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Space-time codes and protocols for point-to-point and multi-hop wireless communications

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Space-time codes and protocols for point-to-point and multi-hop wireless communications

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Content SPACE-TIME CODES AND PROTOCOLS FOR POINT-TO-POINT AND MULTI-HOP WIRELESS COMMUNICATIONS by Madhavan S. Vajapeyam A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2007 Copyright 2007 Madhavan S. Vajapeyam Dedication To my family and friends. ii Acknowledgements I am deeply grateful to my advisor Professor Urbashi Mitra for her support and guidance throughout my Ph.D program at the University of Southern California (USC). I would also like to thank the members of my guidance and dissertation committees - Professor Charles Weber, Professor Bhaskar Krishnamachari, Profes- sor Giuseppe Caire and Professor Peter Baxendale - for their valuable suggestions and feedback. I have greatly bene¯tted from invaluable discussions and collaborations with several researchers, whom I thank profusely. Dr. Jifeng Geng provided fundamen- tal insights that ultimately resulted in Chapter 2 of this thesis; without collabora- tion with Professor Jim Preisig (Woods Hole Oceanic Institute), Professor Milica Stojanovic (MIT), Satish Vedantam (USC) and Dr. Ethem Sozer (MIT), the ¯eld experiments reported in Chapter 4 would not have been possible. I am specially thankful to Professor Keith Chugg. His encouragement was a main factor in my decision to join the Ph.D program, besides his many stimulating lectures which I was fortunate to attend. I would also like to thank the other CSI faculty for their advice, and, in particular, Professor Zhen Zhang. iii I would also like to express my most sincere appreciation and thanks to my friends in the Communication Sciences Institute (CSI): Dr. Stefan Franz, Dr. Ce- cillia Carbonelli, Satish Vedantam, Sridhar Ramanujam, Dr. Majid Nemati, Dr. Jordan Melzer, Terry Lewis, Dr. Wanshi Chen, Nicholas Richard, Dr. Wenyi Zhang, Dr. Greg Dubney, Dr. Jun Yang and Dr. Robert Wilson. Last, but not least, I thank Diane Demetras, Tim Boston and the CSI sta® - MillyMontenegro, MayumiTrasherandGerrielynRamos-fortheiradministrative help and, most importantly, their everlasting patience and friendship. I will miss them all. Madhavan Vajapeyam Los Angeles, January 2007 iv Table of Contents Dedication ii Acknowledgements iii List Of Tables viii List Of Figures ix Abstract xii Chapter 1: Introduction 1 Chapter 2: Performance Analysis and Design of Space-Time Block Codes 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 The Cherno® and Union Bounds. . . . . . . . . . . . . . . . . . . . 15 2.4 Indecomposable Union Bound . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Basic De¯nitions . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Decomposability in Quasi-Static Fading. . . . . . . . . . . . 21 2.4.3 Properties of Indecomposable Error Patterns . . . . . . . . . 24 2.5 The Progressive Union Bound and its Saddlepoint Approximation . 26 2.5.1 PUB Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.2 Saddlepoint Approximation . . . . . . . . . . . . . . . . . . 32 2.6 Code Construction and Isometries . . . . . . . . . . . . . . . . . . . 36 2.7 Code Search Results . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.7.1 Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.2 Hierarchical Design . . . . . . . . . . . . . . . . . . . . . . . 43 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Chapter 3: Cooperative Strategies for Wireless Multihop Communications: Di- versity Analysis and Power Allocation 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Signal Model and Multi-hop AAF (AAF-MH) Scheme . . . . . . . . 54 v 3.2.1 Cooperation Protocol . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Diversity Analysis in AAF-MH: Three-Hop Case . . . . . . . 61 3.2.2.1 Lower Bound on (3.22) . . . . . . . . . . . . . . . . 64 3.2.2.2 Upper Bound on (3.22) . . . . . . . . . . . . . . . 66 3.2.2.3 Main Diversity Result . . . . . . . . . . . . . . . . 68 3.3 Three-Hop Communication via RH and SME Codes . . . . . . . . . 71 3.3.1 RH and SME Cooperation Protocols . . . . . . . . . . . . . 71 3.3.2 Diversity Analysis of RH and SME Protocols . . . . . . . . . 79 3.4 Practical Issues: Asynchronous Relays . . . . . . . . . . . . . . . . 86 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6 Appendix A: Power Allocation . . . . . . . . . . . . . . . . . . . . . 91 3.7 Appendix B: Diversity Analysis in the AAF-MH Scheme: General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.7.1 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.7.2 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Chapter4: DistributedSpace-TimeCooperativeSchemesforUnderwaterAcous- tic Communications 98 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4 Coping with Imperfect Relay Synchronization . . . . . . . . . . . . 117 4.5 Increasing Spatial Diversity with More Relays . . . . . . . . . . . . 118 4.6 Underwater Channel Model . . . . . . . . . . . . . . . . . . . . . . 123 4.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.7.1 Computer Channel Simulations . . . . . . . . . . . . . . . . 125 4.7.1.1 Sensitivity to Imperfect CSI . . . . . . . . . . . . . 127 4.7.1.2 Sensitivity to Channel Time Variations . . . . . . . 131 4.7.2 Experimental Simulations . . . . . . . . . . . . . . . . . . . 132 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Chapter 5: Achievable Rates, Outage Probability and Throughput of Cooper- ative DSTBC Protocols 139 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.2 Received Signal Models for Cooperative Space-Time Protocols . . . 145 5.2.1 Single-Hop (Non-Cooperative) Communication . . . . . . . . 145 5.2.2 The Two-Hop AAF Scheme . . . . . . . . . . . . . . . . . . 145 5.2.3 The Two-hop Rate 1/2 Scheme . . . . . . . . . . . . . . . . 148 5.2.4 The Two-Hop Rate 3/5 Scheme . . . . . . . . . . . . . . . . 150 5.2.5 The Two-Hop Rate 1 D-QSTBC Scheme . . . . . . . . . . . 152 5.3 Outage Analysis of Cooperative Protocols . . . . . . . . . . . . . . 156 5.3.1 Single-Hop Communication . . . . . . . . . . . . . . . . . . 156 5.3.2 AAF Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 158 vi 5.3.2.1 Exact Expression for P (u) o (R) . . . . . . . . . . . . 159 5.3.2.2 Approximate Bounds on P (u) o (R) . . . . . . . . . . 159 5.3.2.3 Approximate Bounds on the OMI . . . . . . . . . . 161 5.3.2.4 Throughput . . . . . . . . . . . . . . . . . . . . . . 164 5.3.3 Rate 1/2, Rate 3/5 and D-QSTBC Protocols . . . . . . . . . 165 5.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.6 Appendix: Exponential-Product Distribution . . . . . . . . . . . . . 172 Chapter 6: Conclusions 175 References 177 vii List Of Tables 2.1 Optimal codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Distance Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 CB codes versus UB codes. . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Hierarchical codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 viii List Of Figures 2.1 UB, IUB and PUB for rate 1.5, 2£2, QPSK, spread system. . . . . 36 2.2 UB, IUB and PUB for rate 1.0, 3£3, BPSK, non-spread system. . 37 2.3 Rate 2, QPSK, 2£2, ½ = 1:0 union bound code versus worst case code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Rate 1.0, 3£3, BPSK, spread system ½=0:3, sensitivity to ½. . . . 44 2.5 Rate 4/3, 3£3, BPSK, ½=0:3, PUB code versus OMM/UB code. 45 2.6 Rate 5/3, 3£3, BPSK, ½=0:3, PUB code versus OMM/UB code. 46 2.7 Rate 5/2, 2£2, 8PSK, ½=0:3, PUB code versus OMM/UB code. . 47 3.1 Cooperative networks for two and three hops. . . . . . . . . . . . . . . 55 3.2 Exponential Integral function. . . . . . . . . . . . . . . . . . . . . . . 67 3.3 AAF-MH scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 RH PEP and diversity slope . . . . . . . . . . . . . . . . . . . . . . . 85 3.5 RH and SME schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.6 AAF-2H, AAF-MH and RH comparison in the presence of path-loss at- tenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.7 Two-hop performance with asynchronous relays . . . . . . . . . . . . . 91 4.1 Two-hop cooperative network with 2 relays. . . . . . . . . . . . . . . . 103 ix 4.2 Two-hop cooperative network with 4 relays . . . . . . . . . . . . . . . 119 4.3 Two-hop channel pro¯le: d = 3 km (range per hop), f c = 15 kHz, depth = 75m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 Two-hop cooperation performance . . . . . . . . . . . . . . . . . . . . 127 4.5 Cherno® Bound for two-hop cooperation . . . . . . . . . . . . . . . . . 128 4.6 Two-hop cooperation performance: sensitivity to channel estimation errors131 4.7 Two-hop cooperation performance: sensitivity to channel time variations 133 4.8 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.9 Source-relay average channel response . . . . . . . . . . . . . . . . . . 135 4.10 Relay-destination average channel response . . . . . . . . . . . . . . . 136 4.11 Source-destination average channel response . . . . . . . . . . . . . . . 137 4.12 Channel responses for asynchronous relays . . . . . . . . . . . . . . . . 137 4.13 Error rate performance of single hop and cooperative transmission . . . 138 5.1 2 hop cooperative network with 2 relays. . . . . . . . . . . . . . . . . . 146 5.2 Outage probability of the AAF protocol . . . . . . . . . . . . . . . . . 162 5.3 10% OMI of the AAF protocol . . . . . . . . . . . . . . . . . . . . . . 163 5.4 10% OMI of AAF and single-hop protocols . . . . . . . . . . . . . . . 164 5.5 AAF throughput, for total energy of 30 dB . . . . . . . . . . . . . . . 166 5.6 AAF throughput, for total energy of 10 dB . . . . . . . . . . . . . . . 167 5.7 Single-hop and AAF throughput, for total energy of 30 dB . . . . . . . 168 5.8 Single-hop and AAF throughput, for total energy of 10 dB . . . . . . . 169 5.9 10% OMI for di®erent cooperation protocols . . . . . . . . . . . . . . . 170 x 5.10 Achievable packet error rate for di®erent cooperation protocols . . . . . 171 5.11 Exponential-product PDF . . . . . . . . . . . . . . . . . . . . . . . . 