xii Acknowledgements xvi Dedication xviii 1 Introduction 1 1.1 Mobile Communications Systems: Past, Present, and Future 1 1.2 From Second- to Third-Generation Multiple Access Schemes 3 1.3 From Third- to Fourth-Generation Multiple Access Schemes 5 1.4 Motivation 8 1.5 Link Adaptation Fundamentals 10 1.6 Subchannel and Power Allocation 12 1.7 This Thesis 17 Contents vi 1.7.1 Objectives 17 1.7.2 Contributions 17 1.7.3 Road Map 20 2 Matched M C - C D M A in Slowly Fading Dispersive Channels 22 2.1 Introduction 22 2.2 System Model 23 2.3 Numerical Results 25 3 Frequency-Hopping Mult icarrier C D M A in Rayleigh Fading 31 3.1 Introduction 31 3.2 F H - M C - C D M A System Model 33 3.3 Numerical Results and Discussions 38 4 Frequency-Hopping Mult icarrier C D M A in Correlated Rayleigh Fading 47 4.1 Introduction 47 4.2 System Model 48 4.2.1 F H - M C - C D M A 48 4.2.2 Channel Model 50 4.3 B E R Analysis of F H - M C - C D M A 51 4.3.1 Uniform F H - M C - C D M A 52 4.3.2 Random F H - M C - C D M A 53 4.3.3 Optimum F H - M C - C D M A 53 4.4 Numerical Results 54 Contents vii 5 A Channel Aware Frequency Hopping Mult ip le Access Scheme . . . 63 5.1 Introduction 63 5.2 System Model 64 5.3 C A F H Scheme 65 5.4 B E R analysis 66 5.4.1 B E R analysis of C A F H for r = l 66 5.4.2 Numerical Results of C A F H for r=l 70 5.4.3 B E R Analysis of C A F H for r >1 72 5.4.4 Numerical Results of C A F H for r >1 74 6 Order Statistics for Correlated Fading Channels 88 6.1 Introduction 88 6.2 Preliminaries 89 6.3 cdf of 7 r : n 91 6.3.1 Correlated Fading 91 6.3.2 Independent Fading 93 6.4 Numerical results 93 6.4.1 Constant Correlation Model 94 6.4.2 Linear Model 96 7 E r r o r Analysis of Order Selection in Correlated Fading Channels . . 102 7.1 Introduction 102 7.2 Channel and System Model 103 Contents viii 7.3 Error Performance for Order Selection B P S K 104 7.4 Numerical Results and Discussion 108 7.4.1 B E R evaluation for Order Selection B P S K 109 7.4.2 Application: B E R evaluation for C A F H 110 8 Conclusions and Future Research 118 8.1 Summary 118 8.2 Future Work 122 B i b l i o g r a p h y 126 ix List of Figures 2.1 (a) Transmitter model (b) Receiver model : 27 2.2 Matched M C - C D M A subcarrier pattern, channel transfer function T F , and integral \&(-) with = 2/BT- The frequency corresponding to the I T H subcarrier is fc + FiBch 28 2.3 B E R as a function of number of users for SNR = 9 dB 29 2.4 B E R as a function of SNR for M = 64 users 30 3.1 (a) Transmitter model (b) Receiver model 40 3.2 F H - M C - C D M A pattern 41 3.3 Theoretical and simulation B E R comparison for N = M 42 3.4 Theoretical and simulation B E R comparison for N < M 43 3.5 B E R comparison between random and coordinated F H - M C - C D M A for M = 32 users ' 44 3.6 B E R comparison between random and coordinated F H - M C - C D M A for M — 64 users 45 3.7 B E R as a function of SNR for M = 64 users 46 4.1 (a) Transmitter model; (b) Receiver model 57 List of Figures x 4.2 Frequency groups when NGj = N V j with /V — L = 4, and F = g = s/N. 58 4.3 B E R versus SNR for M = 32 MS's, and L = 32 59 4.4 B E R versus SNR for M = 32 MS's, and L = 8 60 4.5 B E R versus TV for L = 16, and SNR = 10 dB 61 4.6 B E R versus M for L = 16, and SNR = 10 dB . 62 5.1 A C A F H flow chart describing the proposed subband assignment scheme. 77 5.2 B E R of conventional F H and C A F H with B P S K modulation, L = 32. . . 78 5.3 B E R of conventional F H and C A F H with N C F S K modulation, L = 32. 79 5.4 B E R of C A F H with B P S K for r = 1 and 2, L = 32. , 80 5.5 B E R of C A F H with N C F S K for r = 1 and 2, L = 32 81 5.6 B E R of conventional F H and C A F H with B P S K modulation. 70 = 2 dB and L = 48 82 5.7 B E R of conventional F H and C A F H with B P S K modulation. 7 o = 8 dB and L = 48 83 5.8 B E R vs. average SNR, 70, of C A F H with B P S K for r = 1 and 2, L = 48. 84 5.9 B E R vs. average SNR, 7 o , of C A F H with N C F S K for r = 1 and 2, L = 48. 85 5.10 B E R vs. the number of rounds, r, of C A F H with B P S K . 7 o = 8 dB and L = 48 86 5.11 B E R versus number of MS's of random allocation, C A F H with B P S K for r = 1 and 2, and exhaustive search method. 70 = 4 dB and L — 8 87 6.1 Plot of cdf of 7i:3,72:3 and 73:3 with m — 1 and 7 = 1: theoretical (dotted) and simulation (solid) 98 List of Figures xi 6.2 Plot of cdf of 7i;3,72:3 and 7 3 : 3 with m = 1 and 7 = 1: independent (dotted) and correlated (solid) fading channels 99 6.3 Plot of cdf of 7 1 : 3 with 7 = 1 for different values of m 100 6.4 Plot of cdf of 7 r ; 5 with m = 1 and 7 = 1 for a linear model (solid) and its exchangeable approximation (dotted) 101 7.1 B E R vs. average SNR for Bn = 0.9 112 7.2 B E R vs. average SNR for Bn = 0.5 113 7.3 B E R vs. average SNR for Bn = 0.25. 114 7.4 C A F H B E R vs. number of mobile stations, for 7 = 2 and 8 dB, and Bn = 0.9 115 7.5 C A F H B E R vs. number of MS's, for 7 = 2 and 8 dB, and Bn = 0.25. . . 116 7.6 C A F H B E R vs. number of MS's, for 7 - 2 and 10 dB, and Bn = 0.33. . 117 List of Abbreviations 2G 2nd Generation 3G 3rd Generation 4G 4th Generation A C E Active Constellation Extension A M C Adaptive Modulation and Coding A W G N Additive White Gaussian Noise B E R Bit Error Rate B P S K Binary Phase-Shift Keying BS Base Station C A F H Channel Aware Frequency Hoppin cdf Cumulative Distribution Function C D M A Code Division Multiple Access C F Characteristic Function CP Cycle Prefix CS Channel Sate CSI Channel State Information CQI Channel Quality Information List of Abbreviations xm D A B Digital Audio Broadcasting D - A M P S Digital-Advanced Mobile Phone Service D F T Discrete Fourier Transform D M T Discrete Multitone DS Direct Sequence D S - C D M A Direct Sequence C D M A DSL Digital Subscriber Line D V B Digital Video Broadcasting E D G E Enhanced Data Rates for G S M Evolution E G C Equal Gain Combining F D M A Frequency Division Multiple Access F H Frequency Hopping F H - M C - C D M A Frequency Hopping M C - C D M A GPRS General Packet Radio System G S M Global System for Mobile Communications H D R High Data Rate H - S / M R C hybrid selection/maximal-ratio combining ICI Interchannel Interference IP Internet protocol ISI Intersymbol Interference ITU International Telecommunications Union L A Link Adaptation List of Abbreviations xiv L A N Local Area Network M A Margin Adaptive M A I Multiple Access Interference M C Multicarrier M C - C D M A Multicarrier Code Division Multiple Access MC-SS Multicarrier Spread Spectrum M I M O ' Multiple-Input Multiple-Output MISO multi-input single-output M M S E Minimum Mean Square Error Estimation M R C Maximal Ratio Combining MS Mobile Station M U - O F D M Multiuser O F D M N C F S K Non-Coherent Frequency-Shift Keying O F D M Orthogonal Frequency Division Multiplexin P A P R Peak-to-Average Power Ratio P D C Personal Digital Cellular pdf Probability Density Function P E R Packet Error Rate P F Proportional Fair P G Processing Gain P S A M Pilot Symbol Aided Modulation PSD Power Spectral Density List of Abbreviations xv PTS Partial Transmit Sequence QoS Quality of Service R A Rate Adaptive R F Radio Frequency rv Random Variable SC Selection Combining S F H / C D M A Slow F H / C D M A SINR Signal to Interference plus Noise Ratio SISO Single-Input Single-Output S L M Selected Mapping SNR Signal-to-Noise Ratio S N R r Received Signal-to-Noise Ratio T D L Tapped Delay Line T D M A Time Division Multiple Access TI Tone Injection T R Tone Reservation U M T S Universal Mobile Telecommunications System W C D M A Wideband Code-Division Multiple Access W L A N Wireless Local Area Network W L L Wireless Local Loop XVI Acknowledgements First and foremost, I am obediently thankful to God, praise and glory be to him, for his countless blessings and mercy. Without his guidance and mercy I can do nothing. I would like to express my heartfelt gratitude to my supervisor, Dr. Cyril Leung, for his guidance, technical advice, invaluable feedback, understanding, generous support, and friendship. I have passed through many personal and difficult circumstances during the period of my Ph.D. program. Without Dr. Leung's genuine kindness, I do not think you would be reading these words today. I am very grateful to my co-supervisor, Dr. Robert Schober, for his valuable sugges-tions, and technical assistance. Although he is very busy, I always found him easy to approach and available for help. I would like to express my appreciation to my thesis committee for their valuable comments which enhanced the presentation of this thesis. I have been blessed to come in contact with many wonderful people throughout my study period at U B C . This page is definitely not enough to mention all of them, but, the least I can say to them is: thank you for making my stay at U B C such a pleasant experience. Words cannot describe my feelings toward my parents. I can only pray to God to Acknowledgements xvii reward them for their constant support, persistent encouragement, and most of al l for their unwavering love. x v m Dedicat ion To my beloved parents. 1 Chapter 1 Introduction 1.1 Mobile Communications Systems: Past, Present, and Future The common feature of next generation wireless technologies will be the convergence of multimedia services such a speech, audio, video, image, and data. This implies that a future wireless terminal, by guaranteeing high-speed data, will be able to connect to different networks in order to support various services: switched traffic, IP data packets and broadband streaming services such as video. The development of wireless terminals with generic protocols and multiple-physical layers or software-defined radio interfaces is expected to allow users to seamlessly switch between networks based on different stan-dards. The rapid increase in the number of wireless mobile terminal subscribers, which cur-rently exceeds 1 billion users, highlights the importance of wireless communications in the new millennium. The increase has been sustained through the continuous evolution of standards and products. The adaptation of wireless technologies to the users' rapidly changing demands has been a hallmark of this rapid growth. It is expected that a variety Chapter 1. Introduction 2 of standards and systems will continue to exist. This plethora of wireless communica-tion systems is not limited to cellular mobile telecommunication systems such as Global System for Mobile communications (GSM), IS-95, Digital Advanced Mobile Phone Ser-vice (D-AMPS), Personal Digital Cellular (PDC), Universal Mobile Telecommunications System (UMTS) or cdma2000, but also includes wireless local area networks (WLANs), e.g., H I P E R L A N / 2 , IEEE802. i l and Bluetooth, and wireless local loops (WLL), e.g., H I P E R M A N , H I P E R A C C E S S , and IEEE 802.16 as well as broadcast systems such as digital audio broadcasting (DAB) and digital video broadcasting (DVB). Currently, the research community is focusing its activity on beyond 3G, i.e. J^th generation (4G) systems, with more ambitious technological challenges. The primary goal of next-generation wireless systems (4G) will not only be the introduction of new technologies to cover the need for higher data rates and new services, but also the inte-gration of existing technologies in a common platform. Hence, the selection of a generic air-interface for future generation wireless systems will be of great importance. Although the exact requirements for 4G have not yet been defined, its new air interface should fulfill at least the following requirements [1]: • generic architecture, enabling the integration of existing technologies, • high spectral efficiency, offering higher data rates in a given scarce spectrum, • high scalability with different cell configurations (hot spot, ad hoc), hence better coverage, • high adaptability and reconfigurability, supporting different standards and tech-Chapter 1. Introduction 3 nologies, • low cost, enabling a rapid market introduction, and • future proof, opening the door for new technologies. 1.2 From Second- to Third-Generation Multiple Access Schemes Second Generation (2G) wireless systems are mainly characterized by the transition from analog towards a fully digital technology and comprise the G S M , IS-95, P D C and D-A M P S standards. Work on the pan-European digital cellular standard G S M started in 1982 [2, 3], and now accounts for two-thirds of the world mobile market. In 1989, the technical specifications of G S M were approved by the European Telecommunication Standard Institute (ETSI), and commercialization began in 1993. Although G S M is optimized for circuit-switched services such as voice, it offers low-rate data services up to 14.4 kbit/s. High speed data services up to 115.2 kbit/s are possible with the enhancement of the G S M standard General Packet Radio Service (GPRS) by using a larger number of time slots. GPRS uses the same modulation, frequency band, and frame structure as G S M . However, the Enhanced Data Rates for Global Evolution (EDGE) [4] system which further improves the data rate up to 384 kbit/s introduces a new modulation scheme. Parallel to G S M , the U.S. IS-95 standard [5] (recently renamed cdmaOne) was ap-proved by the Telecommunication Industry Association (TIA) in 1993, and commercial-ization started in 1995. As in G S M , the first version of this standard (IS-95A) offered data services up to 14.4 kbit/s. In its second version, IS-95B, up to 64 kbit/s data ser-Chapter 1. Introduction 4 vices are possible. During the same time period, two other 2G mobile radio systems were introduced: D-AMPS/IS-136, called T D M A in the USA and P D C in Japan [6]. Demands for more capacity for mobile receivers, new multimedia services, new fre-quencies, and new technologies are behind the introduction of 3G systems. A unique international standard was targeted: Universal/International Mobile Telecommunication System (UMTS-IMT-2000) which would enable a new generation of mobile personal com-munications services world-wide. The objectives of the 3G standards, namely U M T S [7] and cdma2000 [8] went far beyond those of 2G systems, especially with respect to [9]: • the wide range of multimedia services (speech, audio, image, video, data) and bit rates (up to 2 Mbit /s for indoor and hot spot applications), • the high quality of service requirements (better speech/image quality, lower bit error rate (BER), higher number of active users), • operation in mixed cell scenarios (macro, micro, pico), • operation in different environments (indoor/outdoor, business/domestic, cellular/cordless), • and finally flexibility in frequency (variable bandwidth), data rate (variable), and radio resource management (variable power/channel allocation). The commonly used multiple access schemes for 2G and 3G wireless mobile com-munication systems are based on either Time Division Multiple Access ( T D M A ) , Code Division Multiple Access (CDMA) or a combinations of T D M A / C D M A in conjunction with an additional Frequency Division Multiple Access (FDMA) component: Chapter 1. Introduction 5 • The G S M standard, employed in the 900 MHz and 1800 MHz bands, first divides the allocated bandwidth into 200 kHz F D M A sub-channels. Then, in each sub-channel, up to 8 users share the 8 time slots in a T D M A manner [3]. • In the IS-95 standard, up to 64 users share the 1.25 MHz channel by C D M A [5]. The system is used in the 850 MHz and 1900 MHz bands. • The aim of D - A M P S ( T D M A IS-136) is to coexist with analog A M P S , and the 30kHz channel of A M P S is divided into three channels, allowing three users to share a single radio channel by allocating unique time slots to each user [10]. • The recent International Telecommunications Union (ITU) adopted standards for 3G (UMTS and cdma2000) are both based on C D M A [7, 8]. For U M T S , the C D M A -F D D mode, which is known as wideband C D M A , employs separate 5 MHz channels for both the uplink and downlink directions. Within the 5 MHz bandwidth, each user is separated by a specific code, resulting in an end-user data rate of up to 2 Mbit/s . 