174 xi Abstract Thepresenceofrandomfadinginwirelesschannelsconstitutesoneofthemostchal- lenging problems for achieving reliable digital communications. At the same time, fading also o®ers opportunities for improved performance via multi-dimensional communications, also known as multiple-input/multiple-output (MIMO) signaling. Given the ever increasing demand for high data rate in real-time applications in- volvingtransmissionofvoice,dataandmultimediacontent,moderncommunication networks need be properly designed in order to fully extract the potential gains of- fered by MIMO schemes. In point-to-point communications, the use of multiple antennas at the transmitter and/or receiver constitutes one of the standard ap- proaches to MIMO signaling, through the use of Space-Time Codes. Such codes enable dramatic performance improvement over single-antenna systems. More recently, techniques for multi-hop distributed communications have also attracted considerable interest. An application where multi-hopping is necessary, for example, are the so called ad-hoc and sensor networks and the 802.16j (Multi- hop Relay) standard currently under development. Theoretical results have shown that, besides the power savings achieved by communicating over shorter distances, xii distributedcommunicationcanalsoachieveasigni¯cantportionoftheMIMOgains via cooperation between terminals. This thesis discusses new approaches for performance analysis of Space-Time Block Codes (STBCs). For point-to-point communication, besides yielding sig- ni¯cantly tighter performance criteria in terms of the error-rate characterization, the new bounds are also used to design better codes. For multi-hop communi- cation, cooperative distributed protocols based on distributed STBCs (DSTBCs) are proposed and analyzed; upper and lower bounds on the performance are also developedforthisscenario, andthediversitygainischaracterizedforseveralmulti- hop communication strategies. The impact of asynchronous communication across the terminals on the performance is also investigated. Techniques for cooperative communicationinthepresenceofmultipathandintersymbolinterference(ISI)envi- ronments are also proposed and investigated. In particular, time reversal DSTBCs (TR-STBCs) are considered. Due to the orthogonal properties of the code, it is shown that maximum likelihood detection can be performed at the receiver with a standardscalarViterbi-typedetector. Experimentalresultsoftheproposedscheme are reported and con¯rm the potential gains predicted in the analysis. Finally, the achievable rates of several half-duplex cooperation strategies are investigated. Up- per and lower bounds on the achievable outage mutual information (OMI), outage probability and throughput are developed. Interestingly, it is shown that multi- hop communication is not always superior to single-hop communication from an xiii achievable rate perspective; while at low transmit energy multi-hop transmission is superior, at high energy it appears that the opposite is true. xiv Chapter 1 Introduction Space-time block codes (STBCs) have attracted considerable recent interest due to theirabilitytotakeadvantageofbothspaceandtimetoreliablytransmitdataover a wireless fading channel and/or increase data rates. This thesis addresses several aspects of point-to-point (multiple-antenna/single hop) and multi-hop communica- tions under the common framework of STBCs. In particular, the work generalizes the application of STBCs from the classical point-to-point communication under asymptotically high SNR to a distributed (multi-hop) setup with ¯nite SNR. In the ¯rst part of this thesis, new and general tools for performance analysis and design of STBCs are developed and analyzed. In the past, STBC design was based on asymptotically tight performance criteria such as the worst-case pairwise error probability (PEP) or the Union bound. However, these quantities fail to give an accurate performance picture, especially at low SNR, where the classical Union bound is known to be loose. This thesis develops tighter performance criteria for STBCs which yield considerably better bounds. In the ¯rst proposed bound, the 1 UnionboundisexpressedastheaverageoftheexactPEP's. BygeneralizingVerd's notion of decomposable errors to matrix modulations, it can be noted that some of the terms in the classical bound are redundant, and, hence, a new bound is obtained by expurgation, which denoted by Indecomposable Union Bound (IUB). Since the IUB in general can still be loose, a tighter bound derived directly from the exact expression for the block error probability is derived, and denoted by the Progressive Union Bound (PUB). Because the PUB cannot be computed in closed forminitsmostgeneralcase,andtoavoidcomputingahigh-dimensionalnumerical integration, its saddlepoint approximation is developed. In addition to the signi¯- cant improvement of the PUB analysis over previous bounding methods, it is also shown via examples that codes designed to optimize the PUB can perform better than those obtained by the looser criteria. Furthermore, besides allowing a more general theoretical understanding of the performance of STBCs, the new bounds greatlyfacilitatenumericalevaluationofcodesbydrasticallyreducingcomputation time compared to Monte-Carlo methods. The second part of this thesis considers the problem of distributed communi- cation over multi-hop networks. This scenario corresponds, for example, to the wireless metropolitan area networks (WMAN), in which multi-hopping results in improved coverage as well as power savings. To this end, the preliminary 802.16j standard for multi-hop relaying is currently being formulated. As shown in this thesis, multi-hop networks can also provide additional gains through cooperation 2 betweenterminalsbytakingadvantageoftheirinherentrichnessinspatialdiversity. Most of the prior work considered simple extensions of existing STBCs to a two- hop communication framework in asymptotically high SNR. This thesis proposes and analyzes novel multi-hop relaying protocols for more than two hops and, for each protocol, derives the achievable diversity in the ¯nite SNR case. It is shown that, although multi-hop (SNR) and (full) diversity gains are still achievable in distributed communications at the limit of in¯nite power, the multi-hop nature of the channels may incur a diversity performance loss in a practical (¯nite power) scenario, due to the Rayleigh-product distribution of the fading channels. On the other hand, this motivates the design of new distributed codes that are shown to mitigate these losses, and, hence, achieve improved performance. In addition, the e®ect of asynchronousrelayson the performance is also analyzed in this thesis, and it is shown that most of the cooperation gains are still maintained, hence yielding a robust distributed system. The third part of this thesis addresses the more general problem of distributed cooperative communication over fading channels with multipath. While most of the STBC designs are for °at fading channels, wireless channels can typically ex- hibit signi¯cant intersymbol interference (ISI), especially at high data rates and/or wideband signaling. In particular, the work's main focus is on underwater acoustic (UWA) channels, which are well known to present several challenges for reliable 3 communications: severe range-dependent attenuation, extensive multipath prop- agation and highly variable propagation delays (due to slow sound propagation). Nevertheless, it is important to emphasize that the techniques presented in this thesis are applicable to any channel with extensive multipath propagation (such as in WCDMA). For point-to-point communication, Lindskog & Paulraj (2000) pro- posed a time-reversal (TR) STBC for two antennas and showed that it achieves full spatial diversity. In this thesis, this approach is generalized to a distributed system with two or more relays cooperating over two hops. It is ¯rst shown that, just as in the dual-antenna STBC case, TR along with the orthogonality of the DSTBC essentially allows for decoupling of the vector ISI detection problem into separate scalar problems, and thus yields strong performance (compared with sin- gle hop communication) with substantially reduced complexity. Furthermore, low complexity decision-feedback equalizer (DFE) structures are developed to decode received symbols. These are shown, both theoretically and via simulations, to ex- hibit performance that is very close to the optimal Viterbi decoder. An interesting feature of the proposed TR-DSTBC scheme is that it is also inherently robust to asynchronous operation at the relays, just as the DSTBC schemes for °at fading discussed earlier. However TR-DSTBC does not require any additional complex- ity at the receiver, which is a signi¯cant practical advantage. Results of ¯eld test experiments performed over measured acoustic channels are reported to provide 4 furthervalidationoftheproposedapproachand, besidescon¯rmingpotentialgains predicted by our analysis, show promising results for future research. Most of the work in this thesis focusses on improving reliability (achieving high diversity) in point-to-point or distributed communications. On the other hand, recent information theoretic results for MIMO channels have shown that, in fact, a tradeo® between diversity and information rate exists. The ¯nal part of this thesis addresses the issue of achievable rates of two-hop cooperative systems. A critical practical assumption is that of half-duplex relaying, in which terminals do not transmit and receive simultaneously, and can be viewed as a rate "penalty" that is incurred for using cooperation. By applying the standard outage analysis of mutualinformation, expressionsarederivedinthisthesisforthep%outagemutual information (OMI) of several cooperative schemes employing orthogonal and non- orthogonalcodes. Asapracticalinterpretation, these arerates thatare guaranteed to be achievable for a given packet error rate of p% over a multi-hop network, and , hence, serve as a benchmark for throughput performance. Theremainderofthisthesisisorganizedasfollows: InChapter2severalbounds on the performance of STBCs are derived and, based on these new bounds, new codes are presented. In Chapter 3 multi-hop protocols for distributed communica- tion under °at fading are presented and analyzed. A diversity analysis is carried out and general results on the e®ect of the number of hops on the diversity are ob- tained. Thee®ectofasynchronousrelaysisalsoconsidered. InChapter4protocols 5 for distributed communications under multipath channels are developed and ana- lyzed. InChapter5theachievablerateandoutageprobabilityofseveraldistributed communication protocols is considered. 6 Chapter 2 Performance Analysis and Design of Space-Time Block Codes Space-time block codes (STBC) have attracted recent interest due to their ability to take advantage of both space and time diversity to reliably transmit data over a wireless fading channel. In many cases, their design is based on asymptotically tight performance criteria such as the worst-case pairwise error probability (PEP) or the union bound. However these quantities fail to give an accurate performance picture, especially at low SNR, because the union bound is known to be loose in thiscase. ThischapterdevelopstighterperformancecriteriaforSTBCswhichyield considerably better bounds. First, we develop the Union Bound as the average of the exact PEP's. By noting that some of the terms in the bound are redundant, a second bound is obtained by expurgation. Since this still yields a loose bound, a tighter bound, denoted by the Progressive Union Bound (PUB), is obtained. 7 Because it cannot be computed in closed form, and to avoid computing a high- dimensional numerical integration, its saddlepoint approximation is developed. In addition to the signi¯cant improvement of the PUB analysis over other bounding methods, it is also shown that codes designed to optimize the PUB can perform better than those obtained by the looser criteria. 2.1 Introduction Space-time block coding has attracted considerable attention recently as a tech- nique that employs diversity to mitigate the adverse e®ects of fading in wireless channels. This has been proven to yield dramatic increases in achievable data rates for a given quality of service requirement in wireless networks. Several STBC schemes have been proposed recently that enforce a certain structure on the code- words to e±ciently take advantage of diversity, such as orthogonal [TJC99] and unitary group [HM00, SHHS01] codes. On the other hand, unstructured designs, such as Optimum Minimum Metric (OMM [Gen00]) and Union Bound [GVM02] codes found by computer-search can o®er large performance gains in comparison to structured approaches. More recently, a hybrid scheme employing limited com- puter searches combined with a hierarchical codeset construction has been shown toenableconstructionofgoodhighratecodesinacomputationallyfeasiblemanner [GM06], yielding codes that o®er improved performance over previously proposed codes. 8 While STBC design has evolved considerably over the recent years, there is still alackofanalyticalresultsforaccurateperformanceassessmentofcodes. Duetothe non-existence of simple expressions for the block error rate of codesets in general, most of the previous work relied on the pairwise error probability (PEP) bound as simple performance criterion [TSC98, GFBK99, SD03]. The ¯rst proposed method for bounding the worst-case PEP of STBCs is the classicalCherno®boundapproach[TSC98,FGS99]. Morerecently,someworkshave proposed tighter upper bounds on the PEP applied to Space-Time Trellis coding schemes. The work in [BL02] presents an upper bound based on Craig's form for theGaussiantailfunction. Althoughtighterthantheboundsin[TSC98,FGS99],it requires a numerical integration to be computed. This is also the case in [HMD03], whereamatched¯lterboundisdevelopedforasysteminfrequencyselectivefading employing binary phase shift keying. The Cherno® bound analysis yields the well-known rank and determinant crite- ria for \optimal" code construction. As the code size increases however, the worst case error probability is not su±cient to characterize the full picture of perfor- mance [GVM02]. Furthermore, the Cherno® bound is loose at moderate SNR even for small codesets. Therefore, union bound (UB) approaches have recently been applied to the analysis of non-linear [GVM02] STBCs. On the other hand, the union bound for encoders employing a linear structure has been studied in [SP01], 9 where it was proved that orthogonal codes yield optimal performance among uni- tary codes by achieving the lowest bound . In [SG01], a union bound criterion for STBC over ISI channels is also considered. We note that the UB has also been investigated in the context of Space-Time trellis codes[BM01, SD03]. Although asymptotically tight, the UB is also quite loose at low SNR, due to the large number of overlapping of decision regions in the PEP computation. We, therefore, present a PEP-expurgation method resulting in a tighter union bounds- theIndecomposableUnionBound(IUB).Theachievableexpurgation,however,can di®er considerably among di®erent codesets. Inthischapter,wedevelop,throughauni¯edapproach,severalupperboundson the performance of STBCs. Rather than considering only the worst case PEP, our bounds take the entire distance spectrum of the codes into consideration, resulting in improved performance assessment. Under this framework, the Union Bound and Indecomposable Union Bound [GVM03] are derived. Moreover, we propose a new,inherentlytighterperformancebound,whichisdevelopedfromtheprogressive union bound (PUB) concept. First presented in [KF97a], the PUB was used to analyze the performance of non-spread transmitter diversity schemes employing intentional frequency o®set and its computation was performed via a numerical integration. Our work applies the PUB to the analysis of spread and non-spread STBC-based systems. We adopt a di®erent approach to the PUB computation, by deriving a saddlepoint approximation of this quantity. This method has the 10 advantageofbeingcomputationallymoreattractivethanthenumericalintegration, while also allowing for a semi-analytical expression to be derived for the PUB. Furthermore, it can be applied to code design by searching for codes that minimize this PUB approximation. Finally, the PUB allows a tradeo® between accuracy and numerical complexity by varying the parameters of the PUB computation . This chapter is organized as follows. In Section 2.2, the STBC system model is presented. Section 2.3 develops the Union Bound on the PEP's of STBCs. The IUBispresentedinSection2.4andsomeofitspropertiesarediscussed. Section2.5 develops the PUB and its Saddlepoint approximation and Section 2.6 presents a brief review of a simple code construction technique. We conclude by applying the analysis criteria discussed here to code design by providing tables of new found codes along with performance comparisons in Section 2.7. Concluding remarks are presented in Section 5.5. 2.2 System Model We consider a general single user system model encompassing both spread and non-spread systems. The terminology spread and non-spread refer to the use or absence, respectively, of possibly distinct spreading codes at each of the transmit antennae. Thus, this very general model is easily extensible to multi-user spread spectrum systems while also having utility for single user systems without signal spreading. We observe that for systems where the transmission bandwidth exceeds 11 the coherence bandwidth of the channel, as often experienced by spread spectrum systems, the appropriate channel model is that of a multipath channel. However, in the current work, for both spread and non-spread systems, we shall focus on channels with a single (dominant) °at fading path component at the receiver. This assumptionismotivatedbyadesiretokeepthenotationsimple(allofourmethods areeasilyextensibletomultipathchannels)andbytheobservationthatcodesopti- mized for °at fading channelsalso provide good performance in multipathchannels [GMF01, Gen00]. Spreading allows for additional signal separation and therefore improved performance. Thetransmitter,equippedwithL t antennae,mapsavector[I(1);I(2);:::;I(k c )] of k c information bits to one of K c = 2 kc space-time codewords, [d i (t)], 1· i· L t and 1 · t · N c . The block length, in terms of bit duration, is N c , resulting in a transmission matrix D of size N c £L t and code rate of R c = kc N c . Each element d i (t)ofD isspreadbyacorrespondingspreadingcodes i (t)andtransmittedviathe corresponding transmit antenna TX i . The receiver is equipped with L r antennae. Note that each row of the codeword matrix is transmitted simultaneously. In this chapter,eachd i (t)isconstrainedtoPSKconstellations,butingeneralcanbetaken from any point on the complex plane. The spreading waveform s i (t) for antenna i is sampled at the chip rate 1 Tc to form a column vector of length L u , denoted by s i (n). Di®erent spreading codes can be used at each antenna. 12 Assuming synchronous transmission, the received signal (output of a matched ¯lter at the receiver) at time n and antenna i can be written as y i (n)= p ¾ t R(n)D(n)h i (n)+m i (n) 2C L t £1 ; (2.1) where ¾ t is signal-to-noise ratio (SNR) normalized by L t (i.e., ¾ t = SNR=L t ), to keep the total transmit power constant and S(n)=[s 1 (n);:::;s L t (n)]2R L u £L t (2.2) R(n)=S(n) T S(n)2R L t £L t (2.3) D(n)=diag(d 1 (n);:::;d L t (n))2C Lt£Lt (2.4) h i (n)=[h i1 (n);:::;h iLt (n)] T 2C Lt£1 (2.5) where R(n) is the spreading code correlation matrix at time n (for non-spread sys- tems, R(n) assumes the form of an all 1's matrix);h i (n)»CN(0;I Lt ) 1 is the chan- nelcoe±cientvectorattimenforthei-threceiveantenna;m i (n)»CN(0;R(n))is the received complex Gaussian noise vector at time n for the i-th receive antenna. Note that D(n) is obtained by diagonalization of the n-th row of codewordD. 1 We use CN(m;K) to denote a circularly symmetric complex Gaussian random vector with mean m and variance matrix K. 13 We assume a quasi-static fading channel, h i (1) =¢¢¢ =h i (N c ). Concatenating y i (n);n=1;:::;N c into a super-vector y i we get y i =[y i (1) T ;:::;y i (N c ) T ] T = p ¾ t 2 6 6 6 6 6 6 4 R(1) 0 0 0 . . . 0 0 0 R(N c ) 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 D(1) . . . D(N c ) 3 7 7 7 7 7 7 5 h i (1)+ 2 6 6 6 6 6 6 4 m i (1) . . . m i (N c ) 3 7 7 7 7 7 7 5 (2.6) , p ¾ t RDh i +m i 2C LtNc£1 : (2.7) where R 2R L t N c £L t N c is the spreading code correlation matrix, D 2C L t N c £L t is the transmitted codeword matrix andh i ,h i (1)2C Lt£1 is the channel coe±cient vector for receive antenna i. We now vertically concatenate the vectors y i , i = 1:::L r , to form the vector ¹ y 2 C LtNcLr£1 . Similarly, we also concatenate vectors h i and m i , obtaining ¹ h 2 C L t L r £1 and ¹ m2C L t N c L r £1 respectively. The resulting signal model becomes ¹ y =[y T i ;:::;y T i ] T = p ¾ t 2 6 6 6 6 6 6 4 R 0 0 0 . . . 0 0 0 R 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 D 0 0 0 . . . 