1.3 From Third- to Fourth-Generation Multiple Access Schemes Besides offering new services and applications, the success of 4G wireless systems will strongly depend on the choice of the concept and technology innovations in architec-ture, spectrum allocation, spectrum utilization, and exploitation [11, 12]. New high-performance physical layer and multiple access technologies are needed to provide high speed data rates with flexible bandwidth allocation. A low-cost generic radio interface, Chapter 1. Introduction 6 operational in in different environments with scalable bandwidth and data rates, is ex-pected to have better acceptance. The technique of spread spectrum may allow the above requirements to be at least partially fulfilled. A multiple access scheme based on direct sequence code division multi-ple access (DS-CDMA) relies on spreading the data stream using an assigned spreading code for each user in the time domain [13]-[16]. The capability of minimizing multiple access interference (MAI) is given by the cross-correlation properties of the spreading codes. In the case of severe multipath propagation in mobile communications, the ca-pability of distinguishing one component from others in the composite received signal is offered by the autocorrelation properties of the spreading codes [14]. The so-called rake receiver should contain multiple correlators, each matched to a different resolvable path in the received composite signal [13]. Therefore, the performance of a D S - C D M A system will strongly depend on the number of active users, the channel characteristics, and the number of branches employed in the rake. System capacity is typically limited by self-interference and M A I , which results from the imperfect auto- and cross-correlation properties of spreading codes. It will be difficult for a D S - C D M A receiver to make full use of the received signal energy scattered in the time domain and hence to handle full load conditions [13]. The technique of multicarrier transmission has recently been receiving wide spread interest, especially for high data-rate broadcast applications. The history of orthogonal multicarrier transmission dates back to the mid-1960s, when Chang published his paper on the synthesis of band-limited signals for multi-channel transmission [17]-[19]. The Chapter 1. Introduction 7 basic principle is to transmit data simultaneously through a band-limited channel without interference between sub-channels (without interchannel interference, ICI) and without interference between consecutive transmitted symbols (without mtersymbol interference, ISI) in the time domain. A major contribution to multicarrier transmission was made in [20] when Fourier transform for base-band processing was proposed in place of a bank of subcarrier oscillators. To combat ICI and ISI, a guard time was introduced between successive transmitted symbols. The main advantages of multicarrier transmission are its robustness in frequency se-lective fading channels and, in particular, the reduced signal processing complexity by equalization in the frequency domain. The basic principle of multicarrier modulation is to divide a high-rate data stream into several low-rate sub-streams. These sub-streams are modulated on to different subcarriers [21]-[23]. By using a large number of subcarriers, a high immunity against multipath dispersion can be provided since the useful symbol duration Ts on each sub-stream will typically be much larger than the channel disper-sion time. Hence, the effects of ISI will be minimized. Since the number of filters and oscillators necessary is considerable for a large number of subcarriers, an efficient digital implementation of a special form of multicarrier modulation, called O F D M , with rectan-gular pulse-shaping and guard time was proposed in [21]. O F D M can be easily realized by using the fast Fourier transform (FFT). O F D M , having densely spaced subcarriers with overlapping spectra of the modulated signals, obviates the use of steep band-pass filters to detect each subcarrier as in F D M A schemes. Therefore, it offers high spectral efficiency. Chapter 1. Introduction 8 O F D M came into prominence in the 1990's as it was the modulation method cho-sen for A D S L in the USA [24], and for the European D A B standard [25]. This success continued with the choice of O F D M for the European D V B - T standard [26] and for the W L A N standards H I P E R L A N / 2 and IEEE802.11a [27, 28] and recently for the interac-tive terrestrial return channel (DVB-RCT) [29]. It is also a candidate for wireless M A N standards, H I P E R M A N and IEEE802.16a [30, 31]. The advantages of multicarrier modulation on the one hand and the flexibility offered by spread spectrum techniques on the other hand have motivated many researchers to investigate the combination of both techniques, known as multicarrier spread spectrum (MC-SS). Such combinations [32]-[38], have resulted in new multiple access schemes called M C - C D M A and M C - D S - C D M A . They offer attractive features such as high flexibility, high spectral efficiency, simple and robust detection techniques, and narrow band inter-ference rejection capacity. M C and MC-SS are today considered promising candidates to meet the requirements of next generation (4G) high-speed wireless multimedia com-munications systems, where spectral efficiency and flexibility will be the most important criteria for the choice of the air interface. 1.4 M o t i v a t i o n The wireless communications industry is growing rapidly for both fixed and mobile ap-plications. The increasing demand for all types of wireless services (voice, data, and multimedia) is fueling the need for higher capacity and data rates. Although improved compression technologies have reduced the bit rate for voice calls, data traffic will re-Chapter 1. Introduction 9 quire much more bandwidth as new services come online. In this context, emerging technologies that improve wireless systems' spectrum efficiency are becoming a necessity, especially in broadband applications. Some examples include smart antennas, multiple-input multiple-output (MIMO) technology, coded multicarrier modulation, power control, channel allocation, link-level retransmission, and adaptive modulation and coding (AMC) techniques [39]. Popularized by cellular wireless standards such as E D G E , A M C techniques that adapt to the time-varying characteristics of wireless channels allow significantly higher data rates, reliability, and spectrum efficiency for future wireless data-centric networks. The set of algorithms and protocols governing A M C is often referred to as link adaptation (LA). While substantial progress has been made in this area to understand the theoretical aspects of time adaptation in L A protocols, more challenges surface as dynamic trans-mission techniques must take into account the additional signaling dimensions explored in future broadband wireless networks. More specifically, the growing popularity of both M I M O and multicarrier (MC) systems that take channel aware algorithms into account create the need for L A solutions that integrate temporal, spatial, and spectral compo-nents together. Currently a few existing and emerging wireless systems such as IS-95B, W C D M A , CDMA2000, HiperLAN/2 and (E)GPRS of G S M have already introduced some kind of L A [40]. While it is desirable to adapt the transmission parameters according to the channel state information (CSI) to capture even small-scale variations, there are Chapter 1. Introduction 10 practical limitations to channel state prediction and link adaptation. Frequent adaptation increases the number of mode-change messages transmitted over the channel, consuming bandwidth, and time resources [41]. Moreover, predicting the future channel quality may also consume significant amount of system resources (e.g., time, bandwidth and power), since it may involve transmission of training-sequences, pilot tones, or feedback messages carrying the CSI. Naturally, the additional cost and complication of predicting CS and L A increase with the desired prediction accuracy. Common channel estimation methods documented in the literature are pilot symbol aided modulation (PSAM) and minimum mean square error (MMSE) estimation. A classical paper on pilot symbol aided modulation for Rayleigh fading was presented in [42] where the optimum Wiener interpolation filter was derived to minimize the variance of the estimation error and system performance for B P S K , QPSK and 16-QAM was studied. Because the optimum Wiener filter requires a priori information of the channel and is computationally complex, several suboptimal interpolation filters have been proposed among which the sine interpolator is most widely used due to its simple implementation and close-to-optimum performance [43]. The bit error rate (BER) of pilot symbol aided M Q A M in Rayleigh fading using the sine interpolator has been analyzed in [44]. 1.5 Link Adaptation Fundamentals The basic idea behind L A is to adapt the transmission parameters to take advantage of the prevailing channel conditions. The fundamental parameters to be adapted include modulation and coding levels, but other quantities can also be adjusted such as power Chapter 1. Introduction 11 level (as in power control and power allocation), channel allocation, spreading factor, and signaling bandwidth. L A is now widely recognized as a key solution to increasing the spectral efficiency of wireless systems. L A techniques are incorporated in current proposals for third-generation C D M A wireless packet data services, such as cdma2000 and wideband CDMA ( W C D M A ) , GPRS and GPRS-136 [39]. In practical L A implementations, the values for the transmission parameters are quan-tized and grouped into what we refer to as a set of modes. An example for a mode might be a pair of modulation level and coding rate. Since each mode has a different data rate (expressed in information bits per second) and robustness level (minimum SNR needed to activate the mode), they are optimal for use in different channel/link quality regions. When using packet-switching technology and multiple modulation and coding levels, the E D G E system employs a link-adaptation technique to adapt packet transmission to one of six coding levels [45] (a recent proposal has nine modulation and coding levels [46]), where the highest data rate can exceed 550 kb/s. When the channel condition is poor, a low modulation level (i.e., few information bits per symbol) and/or heavy coding should be used for packet transmission to enable correct signal detection. On the other hand, if the channel is in a good state, a high modulation level and/or light coding can be used to increase data rate. Due to the unreliable nature of radio links, the provision of quality of service (QoS) such as packet error rate (PER) in wireless networks is challenging. For real-time ser-vices such as Internet protocol (IP) voice, music and video, stringent delay requirements severely limit or even preclude the retransmission of lost packets. Thus, tight delay Chapter 1. Introduction 12 requirements often translate into stringent P E R requirements. The goal of an L A algorithm is to ensure that the most efficient mode is used for given channel conditions, based on a mode selection criterion (maximum data rate, min imum transmit power, min imum P E R , etc). M a k i n g modes available that enable communica-t ion even in poor channel conditions increases system robustness; under good channel conditions, spectrally efficient modes can be used to increase throughput. In contrast, systems without L A are constrained to use a single mode that is often designed to main-tain acceptable performance when the channel quality is poor to get maximum coverage. Such systems are designed for worst-case channel conditions and do not fully exploit the channel when good conditions prevail. In practice, performance can be degraded by problems such as imperfect synchroniza-t ion or quantization effects. However in such cases, these degradations can be reduced to manageable levels. To avoid unnecessary complication of the models, these effects are ignored in this thesis. 1.6 Subchannel and Power Allocation W i t h the introduction of new IP-based multi-rate, mult i -QoS, future networks should be designed for economic packet data transfers [47] wi th decreasing cost per megabyte in order to make them more attractive to the users. These services are highly asymmetrical and require high transmission bandwidth. However, due to the limitations of the available frequency spectrum, its efficient use is crucial to the success of next generation wireless networks. Novel access methods coupled wi th adaptive resource management techniques, Chapter 1. Introduction 13 for both uplink and downlink transmission, are required to improve spectrum efficiency. It is well established that dynamically allocating transmission rate and power are effec-tive techniques to improve the performance of wireless networks. For example, [48]-[52] consider these problems in the context of the information theoretic capacity region of a multi-access fading channel under various assumptions. Multicarrier transmission schemes have been introduced into code-division multiple access (CDMA) systems to enable high data rate transmission. One of the methods is to transmit identical narrowband direct sequence (DS) waveforms in parallel over a number of subchannels using frequency diversity. In [53], a multicarrier C D M A system with an adaptive subchannel allocation method for forward links was proposed. In this system, instead of identical DS waveforms being transmitted over a number of subchannels in parallel, each user's DS waveform is transmitted over the user's preferred subchannel which has the largest fading amplitude among all the subchannels. In [54] subchannel allocation in forward links for M C - C D M A with random signature sequences is studied. A near optimal subchannel allocation policy is formulated by maximizing the total average signal-to-interference-and-noise ratio (SINR). A n iterative algorithm similar to the water filling algorithm was also proposed. In general, efficient subcarrier and power allocation have been shown to reduce performance degradation caused by channel dispersion [55]-[60]. Among the potential performance-enhancing technologies is orthogonal frequency di-vision multiplexing (OFDM). It is suitable for high speed downlink transmission as it can yield high spectral efficiency due to its robust performance over heavily impaired Chapter 1. Introduction 14 links. O F D M has been demonstrated as an efficient way to mitigate the adverse effects of frequency selective multi-path fading by transmitting signals over a number of flat-faded narrow-band channels. The inherent multi-carrier nature of O F D M also allows the use of adaptive modulation and power allocation. Furthermore, dynamic allocation techniques that efficiently use resources such as bandwidth, power and modulation, need to be devised to increase the system spectral efficiency. O F D M is a promising technique for the next generation of wireless communication systems [61], [62]. O F D M divides the available bandwidth into N orthogonal subchannels. By adding a cyclic prefix (CP) to each O F D M symbol, the channel appears to be circular if the C P length is longer than the channel length. Each subchannel can, thus, be modeled as a time-varying gain plus additive white Gaussian noise (AWGN). Besides the improved immunity to multipath fading [22] gained by the multicarrier property of O F D M systems, multiple access is also possible, because the subchannels are orthogonal to each other. In [59] [63]-[67], adaptive bit and power allocation are studied for O F D M . In these studies, optimal allocation of resources requires a centralized controller with knowledge of every user's channel state. Resource allocation for O F D M has been given a lot of attention. Different distributed resource allocation algorithms for circuit-switched traffic in an O F D M multi-cell environ-ment are compared in [68]. For packet-switched traffic, a distributed resource allocation algorithm that employs channel segregation and interference sensing for efficient resource management is proposed in [69]. A non-frequency selective channel is considered in both papers. In [70], a distributed dynamic resource allocation scheme for a downlink O F D M Chapter 1. Introduction 15 packet-based cellular system that adaptively uses channel information and individual rate requirements to determine the subcarrier, bit and power allocation was proposed and its performance was evaluated for real-time multimedia traffic via simulation whereas for non real-time traffic, the performance was evaluated in [71]. In [72], an analytical model of this algorithm is developed to provide insight into the performance gain in terms of achievable system load for voice and data. • Two classes of resource allocation schemes exist, namely: 1) fixed resource alloca-tion [65]; and 2) dynamic resource allocation [59] [63] [66] [67]. Fixed resource allocation schemes, such as T D M A and F D M A , assign an independent dimension, e.g., time slot or subchannel, to each user. A fixed resource allocation scheme may not be efficient since the scheme is fixed regardless of the current channel