0 0 0 D 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 h i . . . h i 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 m i . . . m i 3 7 7 7 7 7 7 5 (2.8) , p ¾ t ¹ R ¹ D ¹ h+ ¹ m 2C LtNcLr£1 : (2.9) 14 where ¹ R2C L t N c L r £L t N c L r and ¹ D2C L t N c L r £L t L r . In the following sections, we develop expressions for performance analysis of STBCs considering the general signal model developed in this section. First, we introduce the classical Cherno® and union bound measures. Subsequently we will show that the Union Bound can be tightened using the notion of indecomposable error patterns, yielding a third criterion - the Indecomposable Union Bound. All these three bounds are asymptotically tight as the SNR increases, but are loose at low SNR. With this motivation in mind, we propose a fourth performance criterion - the Progressive Union Bound - which better predicts code performance at low SNR. 2.3 The Cherno® and Union Bounds We de¯ne the (normalized) di®erence between any pair of transmitted codewords as an error pattern. Thus, the set of error patterns that a®ect the k-th codeword is: E k ,fe kj je kj =( ¹ D k ¡ ¹ D j )=2;8j6=kg; (2.10) For a synchronous system, at high SNR, the average probability of decoding D ¯ whenD ® istransmittedisupperboundedbytheCherno®bound[GFBK99,TSC98] E h [P(D ® !D ¯ jh)]· ³ ¾ t 4 ´ ¡LtLr 1 j©(®;¯;¯)j ; (2.11) 15 which is an asymptotically tight bound. The generalized correlated codeword dif- ference matrix, ©(®;¯;°), is de¯ned as ©(®;¯;°),4(e ®¯ ) H ¹ R(e ®° )2C L t L r £L t L r (2.12) Ifweassume¯xedspreadingcodesareusedwithinoneblockR(1)=¢¢¢ =R(N c ), R, and de¯ne 2 ª(®;¯;°)=(D ® ¡D ¯ ) H (D ® ¡D ° )¯R2C L t £L t ; (2.13) then ©(®;¯;°), can be rewritten as ©(®;¯;°)=diag[ª(®;¯;°);:::;ª(®;¯;°)]2C LtLr£LtLr : (2.14) In contrast to the simple bound given in (2.11), the average pairwise error probability(PEP)canbecalculatedexactlyinclosedformbyE h [P(D ® !D ¯ jh)]= µ(®;¯), where µ(®;¯) is a function of the eigenvalues of ¾t 4 ©(®;¯;¯). For example, in the case of M distinct eigenvalues (see e.g. [Sch96, Ver98]): µ(®;¯)=E h [P(D ® !D ¯ jh)]= 1 2 M X i=1 Ã Y j=1;j6=i ¸ i ¸ i ¡¸ j !Ã 1¡ 1 p 1+2=¸ i ! : (2.15) 2 ¯ denotes Schur product, i.e., element-wise multiplication. 16 We obtain the union bound performance index as UB´ Kc X k=1 Kc X j=1 µ(k;j)= Kc X k=1 X e kj 2E k µ(k;j): (2.16) At high SNR, the Cherno®-based union bound on the symbol error rate (SER) can be used instead SER·UB·CB, 1 K c K c X i=1 K c X j=1;j6=i ³ ¾ t 4 ´ ¡L t 1 j©(i;j;j)j (2.17) Assuming all codeword pairs achieve full diversity, the following quantity is a per- formance index that is equivalent to (2.17), but independent of the SNR CB´ K c X i=1 K c X j=i+1 1 j©(i;j;j)j : (2.18) Note that the equivalence follows from the fact that, for any ¾ t , (2.17) is simply a scaled version of (2.18). Since the union bound is known to be loose at low SNR, a more accurate performance index is desired. We next develop a method for obtaining of a tighter union bound considering the fact that some of the PEP terms in (2.16) can be redundant and thus can be discarded. 17 2.4 Indecomposable Union Bound In the ¯rst part of this section, we introduce the main de¯nitions and extend the notion of error pattern decomposability presented in [Ver86] for additive white Gaussian noise channels to space-time block codes. In the second part, these def- initions are further extended to °at-fading channels and a new bound based on these patterns is developed, which is inherently tighter than (2.16). Finally, some properties of indecomposable patterns under °at-fading channels are presented. 2.4.1 Basic De¯nitions LetC =fD 1 ;D 2 ;:::;D Kc gbethe setofcodewords. Eachrowoftheelementsofthis setisdiagonalized,toformthesetoftransmittedcodewordsC 0 =fD 1 ;D 2 ;:::;D Kc g. We now de¯ne the weighted inner-product between two error patterns e km and e kn of the k-th codeword as S(k;m;n), ¹ h H ©(k;m;n) ¹ h (2.19) The squared norm of an error pattern e kl is thus given by kS(k;l;l)k 2 = ¹ h H ©(k;l;l) ¹ h (2.20) Generalizing the de¯nitions in [Ver86] and [ZB94], we have the following de¯nition: 18 De¯nition 1 For a given ¹ h, an error pattern e kl 2 E k is decomposable into pat- terns e km 2E k and e kn 2E k (denoted by e kl : =e km +e kn ) if: i) e kl =e km +e kn ii) [e kl ] ij =0,[e km ] ij =[e kn ] ij =0 iii) <hS(k;m;n)i¸ 0 where<hi denotes the real part. We also de¯ne w(e), X i X j I([e] ij 6=0) (2.21) where I(¢) is the indicator function, taking the value 1 when its argument is true, and0otherwise. Notethatife kl : =e km +e kn holdsthenanimmediateconsequence of condition (ii) is that w(e km )<w(e kl ) and w(e kn )<w(e kl ). We denote the set of indecomposable patterns in E k by F k ( ¹ h). Thus, F k ( ¹ h),f indecomposable patterns under channel ¹ hgµE k (2.22) The de¯nition of error patterns presented here di®ers from that given in [Ver98] and [ZB94] (in the context of multiuser detection) in the sense that all symbols in a codeword are taken into consideration when de¯ning allowable error patterns, instead of only the symbol of the user of interest. With the above de¯nitions, we 19 can develop an expression for the union bound on pairwise error probability of a codeword error, for a given channel realization. Following a procedure similar to that of[Ver98], the union bound of the PEP for codeword k conditioned on the channel is P k j ¹ h· X e kl 2E k Q ¡p 2¾ t kS(k;l;l)k ¢ (2.23) Furthermore, this bound can be tightened by expurgating the decomposable error patterns, resulting in P k j ¹ h· X e kl 2F k ( ¹ h) Q ¡p 2¾ t kS(k;l;l)k ¢ (2.24) Note that in this case we sum over the smaller set of indecomposable patterns only. We now average (2.24) over the channel statistics, resulting in P k ·E¹ h 8 < : X e kl 2F k ( ¹ h) Q ¡p 2¾ t kS(k;l;l)k ¢ 9 = ; (2.25) As observed in [ZB94], the averaging in the right-hand side of (2.25) is intractable due to the fact that each set F k ( ¹ h) depends on the particular channel realization ¹ h. Thus, our next step is to develop a channel-independent criterion for decompos- ability, which allows us to interchange the expectation and summation and obtain a closed form upper bound expression. 20 2.4.2 Decomposability in Quasi-Static Fading In order to obtain a channel-independent criterion for decomposability, we place a stricter de¯nition of decomposable sets by modifying condition (iii) for decompos- ability to: <hS(k;m;n)i=<h ¹ h H ©(k;m;n) ¹ hi¸ 0;8 ¹ h (2.26) Notethatthisde¯nitionis,ingeneral,lessstrictthantheorthogonaldecomposability condition proposed in [ZB94] for multiuser systems in °at fading channels, which in this case would be, <hS(k;m;n)i=0;8 ¹ h (2.27) Atthispointwerecallthefact(see[HJ96])thatanymatrixwithcomplexentries A can be written uniquely as A = A S + jA T , where A S and A T are Hermitian matrices and given by A S = (A+A H )=2 and A T =¡j(A¡A H )=2. Thus, we can write ©(k;m;n)=© S (k;m;n)+j© T (k;m;n); (2.28) where © S (k;m;n) and © T (k;m;n) are Hermitian. Substituting (2.28) in (2.26) we obtain a decomposability condition for single path fading as © S (k;m;n)¸0 (2.29) 21 which, in other words, states that © S (k;m;n) has to be positive semi-de¯nite. This condition is clearly weaker than the orthogonal condition, which requires © S (k;m;n) = 0, thus allowing a larger number of patterns to be treated as de- composable. However, in the particular case of quasi-static fading, we can apply (2.13) and write ª(k;m;n) as ª(k;m;n)=¢D H km ¢D kn ¯R (2.30) where we denote ¢D ®¯ =D ® ¡D ¯ . From (2.14) and condition (ii), it immediately follows that tr[©(k;m;n)]=tr[© S (k;m;n)]=0, and (2.26) becomes © S (k;m;n)=0 (2.31) Thus for the quasi-static case, (2.26) and (2.27) are equivalent. We are now ready to state channel-independent conditions for decomposability of error patterns. De¯nition 2 An error pattern e kl 2 E k is decomposable into patterns e km 2 E k and e kn 2E k (denoted by e kl : =e km +e kn ) if: i*) e kl =e km +e kn ii*) [e kl ] ij =0,[e km ] ij =[e kn ] ij =0 iii*) © S (k;m;n))¸0 22 Note that (i*) and (ii*) are exactly the same as (i) and (ii) respectively, but (iii) has been replaced by the stronger condition (iii*). The set of channel-independent indecomposable error patterns in E k is denoted by F ¤ k . Clearly, F k ( ¹ h)µ F ¤ k µ E k . By summing over F ¤ k instead of F k ( ¹ h), we can upper-bound (2.25) in closed form P k ·E¹ h 8 < : X e kl 2F ¤ k Q ¡p 2¾ t kS(k;l;l)k ¢ 9 = ; = Kc X k=1 X e kj 2F ¤ k µ(k;j) (2.32) Finally, by averaging (2.32) over all possible transmitted codewords, we obtain a third performance bound IUB´ Kc X k=1 X e k;j 2F ¤ k µ(k;j) (2.33) Note that, by construction, IUB · UB. Thus the SER an and the performance bounds can be ordered as SER·IUB·UB·CB (2.34) We now turn our attention to some simple properties of indecomposable patterns as de¯ned by (i*-iii*) above. These properties have practical importance mainly in reducing the amount of computations necessary to ¯nd indecomposable error patterns or compute the union bound in (2.33). 23 2.4.3 Properties of Indecomposable Error Patterns Property 1: For any k and l, an error pattern e kl with w(e kl ) = 1 is always indecomposable. Property 2: For any k and l, if an error pattern e kl = 2 F ¤ k , then e lk = 2 F ¤ l . Consequently, if e kl 2F ¤ k , then e lk 2F ¤ l . Theimmediatepracticalconsequenceofthispropertyisthatthecomputational cost of searching for indecomposable sets can be cut by half, since only half of the possible error patterns needs to be tested for conditions (i*-iii*). Proof: First we note that, for any choice of codewords x;y;z, it is always true that e xy =¡e zx +e zy (2.35) It is clear that e lk 2 E l , by de¯nition. If e kl = 2 F ¤ k , then e kl : = e km +e kn for some m and n. Now, to prove the property, we need to show that e lk : = e l® +e l¯ for some ® and ¯. Note that the ordering of the subscripts is crucial. Observing that e lk = ¡e kl , we have that e lk = ¡e km ¡e kn , however, this is not a valid decomposition for e lk in the sense of (i*). Thus, ¡e km =¡e kl +e kn : (2.36) We now observe that the right side of (2.36) is of the same form as the right side of the general form given in (2.35). Thus, comparing both expressions, it is clear 24 that¡e km =¡e kl +e kn =e ln . Similarly,¡e kn =¡e kl +e km =e lm . Thus, we have that e lk =e ln +e lm which is condition (i*) for decomposability of e lk . Conditions (ii*) and (iii*) are also satis¯ed, since e ln = ¡e km and e lm = ¡e kn . This proves that e kl = 2F ¤ k !e lk = 2F ¤ l and consequently, that e kl 2F ¤ k !e lk 2F ¤ l . Property 3: F ¤ k is invariant to channel or spreading codes for any k if the following three conditions are met: (1) Fixed spreading codes are used within each block, ie, R(n)=R 8n; (2) the channel is quasi-static; (3) [R] ij 6=0 8i;j. The property allows reduction of computations in performance analysis of code sets. Once the indecomposable patterns of a code set are found, (2.33) can be used to compute the union bound for di®erent R, without the need for recomputing the indecomposable sets F ¤ k , k =1:::K c . Proof: Since (i*) and (ii*) are independent of R, we only need to check (iii*). If e kl : =e km +e km , we have, from (iii*) and (2.30), that ª(k;m;n)=¢D H km ¢D kn ¯R =0: (2.37) From (2.37), if R does not contain zero elements, the decomposability criterion becomes [¢D H km ¢D kn + ¢D H kn ¢D km ] = 0, which is independent of R, and thus proves the property. 