condition. On the other hand, in dynamic resource allocation, resources are adaptively assigned to users based on their channel gains. Due to the time-varying nature of the wireless channel, dynamic resource allocation makes full use of multiuser diversity to achieve improved performance. Two classes of optimization techniques have been proposed for multiuser O F D M , namely: 1) margin adaptive (MA) [63]; and 2) rate adaptive (RA) [66][67]. The M A objective is to achieve the minimum overall transmit power given the constraints on the users' bit error rate (BER). The R A objective is to maximize each user's error free capac-ity with a total transmit power constraint. These optimization problems are nonlinear and, hence, computationally intensive to solve. In [59], the nonlinear optimization prob-lems were transformed into a linear optimization problem with integer variables. The optimal solution can be achieved by integer programming. However, even with integer Chapter 1. Introduction 16 programming, the complexity increases exponentially with the number of constraints and variables. In [66], it is shown that for R A the sum capacity is maximized when each subchannel is assigned to the user with the best subchannel gain and power is then distributed by the water-filling algorithm. However, fairness is not considered in [66]. When the path loss differences among users are large, it is possible that users with higher average channel gains will be allocated most of the resources, i.e., subchannels and power, for a significant portion of the time. The users with lower average channel gains may be starved. In [67], a max-min problem approach is used to ensure that all users achieve a similar data rate. However, the maxmin optimization problem can only provide maximum fairness among the users. In most wireless systems of interest, different users require different data rates, which may be accommodated by allowing users to subscribe to different levels of service. In [73], an adaptive subchannel allocation and bit loading scheme for the multiuser M I M O / O F D M A system is proposed through some modification of the approach for O F D M A systems in [74]. A distributed power control scheme involving iterative water-filling is proposed [75] for the digital subscriber line (DSL) interference channel. An iterative power control algorithm is proposed [76] for an M I M O interference system with feedback. The capacity of M I M O interference systems without feedback is treated in [77], and it is shown that putting all power into one antenna is optimal when the interference is sufficiently strong. Optimum signaling for M I M O interference systems with feedback for flat Rayleigh fading channels is treated in [78]. In [79], long-term proportional fairness resource allocation with dumb antennas is studied. It was pointed out that in multiuser Chapter 1. Introduction 17 systems, channel fading can be exploited as a source of randomness, i.e., multiuser d i -versity. However, in some scenarios, due to slow channel variation, the dynamic range of channel fluctuations i n the time scale of interest may be small . 1.7 This Thesis 1.7.1 Objectives This thesis has the following main objectives. To develop new methods for M C - C D M A that involve efficient subcarrier allocation to improve system capacity for cellular systems. Subcarrier correlation is also considered. To introduce and study an adaptive subchannel allocation algorithm based on order statistics suitable for channel-aware systems. To in-vestigate the effects of correlation on the error performance of order selection, a multiuser diversity method that selects the r - th order channel gain for transmission. 1.7.2 Contributions The main contributions of this thesis are: • A channel-matched multicarrier code division multiple access (matched M C - C D M A ) scheme is proposed for downlink mult ipath slowly fading channels. The design over-comes the performance degradation caused by the channel dispersion. In contrast to a conventional M C - C D M A scheme, subcarriers are chosen so as to be located i n frequency subbands characterized by favorable channel transmission gains. T w o diversity reception techniques, equal gam combining ( E G C ) and maximum ratio Chapter 1. Introduction 18 combining (MRC), are considered. The bit error rate performance improvement over conventional M C - C D M A is examined. The results have appeared in [80]. • A new transmission scheme, frequency hopping multicarrier code division multi-ple access ( F H - M C - C D M A ) , is proposed and investigated. This scheme is com-patible with existing narrowband second-generation frequency hopping and third-generation wideband C D M A systems. An analysis of the downlink bit error rate with E G C and M R C in slow frequency-selective Rayleigh fading is provided. Two frequency hopping (FH) schemes, namely random and coordinated F H , are used. The performance improvement of F H - M C - C D M A over conventional frequency hop-ping is demonstrated, with a capacity gain approaching that of M C - C D M A but using a much lower number of subcarriers. The results have appeared in [81]. • The F H - M C - C D M A scheme is modified for operation in correlated fading channels. The bit error rate (BER) performance of the modified F H - M C - C D M A scheme is compared to those of conventional F H and M C - C D M A , respectively, assuming a tapped delay line (TDL) channel model in which the subcarriers undergo correlated fading. Numerical results indicate that the proposed F H - M C - C D M A scheme can yield a much lower B E R than conventional F H ; with a proper choice of parameter values, the modified F H - M C - C D M A also outperforms M C - C D M A . This work has been reported in [82]. • A channel aware multiple access scheme based on frequency hopping (CAFH) is proposed. In contrast to conventional F H , which uses a channel state independent Chapter 1. Introduction 19 hopping sequence, the transmitter in the channel aware scheme hops to the available frequency subband which currently has the largest transmission gain. It is shown that the proposed scheme can offer large performance gains over the conventional F H scheme [83] [84]. A n efficient method is presented to evaluate the performance of C A F H with r rounds. The resulting closed-form expressions are used to investigate the effect of the number, r, of rounds on the error probability for a cellular com-munication system. Numerical examples are presented to illustrate the application of the new analysis. This work has been reported in [85]. • A procedure for computing the cumulative distribution function (cdf) of the r-th order statistic of a set of arbitrarily correlated fading channel gains is studied. The method should prove useful in the study of various communication problems. For example, it can be applied to the performance analyses of diversity systems operat-ing over correlated Nakagami-m fading channels. Numerical results are presented to illustrate the effect of fading correlation and the fading severity parameter as well as the accuracy of an "exchangeable" approximation. The results have been reported in [86]. • A systematic characteristic function (CF)—based methodology has been developed for computing the B E R of the r-th largest channel gain for correlated Nakagami fading channels for any given covariance matrix. The method should prove useful in the study of various communication problems. Numerical results are presented to point out the application of the new analysis. The expressions derived are then used to evaluate the error performance for C A F H . The effect of the fading correlation Chapter 1. Introduction 20 on the B E R performance is illustrated. 1.7.3 R o a d M a p In Chapter 2, matched M C - C D M A is proposed as an adaptive subchannel allocation scheme with-non-uniformly spaced subcarriers matched to the channel transfer function to improve B E R performance. A new scheme, Frequency-Hopping Multicarrier C D M A ( F H - M C - C D M A ) , is de-scribed in Chapter 3. The performance of F H - M C - C D M A used over a Rayleigh fad-ing channel is analyzed. F H - M C - C D M A may be viewed as a combination of F H and M C - C D M A in which M C - C D M A is applied to portions or subbands of the available bandwidth. A new analytical model for correlated F H - M C - C D M A is presented in Chapter 4. The effect of equalizing the interference in each subcarrier group while maintaining a reduced correlation among the subcarrier fading processes for a group is studied. A novel channel aware frequency hopping (CAFH) multiple access scheme is described in Chapter 5. The B E R of C A F H is analyzed assuming independently Rayleigh faded subbands. It is shown that C A F H can offer significant performance improvements over conventional F H . The resulting channel assignment algorithm is iterative in nature. The performance analysis of C A F H with r rounds is also considered. Expressions for the cumulative distribution function (cdf) of the r-th order statistic, r = 1, 2 , . . . , n, for correlated Nakagami fading branches are obtained in Chapter 6. The B E R of a channel aware diversity system, hereafter referred to as order select-ion, is Chapter 1. Introduction 21 evaluated in Chapter 7 for correlated Nakagami fading channels, when the r-th order branch is selected for transmission. Chapter 8 concludes the thesis and outlines future research problems. 22 Chapter 2 Matched M C - C D M A in Slowly Fading Dispersive Channels 2.1 Introduction Frequency selective multipath fading is common in urban and indoor environments and can significantly degrade the performance of wideband, high rate mobile communication systems. M C - C D M A , a combination of O F D M and C D M A , has been proposed as a viable transmission technique [32, 33, 35, 38, 87]. With M C - C D M A , the conventional serial transmission of a data stream is converted into a parallel transmission of data symbols using a large number of narrowband orthogonal subcarriers. In this chapter, a modified M C - C D M A technique, hereafter referred to as matched M C - C D M A , with non-uniformly spaced subcarriers matched to the channel transfer func-tion is proposed to improve the B E R performance. Two receiver diversity combining techniques, E G C and M R C , are considered for combining subcarrier signals. °A version of this Chapter has been published [80], Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 23 2 . 2 S y s t e m M o d e l The matched M C - C D M A transmitter spreads the original data stream over different subcarriers in the frequency domain using a given spreading code. Subcarriers are con-centrated in the regions of the channel with low attenuation, thereby reducing the per-formance degradation caused by the frequency-selective channel. A n appropriate channel measurement technique is assumed. The channel is modeled using a two-path transfer function developed by Rummler of the form [88], [89] where B? is the total available bandwidth, a and (3 are the overall attenuation parameter and shape parameter respectively, f0 is a fade minimum frequency, and T(I is the relative time delay between the two paths. The effect of the multipath component is to create deep fades at / = f0 + i/r^, % = 0, ± 1 , ±2 , • • •. Because of the reduced M C - C D M A symbol rate, a synchronous downlink channel is assumed [87]. The distribution of the matched frequency subcarrier locations is proportional to the square of the magnitude of the channel transfer function, | C ( / ) | 2 , and is obtained from the cumulative probability distribution function (cdf) of | C ( / ) | 2 as in [55] A matched M C - C D M A signal is generated by replicating a single data bit into L C(f) = a[l - Be-**V-MT*], f £ (/c, fc + BT) (2.1) elsewhere. (2.2) Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 24 copies, each multiplied by a different element (chip), Cj[l] € — 1, 2, •• •, L of the spreading code assigned to user j and then binary phase-shift keying (BPSK) modulated onto a subcarrier. The transmitted matched M C - C D M A signal can be written as sj(t) = ^ J Z ^ - o o Ef=i • cos ( 2 T T ( / c + g ) i ) • p(t - kTb), FeZ (2-3) where aj[k] € {—1, +1} is the A;"1 data bit of user j, Tb is the bit duration, p(t) = rect(£/Tb), and {Fi} are the L subcarrier indices selected for channel matching. The total number of available orthogonal subcarriers, uniformly located within BT, is denoted by N. Each subcarrier has a bandwidth of BCB. Conventional M C - C D M A can be viewed as a special case of matched M C - C D M A with Fi = IF, F 6 Z. The received signal to noise ratio (SNR r ) for the Ith subcarrier is where fi — fc + ^ and At = \Ai\ej&Ai = C(fc + | f) is the Ith subcarrier complex channel gain and Eb is the bit energy. Matched M C - C D M A is expected to provide an improved performance over conventional M C - C D M A since L L \Al,match\2 > ^2 \Al!COnv\2 > 1 = 1,- ••,£/. (2-5) ( = 1 1=1 where A^match and AiiConv are the complex channel gains corresponding to the Ith sub-carrier in the matched and conventional M C - C D M A schemes respectively. The received Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 25 signal with M active transmitters can then be written as r(t) = i Et°°-oo Ef=l KIU'M*] ' C O S (2TT(/c + g)t + 6A) • P(t - kTb) + n{t) (2.6) where n(t) is a sample function of a white Gaussian noise process with zero mean and power spectral density N0/2. where Gjti is the gain factor for user j at subcarrier I and depends on the combining scheme used, and {ni(kTb), I = 1,2, • • - ,L} are sample values of independent Gaussian random variables with mean zero and variance The Walsh-Hadamard sequences with a processing gain (PG) of 64 are assigned to users as follows: the reference user (user 1) is assigned each of the 64 sequences in turn. Assume that user 1 is assigned sequence number j. If j = 1 or j > M, user i, i = 2, • • •, M is assigned sequence number i. If 2 < j < M, then user i is assigned sequence number i if i < j, and i + 1 if i > j. Numerical results show that the proposed matched M C - C D M A scheme can provide a much lower B E R than conventional M C - C D M A . As shown in Fig. 2.2, the number, Assuming perfect phase recovery, the decision variable for the kth symbol of user j is DJMC(kTb) - Ef=i {Giti [E!=i +nl{kTb)) (2.7) 2 . 3 N u m e r i c a l R e s u l t s Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 26 N, of available subcarrier channels is 128 from which L = 64 matched subcarriers are selected. The channel parameters are a — —1 dB, (3 = 0.5, f0 = 0.2BT and = 2/BT-Fig. 2.3 shows the B E R as a function of the number of users for matched M C - C D M A and conventional M C - C D M A with F = 2, and SNR = Eb/N0 of 9 dB, where Eb is the transmitted energy per bit. The figure shows that the proposed matched M C - C D M A scheme can yield up to a 10-fold reduction in B E R compared to conventional M C - C D M A . A B E R comparison as a function of SNR for M=64 users is illustrated in Fig. 2.4. For the same combining scheme, the SNR difference between matched and conventional M C -C D M A required to achieve a given target B E R increases as this target B E R decreases. For 64 users and a B E R of I O - 2 , matched M C - C D M A with E G C provides a gain of about 4 dB compared to conventional M C - C D M A with E G C . It should be noted that for a small number of users (i.e., in a noise limited environment), M R C outperforms E G C . However, for a large number of users (i.e., in a interference limited environment), E G C has a superior performance. This is likely due to the fact that the orthogonality of the codes is affected to a greater extent by M R C . M R C is only optimal in an additive white Gaussian noise (AWGN) environment, which is the case for an uplink scenario. Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 27 (b) Figure 2.1: (a) Transmitter model (b) Receiver model Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 28 21 1 1 r 1 .75 -S u b c a r r i e r n u m b e r , / Figure 2.2: Matched M C - C D M A subcarrier pattern, channel transfer function T F , integral \T/(-) with Td — 2/BT- The frequency corresponding to the Ith carrier is fc + FtBch. and sub-Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 29 Matched M C - C D M A (EGC) Conventional M C - C D M A (EGC) Matched M C - C D M A (MRC) o Conventional M C - C D M A (MRC) 10 20 30 40 Number of users, M 50 60 70 Figure 2.3: B E R as a function of number of users for S N R — 9 d B . Chapter 2. Matched MC-CDMA in Slowly Fading Dispersive Channels 30 i o u SNR, dB Figure 2.4: B E R as a function of S N R for M = 64 users. 31 Chapter 3 Frequency-Hopping Multicarrier C D M A in Rayleigh Fading 3 . 1 I n t r o d u c t i o n Multicarrier modulation is an attractive scheme for high data rate digital transmission because it allows a reduction of the symbol rate, at the cost of increasing the number of subcarriers used to transmit the information. In C D M A , multiple users can transmit si-multaneously within the same frequency band by using a number of different pseudo-noise sequences. M C - C D M A is a combination of these two techniques. Unlike D S - C D M A , in M C - C D M A the input data is duplicated and transmitted through a number of narrow-band subcarriers. The signaling period can thus be much larger than the delay spread of the channel making the system less susceptible to ISI due to multipath propogation. On each subcarrier, the transmitted bit is binary antipodally modulated based on the spread-ing codes assigned to the user. At the receiver side, the received signals at the different subcarriers are weighted and summed to produce a decision variable. For the downlink system, orthogonal spreading codes can be used to reduce multiuser interference. How-°A version of this Chapter has been published [81]. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 32 ever, since the channel gains for different subcarriers are different, strict orthogonality among users is lost and multiuser interference is introduced. The two principal types of C D M A are DS and F H . A major advantage of F H is that it can be implemented over a much larger frequency band than direct-sequence spreading, and the band can be noncontiguous [90]. A second advantage is that frequency hopping provides resistance to multiple-access interference while not requiring power control to deal with the near-far problem. In D S - C D M A systems, accurate power control is critical on the uplink [91]. A form of F H which allows simultaneous transmission in several frequency bins has also been investigated [92]. Because at each frequency hop, a frequency-hopping multicarrier CDMA (FH-MC-C D M A ) user transmits in only a small portion of the band at a given time, multiple transmissions may occur simultaneously. In addition, users are separated through the use of different spreading codes in the frequency domain thereby reducing the effects of hits. In a synchronous down-link mobile radio communication channel, we may use orthogonal Walsh Hadamard codes. The proposed F H - M C - C D M A may be viewed as a combination of F H and M C - C D M A in which M C - C D M A is applied to portions or subbands of the available bandwidth, to which users may hop to. M C - C D M A has disadvantages [93] and problems associated with nonlinear amplification [94] and high peak-to-average power ratios (PAPRs), which result from the fact that the M C - C D M A signal is composed of many subcarriers, are reduced, as F H - M C - C D M A generally uses a smaller number of subcarriers than M C - C D M A . In an F H - M C - C D M A receiver, at the different hops, the received signal is combined in the frequency domain, therefore, the receiver can employ Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 33 all the received signal energy scattered in the frequency domain. 3.2 F H - M C - C D M A S y s t e m M o d e l Consider a system where M users are transmitting over the channel. Each user transmits one message bit per hop in one of g available frequency groups. A synchronous frequency-hopping system is assumed, with the hop intervals (of period Tb), of all transmitters aligned at the receiver. The users' hop sequences and message bits are assumed to be independent random sequences, uniformly distributed over the g frequency groups and the set {-1,1}, respectively. To evaluate the performance of the system, it suffices to consider one of the users, hereafter referred to as the reference user. The proposed transmitter and receiver are illustrated in Fig. 3.1. The Ith bit dm[l] of the mth user is first mixed with a frequency-hopping tone of frequency fh*1'™^• M C - C D M A processing with a spreading code of length N is then performed. The transmitted signal for the Ith bit of the mth user can be written as Sm(t) = J2ll dm[l]Cm\i\ • COS ^ ( / ^ + f)t) , IT„ < t < (I + l)Tb (^) with the subcarrier tone frequency f — where F and i are integers. In F H - M C - C D M A , the total radio frequency (RF) bandwidth (BWRF) is divided into s subcarriers which are organized into g frequency groups. Each frequency group consists of N subcarriers (s — gN). The R F signal from a given transmitter is hopped from group to group by changing the carrier frequency as illustrated in Fig. 3.2. An F H - M C - C D M A Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 34 system can be viewed at one extreme as a M C - C D M A system when 5 = 1, and at the other extreme as a conventional F H system when N = 1, for a fixed total bandwidth, i.e., a fixed value for s. A pseudo-FH system is obtained by choosing a large number of frequency groups, each with a small number of subcarriers. For example, with s — 128, we might choose g = 64 and N = 2. In this way, some of the advantages of both F H and M C - C D M A systems may be obtained in a single hybrid scheme with those of the F H dominating. On the other hand, a pseudo-MC-CDMA is obtained by choosing a large number of subcarriers and a small number of frequency groups, e.g., N = 64 and 5 = 2. Assuming that signals in different groups do not interfere with each other and that delay spread effects are negligible, the received signal corresponding to the transmission of the Ith message bit is given by where Mh is the number of users in a given group, pm<i is the channel gain at the ith subcarrier of the mth user, 9^m) is a random phase offset and n(t) is a sample function of a white Gaussian noise process with zero mean and power spectral density N0/2. It is assumed that {pm ii}m=i,...,M / l,i=i,...,iv are outcomes of statistically independent Rayleigh distributed random variables and {fV',m^}m=i,...,Mh,i=i,...,JV are statistically independent random variables, each of which is uniformly distributed in [0, 27r). The probability that Mh — 1 other transmitters (interfering users) share the same r(t) = £. (3.2) lTb<t<(l + l)Tb Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 35 frequency group as the reference user is given by ^ M - 1 ^ P{Mh Mh-l Pf?»-\\-Ph)M-M\ (3.3) where Ph — ^ is the probability of another transmitter hopping to the reference frequency group. Two F H - M C - C D M A schemes are considered, namely random and coordinated F H - M C - C D M A . In random F H - M C - C D M A each user randomly selects a spreading code (out of the N possible Walsh Hadamard codes) and a frequency group (out of the g possible groups). In coordinated F H - M C - C D M A , the M users are assigned to the g frequency groups in such a way as to equalize as much as possible the number in each group. For each frequency group, each of the Mh users is assigned a distinct code, so as to avoid multiple users sharing the same code and group simultaneously (i.e., hit). This is done by picking at random a set of Mh distinct codes from the N available codes. There are a total of such sets. \Mh) Assuming that the receiver is synchronized with the desired reference user {m — 1), we can express the decision variable for the Ith bit as follows Z = % I i n m r(t) E i = i • cos {2,{h[l'l) + fi)t + * i U ) ) dt, (3-4) where is the reference receiver gain factor (which depends on the diversity reception combining scheme) at subcarrier i, and df'1^ is the estimated phase at the ith subcarrier of fh~l,r>• Assuming perfect phase correction (of — O^1'1^), the decision variable reduces Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 36 to Z = M1} T,i=l Pl,iCl[i]Ql,i + Emi2 dm[l] HiLl Pm,iCm[i]qi,i • COs(0f m ) ) + p, (3-5) where 0J' , m ) = 9i{l,1) - ^ ( ' , m ) and p. is & noise term. With E G C , the gain factor at the ith subcarrier is given as q\^ = C\[i], and the decision variable becomes ZEGC = dX[l} Ef= 1 Pl,i + E ^ = 2 dm[l] Ei l l PrnjCmlilCllil • COs(0<''m)) + p , E G C . (3-6) The three terms in (3.6) may be expressed as I$2° = E ^ 2 drnll] E i = l / V i ^ W c x W • COS(ff ^ Msec = i C1)Tt »(*) Ef=i c i W • c o s ( 2 T ( A ( M ) + /<)* + A . The mean and variance of the random variable for the interference term are given by a^oc = E[{l^f] (3-7) = 2 ( M , - l ) A ( l - f ) , where Pm — N^f^- represents the mean power of the mth user. The mean and variance Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 37 of the noise is given by E \ P B G C \ = 0 GlEGc = E ^ E G C P E G C \ = %N. With M R C , the gain factor at the ith subcarrier is given as q\ti = C\ \i\p\,i, and the decision variable becomes ZMRC = di [I] Ei l i Pl,i + £„=2 dm[l] E,=l Pm,iP\,iCm[i)Cl [i] • COs(0J' ' M ) ) + PMRC-(3.8) The three terms in (3.8) may be expressed as < B C = di[/]E^1P?,i = E^2 dm[l] Til Pm.Pl^cM • C08(f l i ' ' m ) ) % f l C = t C i m n(t) £ f = 1 Pi,iCi[i] • cos (27r(/hC-1) + ft)t + 0fl)) dt. The mean and variance of the random variable for the interference term are given by o)u,c = E[I%r2} (3-9) int .2 ]\2] = (Mh — l)N[E[pfj] - (E[plti\) Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 38 The mean and variance of the noise are given by E\PMRC\ = 0 No alMRC - EI^MRCVMRC] — 2 - P i The B E R for a F H - M C - C D M A system can be calculated as BER7ZMC = E t f t = l PiMh ~ l)BEP\MC-CDMA),EGC (3.10) BERFH-MC = S M „ = 1 P(Mh ~ 1)BER{tMC-CDMA),MRC where BER$£_CDMA)>EGC and BER^_CDMA)MRC are the downlink BER's of an M C -C D M A system employing E G C and M R C respectively. Approximating the interference, Iint, from other users by a Gaussian random variable results in an M C - C D M A B E R given by [38] BER^_CDMA)EGC ~ lerfc {J^^ ^ J ^ p ^ + NqJ ' ^ 7Mh,N ^ 1 r_ I / PoTb BEB(MC-CDMA),MRC ~ {\j 2^P0Tb + N0 J ' ^ 3 . 3 N u m e r i c a l R e s u l t s a n d D i s c u s s i o n s The B E R values for coordinated F H - M C - C D M A calculated from (3.10) and obtained from computer simulations for N — M, are plotted as a function of the average SNR in Fig. 3.3. The SNR is defined as PiTb/N0. The simulation results are represented by the Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 39 symbols and the theoretical results'by the dotted curves with symbols. Fig. 3.4 shows the theoretical and simulation results for coordinated F H - M C - C D M A with N < M. We find that simulation results are well approximated by the analytical results when the number of users is large. For a small number of users the Gaussian approximation of (3.10) is not as accurate, e.g., 32 users using E G C . The B E R performance of random F H - M C - C D M A and coordinated F H - M C - C D M A using E G C for different number, N, of subcarriers is.compared in Fig. 3.5, for M = 32 users. A B E R comparison of random and coordinated F H - M C - C D M A for M = 64 users for different.subcarriers and employing E G C is illustrated in Fig. 3.6. It is clear that coordinated F H - M C - C D M A provides a lower B E R than random F H - M C - C D M A . The B E R comparison between conventional F H employing 128 subbands, M C - C D M A having 128 subcarriers and coordinated F H - M C - C D M A employing different number, N, of subcarriers for M = 64 using E G C and M R C are shown in Fig. 3.7. As expected, for a given average SNR, the B E R decreases with N, due to the bigger diversity gain. The figures show that the proposed F H - M C - C D M A scheme can yield up to a 10-fold reduction in B E R compared to conventional F H when employing two subcarriers in the interference-limited region. In the noise-limited region F H - M C - C D M A becomes comparable with M C - C D M A . We find that the performance difference between a F H - M C - C D M A scheme employing 64, 32, 16, 8 and 4 subcarriers and a 128 subcarrier M C - C D M A is small. The performance difference is even smaller at low SNR values. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 40 4sn (l.m) COS(2JC/A )t Frequency Synthesizer [ 1 ] COS 2K -7f 1 c [2] COS 271 "= ' NF [N] COS 27C t Band Pass Filter (a) r(0 (l.m) • cos(2nf )t Frequency Synthesizer Band Pass Filter (l.m) Band Pass Filter C O S ( 2 7 r ^ t + 9 ; ) ^ , a™; COS(27t— f + 9,) Band Pass Filter j / , m ; cos(2n—Ft + 6 ) <7 fc 4jn Integrator (b) Figure 3.1: (a) Transmitter model (b) Receiver model Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 41 Frequency Time Hop interval Figure 3.2: F H - M C - C D M A pattern. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 42 - i f " * * * + MRC - 64 users * MRC - 32 users A EGC - 64 users • EGC - 32 users & k o 10 average SNR, dB 15 20 Figure 3.3: Theoretical and simulation B E R comparison for N = M Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 43 H T ' t I. + MRC - 64 users, 32 subcarrier * MRC - 32 users, 16 subcarriers A EGC - 64 users, 32 subcarrier • EGC - 32 users, 16 subcarriers A 0 10 average SNR, dB 15 20 Figure 3.4: Theoretical and simulation B E R comparison for N < M. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 44 i o u 10 -3 10 * 16 subcarrier coordinated F H - M C , EGC -*— 16 subcarrier random F H - M C , EGC o 8 subcarrier coordinated F H - M C , EGC -a- 8 subcarrier random F H - M C , EGC A 4 subcarrier coordinated F H - M C , EGC - A - 4 subcarrier random F H - M C , EGC ' A-10 average SNR, dB 15 4 A A D n • * • . . " ' • * 20 Figure 3.5: B E R comparison between random and coordinated F H - M C - C D M A for M 32 users. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 45 10" 10" 32 subcarrier random F H - M C , E G C 32 subcarrier coordinated F H - M C , E G C 16 subcarrier random F H - M C , E G C 16 subcarrier coordinated F H - M C , E G C 8 subcarrier random F H - M C , E G C 8 subcarrier coordinated F H - M C , E G C 4 subcarrier random F H - M C , E G C 4 subcarrier coordinated F H - M C , E G C A ••• *• O • 10 average SNR, dB 15 A •Q. • * • o • 20 Figure 3.6: B E R comparison between random and coordinated F H - M C - C D M A for M 64 users. Chapter 3. Frequency-Hopping Multicarrier CDMA in Rayleigh Fading 46 i o u 10 -11 04 W CQ '•A. 10" 10 128 subcarrier M C - C D M A , 128 subband F H 64 subcarrier F H - M C , EGC 32 subcarrier F H - M C , E G C 16 subcarrier F H - M C , E G C 8 subcarrier F H - M C , EGC 4 subcarrier F H - M C , E G C 2 subcarrier F H - M C , EGC E G C A - A • ••• • #• •q>. • • *• ••6> : 10 average SNR, dB 15 20 Figure 3.7: B E R as a function of S N R for M — 64 users. 47 Chapter 4 Frequency-Hopping Multicarrier C D M A in Correlated Rayleigh Fading 4.1 Introduction In M C - C D M A , the available system bandwidth is partitioned into s subbands, each asso-ciated with a subcarrier frequency. Each data bit of a mobile station (MS) is multiplied by a chip sequence of length s; each of the resulting s chips is then transmitted using a different subcarrier so as to exploit the diversity available in frequency-selective channels. In [81], F H - M C - C D M A was proposed for use in a frequency-selective channel in which different subbands fade independently. In F H - M C - C D M A , in each hop, a MS transmits simultaneously over a predefined set of frequency subcarriers, referred to as a group. In this chapter, we propose a modification of the F H - M C - C D M A scheme in [81] that is suitable for use in correlated fading channels. In particular, with the appropriate choice of the number of subcarriers in each group and subcarrier interleaving, M A I and °A version of this Chapter has been published [82]. Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 48 correlation among the subcarrier fading processes for a group are largely reduced. We compare the B E R performance of the modified F H - M C - C D M A scheme with those of conventional F H and M C - C D M A , respectively. The results show that F H - M C - C D M A with appropriate subcarrier interleaving and an appropriate number of subcarriers for each group can provide a significantly lower B E R than M C - C D M A . 