25 2.5 TheProgressiveUnionBoundanditsSaddlepoint Approximation Asnotedearlier,theperformanceboundsdevelopedsofarareasymptoticallytight, but fail to give an accurate performance prediction under low SNR scenarios. This is true even for the IUB in general, since the amount of possible expurgation in the Union Bound expression varies highly between di®erent codesets. In this sec- tion, we ¯rst address this issue by developing a generalization of the Union Bound, denoted by Progressive Union Bound (PUB)[KF97b]. As will be shown, its major advantage over the other bounds is the facilitation of a tradeo® between its com- putational complexity and tightness. Its drawback, however, is that it cannot be computed in closed form, except for a very special case. Therefore, we obtain a semi-analytic expression for the PUB by developing its saddlepoint approximation [Nut00, Nut01]. 2.5.1 PUB Derivation Denoting by ¹ y i the matched ¯lter received signal corresponding to codeword D i being transmitted, we recall that ¹ y i = p ¾ t ¹ R ¹ D i ¹ h+ ¹ m (2.38) 26 Therefore, the e®ective log-likelihood of codeword D j if D i is transmitted, denoted by T jji , is given by T jji =¡(¹ y i ¡¾ t ¹ R ¹ D j ¹ h) H ¹ R ¡1 (¹ y i ¡¾ t ¹ R ¹ D j ¹ h) (2.39) Usingatechniquesimilarto[KF97b], wenowdetermineanexactexpressionfor the performance of a STBC employing maximum-likelihood detection. First, we de¯ne the event M j;kji ,fD j more likely than D k when D i is transmittedg=fT jji >T kji g (2.40) where the right hand side of (2.40) follows from (2.39). Now, from (2.40), the probability of detecting D j when D i is transmitted cor- responds to PfD i !D j g=P ( \ k6=j M j;kji ) =P ½ T jji ¸max k6=j [T kji ] ¾ (2.41) We denote the codeword di®erence matrix ¢ ij by ¢ ij = ¹ D i ¡ ¹ D j . Substituting (2.38) and (2.39) into (2.40), and performing some simpli¯cations, we can show that M j;kji =fw H Q (i) j;k w·0g,fz (i) j;k ·0g (2.42) 27 where z (i) j;k ,w H Q (i) j;k w (2.43) w =[ ¹ h T ; ¹ m T ]2C L t L r (N c +1)£1 (2.44) w»CN(0;K w ) (2.45) K w = 2 6 6 4 I 0 0 ¹ R 3 7 7 5 2R L t L r (N c +1)£L t (N c +1) (2.46) Q (i) j;k = 2 6 6 4 ¾ t ©(j;i;i)¡¾ t ©(k;i;i) p ¾ t (¢ H ji ¡¢ H ki ) p ¾ t (¢ ji ¡¢ ki ) 0 3 7 7 5 (2.47) where Q (i) j;k 2C L t L r (N c +1)£L t (N c +1) . Thus, z (i) j , [z (i) j;1 ;z (i) j;2 :::z (i) j;j¡1 ;z (i) j;j+1 :::z (i) j;K c ] T is a (K c ¡1)£1 vector of complex Gaussian quadratic forms consisting of a su±cient set of metrics for the exact determination of PfD i !D j g. Its pdf is given by f(z (i) j )= 1 (2¼j) K c ¡1 Z C ¹ z (¸ j )exp[¡(¸ j ) T z (i) j ]d¸ j (2.48) where ¸ j = [¸ 1 :::¸ j¡1 ;¸ j+1 :::¸ Kc ] and ¹ z (¸ j ) is the moment generating function of z (i) j and is given by [Nut00] ¹ z (¸ j )= 1 det[I¡ P K c m=1;m6=j ¸ m C m ] , 1 det[I¡D(¸ j )] , 1 detP(¸ j ) (2.49) 28 where C m , SQ (i) j;m S H and S H S = K w . We also de¯ne D(¸ j ), P K c m=1;m6=j ¸ m C m and P(¸ j ),I¡D(¸ j ). The ROA (Region of Analyticity) of ¹ z (¸ j ) is given by ROA(¹ z )=f¸ j jlargest eigenvalue of D(¸ j ) is <1g: (2.50) Using (2.42), PfD i !D j g can be expressed as PfD i !D j g=Pfz (i) j ·0 (Kc¡1)£1 g: (2.51) We compute the probability of this error event by integrating the pdf in (2.48): PfD i !D j g= Z 0 ¡1 ::: Z 0 ¡1 f(z (i) j )dz (i) j ; (2.52) where (K c ¡1) integrals have to be performed. By substituting (2.48) and (2.49) into (2.52) and switching the integration order we achieve, PfD i !D j g= 1 (2¼j) Kc¡1 Z C ¹ z (¸ j ) Q K c m=1;m6=j (¡¸ m ) d¸ j (2.53) with integration contour chosen so that C :f<(¸ m )<0 ;8mg\ROA(¹ z ): (2.54) 29 The evaluation of the event probability in (2.41) given by the integral in (2.53) is analytically unsolvable due to the number of necessary metric comparisons. If however, only one comparison is performed, the error probability can be bounded via an exact expression. This bound is obtained from (2.41) as: PfD i !D j g=P ( \ k6=j M j;kji ) ·P 1 fD i !D j g,P © M j;iji ª =P © T jji ¸T iji ª (2.55) where the subscript in P 1 fD i ! D j g indicates that only one metric comparison is performed. Not surprisingly, this yields the expression for the pairwise error probability given in (2.15). While (2.41) gives the exact error probability, (2.55) considers only the event M j;kji , for k = i and is a common method for bounding thisprobabilityoferror. Theclassicalunionboundforthesymbolerrorprobability given in (2.16) can thus be rewritten as SER·UB = 1 K c K c X i=1 K c X j=1;j6=i P 1 fD i !D j g (2.56) while the exact expression is SER = 1 K c K c X i=1 K c X j=1;j6=i PfD i !D j g (2.57) 30 A bound that is tighter than (2.56) can be obtained by performing more metric comparisons in (2.55) instead of just one. To achieve a compromise between com- putational complexity and tightness, we consider M comparisons (M > 1) instead of all as in (2.57) or only one as in (2.56). For instance, if M =2, we modify (2.55) as PfD i !D j g·P 2 fD i !D j g,P © M j;iji \M j;ijl ª (2.58) which performs two comparisons since a third codeword D l is taken into consid- eration. Clearly, the choice of D l impacts the tightness of the resulting bound (although P 2 · P 1 always). Therefore, we use the PEP expression in (2.15) to select the codeword D l (l6= j) which is most likely to have the highest impact on the bound. This criterion can be stated as l =argmax ® µ(i;®) (2.59) Denoting by P M fD i !D j g the resulting error probability bound for M metric comparisons, we can write the PUB as SER·PUB(M), 1 K c K c X i=1 K c X j=1;j6=i P M fD i !D j g (2.60) Although(2.57)and(2.60)giveus,respectively,theexactsymbolerrorprobabil- ity and its progressive union bound, they cannot be computed in closed form since PfD i ! D j g and P M fD i ! D j g cannot be analytically computed. In [KF97b], a 31 multidimensionalnumericalintegrationwasproposedtocomputeboundsofsimilar form. Computationally, however, this is a highly non-trivial task especially when the codesets and/or block sizes become larger. In this chapter, we employ the sad- dlepoint (SP) technique presented in [Nut00, Nut01] to obtain a semi closed-form expression whichcanapproximate(2.60) closely evenat lowSNR values, and with- out the need to perform a numerical integration. The next section describes this technique. 2.5.2 Saddlepoint Approximation For notational simplicity, we derive the SP approximation for the exact expression in (2.53). Its extension to the progressive error probability is straightforward. Our approach for obtaining the SP approximation extends the method described in [Nut00, Nut01]. The SP approximation consists of ¯rst determining a real SP for the integrand expression in (2.53), then, a Taylor series expansion of the integrand is carried out around the SP. By truncating this expansion at the second order term, the integra- tion can be performed analytically and a closed form approximation is obtained. We start by rewriting (2.53) as P(D i !D j )= 1 (2¼j) K c ¡1 Z C exp[¤(¸ j )]d¸ j (2.61) 32 where ¤(¸ j )=¡log(detP(¸ j ))¡ K c X m=1;m6=j log(¡¸ m ) (2.62) The real SP ^ ¸ has to satisfy the simultaneous equations: · @¤(¸ j ) @¸ m ¸ ^ ¸ =0; for m=1:::K c ; m6=j: (2.63) Therefore, eachintegrationcontourinC istakentopassthroughthisrealSPwhich mustlieintheregionspeci¯edby(2.54). Amultidimensionalsearchtechnique,such as Newton-Raphson search, may be used to ¯nd the SP, ^ ¸. Once it is found, we expand ¤(¸ j ) around the SP ^ ¸: ¤(¸ j )=¤( ^ ¸)+ Kc X m=1;m6=j · @¤(¸ j ) @¸ m ¸ ^ ¸ (¸ m ¡ ^ ¸ m )+ Kc X m=1;m6=j Kc X n=1;n6=j ¤ 2 (m;n)(¸ m ¡ ^ ¸ m ) 2 2 +::: (2.64) where ¤ 2 (m;n)= · @ 2 ¤(¸ j ) @¸ m @¸ n ¸ ^ ¸ ; m;n=1:::K c and m;n6=j: (2.65) The (K c ¡1)£(K c ¡1) Hessian matrix ¤ 2 at ^ ¸ is ¤ 2 ,[¤ 2 (m;n)] m;n=1:::K c and m;n6=j (2.66) 33 By truncating (2.64) at second order, and substituting (2.63) in (2.61) we have: P(D i !D j )¼ 1 (2¼) K c ¡1 £ Z exp " ¤( ^ ¸)+ Kc X m;n=1;m;n6=j ¤ 2 (m;n)(¸ m ¡ ^ ¸ m ) 2 2 # d¸ j (2.67) By performing the change of variables: ¸ m = ^ ¸ m +js m m=1:::K c and m6=j (2.68) and de¯ning s j =[s 1 :::s j¡1 ;s j+1 :::s K c ] we have P(D i !D j )¼ exp[¤( ^ ¸)] (2¼) K c ¡1 £ Z +1 ¡1 exp " ¡ 1 2 X m;n ¤ 2 (m;n)s m s n # d^ s (2.69) which ¯nally gives P(D i !D j ) (0) ¼ exp[¤( ^ ¸)] (2¼) (Kc¡1)=2 [det(¤ 2 )] 1=2 (2.70) which is known as the 0-th order approximation to the integral in (2.53). Since ¤ 2 is always positive de¯nite in our problem [Nut00, Nut01], the approximation is guaranteed to always yield a positive value. A ¯rst order approximation is obtained by PfD i !D j g (1) =e c t P(D i !D j ) (0) = exp[c t ¤( ^ ¸)] (2¼) (K c ¡1)=2 [det(¤ 2 )] 1=2 (2.71) 34 where c t is a correction term, which is a function of third and fourth order partial derivativesof¤(¸ j )andneedonlybecomputedaftertherealSPisfound. Detailed expressions for c t can be found in [Nut00]. The ¯nal step is to write the SER and PUB approximations SER¼ 1 K c Kc X i=1 Kc X j=1;j6=i PfD i !D j g (1) (2.72) and PUB(M)¼ 1 K c Kc X i=1 Kc X j=1;j6=i P M fD i !D j g (1) : (2.73) Figure 2.1 compares the PUB approximations with the exact UB for a 2£2 QPSK STBC with eight codewords, (see [GVM02]). The UB and IUB for the code are also shown for comparison. Note that the PUB approximation is signi¯cantly more accurate than the IUB which in turn is tighter than the UB, as expected. Since it takes more decision metrics into consideration, the PUB clearly gives a much better prediction for the code performance. Furthermore, the choice of M yields a tradeo® between approximation accuracy and numerical complexity. Figure 2.2 displays similar approximations for a 3£3 BPSK code. We note that in this case, the IUB curve turns out to be exactly the same as the UB curve (due to the fact that the code contains no decomposable patterns). 