4 .2 System Model 4.2.1 F H - M C - C D M A Let the total number of subcarriers which can be accommodated in the available sys-tem bandwidth, W, be denoted by s. The subcarrier spacing can be chosen to be approximately 1/Tj,. Thus, for a system which fully utilizes the available bandwidth, W ~ s/Tb. In F H - M C - C D M A , the s subcarriers are partitioned into g frequency groups Gi, G2, • • -,Gg, each consisting of Ncj subcarriers so that In each hop of bit duration Tb, one bit from each MS is sent using one of the g groups. The M G . MS's in a given group Gj transmit data bits simultaneously to the base station (BS). Each of the Mci MS's is assigned a distinct spreading code of length Ncr The F H - M C - C D M A scheme in [81] was proposed for the case in which the fading gains in the s subbands are uncorrelated. For the case when these gains are correlated we propose a modified F H - M C - C D M A scheme in which the frequency separation between Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 49 adjacent subcarriers in a given group Gj is increased so as to exceed the coherence bandwidth, A / c , by interleaving the subcarriers of group Gj with those of other groups. This allows subcarriers assigned to a given MS to experience largely uncorrected fading. We note that an M C - C D M A system with a larger subcarrier spacing would require a larger system bandwidth when compared to that of the proposed F H - M C - C D M A scheme. Based on MS allocation and subcarrier assignment, three F H - M C - C D M A schemes are considered: 1) In uniform F H - M C - C D M A , the M MS's are assigned to the g groups in such a way as to equalize as far as possible the number of MS's in each group. Each group consists of Nn = N — - V j subcarriers. 3 9 2) For comparison purposes, we also consider random F H - M C - C D M A , where each MS randomly selects one frequency group (out of the g possible groups) for use. Each group Gj has NCj = N = * V j subcarriers, and MGj G {1, • • •, N} MS's. 3) The numerical results in Section 4.4 suggest that optimum error performance is achieved when choosing MGi = 1 V j . In this case, hereafter referred to as optimum F H - M C - C D M A , we have g = M. The proposed system is illustrated in Fig. 4.1. For a given MS m, each component of the spreading code (cm[0], c m [ l ] , . . . ,cm[NG.-l)), where cm[i\ € { - 1 , +1}, is multiplied by the kth data bit, dm[k] G {-1, +1}. The ith product signal, where i = {0,1, • • •, NG. - 1}, is used to modulate a baseband subcarrier of frequency fi = TJT, where F is an integer that determines the subcarrier separation. The NGj modulated baseband subcarriers are then summed to form the composite baseband signal. The R F signal is formed by frequency Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 50 shifting the composite baseband signal by an amount equal to the carrier frequency, fh(k,m)_ Yox F H - M C - C D M A , the transmitted signal of MS m for bit k can be written as sm(t) = E £ O ~* dm[k}cm[i\ • cos (2n(fh^ + /<)*) , kTb < t < (k + 1)T„. (4-2) 4.2.2 Channel Model The channel is modeled as a tapped-delay line (TDL) with a baseband equivalent impulse response [95] -^)(i)=x:^ 'm) -^^ )I (4.3) (=o where S(-) is the unit impulse function, and the complex channel coefficients, h^'m\ for path / are independent, identically distributed zero-mean complex Gaussian random variables, and L is the total number of resolvable paths. The complex subband coefficient of channel subcarrier i, located at / = fmin + jr, i — 0,1, • • •, s — 1, with fmin representing the lowest subcarrier frequency in the available transmission bandwidth, W, is L-1 hf,m) = ^2 e-j27Tf'^wh<ik'm)e-j2viK (4.4) (=0 where h\k'm^ — p^k'm)e^ei(k'm)', with p^k'm^ representing the Rayleigh distributed channel amplitude gain and 9^k'm^ representing the random phase offset. For F H - M C - C D M A , the signal received by the BS from group Gj, over a slowly time varying, frequency selective Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 51 Rayleigh fading channel can be expressed as r(t) = Em=! dm[k] ES" 1 Pm(k'm)cm\i] • cos ^ ( / ^ + f)t + %) ( f c 'm )) + n(t), fcTb < t < [k + l)Tb where = (fh~k'm^ ~ fmin + fi) • ^1, and n(t) is a sample function of a white Gaussian noise process with power spectral density No/2. Assuming that L divides s, subcarrier indices separated by nf, n G {1, 2, • • •, L — 1}, experience uncorrected fading since = ^ E/TQ1 e?2™± = 0 (4.5) where A = E[\h[\2} and E[-] denotes expectation. 4.3 BER Analysis of FH-MC-CDMA Assuming that the BS is phase synchronized with the target MS, m, we can express the decision variable for the kth bit as z = i &T)Tb r W _ 1 ^ • c o s ( 2 7 r ( A ( f c , m ) + /*)* + 0m{k'm)) dt> ( 4 6 ) where qm<l — p\k^cm[i] is the receiver weighting factor for subcarrier when maximum ratio combining (MRC) is used by the BS. Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 52 4.3.1 Uniform F H - M C - C D M A In this case, each group consists of Nc3 — N V j subcarriers. Improved error performance is achieved by setting N < L, F = g = jj, and fh{k'm) - fmin G{0, Yb>Tb>" '> Efb1} a s shown in Fig. 4.2. This may be explained by the fact that a group Gj will experience larger M A I for larger values of N (i.e., larger values of MGj in a given group Gj), and correlated fading among its subcarriers as N increases above L. Here, we choose j groups to contain % + 1 MS's and the remaining g — j groups to contain i MS's, i.e., M = j.(i + l) + (g-j).i = i.-g + j, (4.7) where i = and j = M mod g; [x\ and "mod" denote the largest integer smaller than x and the modulo-operator, respectively. The average B E R can then be calculated as p _ M-(i+l)j pi,N (j+l)j pi+1,. uniform M c M c (4.8) where P ' ' is the average B E R on the uplink of an M C - C D M A system using M R C , when having i MS's and N subcarriers [89], [38], [95] ^ N - 1 + n ^ n Q . 27 r LLzL~1e-Lz J (L-1)! 0 N < L, i = 1 N < L, i > 1 N > L, i > 1 (4.9) Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 53 where Q(z) = -j=e~y2/2dy, 7 = NE^PH%No * ^  is the average signal-to-noise ratio (SNR) of the mth MS, and p, = \/j^=, where 70 = ^ is the average SNR per channel. 4.3.2 Random F H - M C - C D M A As in uniform F H - M C - C D M A , each group consists of NGj =-N V j subcarriers. In this case, each MS randomly selects one frequency group for use (i.e., MGi € {1,- • •, N}). The average B E R for random F H - M C - C D M A can be calculated as P D =T?1P(i-i)P T- N , ( 4- 10) random Z_- /2=l \ / o ' where P(i — 1) is the probability that i — l interfering MS's are in the same group as the desired reference MS, and is given by ^ M - 1 ^ P(i - 1) = 1 Y ~ * 1 (1--)M-1 . (4.11) 4.3.3 Optimum F H - M C - C D M A The total number of MS's may be written as M = 2l + j , (4.12) where j = {0,1,' • -,2l — 1} and N = |f. Our results in Section 4.4 suggest that optimum error performance is achieved by choosing (2l — j) groups to have TV sub-Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 54 carriers and the remaining M — (2l — j) groups to have y subcarriers. In this case, f (fc.m) _ t p f n i 1 , , . E=l 2. 1 + J_ £ - L - 2 _ . . . s + E=l\ F — = Mr = 1 V j and g = M. Using this method, M A I is eliminated by having one MS in a group, while maximizing the diversity for a group. Reducing the number of subcarriers in a group below the proposed number will result in diversity reduction, while increasing the num-ber of subcarriers in a group beyond the proposed number will result in more MS's in a group, thus, M A I is introduced. The average B E R can then be calculated as = M _ ( 2 < _ j ) l . g &-j)pi.TT (4.13) op tirnurn M o M e 4.4 Numerical Results Numerical results are now presented to compare the performance of the proposed modified F H - M C - C D M A with that of M C - C D M A and conventional F H . Although not shown, simulation results were obtained and found to agree closely with the theoretical curves. In Fig. 4.3, the number of resolvable paths, the total number of subcarriers, and the total number of MS's are chosen as L — 32, s = 256, and M = 32, respectively. When operating at N = 8 subcarriers, and F = g = s/N set to 32, uniform F H - M C - C D M A reduces to optimum F H - M C - C D M A having a single MS (MGj = 1 V j) in each group. In this case, F H - M C - C D M A may be viewed as an optimum access scheme with no M A I , thus, outperforming all other schemes. This result is expected due to the fact that a Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 55 group Gj will experience larger M A I if N > 8, and a diversity gain reduction if N < 8. When operating at N = 16, uniform F H - M C - C D M A has MGj = 2 V j. As expected, the B E R performance degrades with an increase in N. The error performance rapidly worsens when N > L due to the correlated fading among the subcarriers of a group. M C - C D M A which is a special case of F H - M C - C D M A when N = s and g = 1, has the highest M A I and the largest correlation among its s = 256 subcarriers. The figure clearly shows that F H - M C - C D M A outperforms M C - C D M A . In particular, smaller values of MG. result in a lower B E R . To demonstrate the effect of lower diversity orders, L = 8 is chosen in Fig. 4.4. When operating at N = 8, uniform F H - M C - C D M A achieves independent fading among its N subcarriers with no M A I (i.e., MG. = 1 V j). Comparing Figs. 3 and 4 for N — 8, the error performance is the same for both cases (L — 8 and 32). For N > 8, it is clear that a smaller value of L results in a significant performance deterioration due to the increasing correlation among the subcarrier fading processes. Fig. 4.5 depicts the B E R of uniform F H - M C - C D M A as a function of N when L = 16 and SNR = 10 dB. For a given M, the figure clearly indicates that N = 1 (i.e., FH) and N = s (i.e., M C - C D M A ) are not favorable choices for N. The B E R performance improves when N is varied between 1 and s. In particular, smaller values of MGj result in a lower B E R . This is due to the fact that for a given M, a reduction in the number of MS's in each group results in smaller M A I . Optimum error performance is achieved when MG. = 1 V j (i.e., optimum F H - M C - C D M A ) . Finally, in Fig. 4.6, we show the B E R as a function of M when L — 16 and SNR = 10 Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 56 dB. Clearly, F H - M C - C D M A outperforms all other schemes over a wide range of active MS's. More specifically, optimum F H - M C - C D M A provides the lowest B E R . Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 57 d[k] c [0] - OF COS(ZK — t ) ' b \F | i ) COS(2K Y, I) (NC-\)F c IN - l ] cos(2n - „ <" j TH t) Sfi) Band Pass ) Filter (km) 2cos(2nf t ) Frequency Synthesizer Transmitted signal for group Gj (a) r(0 2 ft"") — cos(2nf t) Frequency Synthesizer Band Pass Filter 'h Band Pass Filter 1 C W Band Pass Filter 41*1 Integrator 7i ' (b) Figure 4.1: (a) Transmitter model; (b) Receiver model Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 58 0 mm 1 jj • group 1 H group 2 1 J W 7VF frequency • g r o u P ^ r Figure 4.2: Frequency groups when NGj = N V j with N — L = 4, and F = g = s/N. Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 59 Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 60 Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 61 2 4 8 16 32 64 128 256 Number of subcarriers, /V Figure 4.5: B E R versus N for L = 16, and S N R = 10 d B . Chapter 4. Frequency-Hopping Multicarrier CDMA in Correlated Rayleigh Fading 62 63 C h a p t e r 5 A C h a n n e l A w a r e F r e q u e n c y H o p p i n g M u l t i p l e A c c e s s S c h e m e 5.1 Introduction The B E R of F H systems has been studied in [96]. Although a great deal of emphasis has been placed on DS, recent developments in F H technology [97, 98, 99, 100] have resulted in commercial applications such as mobile cellular communications, personal communi-cations and wireless local area networks (LANs). On the other hand, F H systems have been considered for a variety of military applications due to their frequency diversity and resistance to near-far problem [91, 101]. In an F H system, the total available bandwidth is divided into a number of subbands. F H allows many mobile stations (MS's) to share a common channel by employing user-specific hopping patterns which specify the subband occupied by each MS at a given time and allow the data of different MS's to be recov-ered at the base station (BS). Thus, the'transmitted signal appears as a data-modulated carrier that is hopping from one frequency to the next. A n advantage of F H over DS is that it can be implemented over a larger frequency band, and the band can be non-°A version of this Chapter has been published [83, 84, 85]. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 64 contiguous [90]. A second advantage is that F H provides resistance to multiple-access interference while not requiring power control to deal with the near-far problem. In [55], matched F H was described for a single-user system to provide performance improvement on a time-invariant frequency selective channel. The subbands were designated as either usable or unusable (i.e., very poor) and hops were only allowed to usable subbands. . In this chapter, a channel aware frequency hopping (CAFH) multiple access scheme is described. In this scheme, a suitable channel measurement technique is assumed. The utilization of one of the recently developed blind identification techniques [102] would be useful in this aspect. In C A F H , the BS monitors the channel to estimate the subband gains for each MS. The BS then uses this knowledge to assign to each MS the subband in which the MS has the highest gain, while ensuring that at most one MS is assigned to each subband. The B E R of C A F H is analyzed assuming independently Rayleigh faded subbands. It is shown that C A F H can offer significant performance improvements over conventional F H . 5 . 2 S y s t e m M o d e l We adopt a slowly time-varying, frequency-selective Rayleigh fading channel model which is commonly used for wideband systems [89, 103]. The instantaneous SNR, F, on each subband of the channel is distributed according to an exponential distribution given by 1 _ J L Prh) = — e •«>, 7 > 0 (5.1) 7o Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 65 where 70 is the average value of T. The slowly varying nature of the channel implies that it can be treated as time-invariant over the duration of a few symbols. Consider the uplink of a system in which K MS's transmit over L subbands to the BS. The signal of MS k in signaling interval i is skti{t) = ^Y~dkti cos(27r(AM + fc)t), (i-l)T<t< %T (5.2) where T is the bit duration, fc is the carrier frequency, fhk i is the frequency offset, E is the bit energy and <4 , i € {—1, +1} depends on the information bit to be sent. The signal received during the ith symbol interval at the BS is K r(t) = 9k,iSktl{t - T F C) + n(t) (5.3) k=l where rk is the kth MS time delay, n(t) represents additive white Gaussian noise (AWGN) with mean zero and two-sided power spectral density (PSD) N0/2 and gk,i is the kth MS subband gain during the ith signaling interval. The subband gain, gk>i, depends on the scheme that is used to assign subbands to MS's. 5.3 C A F H Scheme In the C A F H scheme with r rounds, the BS assigns subbands to active MS's as follows. The round number is initially set to 1. In round j , j = 1,2,- • -,r, for each MS, k, which has not yet received its subband assignment, the subband fk G {1, • • -,L}, with Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 66 the jth best subband gain, g^\, out of L subbands is determined. If a given subband has the jth best gain for exactly one MS, the subband is assigned to that MS and the scheme enters round j +1. If an available subband has the jth best gain for several MS's, the subband is assigned to the MS with the largest gain and the scheme enters round j + Once a subband is assigned to a MS, it is not available in subsequent rounds. The subband assignment procedure continues until all K MS's have been assigned subbands or round r + 1 is reached. In the latter event, i.e., there are still unassigned MS's after round r, the BS chooses each such MS in turn in a random order and assigns to it the unoccupied subband for which its gain is highest. A flow chart describing the C A F H scheme is provided in Fig. 5.1. It might be noted that the number of rounds required is at most min{r, K}. 5 . 4 B E R a n a l y s i s 5.4.1 BER analysis of C A F H for r=l For simplicity of the analysis, we consider a synchronous system, i.e., T\ = T 2 = . . . rK = 0 [104], with r = 1 round. The B E R , PCAFH, for the C A F H scheme can be written as PCAFH = A + B , (5.4) Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 67 where K A = ll lZ(K)P^L (5.5) fc=l represents the B E R for the case when a subband is assigned to the target MS in round 1, and z(k) K - l \ / ^\k-l / T -, \ K-k (5.6) L represents the probability that k — 1 other MS's occupy the subband of the target MS in round 1. For the case in which a subband is not assigned to the target MS in round 1 (i.e., the highest gain among all the L subband gains of the target MS was smaller than that of some other MS that shared a common highest gain subband as the target MS), the B E R can be approximated by 1 K k=2 where K Z = Y\z{k) (5.8) fc=i is the probability that the target MS is selected in the first round. Let G^ be the random variable corresponding to the jth largest gain out of h gain samples. The term Pe,j,h represents the B E R when the transmitted signal from an MS experiences a subband gain G^ and is contaminated by A W G N with two-sided PSD N0/2 and is given by P e T K = / Q(V^l)Pr(dl)dl (5.9) Jo Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 68 for B P S K modulation, where Q(z) = -^e^^dy. For non-coherent FSK (NCFSK) modulation PS?3*' = [3 \e-l/2vMl)dl- (5.10) where = [G^]2E/N0 is the jth largest SNR out of h samples. For independent, identically distributed Rayleigh gain samples, the probability density function, p^il) of can be obtained using a standard result in ordered statistics [105] as Pro»(7) - WMI^y. (l - e~lh°)h~3 • e - ^ - ^ e - 7 7 7 0 ' ( 5 ' n ) From (5.9) and (5.11), the single-user B E R when using B P S K may be written as P ^ h S K = (^jnyr JT <?