35 0 1 2 3 4 5 6 7 8 9 10 10 −2 10 −1 10 0 Rate 1.5, 2x2 QPSK ρ=0.3 SER SNR PUB, M=2 PUB, M=3 Simulation UB IUB Figure 2.1: UB, IUB and PUB for rate 1.5, 2£2, QPSK, spread system. 2.6 Code Construction and Isometries In the previous sections, we presented several techniques (Cherno®, Union, Inde- composable and Progressive Union Bounds) for use in evaluating the performance of a space-time block coded system. With this performance criteria in hand, we can, in principle, search for code sets which optimize these quantities. However,duetothecomplicatednatureofthecodewordspace(non-linear,non- metric), constructing codeword sets that optimize any of the performance criteria described in the previous sections is a di±cult task. This is especially the case for thePUBcriterion, sinceitinvolvessearchingnumericallyforasaddlepointforeach pairwise error in the codeset. Thus, as these sets become large (for higher rates), 36 0 1 2 3 4 5 6 7 8 9 10 10 −3 10 −2 10 −1 10 0 Rate 1.0, 3x3 BPSK ρ=1.0 SER SNR Simulation PUB (M=1)≈ UB =IUB PUB (M=2) Figure 2.2: UB, IUB and PUB for rate 1.0, 3£3, BPSK, non-spread system. not only does the search space increase, but the PUB computation for each set becomes more challenging. Recently,in[GM06],acodeconstructionmethodforspace-timeblockcodeswas proposed that takes advantage of certain \distance preserving" transformations in order to hierarchically build higher rate codes from smaller sets. These transfor- mations are called isometries. An isometry is an operation Á over the codewords of a setC which results in a new set C ¤ such that the \distance" (e.g. rank criterion or coding gain) between codewords is preserved. Speci¯cally, we denote C ¤ =Á(C),fÁ(D i )jD i 2Cg: (2.74) 37 In [GM06], isometries were applied to the design of codes by optimizing a worst-case performance criteria. We instead apply isometries to PUB optimiza- tion. Therefore, Á must satisfy P M fD i ! D j g = P M fÁ(D i ) ! Á(D j )g 8i;j. Denoting by D any codeword in C, the following operations can be shown to be valid isometries for PUB: I1) Á(D)=UD, U being a unitary matrix U H U =I. I2) Á(D)=DP, P being a unitary matrix P H P =P. Note that, for STBCs formed via PSK constellations, U and P are required to have only one nonzero element in each row and column and these nonzero elements should be drawn from the same PSK alphabet as D. For non-spread systems, both (I1) and (I2) constitute isometric transformations, whereas only (I1) applies to spread systems, in general [GM06]. ItcanbeshownthattheunionboundmeasuresUB,CB,IUB andPUB forC ¤ arethesameasinC. Thus,isometricoperationsenableustoreuseagoodcodeword structure found for lower cardinality sets and duplicate it, consequently doubling the cardinality of the code. In order to employ the union bound to isometric code constructionsfornonspreadsystems,wehereinpresentamodi¯cationofthegreedy algorithm proposed in[GM06]: 38 1. Start with a good setC k 2. Generate the setC ¤ k fromC k such that C ¤ k =fU m C k P n jmin m;n SER(C k [U m C k P n )g (2.75) 3. C k+1 =C k [C ¤ k . If desired rate is achieved, stop. Otherwise, go to step 2. We use SER(C) to denote any of the union bound performance measures dis- cussed before for a given code-set C. If a design for spread system is desired, we simply replace (2.75) by C ¤ k =fU m C k jmin m SER(C[U m C)g (2.76) Although suboptimal in general, designs obtained via isometries have been shown to yield very good codes, sometimes in exact agreement with the optimal codes obtained via full search [GM06]. 2.7 Code Search Results Our results are divided in two parts. First, we present exhaustive search results for small cardinality codes that optimize UB and CB. Due to its higher complexity, 39 we don't consider exhaustive search using the PUB criterion. Our results are com- pared with optimum minimum metric codes [GMF01] that optimize the worst case pairwise error probability. Searches using UB and PUB are performed assuming that SNR = 1 dB. It is important to stress that all the performance criteria de- veloped in this chapter take into consideration the entire distance spectrum of the codes, whereas the classical approach of rank and determinant maximization only accountsfortheworstcase errorprobabilityscenario. Wecharacterizethe distance spectrum of a code with cardinality K by a matrix ¨ with entries ¨ ij = µ(i;j) or ¨ ij =j©(i;j;j)j, with 1·i;j·K. In the second part, we present search results employing a hierarchical construc- tion via isometries by optimizing PUB. Throughout this section we consider OMM codes as the baseline for comparison against new found codes. This is justi¯ed because OMM codes are already optimized (in the sense of worst-case PEP) and have already been shown to perform better than other classical designs, such as orthogonal and unitary group codes [GMF01]. 2.7.1 Exhaustive Search We denote the Optimal Minimum Metric, Union Bound and Cherno® Bound codes byC OMM ,C UB andC CB respectively. Followingtheargumentsof[GMF01,GVM02], we assume spreading code sets which are equi-correlated which enables the modi¯- cation of the correlation in a controlled manner. For spread systems, a reasonable 40 correlation value of ½ = 0:3 is used, for non-spread systems ½ = 1. We use the term \distance uniform" (DU) to denote when the distance spectrum of a code set is symmetric such that from each codeword point of view, the distance (jdet(©)j or µ) distribution of all other codewords are identical. For ease of representation, a unique integer number, the code index, is used to represent a codeword. If nPSK is employed, u is the n th root of unity u = e j 2¼ n , then d i (t) is a power of u, say d i (t) = u k i (t) , thus a block code matrix of size N c £L t can be represented by the index: k L t (N c )+k L t ¡1 (N c )n+¢¢¢+k 2 (1)n N c L t ¡2 +k 1 (1)n N c L t ¡1 : (2.77) ThefoundcodesarelistedinTable2.1. Foreachfoundset,wesort1=j©(i;j;j)j;i6= j in descending order, denoting them accordingly by f° 0 ;° 1 ;:::g, and the number of codeword pairs achieving ° k by f¯ 0 ;¯ 1 ;:::g. Clearly, CB = P k ¯ k ° k . We use (¯ k ;° k ) as a shorthand note to represent ¯ k pairs of codewords achieving ° k . This yieldsanotherdescriptionofthecodes,intermsoftheirdistancespectra,whichare listedinTable2.2. Forrate12£2codes(1,2,3,7,8,9inTable2.1),eachsethasonly 4 codewords, thus C CB is either identical or slightly better than C OMM for both spread and non-spread systems. For rate 1 3£3 BPSK codes (4,10), each set has 8 codewords. The CB search yields slightly better code sets for spread systems, but theidenticalcodesetfornon-spreadsystems. Forrate1.52£2QPSKcodes(5,11), C CB is slightly better than C OMM for spread systems, but identical for non-spread systems. Interestingly, the worst case approach yields the same code set for both 41 spreadandnon-spreadsystems, andthiscodeisindeedtheorthogonalcode with a uniform distance spectrum [GDF91]. Therefore, the performance of this code is independent of the spreading code correlation ½ (due to orthogonality), further search veri¯es that this code is optimal for all ½ by the worst case criterion. For rate 2 2£2 QPSK codes (6,12), for non-spread systems, C CB exhibits larger gains over C OMM , they both have ° 0 = 0:25 and ¯ 0 = 32 for C CB , ¯ 0 = 26 for C OMM . Despite the disadvantage of ¯ 0 , the simulation results in Figure 2.3 con¯rm that C CB has about 0.4dB gain over C OMM . This illustrates the fact that a worst case analysisfailstogiveacompletedescriptionofperformancebynottakingtheentire distance spectrum into consideration. For spread systems, C CB shows slight gain overC OMM . AninspectionofthedistancespectrumrevealsthatC CB 'saredistance uniform for both spread and non-spread systems. Results for code searches based on the UB criterion are provided in Table 2.3. The CB codes are also shown for comparison. The UB codes yield slightly better performance. It is interesting to note that the UB code (11) for rate 1.5 QPSK 2x2 is distance uniform, while the CB code is not. This is also true for spread and non-spread rate 1 BPSK 3x3 codes (4,10). The UB codes also appear to be more robust to changes in the correlation value ½, i.e., they perform better under di®erent values of correlation. This is illustrated in Figure 2.4, which shows the performance of codes in (10). For code group (4), 42 0 2 4 6 8 10 12 14 16 10 −3 10 −2 10 −1 10 0 SNR SER Rate 2, 2X2, QPSK, ρ=1 OMM UB Figure 2.3: Rate 2, QPSK, 2£2, ½=1:0 union bound code versus worst case code. simulation results con¯rm the advantage of UB codes versus CB codes at SNR =1 dB. 2.7.2 Hierarchical Design Exhaustive searches for codes optimizing PUB are computationally infeasible due to the number of operations required to compute the saddlepoint approximation. Therefore, we employed a hierarchical approach to design a few sporadic codes. Another important application of hierarchical searches is the design of large car- dinality code sets, since the computational cost for exhaustive search is naturally very high in this case. 43 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.32 0.34 0.36 0.38 0.4 0.42 Union Bound vs. ρ for SNR=1dB. Design Parameters: 3x3 BPSK, ρ=0.3, rate 1, SNR=1dB ρ Union Bound UB low SNR code CB code Worst case code Figure 2.4: Rate 1.0, 3£3, BPSK, spread system ½=0:3, sensitivity to ½. Our PUB-optimized code search results are summarized in Table 2.4. A spread system with (½ = 0:3) is assumed in all cases. It turns out that the hierarchical construction based on OMM and UB (or IUB) criteria yields the same codes and thesearealsoshowninTable2.4. Figures2.5,2.6and2.7comparetheperformance of PUB and OMM/UB codes of di®erent block sizes, cardinality and rates. An improvement of around 0.5 dB can be observed. A closer inspection of the codes revealsthattheOMMcodesaredistanceuniformwithrespecttothePUBdistance measure. On the other hand, the optimal PUB codes are not. This illustrates that, although distance uniformity is characteristic of many \good" codes, enforcing it might, in some cases, entail some loss in performance. 