(V27)- (! ~ e^)h~j ' " i ^ l h ° d ^ ( 5 ' 1 2 ) h-3 0 V 1 J (5.13) Integration by parts as pBPSK _ ft! v ' 1 - ^ — I s ! /0°° Q U ^ ) • | l e - ^ ( ^ ) | d 7 , (5.14) Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 69 yields pBPSK _ h\ srh~i re,j,h (h-j)!(j-l)\ Z^i=0 i+j h-3 Using lno-1 Jo 2 with CT2 = 2{lJli+j) in (5.14), we have pBPSK h\ S^h-j (-1)' e,j,h 2 ( / i — — 1 ) ! Z^i=0 i + j 1 - . / - S lo+i+j For the best SNR case (i.e., j = 1), pBPSK h Y^/I-I ( - i ) j 70 70+J+l (5.15) (5.16) (5.17) (5.18) From (5.10) and (5.11), the single-user B E R when using N C F S K may be computed as follows pNCFSK _ h\ ST^h-j, 1 \i re,j,h — (h-j)\{j-l)\ Z^i=0 V L ) (5.19) can be simplified to V * J 70+2(i+j) • (5.19) P. NCFSK _ h\(h-j + l)< e,J,h. + 2 n t 0 J ' ( ^ + i + j ) ' (5.20) Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 70 For j — 1, we have the result given in [106], i.e. pNCFSK = V _ ( 5 2 1 ) 2^t 1 (?+^) For comparison purposes, we note that the B E R , P c o m ) , for conventional F H in Rayleigh fading on the uplink is given by [107] Pconv,BPSK = \{1- JT+YO) ^ ~ ^ + \ P H ( 5 ' 2 2 ) for B P S K modulation. For N C F S K modulation it is given by Pconv,NCFSK = (1 ~ Ph) + TT-P/i- (5.23) 2 + 70 2 where Ph = 1 — ( l — 1 is the probability of a hit. 5.4.2 Numerical Results of C A F H for r=l The B E R curves on an uplink channel for C A F H with r — 1 are compared with conven-tional F H in Fig. 5.2 assuming B P S K modulation. The curves are plotted using (5.4) assuming that the L = 32 subbands undergo independent Rayleigh fading. The results show that C A F H / B P S K can provide a much lower B E R than conventional F H / B P S K over a wide range of number of MS's. The B E R of C A F H / B P S K decreases rapidly with SNR. At an average SNR, 7 o , value of 2 dB, C A F H / B P S K provides roughly a 100-fold Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 71 reduction in B E R compared to conventional F H / B P S K . The B E R difference between C A F H / B P S K and conventional F H / B P S K increases with 70. Fig. 5.3 shows the B E R curves of C A F H and conventional F H for N C F S K modu-lation. Comparing Figs. 5.2 and 5.3, it can be seen that C A F H / B P S K outperforms C A F H / N C F S K . The B E R of C A F H / B P S K is one to three orders of magnitude smaller than that of C A F H / N C F S K for a given SNR and a small number of MS's. With a large number of MS's, C A F H / B P S K provides roughly an order of magnitude reduction in B E R . C A F H / B P S K and C A F H / N C F S K performance differences tend to increase with SNR. Computer simulation results were obtained in order to ascertain the accuracy of the analysis and are shown as dotted curves in Figs. 5.2 and 5.3. It can be seen that the simulation curves agree closely with the (approximate) analytic results. It might be noted that the analytic B E R results are closer to the simulation results when the network is lightly loaded. The performance difference increases with the number of MS's. Fig. 5.4 shows the B E R of C A F H / B P S K for r = 1 and 2. A larger value of r results in a substantial decrease in B E R as the number of MS's increases. The performance difference between r = 1 and 2 increases with SNR. For example when K = 8 MS's are transmitting, the performance difference is roughly 3-fold, 4-fold, 10-fold and 40-fold when operating at an average SNR of 70 = 2,4,6 and 8 dB, respectively. The B E R curves of C A F H / N C F S K for r = 1 and 2 are shown in Fig. 5.5 and are qualitatively similar to the C A F H / B P S K curves. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 72 5.4.3 B E R Analysis of C A F H for r >1 The C A F H B E R with r rounds can be obtained as PCAFH = Pi + (1 - Z1){P2 + (1 - Z2){P3 + (1 - Z3){ • ' ' {Pr-1 + (1 ~ Zr-i){Pr + (1 — Zr)Prandom}} ' ' ' }}}, (5.24) where Zj, Pj, 1 < j < r, and Prandom denote the probability that the target MS is selected in round j, the B E R when a subband is assigned to the target MS in round j, and the B E R when a subband is assigned to the target MS after round r, respectively. The terms Zj, Pj, and Prandom can be expressed as k=l K-j+l K~k E I E HKn)P^\n)T^{n), j>l fc=l n=j—1 (5.25) and K Ei*(fe,o)p e , i ,fct. P = I fc=1 J i A:-J+I K-k E I E ^ ( ^ n j p w ^ r w w P e , ! ,/s(L-n)) J' > 1 fc=l n—j—1 (5.26) random l ^ (5.27) respectively. For the sake of mathematical tractability, we assume in (5.27) that when a subband is not assigned to the target MS in round j < r, one of the subbands with the {r + 1, r + 2, • • -,K} largest gains is assigned at random to the target MS. This procedure Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 73 is only an approximation of the C A F H assignment scheme described in Section 5.3, of course. However, as our results will show, (5.27) is a very accurate approximation of the true error rate. In (5.25) and (5.26), T^\n) is the probability that the subband with the jth i a r g e s r j gain is available when n> j — 1 subbands have been occupied by MS's in the previous j — 1 rounds. T^\n) is given by Furthermore, the probability, $(fc,n), that k — 1 other MS's occupy the subband of the target MS in round j when n subbands have been assigned in the previous j — 1 rounds is given by ( K - T7. - 1 \ ®(k,n) In obtaining (5.29), it is assumed that the jth largest gain of each of the K — n — 1 other remaining MS's is randomly located in one of the L — n remaining subbands. In (6.25) and (5.26), P^\n) denotes the probability that n subbands have been assigned in the previous j — 1 rounds. P^\n) may be obtained recursively as n - l pU)(n) = P{i~l\i)q{L-i,n-i), j > 2, (5.30) i=j-2 with the initial condition P ( 2 ) (n ) = q(L,n), > (5.31) Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 74 where q(l, m) is the probability that m subbands are assigned in the previous round (i.e., round j —1) when I subbands were available. The term q(l,m) can be expressed as follows q{l,m) = M-l m - 2 M - 2 - £ ij M-m M-m+l—ii M—m+2—ii—i2 3 = 1 (*) E E E ••• E i l = l » 2 = 1 1 3 = 1 ! m - l = l V J M-l-ix «2 M - 1 - ix - i2 \ ( m-2 \ j=i V ^m— 1 (5.32) where M = K — (L — I) is the number of unassigned MS's at the beginning of round j — 1. In (5.32), the summation indices ii,i2,' • ^hn-i represent the number of MS's that could be assigned to each of the first m — 1 subbands and the combinatorial terms represent the number of ways of selecting the MS's in each subband from the available set of unassigned MS's. Let be the random variable corresponding to the jth largest gain out of h gain samples. In (5.26) and (5.27), Pe,j,h denotes the B E R of a MS that experiences a subband gain Gij) and is impaired by A W G N with two-sided PSD N0/2. 5.4.4 Numerical Results of C A F H for r >1 The B E R curves for the uplink of a system using C A F H with r = 1 and 2 are compared to conventional F H in Fig. 5.6 assuming B P S K modulation and an average SNR of 7o = 2 dB. The results show that C A F H / B P S K can provide a much lower B E R than conventional F H / B P S K over a wide range of number of MS's. At 70 = 2 dB, it is observed Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 75 that C A F H / B P S K can yield over a 100-fold reduction in B E R compared to conventional F H / B P S K . The curves are plotted using (5.24) assuming that the L — 48 subbands undergo independent Rayleigh fading. Computer simulation results were obtained in order to ascertain the accuracy of the analysis and are shown as dotted curves. It can be seen that the simulation curves agree closely with the analytic results. It is evident from Fig. 5.6 that for a fixed target B E R , the number, K, of MS's that can be supported by a r = 2 C A F H can be significantly larger than in the r = 1 case. At 70 = 2 dB, and a fixed target B E R of 10~3, it is found that the system capacity is 18 MS's for r = 2 compared to only 12 MS's for r — 1. Moreover, the B E R ratio between r = 1 and r — 2 increases with K, for K < 15. Beyond K = 15, the B E R ratio is fairly constant. The curves for 70 = 8 dB are plotted in Fig. 5.7 and are qualitatively similar to the 70 — 2 dB results. The B E R ratio between conventional F H / B P S K and C A F H / B P S K increases with 70. Fig. 5.8 illustrates the B E R versus average SNR for C A F H / B P S K with L = 48 subbands. Clearly, the performance difference between the different numbers of MS's {K = 1,10,20) increases with 7 o . The curves for C A F H / N C F S K are plotted in Fig. 5.9. It is evident that as 70 increases, so does the improvement offered by r = 2 compared to r = 1. Fig. 5.10 shows the B E R versus the number, r, of rounds for C A F H / B P S K with 70 = 8 dB and L — 48 subbands. As expected, the performance difference between the different rounds gradually decreases with r as the C A F H scheme reaches saturation at a B E R of 10~8 and 2 x 1 0 - 6 for 10 and 20 MS's, respectively. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 76 To compare the performance of C A F H with an optimal allocation scheme, an exhaus-tive search method that searches over all the possible L(L—l)(L — 2)---(L—K+l) channel allocation permutations is used. The exhaustive search tries all the different channel al-location possibilities and selects the one that gives the lowest B E R , on which the current information bit is transmitted. It can be seen from the results in Fig. 5.11 that the difference in performance between C A F H and the optimal exhaustive search method is very small. However, the channel utilization of C A F H is much higher than that of the exhaustive search method. This can be explained by noticing that C A F H requires the knowledge of the channel gains at each information bit transmission, which requires a single channel measurement session per information bit, regardless of the number of MS's and number of channels. On the other hand, the exhaustive search method requires the knowledge of the B E R for each of the different channel allocation permutations. This requires L(L — 1)(L — 2) • • • (L — K +1) channel measurement sessions per information bit. Since the exhaustive search calculates the B E R , a sufficient number of information bits is required for transmission. Moreover, system delay should be considered. The time taken to measure the channel, and to transmit the information bits for B E R calculation must be less than the channel coherence time, i.e., constant channel behavior. This becomes extremely difficult to achieve for the exhaustive search method. The exhaustive search is a hypothetical approach used only for comparison purposes. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 77 Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 78 10 I 1 i 1 1 1 1 ' r Number of MS's Figure 5.2: B E R of conventional F H and C A F H wi th B P S K modulation, L = 32. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 79 10 10" 10" 04 3 W 10 CQ 10 -4 10" 10 V - • - X • • • x • • * x•• - X FH (Y0 = 6 dB) FH (yo - 8 dB) CAFH (Yo - 2 dB) CAFH (Yo = 4 dB) CAFH (Yo = 6 dB) CAFH (Yo = 8 dB) 6 8 10 Number of MS's 12 14 16 18 Figure 5.3: B E R of conventional F H and C A F H with N C F S K modulation, L = 32. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 80 • X- • r = 1 ( Y o = 2 d B ) • • + • r = 1 ( Y 0 = 4 d B ) • Q • r = l ( y 0 = 6 d B ) -r = 1 (Yo=8dB) - * - r = 2 ( y 0 = 2 d B ) —t— r = 2(Yo=4dB) - B - r = 2(Yo=6dB) —•— r = 2(Yo=8dB) 6 8 10 Number of MS 's 12 14 16 18 Figure 5.4: B E R of C A F H with B P S K for r = 1 and 2, L = 32. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 81 10 10 % 10" Pi w 10 -4 10 10' • • X' • r = 1 (To = 2 dB) • • + • r = K Y J = 4 dB) D r = Hl6 = 6 dB) r = 1C» - 8 dB) . -*- r = 2(7, = 2 dB) r - 2 ^ = 4 dB) r = 2(16 = 6 dB) 1 r = 2(7) i = 8 dB) i 6 8 10 12 Number of MS's 14 16 18 Figure 5.5: B E R of C A F H wi th N C F S K for r = 1 and 2, L = 32. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 82 10 —*— Conventional F H - e - C A F H <>=1) o Simulation C A F H (r = — C A F H ( r = 2) Simulation C A F H (r = 2) Number of MS's Figure 5.6: B E R of conventional F H and C A F H with B P S K modulation. -y0 = 2 dB and L = 48. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 83 10 Number of MS's Figure 5.7: B E R of conventional F H and C A F H with B P S K modulation. 70 = 8 dB and L = 48. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 84 Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 85 Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 86 Number of rounds, r Figure 5.10: B E R vs. the number of rounds, r, of C A F H wi th B P S K . 70 = 8 d B and L = 48. Chapter 5. A Channel Aware Frequency Hopping Multiple Access Scheme 87 Number of MS's Figure 5.11: B E R versus number of MS's of random allocation, C A F H with B P S K for r = 1 and 2, and exhaustive search method. 70 — 4 dB and L = 8. 88 C h a p t e r 6 O r d e r S t a t i s t i c s fo r C o r r e l a t e d F a d i n g C h a n n e l s 6.1 Introduction In the design of wireless communication systems, much research effort has been devoted to the study of diversity schemes for mitigating the effects of channel fading and improving the received SNR. The theory of order statistics underpins the performance analysis of many diversity reception techniques. Selection combining (SC) [108] is a simple diversity scheme in which the signal from only one of the diversity branches (the branch with the largest SNR) is processed. In signal detection and estimation problems, the cdf of the r-th order statistic is often required. For example, it is useful in calculating the performance loss when the receiver makes an error in selecting the largest SNR branch. It is also applicable to the calculation of the average symbol error rates for various modulation schemes. Most studies on the ordering of channel gains assume that the fading processes on the different branches are uncorrelated. This is partly due to the lack of a simple procedure °A version of this Chapter has been published [86]. Chapter 6. Order Statistics for Correlated Fading Channels 89 for analyzing order statistics for fading envelopes with arbitrary cross-correlations. For correlated fading, expressions for the pdf of the highest SNR for two and three branch diversity systems are available [109, 110, 111, 112]. For a larger number of diversity branches, [113] and [108] present an expression for the correlated SC output pdf. In [114], the performance of a hybrid selection/maximal-ratio combining (H-S/MRC) diversity scheme over an exchangeable1 correlated Nakagami fading channel is analyzed. . Expressions for the cdf of the r-th order statistic, r = 1, 2 , . . . , n, for correlated Nak-agami fading branches are not available in the literature except for r = n (i.e. the largest order statistic). In this chapter, we apply a result from the theory of order statistics [115] for evaluating the cdf of the r-th order rv out of a set of n arbitrarily correlated rv's to the case of correlated Nakagami-m fading. Specifically, numerical results are obtained which illustrate the effects of the fading parameter m and the branch correlation matrix. For a large number of branches, the numerical procedure can be quite computationally intensive. Thus, the accuracy of using a simplifying exchangeable approximation is also investigated. 6 . 