44 0 1 2 3 4 5 6 10 −2 10 −1 10 0 Rate 4/3, 3x3 QPSK ρ=0.3 SER SNR PUB code high SNR code Figure 2.5: Rate 4/3, 3£3, BPSK, ½=0:3, PUB code versus OMM/UB code. 2.8 Conclusions Inthischapter, wedevelopedseveralindicesforperformanceassessmentofSTBCs. The Cherno® and Union Bounds expressions were obtained by simple averaging of the pairwise error probabilities of the set. Subsequently, it was shown that some terms in the Union Bound summation were redundant, and therefore could be expurgated. Further analysis of decomposable error patterns allowed us to obtain atighterversionoftheUnionBound-theIndecomposableUnionBound. Allthese bounds still revealed to be quite loose at low SNR and therefore we also proposed the progressive union bound as a performance index for STBCs. A semi-analytic approximation for it was derived by applying a saddlepoint technique and shown 45 0 1 2 3 4 5 6 7 8 9 10 10 −1 10 0 Rate 5/3, 3x3 BPSK ρ=0.3 SER SNR PUB code High SNR code Figure 2.6: Rate 5/3, 3£3, BPSK, ½=0:3, PUB code versus OMM/UB code. to match the simulated code performance more closely than the other bounds. As another advantage, it was noted that PUB allows a tradeo® between numerical complexity and approximation accuracy. Finally, we showed that code searches performed by optimizing the new criteria can show signi¯cant improvement over worst-case designs. Our results also indicate that optimizing tighter bounds during the searches can yield better codes in general. As a future work, we will consider the application of the new found codes as inner codes for a serially concatenated system. 46 0 1 2 3 4 5 6 7 8 9 10 10 −2 10 −1 10 0 SER SNR Rate 2.5 3 × 3 8PSK codes, ρ=0.3 PUB code OMM/UB code Figure 2.7: Rate 5/2, 2£2, 8PSK, ½=0:3, PUB code versus OMM/UB code. index ½ rate constl size OMM CB UB 1 1 1 2£2 BPSK [1,7,8,14] ditto OMM ditto OMM 2 1 1 2£2 QPSK [1,41,156,247] [2,42,156,180] ditto CB 3 1 1 2£2 8PSK [3,1314,2167,3414] [4,803,2224,2967] ditto CB 4 1 1 3£3 BPSK 1 84 166 248 282 335 429 499 ditto OMM 1 84 189 232 282 335 422 499 5 1 1.5 2£2 QPSK 2 42 93 117 128 168 223 247 ditto OMM ditto OMM 6 1 2 2£2 QPSK 2 6 6 4 0 20 42 62 70 82 107 127 133 145 173 185 195 215 236 248 3 7 7 5 2 6 6 4 2 30 42 54 65 93 105 117 128 156 168 180 195 223 235 247 3 7 7 5 ditto CB 7 0.3 1 2£2 BPSK [0,6,11,13] ditto OMM ditto OMM 8 0.3 1 2£2 QPSK [1,87,174,248] [1,88,191,230] [1,42,156,183] 9 0.3 1 2£2 8PSK [3,1314,2447,3246] ditto OMM ditto OMM 10 0.3 1 3£3 BPSK 0 31 99 124 421 442 454 473 1 30 232 247 331 340 418 445 10 87 180 233 317 352 387 478 11 0.3 1.5 2£2 QPSK 2 42 93 117 128 168 223 247 2 8 87 110 145 185 236 247 1 47 84 122 131 173 214 248 12 0.3 2 2£2 QPSK 2 6 6 4 0 10 37 47 82 88 119 125 133 143 160 170 215 221 242 248 3 7 7 5 2 6 6 4 2 20 42 60 75 93 99 117 128 150 168 190 201 223 225 247 3 7 7 5 [ditto CB] Table 2.1: Optimal codes. 47 rule DU CB (¯ 0 ;° 0 ) (¯ 1 ;° 1 ) (¯ 2 ;° 2 ) (¯ 3 ;° 3 ) (¯ 4 ;° 4 ) (¯ 5 ;° 5 ) (¯ 6 ;° 6 ) 1 OMM Y 0.2812 (2,0.0625) (4,0.0327) UB Y 0.2812 (2,0.0625) (4,0.0327) 2 OMM N 0.2903 (1,0.0625) (4,0.0500) (1,0.0278) UB Y 0.2361 (2,0.0625) (4,0.0278) 3 OMM Y 0.2321 (4,0.0397) (2,0.0366) UB Y 0.2189 (2,0.0467) (2,0.0341) (2,0.0278) 4 OMM N 0.3672 (22,0.0156) (6,0.0039) UB N 0.3672 (22,0.0156) (6,0.0039) 5 OMM Y 1.5625 (24,0.0625) (4,0.0156) UB Y 1.5625 (24,0.0625) (4,0.0156) 6 OMM N 14.6285 (24,0.2500) (52,0.1250) (26,0.0625) (8,0.0312) (8,0.0278) (2,0.0156) UB Y 12.0139 (32,0.2500) (48,0.0625) (32,0.0278) (8,0.0156) 7 OMM Y 0.2259 (4,0.0625) (2,0.0156) UB Y 0.2259 (4,0.0625) (2,0.0156) 8 OMM N 0.2278 (4,0.0423) (1,0.0305) (1,0.0281) UB N 0.2272 (4,0.0423) (1,0.0302) (1,0.0278) 9 OMM Y 0.2151 (4,0.0366) (2,0.0343) UB Y 0.2151 (4,0.0366) (2,0.0343) 10 OMM N 0.1309 (4,0.0082) (4,0.0055) (4,0.0046) (8,0.0043) (4,0.0029) (4,0.0028) UB N 0.1271 (4,0.0082) (4,0.0055) (4,0.0043) (4,0.0042) (4,0.0039) (4,0.0029) (4,0.0028) 11 OMM Y 1.5625 (24,0.0625) (4,0.0156) UB N 1.5359 (2,0.0687) (8,0.0654) (10, 0.0625) (8,0.0312) 12 OMM N 8.8273 (64,0.0859) (32,0.0687) (16,0.0625) (4,0.0156) (4,0.0172) UB Y 8.7988 (64,0.0887) (48,0.0625) (8,0.0156) Table 2.2: Distance Spectrum design ° 0 UB 8 CB 0.0423 0.8031 UB 0.0430 0.8028 11 CB 0.0687 4.2395 UB 0.0654 4.2268 10 CB 0.0082 2.8080 UB 0.0086 2.8047 4 CB 0.0156 3.0522 UB 0.0156 3.0473 Table 2.3: CB codes versus UB codes. index rate size constl PUB OMM/UB 1 4/3 3£3 BPSK 2 6 6 4 10 501 13 498 134 377 326 185 87 424 111 400 53 458 50 461 3 7 7 5 2 6 6 4 10 501 13 498 122 389 125 386 50 461 53 458 66 445 69 442 3 7 7 5 2 5/3 3£3 BPSK 2 6 6 6 6 6 6 6 6 6 4 10 501 13 498 134 377 326 185 87 424 111 400 53 458 50 461 69 442 66 445 17 494 41 470 136 375 328 183 386 125 389 122 3 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 4 10 501 13 498 122 389 125 386 50 461 53 458 66 445 69 442 17 494 41 470 87 424 111 400 22 489 46 465 80 431 104 407 3 7 7 7 7 7 7 7 7 7 5 3 5/2 2£2 8PSK 2 6 6 6 6 6 6 6 6 6 4 4 2336 292 2048 1202 3478 1426 3254 607 2939 895 2651 1741 4073 2029 3785 2633 877 589 2921 2011 3839 4063 1787 3236 1408 1184 3460 2102 274 50 2326 3 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 4 4 2336 292 2048 1202 3478 1426 3254 607 2939 895 2651 1741 4073 2029 3785 22 2354 310 2066 1156 3488 1444 3200 617 841 1759 2047 2669 2893 3803 4091 3 7 7 7 7 7 7 7 7 7 5 Table 2.4: Hierarchical codes. 48 Chapter 3 Cooperative Strategies for Wireless Multihop Communications: Diversity Analysis and Power Allocation In resource limited, large scale sensor networks, cooperative communication over multiplehopso®ersopportunitiestosavepower: intermediatenodesbetweensource and destination act as cooperative relays. In order to exploit spatial diversity, pro- tocols coupled with space-time coding strategies are proposed herein and analyzed for distributed cooperative communication. In contrast to prior work, multi-hop (versu two-hop) schemes are developed and analyzed. Amplify-and-forward type protocols are considered, along with a derived optimal power allocation strategy. First, the Alamouti-based two-hop scheme proposed in (Hua et al, 2003) and ana- lyzed in (Jing & Hassibi, 2006) is generalized to an arbitrary number of hops and a general expression for the pairwise error probability (PEP) is obtained, as well 49 as its achievable diversity. It is further shown that the diversity decreases with L, for large, but ¯nite signal-to-noise ratio (SNR). This diversity loss is quanti¯ed in the general multi-hop case. This motivates the development of novel distributed multi-hop protocols to mitigate the diversity losses and, hence, yield improved per- formance. To this end, two new strategies are proposed, for the speci¯c case of L = 3 hops, and their diversity gains are also characterized. These schemes are based on the structure of the rate-half codes (Tarokh, 1999) and the square-matrix embeddable codes (Tirkkonen & Hottinen, 2002) . 3.1 Introduction In recent years, wireless ad-hoc networks are being increasingly considered for many commercial and military applications in order to increase reliability, cov- erage, throughput and capacity (see [GB03, jsa99, per01, BFY04] and references therein). A common feature among many of these networks is their multi-hop na- ture: communicationisperformedbetweensourceanddestinationviaintermediate terminals. This provides several advantages over single hop schemes[GB03]: a) combating the severe electromagnetic signal decay over long distances, and there- fore, saving transmission power; b) providing signal paths between terminals which do not have a direct line of sight between them; and c) providing multiple commu- nication links for applications with a high data rate requirement which cannot be satis¯ed via a single link. 50 Multihop networks can also provide additional gains through cooperation be- tween terminals. Recent information theoretic results show that cooperation can increase the overall capacity of these networks by taking advantage of their inher- entrichnessinspatialdiversity[SEA03a,SEA03c,LW03,LTW04]. Hence,anatural way to exploit this diversity is via Distributed Space-Time Block Coding (DSTBC) originally proposed in [LW03]. The goal of a DSTBC-based protocol is to allow the cooperatingterminalstoact,fromthedestinationpointofview,asamulti-antenna array employing a well designed Space-Time Block Code (STBC)[TJC99]. Several DSTBC schemes have been recently proposed [LW03, HMC03, ALK03, NBK04, JH06]: in [LW03] a repetition coding protocol was suggested; it however, requires the terminals to transmit over orthogonal channels and hence is bandwidth inef- ¯cient for large networks. In [HMC03] and [ALK03], DSTBC protocols based on the Alamouti STBC [Ala98] for a two-hop network are proposed. A similar idea is shown in [NBK04], applied to a single-relay protocol. The work in [JH06] considers a two-hop network with arbitrary number of relays employing a distributed linear dispersion code. A key feature of the above schemes and including those presented in this chapter, is that decoding is only performed at the destination node. This can be an attractive feature, since obviating the need for symbol decoding and channel estimation by the sensors can potentially simplify their signal processing requirements and, hence, reduce their cost. 51 In this chapter, DSTBC protocols for networks with two or more hops are con- sidered. Tothebestofourknowledge,priorworkonDSTBCprotocolshasfocussed two-hop schemes. In contrast, the new schemes proposed herein are designed for communication over more than two hops, hence, yielding further performance im- provement due to the smaller signal attenuation per hop. We assume a single source communicating to a single destination and DSTBC protocols based on the Alamouti[Ala98], rate-half (RH) orthogonal[TJC99] and square matrix embeddable (SME) codes [TH02]. We emphasize the fact that, un- like other proposed DSTBC cooperative schemes [CG79, ALK03, NBK04] a di- rect source-destination link is not available, and thus all communication is carried out hop by hop over (possibly) multiple relay stages. As we show subsequently, these codes can be implemented without requiring symbol decoding by the relays and, due to their inherent structure, enable low complexity maximum likelihood decoding[TJC99, TH02] at the destination. Such schemes would be applicable, for example, in a multi-sensor ad-hoc network, where individual sensors or their rout- ing protocols only have knowledge of their neighboring nodes. Another potential application involves the mobile multi-hop relay (MMR) project currently under development by the IEEE 802.16 Relay Task Group[Rel06]. Performanceanalysis ofthe DSTBC schemesproposed in this chapter is carried outviaanextensionoftheCherno®boundanalysisusedformulti-antennasystems. However, unlike the multiple antenna case, the proposed cooperation schemes yield 52 an overall source-destination channel of Gaussian-product statistics, which makes the explicit derivation of error probability expressions highly challenging. There- fore, upper