2 P r e l i m i n a r i e s Let gi,- • -,gn be n arbitrarily correlated rv's which represent the branch gains in a diversity communication system. If g1: • • -,gn are arranged in increasing order of their magnitudes and written as 9\:n <••• < 9n:n, (6-1) X A set of random variables A\, • • •, An are said to be exchangeable if the joint distribution of AWl, • • •,Ann does not depend on the permutation ir. Chapter 6. Order Statistics for Correlated Fading Channels 90 we refer to gr:n, as the r-th order branch gain. In this chapter, g\,---,gn are assumed to be statistically dependent and not necessarily exchangeable branch gains. The corresponding instantaneous branch SNR's are denoted 71,72, • • •, 7n with 7 l = [gi]2E/N0,i = l,2,...,n (6.2) where E is the transmitted bit energy and No is the one-sided noise PSD. The r-th order branch SNR is l r : n = {9r:n]2E /7V0. (6.3) The joint cdf of 7 l , • • - ,jn can be expressed as [116, p. 140] where (p{t\, • • -,tn) is the joint characteristic function (CF) of 71, • • - ,7 n . Hence, the cdf of the highest order statistic 7 n : n = max{7i, • • - , 7 „ } can be written as = Pr{all 7i < 1 (27r)" j_ fc=i 1 - e - j t k v dt\ • • • dtr, (6.5) (6.6) (6.7) Chapter 6. Order Statistics for Correlated Fading Channels 91 6.3 cdf of 7 r : n 6.3.1 Correlated Fading In this section, we derive an expression for the cdf, F 7 r ; n , of the r-th order branch SNR, 7 r : n , in terms of {Flj:.(v), j = r , r + 1, • • • , n}. Recall that the probability, pr,n, of the occurence of at least r out of n events {Ai, • • -,An} is given by [117] [11, pp. 109] Pr,n = 2(-1)J'"rCll) '^r = 1'2'"- ^ (6-8) where Sj= ]T P l ' { - ' ' 1 ^ - - - : A , i } . (6.9) 1 <ii <---<i-j<n In (6.9), the summation is over all different ways of selecting j out of n rv's. Letting Ai,i = 1, • • -,n denote the event {% < v}, (6.9) can be re-written as l<ii<--<ij<n £ P r { m a x ( 7 i l ) - - - , 7 i J - ) < « } ( 6 - n ) l<ii<---<i3-<n _ Yl i M - x r " ' ^ ( ; ) ( 6 - 1 2 ) 1 < i i <•• <ij<n K i i < - < i j < n In (6.13), the sum is over all possible ways of selecting j out of n rv's and the superscript notation in F^f''1^ indicates that only jiit- • • ,7^. are included in the selection. It is Chapter 6. Order Statistics for Correlated Fading Channels 92 assumed that the sum is equal to Flrvn{y) for j = n. The term F-£- (f) is obtained using (6.7). From the definition of Ai} we can write Pr,n = Pr{7l:n <V,-- - , 7 r : n < v} (6-14) = P r { 7 r : n < v} (6.15) = F 7 r : » . (6.16) Following the result in [115, p. 99], and substituting (6.13) and (6.16) into (6.8), we can obtain the cdf of the r-th order statistic KJv) = t When n is large, the evaluation of (6.17) may be time-consuming. However, if the 7J'S are exchangeable, (6.17) can be easily evaluated as it reduces to KJV) = B-irr(Jr: J) (6.i8) \<i\<---<ij <n ly.j (6.17) Chapter 6. Order Statistics for Correlated Fading Channels 93 6.3.2 Independent Fading Suppose that gi,---,gn are n independent, but not necessarily identically distributed rv's, (6.7) simplifies to Fjn:n(v) = ^ {ll , ' ' ', In < V} = P r { 7 l <v}--- P r { 7 n < v} k=l B y substituting (6.19) in (6.17) we get (6.19) E (-iy j - i r - 1 E UF%+I'-MM l<ij+l<---<in<n k=\ (6.20) If the channel gains are now assumed to be independent, identically distributed (i.i.d) rv's, using the property of exchangeability, (6.20) reduces to * V : » = i > i : 3=r (6.21) 6 . 4 N u m e r i c a l r e s u l t s To illustrate the application of the results, we evaluate the cdf of the r - th order statistic for correlated Nakagami-m fading branches. In this case, the joint C F is given by [118, p. 359] tn) = | I - j T S | - m (6.22) Chapter 6. Order Statistics for Correlated Fading Channels 94 where I is the identity matrix of size n x n, | • | denotes the determinant, T = diag{t 1 ; t2, • • •,tn}, m € [0.5,oo), and S is a symmetric matrix with elements In (6.23), Rk,e = Cov(7fc,7^) is the element in row k and column I of the correlation matrix R. Of particular interest for wireless communication systems are the values m = ~, m = 1, and m > 1. The Nakagami distribution is a one-sided Gaussian distribution to a Rician distribution when m > 1. 6.4.1 Constant Correlation Model The constant correlation model, discussed in [119, 120, 121], may be used either to approximate closely spaced antennas or to perform a first-order analysis using the average value of the correlation coefficients for all the off-diagonal entries of the correlation matrix [121]. Suppose we have a diversity receiver with an array of n = 3 antennas placed at the vertices of an equilateral triangle. Then the branch gains are exchangeable Nakagami-m rv's and the cdf can be obtained using (6.18). As an example, we use the correlation (6.23) (severe fading) for m = |, a Rayleigh distribution for rn — 1 , and a good approximation Chapter 6. Order Statistics for Correlated Fading Channels 95 matrix from [121] 1.0 0.6 0.6 0.6 1.0 0.6 (6.24) 0.6 0.6 1.0 where 7 denotes the average SNR. Fig. 6.1 shows the cdf of the r-th order statistic in a correlated Nakagami-m fading environment with the correlation matrix in (6.24), m — \ and 7 = 1. Computer sim-ulation results (shown as dotted curves) were used to verify the analysis results (solid curves). As to be expected, it can be seen that a lower order statistic, e.g. 71:3, has a steeper curve than a higher order statistic, e.g. 7 3 : 3 . In a SC system, this provides the performance loss when the receiver incorrectly selects a branch with a lower gain instead of the highest gain branch. Fig. 6.2 shows the cdf of the r-th order statistic in a correlated Nakagami-m fading environment (solid curves) with the correlation matrix in (6.24), m — 1 and 7 = 1. The cdf curves for independent fading (dotted curves) are obtained using (6.21) and are shown for comparison. When the highest SNR branch is selected (i.e. 73:3), the performance is better when the branches are independently fading; when the lowest SNR branch is selected (i.e. 7 i : 3 ) , correlated fading yields better performance. The cdf of 7^3 is plotted in Fig. 6.3 for different values of m in correlated fading using the correlation matrix in (6.24). It can be seen that the performance degrades as fading severity increases (i.e. m decreases). Calculating the cdf of the r-rth order statistic can be applied to evaluating the average Chapter 6. Order Statistics for Correlated Fading Channels 96 error performance of various modulation schemes in correlated fading channels. In diver-sity applications, this provides the performance loss when the receiver incorrectly selects a channel with a lower gain. In multiuser diversity applications, this is a good measure for the channel gain loss from one user to another, which implies a total throughput improvement when selecting only the user(s) with stronger instantaneous SNR. 6.4.2 Linear Model This type of correlation model applies to linear antenna arrays. We consider the correla-tion matrix example for a five-element array in [122] 1.000 0.795 0.605 0.375 0.283 0.795 1.000 0.795 0.605 0.375 R = l\ 0.605 0.795 1.000 0.795 0.605 | - (6.25) 0.375 0.605 0.795 1.000 0.795 0.283 0.375 0.605 0.795 1.000 The correlation coefficient value decreases as the distance between array elements in-creases. In this case, the channel SNR rv's are not exchangeable and the cdf has to be obtained using (6.17). Using the correlation matrix in (6.25), Fig. 6.4 shows the cdf for 7i:5,72:5, • • • ,75:5 with m — 1 and 7 = 1 (solid curves). Approximate cdf curves (dotted curves), obtained using (6.18) and a 5 x 5 exchangeable matrix in which an average value of 0.514 is used for all off-diagonal entries, are also plotted. It can be seen that there is a significant difference between the exact and exchangeable approximation cdf curves for Chapter 6. Order Statistics for Correlated Fading Channels 97 7i:5 and 7 5 : 5 . Chapter 6. Order Statistics for Correlated Fading Channels 98 Figure 6.1: P lo t of cdf of 71:3,72:3 and 7 3 : 3 w i th m = 1 and 7 = 1 : theoretical (dotted) and simulation (solid). Figure 6.2: Plot of cdf of 71:3,72:3 and 7 3 : 3 with m = 1 and 7 = 1: independent (dotted) and correlated (solid) fading channels. Chapter 6. Order Statistics for Correlated Fading Channels 100 v Figure 6.3: Plot of cdf of 7i:3 with 7 = 1 for different values of m. Chapter 6. Order Statistics for Correlated Fading Channels 101 Figure 6.4: Plot of cdf of j r : 5 with m = 1 and 7 = 1 for a linear model (solid) and its exchangeable approximation (dotted).. 102 C h a p t e r 7 E r r o r A n a l y s i s o f O r d e r S e l e c t i o n i n C o r r e l a t e d F a d i n g C h a n n e l s 7 . 1 I n t r o d u c t i o n In order to increase system's capacity and to improve the offered QoS in wireless commu-nication systems, several techniques have been proposed to mitigate the effects of channel fading and to improve the received SNR. Such techniques are diversity reception, dynamic channel allocation, and power control. The theory of order statistics underpins the per-formance analysis of many diversity reception techniques that involve efficient channel allocation and signal processing algorithms for signal detection and estimation. Expressions for the cdf of the r-th order statistic, r = 1,2, . . . , L , for correlated Nakagami fading branches have appeared in [86]. In this chapter, we apply a general result from the theory of order statistics [86] and [115] to evaluate the B E R of a channel aware diversity system, hereafter referred to as order selection, when the r-th order branch is selected for transmission when using B P S K modulation. The expressions derived are then used to evaluate the error performance for C A F H . The C A F H evaluation is carried out assuming correlated Nakagami-m fading when the coherence bandwidth notably exceeds Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 103 the transmission bandwidth, which is the real scenario in practical C A F H systems. 7.2 Channel and System Model Let <7i,<72,-' - , 9 i be L arbitrarily correlated rv's which represent the branch gains in a diversity communication system. The corresponding branch SNR's are denoted by 7i>72,' • • ,7i» w i t h 7i^{gi}2E/No,i = l,2,...,L (7.1) where E is the transmitted bit energy and N0 is the one-sided noise PSD. Consider an L-branch diversity receiver of a cellular system in which K MS's transmit over L, L > K, branches to the BS. The received base-band signal on the 7r fc-th branch of the /c-th MS at the BS is ^ f c(t) = ^ s f c ( t ) e - ^ + n ( t ) , (7.2) where Sk(t) is the transmitted signal of the /c-th MS and n(t) is independently identically distributed A W G N with zero mean and one-sided spectral density N0. The random phase ' ipnk is uniformly distributed over the range [0, 27r) and g„k is the random magnitude of the 7Tfc-th diversity branch gain. Here we model the fife's as correlated rv's with a marginal Nakagami pdf given by [123] J9nk\ ) r(m„ f c) V f i"fc/ j - 7 3^ k = l,2,--;K where the fading parameter mWk is assumed to be a positive integer, T(-) is the Gamma Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 104 function defined by T(p)= JQ°° vp~le~vdv, and ClVk = E[(gnk)2}, E[-] being the expectation factor. Let j „ k denote the instantaneous signal-to-noise ratio (SNR) on the ^ - t h branch, expressed as 7xfc = ^r{9,k)\ (7.4) N, o where E/N0 is the ratio of the bit energy to the Gaussian noise spectral density. It is well known that the marginal pdf of 7 ^ follows the Gamma distribution f (v) = 211^." e-m*k(vfrV v > 0 (7.5) where the average received SNR on the 7Tfc-th branch is given by 7 ^ =E[7 ? RJ=^|fi 7 r (.. 7.3 Error Performance for Order Selection B P S K If the rv's <7i, • • •, are arranged in increasing order of their magnitudes and written as 9l:L <•••< 9L:L, (7.6) we refer to the r-th order rv, gr:L, as the r-th order statistic (r = 1, • • •, L). In this section, gi, • • •, gL are assumed to be statistically dependent and not necessarily exchangeable branch gains. Accordingly, the SNR of the r-th order statistic is then 7r:L = [gr:L}2E/N0. (7.7) Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 105 The average error probability when transmitting on the r-th order branch out of L branches for a B P S K modulation over an A W G N channel with two-sided PSD N0/2 is obtained by averaging the conditional error probability over the pdf of jr-i, i.e., poo PlL= \ P(e\v)flTJv)dv, Jo (7.8) where P(e\v) is the conditional error probability. By differentiating (6.17), we can write the pdf of 7 r : £ as U M = dv L = E i=r L = E • i ^ C l l ) E ^ l<7Tl<-<7Ti<L ( - i ^ C - i ) E U-l < 7 T l < - < 7 r j < i , dv (fl,•••,*•<)/ (7.9) where the pdf of the largest order statistic, ^L-.L, out of L rv's is given by [108] U M = j^yz JZo • • • JZo^i, • • ;tL)h{y,t)dti • • • dtL, (7.10) with h(v,t) dv T - T / \-e-3Hv hut E(-i) n=l n+1 E b H \-bL=n j(b1t1+-+bLtL) exp(jv\biti + ---+bLtL]) (7.11) Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 106 Inserting (7.9) into (7.8) yields p ; : L = E ( _ 1 ) i - ^ - l ) x E H ' ' • I-oo^1'-^^- • •,Ui)w^-^KU1,- • ;UAdh • • • dtni l<7r i<- - -<7Ti<i (7.12) wi th 1 (27T)* Jo P(e\v)h{7I1'-'^){v,t)dv. (7.13) In (7.12) the integrand is decomposed into two factors. The first factor 4>(t) is the C F , dependent solely on the channel characteristics. The second is the weighting function w(t); it depends solely on the modulation scheme. The conditional error probabili ty for B P S K is given by [89] P(e| 7) = Q(v^7)- (7-14) It follows from inserting (7.14) into (7.13) that w 7Tl ) ' ' ') t-JTi ) (2TT)< POO / Q(V2v)h{nw",nt){v,.t)dv. (7.15) Jo Integration by parts as w (Tl,-,*i)(t • • • t ) 1 e=i x E ( - i ) n+l E /o00QW*J)• j^!iri + '" + b,riS: „dv. favrj H 1-6*4 =n exp(jv[bnitn H VK^]) ds (7.16) Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 107 yields w UUU) x E ( - i ) n + 1 n=l bit, H h6 w . =n E 2 1 V5F Ji) 1 P O O e - « ( 1 + J ' l l ' i r i ' irj "I H>7rj'irj]) (7.17) Let t ing x — y/v, we obtain w \u7Tl , J 7T] / 1 (2TT)' T -1 UUU) x E ( - i ) n + 1 n=l i L J " ° ° e - a ; 2 ( i + j K ^ i + - + ^ ^ 1 ) ^ . bjrjH H%j=n (7.18) Using in (7.18) wi th u2 = 2 ( l + j [ 6 T l ^ 1 + - - - + f c „ i t T . ] ) » we have (7.19) w 1 ( 2 ^ uuu) x E ( - i ) n + 1 n=l • E I birjH 1 -6^=" ^ / l + j f f c i r ^ T T j H h b i r j t T T j ] (7.20) The error performance can then be obtained by inserting i f / 7 1 " 1 ' ' " ' " "^^ , , • • shown in (7.20) and <j>^<"'^(£W1, • • -,tWi) for a given operational environment into (7.12). Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 108 7.4 Numerical Results and Discussion In this section, numerical examples are provided for several correlation models, well known from practical diversity systems, in order to provide the applicabil i ty and check the usefulness of the proposed analysis. We consider the uplink of a cellular system in which a M S transmits over L — 8 subbands to the B S . For a given M S , we assume that the various subbands are subject to correlated fading, where the amount of correlation depends on, among other things, the subband frequency separation. The correlation at any instant of time between the fade envelopes, gi and gj of the ith and jth subbands, respectively, is taken to be [124] where Bc is the coherence bandwidth of the channel. Therefore, the L x L covariance mat ix R is given by E{gi,gj} 1 (7.21) l + ( ^ ) 2 1 Pl,2 Pl,L P2,l 1 P2,L R = 7 Pl,l PL,2 1 Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 109 where Pi,i = 1, 1 < i < L, and R is symmetric and positive definite [106]. For correlated Nakagami-m fading chan-nels, the joint C F is given by (6.22). We shall assume that each subband experiences the same extent of fading, i.e., me — m for £ — 1, 2, • • •, L. 7.4.1 B E R evaluation for Order Selection B P S K Substituting the component correlation coefficients into the joint C F expression (6.22) and using B E R expression (7.12), we will see how those parameters affect B E R per-formance in a correlated Nakagami-m fading channel. To illustrate the application of the presented mathematical model, we evaluate the B E R , PgL, when transmitting on the r-th order subband out of L available subbands for a B P S K modulation in m — 1 correlated fading environment. » In Fig. 7.1 we select Bn — Bs/Bc = 0.9, where Bn is the subband bandwidth nor-malized by the coherence bandwidth of the channel, and the bandwidth of each subband is given by Bs. Fig. 7.1 shows the B E R , P e r : i , of B P S K when r = 8, 7, 4,1 for L = 8 subbands . In Fig. 7.1, PreL is plotted versus the SNR in correlated fading environment (solid line). In the same figure, Preh is depicted for an uncorrelated fading environment (dotted line). For comparison, the B E R of conventional B P S K without ordering is plotted (dashed line). When the highest SNR subband is selected (i.e. 78:s)> the performance is better when the subbands are independently fading. On the other hand, if the lowest SNR subband Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 110 is selected (i.e. 7i :s), correlated fading yields better performance. It can also be seen that the performance difference between correlated and uncorrelated fading subbands increases with r. At a SNR of 10 dB, operating in an uncorrelated fading environment provides roughly a 20-fold, 7-fold and 2-fold reduction in B E R compared to a correlated case when operating at r = 8,7 and 4, respectively. At r = 1 the performance difference is quite negligible. To demonstrate the impact of the fading correlation to the error performance, Bn = 0.5 is chosen in Fig. 7.2. It is evident that the effect of the correlation to the B E R performance is greater for larger correlation coefficients. As expected, the B E R worsens for smaller Bn values when the correlation coefficient increases. For example when trans-mitting at a SNR of 10 dB, the performance difference is roughly 100-fold, 30-fold and 4-fold between uncorrelated and correlated fading channels at r = 8, 7 and 4, respectively. At r = 1 the performance difference is more noticeable. Following the same trend, the B E R for Bn = 0.25 with even a larger performance difference between the corrrelated and uncorrelated cases is depicted in Fig. 7.3. 7.4.2 Application: BER evaluation for C A F H The presented C A F H analysis is carried out assuming correlated Nakagami-m fading when the coherence bandwidth notably exceeds the transmission bandwidth, which is the real scenario in practical C A F H systems. The B E R for C A F H with one round can Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 111 be calculated as [83] PCAFH — ) K - l 1 i=L-K E (7-22) L - 1 where the first term in (7.22) represents the B E R for the case when a subband is assigned to the target MS in round 1. Accordingly, when the target MS was not considered in round 1 (i.e., a MS with a larger gain occupied the same subband as target MS), the target MS is chosen by the BS in a random order and is assigned to an unoccupied subband, this constitutes the second term in (7.22). And represents the probability that k — 1 other MS's occupy the subband of the target MS in round 1. To illustrate the sensitivity of C A F H to the fade envelopes correlation, Figs. 7.4 and 7.5 plot the B E R performance for C A F H versus number of MS's for Bn = 0.9 and Bn — 0.25, respectively when L = 8. For smaller 'L, Fig. 7.6 plots the B E R performance for Bn = 0.33 and L = 3. It is observed that the performance difference between correlated and uncorrelated fading channels increases with 7. For a fixed 7, the performance difference decreases with the number of MS's. fc-i K-k (7.23) Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 112 BPSK with ordering (uncorrelated fading channels) I i i i i I 0 2 4 6 8 10 Average SNR (dB),Y Figure 7.1: B E R vs. average SNR for Bn = 0.9. Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 113 1(T 10 Pe' - I t - - - - -10" ,4 :8 Pi -3 H 10 PQ 10" 10" 10 -6 Conventional BPSK without ordering BPSK with ordering (correlated fading channels) BPSK with ordering (uncorrelated fading channels) 0 2 4 6 8 Average SNR (dB), y" Figure 7.2: B E R vs. average SNR for Bn = 0.5. 10 Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 114 10U 10" 10" „7:8: Pi -3 w 10 CQ ,7:8 p4:8i 10" 10" 10" Conventional BPSK without ordering BPSK with ordering (correlated fading channels) BPSK with ordering (uncorrected fading channels) 0 2 4 6 8 Average S N R (dB) , J Figure 7.3: B E R vs. average SNR for Bn - 0.25. 10 Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 115 1 0 u 10" -2 1 0 -3 w 1 0 CQ 1 0 -4 1 0 10" -5 I I I I I 1 7= 2 dB 7= 8 dB ^ 7=8dB 1 1 1 correlated fading channels uncorrelated fading channels • • i i 3 4 5 6 Number of mobile stations Figure 7.4: C A F H B E R vs. number of mobile stations, for 7 = 2 and 8 d B , and Bn = 0.9. Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 116 1 0 u 1 0 " pi -3 8 1 0 CQ 1 0 " 10"' 10" I 1 - 1 1 1 1 1 l" ' y = 2 dB ; 7= 8 dB 7=8dB 1 1 correlated fading channels uncorrelated fading channels • I I i i 3 ' 4 5 6 Number of mobile stations Figure 7.5: C A F H B E R vs. number of MS's, for 7 = 2 and 8 dB, and Bn = 0.25. Chapter 7. Error Analysis of Order Selection in Correlated Fading Channels 117 1 0 I I I . 7= 2 dB y=10dB 7=10dB i correlated fading channels uncorrelated fading channels Number o f mob i le stations Figure 7.6: C A F H B E R vs. number of MS's, for 7 - 2 and 10 dB, and Bn = 0.33. 118 Chapter 8 Conclusions and Future Research We conclude this thesis with a summary of our contributions and suggestions for future work. 8.1 Summary The main objectives of this thesis were: • The proposal of new methods for the improvement of M C - C D M A as an effective signaling scheme. • The development of new channel aware systems with adaptive subchannel alloca-tion. • The investigation of statistics of general order selection in correlated Nakagami fading channels. The following is a summary of Chapters 2-7. In Chapter 2, we have presented a new M C - C D M A scheme, matched M C - C D M A . In matched M C - C D M A , preference is given to the use of subcarriers with low channel attenuation. The B E R as a function of the number of users and the SNR for the proposed scheme and conventional M C - C D M A Chapter 8. Conclusions and Future Research 119 was investigated. It was found that matched M C - C D M A provides a lower B E R . The performance difference between matched and conventional M C - C D M A increases as the target B E R decreases. The results in this chapter also have appeared in [80]. The performance of a new F H - M C - C D M A scheme for transmission over a Rayleigh fading channel has been analyzed in Chapter 3. The proposed system can combine efficiently the techniques of slow F H , O F D M , and M C - C D M A . Both F H and M C - C D M A can be viewed as special cases of the proposed scheme. The performance of F H - M C -C D M A using E G C and M R C was evaluated and compared with those of corresponding F H and M C - C D M A systems. The results show that F H - M C - C D M A is an attractive wireless multiple access candidate, which can be integrated into existing 2G and 3G C D M A systems. F H - M C - C D M A provides a lower B E R than conventional F H and a comparable B E R to M C - C D M A while employing a lower number of subcarriers. This can avoid some of the disadvantages of M C - C D M A . Results of this work have been published in [81]. Chapter 4 has presented a detailed architecture for a modified F H - M C - C D M A system as a new transmission scheme for wireless communications. We propose a modification of the F H - M C - C D M A scheme in [81] that is suitable for use in correlated fading channels. Both F H and M C - C D M A can be viewed as special cases of the proposed generalized F H - M C - C D M A when the number of subcarrier in a group j , NGj = 1, and NQ0 — s, respectively. It is found that NGj = 1 and NGJ = s are not favorable choices for NQJ-Optimum error performance is achieved by varying NGj between 1 and s while keeping the number of MS's in a group j, MGj = 1, V j . The B E R performance of F H - M C -Chapter 8. Conclusions and Future Research 120 C D M A using M R C on an uplink correlated Rayleigh fading channel has been compared with that of F H and M C - C D M A , respectively. The results show that F H - M C - C D M A can achieve a much lower B E R than F H and M C - C D M A whilst employing a smaller number of subcarriers in a group. This work has been published in [82]. In Chapter 5, we have presented an effective access scheme for slowly time-varying, frequency selective channels. A C A F H multiple access scheme was proposed for inde-pendently Rayleigh faded subbands. C A F H involves the ordering of subband gains of the MS's. In C A F H , the channel status is monitored at the base station (BS). The BS then determines the subband gains for each MS and uses this knowledge to assign to each MS the subband that enjoys the highest gain,'while ensuring that no more than one MS is assigned to the same subband. It was shown that C A F H can provide a much lower B E R than conventional F H over a wide range of SNR values and number of MS's. This is achieved by exploiting information about the MS's subband gains. The perfor-mance difference between conventional and C A F H increases with SNR. For a fixed target B E R , the number of MS's which can be supported by C A F H is much larger than that with conventional F H . This work has also appeared in [83] and [84]. The performance of C A F H with r rounds over slowly time-varying, frequency selective channels is ana-lyzed. In [83], the analysis was limited to just one round. In this chapter, closed-form expressions that are recursively-based are derived to evaluate the B E R for C A F H with r rounds and its performance in a cellular communication system is studied. The B E R improvement achieved when using r > 1 is depicted. Numerical results show that the improvement obtained using r > 1 is considerable compared to the r = 1 case. Moreover, Chapter 8. Conclusions and Future Research 121 it is found that C A F H reaches saturation at larger r values; the dependence of the system performance on r is less significant for large r. This work has been reported in [85]. In chapter 6, the cdf (and hence outage probability) of the r-th order branch SNR in correlated Nakagami—m fading has been studied. The accuracy of a simple exchangeable approximation to reduce the computational load has been examined. Results from this work has been reported in [86] and [125]. Chapter 7 investigates the effects of correlation on the error performance of channel aware systems that involve the ordering of channel gains in correlative Nakagami fading environments. A useful analytical formula for the B E R of the r-th order statistic of a set of arbitrarily correlated and not necessarily exchangeable diversity branch gains is described and may be efficiently applied in the performance analysis of various diversity systems. In the case of coherent B P S K , a closed-form expression is extracted for the B E R when transmitting on the r-th order branch. This result is then employed for the B E R calculation of C A F H . Depending on the level of correlation, it can be observed from the results in Figs. 7.1, 7.2, and 7.3 that the performance over the r-th order branch is significantly degraded under correlated fading conditions when compared to the uncorrelated fading case, as a consequence, the average B E R performance of C A F H is noticeably affected. This work has been submitted for publication [126]. In conclusion, among the main contributions of this thesis, we proposed: • Matched C D M A and F H - M C - C D M A for wireless communications. • Channel aware frequency hopping (CAFH) as an effective access scheme for corre-lated and independently faded subbands. Chapter 8. Conclusions and Future Research 122 • B E R of order selection when the r-th order branch is selected for transmission. 8 . 2 F u t u r e W o r k We present below a list of research problems that can be investigated as possible exten-sions for research work reported in this thesis. 1. The use of multiple antennas to provide antenna diversity over a point-to-point wireless link has been studied extensively since the early 1960s. It is still a popu-lar research topic for improving energy efficiency through the use of both transmit and receive diversity. Recently, with the increasing demand for high-speed wireless data services, antenna diversity in a multiuser data network has received particular attention [127] [128], as it requires a broader multilink perspective for the spatial diversity gain in the system [79]. Recent progress in information theory [49] [129] has shown that in a data network with multiple active users requesting services simultaneously, multiuser diversity can be obtained by a packet scheduler, which always allocates the common radio resource to the user having the best channel quality, on the condition that the instantaneous user channel quality information (CQI) in terms of SNR is available to the scheduler. Multiple transmit antennas can be used either to obtain transmit diversity or to form M I M O channels. We may refer to this multiuser scheduling scheme as greedy scheduling. Adaptive loading (or adaptive modulation) technique is an efficient technique to achieve power and rate optimization based on the sub-channel gains. Adaptive bit allocation using the simple Greedy method gives an optimal solution for a single user system. Adaptive Chapter 8. Conclusions and Future Research 123 bit allocation using the Greedy method cannot give an optimal solution for mul-tiuser cases. It is probable that the sub-channels with the largest channel gain for one user also be the largest for another user. A practical channel-aware scheduling algorithm, proportional fair (PF) scheduling [79] [130], has been used in the down-link of cellular packet data system IS-856 [131] [also known as l x E V - D O or high data rate (HDR)], which trades some total system throughput for resource fairness among the active users. When users cannot share the same sub-channel at the same time, it is a difficult problem to find the best way to assign the sub-channels to the users. It would be interesting to study the improvement in average throughput due to multiuser diversity in M I M O cellular systems by assigning the transmit antennas to the users using the channel aware algorithm described in Chapter 5. 2. Multiuser diversity helps on two accounts. First, one can pick the user(s) that is (are) in a favorable channel condition, thereby increasing the network throughput for the forward link. Second, one can exploit the multiuser diversity to decrease feedback requirements in the reverse link. This novelty has been introduced in [79], where the transmitter pre-codes in a pseudo random fashion and that user is picked whose ratio of instantaneous data rate to its own average data rate is the largest. This idea can be extended to include users with different ratios. Hence, studying the effect of choosing users with different SNR levels on the network throughput. Using the concepts covered in Chaper 7, expressions for the B E R and throughput may be obtained. 3. High P A P R of the transmit signal is a major drawback of multicarrier transmis-Chapter 8. Conclusions and Future Research 124 sion such as O F D M , M C - C D M A and discrete multitone (DMT). It is possible to avoid high P A P R signals by employing a technique named clustered O F D M [132] [133] [134]. In this technique the subcarriers are clustered into several smaller blocks and transmitted over separate antennas. The P A P R is reduced since there are fewer subcarriers per transmitter. F H - M C - C D M A is another method that may be applied to P A P R reduction in multicarrier systems, as F H - M C - C D M A generally uses a smaller number of subcarriers. Since F H - M C - C D M A requires a single an-tenna for transmission, it may provide an effective method for reducing both cost and complexity when compared to clustered O F D M . 4. Multiple transmit and receive antennas can be used to improve the performance and increase the capacity of wireless communications systems. It is shown that when multiple transmit and receive antennas are used to form a M I M O system, the system capacity can be improved by a factor proportional to the minimum of the number of transmit and receive antennas compared to a single-input single- output (SISO)' system over a flat narrowband Rayleigh fading channel [135] [136]. However, for wideband channels O F D M has to be used with M I M O techniques for intersymbol interference mitigation and capacity improvement. This M I M O - O F D M system is investigated in [137] [138] [139]. An alternative to O F D M is F H - M C - C D M A that can be applied to M I M O technologies. In principle, F H - M C - C D M A and M I M O can be synergistically integrated to offer the benefits of both system simplicity and high performance. M I M O - F H - M C - C D M A could be effective in reducing ICI as well as in obtaining a diversity gain even for highly-correlated fast fading channels. Since Chapter 8. Conclusions and Future Research 125 M I M O - O F D M systems are based on O F D M , they also suffer from the problem of inherent P A P R . M I M O - F H - C D M A can reduce such drawbacks. To deal w i t h the increasing number of system users and fading correlation, a sub-carrier allocation scheme may be applied to M I M O systems w i t h channel state in-formation (CSI) at the transmitter or receiver. C A F H for M I M O channels ( M I M O -C A F H ) can be considered for wideband transmission to mitigate intersymbol in-terference and enhance system capacity. Studying the capacity of M I M O enabled C A F H system under a generic multiuser multicarrier framework may reduce I C I caused by high-speed mobiles i n cellular environments. M u l t i p l e transmit and re-ceive antennas can be used w i t h C A F H to further improve system performance. Using M I M O - C A F H , P A P R related problems can be dramatical ly reduced. 126 Bibliography [1] S. Hara and R. Prasad. 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