and lower bounds on the error probability along with suitable approx- imations are obtained, in order to yield insight on the performance. Furthermore, by appropriately de¯ning the achievable diversity as an SNR-dependent quantity characterizing the error probability slope of a given scheme, a simple, high SNR diversity performance criterion is obtained. Our analysis yields the somewhat surprising result: from a diversity point of view, multi-hopping is apparently not always bene¯cial. As shown in this chapter, the product-channel nature inherent to the proposed protocols yields a loss in di- versity (compared to the standard Rayleigh fading). In the ¯rst proposed protocol, the Alamouti multi-hop (AAF-MH) cooperation, we show that this loss increases linearly with the number of hops. A result of similar nature was shown in [JH06] for a two-hop network. We also show that the diversity loss can be mitigated by employing appropriate cooperation strategies and illustrate this idea by proposing novel schemes based on RH and SME codes. Furthermore, we also extend the opti- malpowerallocationresultfortwo-hops(givenin[HMC03,JH06])toourmulti-hop protocols. Giventhepotentialdi±cultiesinmaintainingfullsynchronizationbetweentrans- mitting relays in a practical setting, we shall also consider the e®ect on asyn- chronous transmissions on the performance. To this end, we extend the work in 53 [MHSD05], which considers the problem of distributed terminals communicating asynchronously to a single destination. However, unlike the relay scenario consid- ered here, in [MHSD05] the terminals are assumed to have full knowledge of the transmitted symbols. For the cooperative relay setting, we show that, although a loss in e®ective SNR is incurred due to asynchronous relays, the diversity gains due to cooperation are still maintained. This chapter is organized as follows. Section 3.2 presents the multi-hop Alam- outi Amplify and Forward (AAF-MH) cooperation protocol along with its perfor- mance analysis. Section 3.3 presents and analyzes two new cooperation protocols which achieve improved diversity. Section 3.4 addresses the issue of cooperation with asynchronous relays. An optimal power allocation strategy among nodes is derived in Appendix A (Section 3.6) and conclusions are outlined in Section 3.5. 3.2 SignalModelandMulti-hopAAF(AAF-MH) Scheme In this section, we present the general signal model and propose a multi-hop coop- eration scheme denoted by AAF-MH. For the speci¯c case of two-hop cooperation, AAF-MHisequivalenttotheschemebyHua[HMC03](denotedbyAAF-2Hinthis chapter). Furthermore, expressions for the pairwise error probability and diversity of the AAF-MH scheme are derived analytically. 54 Stage1 Stage2 Source Destination 2 - hop communication 3 - hop communication R 1 R 2 R 3 R 4 Stage1 Stage2 Source Destination 2 - hop communication 3 - hop communication R 1 R 2 R 3 R 4 Figure 3.1: Cooperative networks for two and three hops. Complex baseband notation is used throughout this chapter. We shall assume slotted communication protocols with coherent reception. Except for the analysis in Section 3.4, perfect synchronization is also assumed. The practical constraint of half-duplextransmissionsisenforced,i.e.,theterminalsdonottransmitandreceive simultaneously. We adopt a symmetric topology, in which each stage of relays is located at the exact midpoint between the previous and the following stages. As a result, within each hop, transmitted signals are received simultaneously by the relays. Assumingequalprocessingtimeforeachrelayinagivenstage,transmissions arealsoperfectlysynchronous. Anillustrationofthebasicsetupfortwo-hop(AAF- 2H) and three-hop cooperation is shown in Figure 3.1, which illustrates a single source (S) communicating to a single destination (D) via one or two intermediate relay stages. Nodes are numbered according to their distance from the source and from top to bottom. Hence, node 0 is the source, node 1 is the top relay of the ¯rst stage, node2isthebottomrelayandsoon. Index dalwaysdenotesthedestination node. 55 3.2.1 Cooperation Protocol In the AAF-MH protocol, each transmission block s , [s(1) s(2)] T from the source consists of 2 symbols s(1) and s(2) taken from a complex alphabet (QAM or PSK). The i-th relay receives a noisy and distorted version s i ,[s i (1) s i (2)] T of the source block, i.e., s i (k)= p 2´ 0 h 0;i s(k)+w i (k);k =1;2 and i=1;2 (3.1) where the h 0;i are the fading channel coe±cients between source and relay i, 2´ 0 is the source-relay average SNR. We will always assume a quasi-static fading channel throughoutthischapter,andtherefore,h j;i isconstantovertheentireblockduration and independent from block to block. We de¯ne h 0 = [h 0;1 h 0;2 ] T as the vector of channel coe±cients for the source with distribution h 0 »CN(0 2 ;I 2 ), where 0 n is the n£1 zero vector and I n is the n£n identity matrix. We useCN(m;K) to denote a circularly symmetric complex Gaussian random vector with mean m and variance matrix K. Thenoiseateachrelayisalsocircularlysymmetric,temporallywhite,andinde- pendentfromrelaytorelay. Thus,w i (k)»CN(0;1)andfurthermoreEfw i (k) ¤ w j (l)g= ±(i¡j)±(k¡l) where ±(k) is the Kronecker delta function. 56 We now describe the processing performed by the relays. Representing the transmission of relay i by a column vectorc i , the joint transmission of relays 1 and 2 over two time slots can be represented by the matrix C 1;2 , C 1;2 =[c 1 c 2 ], r ´ 1 P 1 2 6 6 4 s 1 (1) ¡s ¤ 2 (2) s 1 (2) s ¤ 2 (1) 3 7 7 5 , 2 6 6 4 x T 1;2 (1) x T 1;2 (2) 3 7 7 5 (3.2) where x T 1;2 (k) denotes the k-th row of C 1;2 , ´ 1 is the relay-destination amplifying factor. The power normalization factor P 1 ,1+2´ 0 ensures that each relay trans- mits an average total power ´ 1 . Note that (3.2) resembles the Alamouti space-time block code; however, a key di®erence is that the two relays are forwarding not the desired signals s(1) and s(2), but rather, their noisy versions s i (k);k = 1;2 (see (3.1)). The received signal at relay j =3;4 and time k can be written as s j (k)=x T 1;2 (k)g j +w j (k); j =3;4 (3.3) where the vector of channel coe±cients "arriving" at relay j is g j , [h 1;j h 2;j ] T , w j (k) » CN(0;1) is the relay j AWGN at time k. Concatenating s j (k);k = 1;2 into a vector s j , we get s j = 2 6 6 4 s j (1) s j (2) 3 7 7 5 = 2 6 6 4 x T 1;2 (1) x T 1;2 (2) 3 7 7 5 g j +w j ; j =3;4: (3.4) 57 Substituting (3.1) and (3.2) in (3.4), yields s j = r 2´ 0 ´ 1 P 1 2 6 6 4 s(1) ¡s ¤ (2) s(2) s ¤ (1) 3 7 7 5 2 6 6 4 h 0;1 h 1;j h ¤ 0;2 h 2;j 3 7 7 5 +n j , p ° 2h Sf 2h +n j ; j =3;4 (3.5) where ° 2h is the signal power term at each of the second-stage relays, n j is the overall noise at relay j, given by n j =w j + r ´ 1 P 1 (w 1 h 1;j + ~ w 2 h 2;j ); (3.6) with ~ w 2 , [¡w ¤ 2 (2) w ¤ 2 (1)] T . Conditioned on h 1;j and h 2;j , n j has distribution CN(0;K j ) with K j = ( ´ 1 P 1 jjg j jj 2 + 1)I 2 , ¾ 2 j I 2 . The vector f 2h is the equivalent two-hop source-to-destination channel, and S is the Alamouti codeword matrix S, 2 6 6 4 s(1) ¡s ¤ (2) s(2) s ¤ (1) 3 7 7 5 : (3.7) In the proposed three-hop AAF-MH scheme, the second stage relays normalize (to unit average power) and amplify their received signals. Hence, the three-hop received signal at D, denoted by y 3h , is y 3h = r ´ 2 P 2 (s 3 h 3;d +s 4 h 4;d )+w d (3.8) 58 where P 2 =1+2´ 1 and w 3 is the destination AWGN. Substituting (3.5), y 3h = r 2´ 0 ´ 1 ´ 2 P 1 P 2 [S S]f 3h +n d , p ° 3h Sf 3h +n d (3.9) where S,[S S]. The overall channel f 3h and destination noise n d are given by f 3h = · h 0;1 h 1;3 h 3;d h ¤ 0;2 h 2;3 h 3;d h 0;1 h 1;4 h 4;d h ¤ 0;2 h 2;4 h 4;d ¸ T (3.10) n d = w 3 + r ´ 2 P 2 (n 3 h 3;d +n 4 h 4;d ): (3.11) Substituting n j in (3.6), n d =w d + r ´ 2 P 2 [w 3 h 3;d +w 4 h 4;d + r ´ 1 P 1 [w 1 (h 1;3 h 3;d +h 1;4 h 4;d )+ ~ w 2 (h 2;3 h 3;d +h 2;4 h 4;d )] ¸ (3.12) which, conditioned on the channels h i;j , is complex Gaussian and white with com- ponents having variance ¾ 2 3h =1+ ´ 2 P 2 µ jh 3;d j 2 +jh 4;d j 2 + ´ 1 P 1 ¡ jh 1;3 h 3;d +h 1;4 h 4;d j 2 +jh 2;3 h 3;d +h 1;4 h 4;d j 2 ¢ ¶ : (3.13) Averaging ¾ 2 3h over h i;j , yields the average noise power for L=3 hops, denoted by ¹ ¾ 2 3h ¹ ¾ 2 3h =1+2 ´ 2 1+2´ 1 +4 ´ 1 ´ 2 (1+2´ 0 )(1+2´ 1 ) : (3.14) 59 This completes the characterization of the AAF-MH protocol for three hops. It is straightforward to show that the amplify-and-forward procedure of (3.8) can be repeated for any number of hops L. Hence, with two relays in each stage, the following recursion holds y Lh = r ´ L¡1 P L¡1 (s 2L¡3 h 2L¡3;d +s 2L¡2 h 2L¡2;d )+w d : (3.15) The following remarks are in order: ² The AAF-2H scheme[HMC03] is a particular instantiation of the general AAF-MH proposed here. The received signal model in this case can easily be obtained by replacing j by D in (3.5). ² In general, amplify and forward operations incur noise enhancement at the relays, which in turn gets forwarded to next stages. This results in several noisecomponentsatthereceiver: the¯rstduetothepurereceivernoisefrom the destination, along with others due to noise-channel coupling from each of the relay transmission stages. ² Although the AF scheme described here applies to the special case of 2 relays, generalized orthogonal schemes can be used when this number is larger[TJC99, HMC03]. Given the received signal in (3.15), maximum likeli- hooddecodingcanbeperformedtodeterminethedecodedcodeword ^ S,under 60 the assumption of perfect channel state information (CSI) of all communica- tionlinksatD 1 . NotethattheschemedoesnotrequireanyCSIatthesource orrelays. DuetotheorthogonalstructureofS,maximum-likelihood(ML)de- codingofeachofthetransmittedsymbolscanbedecoupledandperformedby a simple linear processing receiver[TJC99]. Similarly, all cooperative schemes presented in this chapter also present this desirable property. In the following subsections, we apply the Cherno®-Bound approach to provide a diversityanalysisfortheproposedAAF-MHscheme. Forthesakeofclarity,we¯rst obtain the results in the speci¯c case of three-hop cooperation and subsequently generalize them to arbitrary number of hops in Appendix B (Section 3.7). 3.2.2 Diversity Analysis in AAF-MH: Three-Hop Case Recall, from (3.9) that the received signal is y 3h = p ° 3h Sf 3h +n d (3.16) 1 In practice, channels can be estimated by sending known training symbols from the relays and from the source. 61 where S, [S S], f 3h is given in (3.10), S denotes the Alamouti codeword in (3.7), ° 3h = 2´ 0 ´ 1 ´ 2 P 1 P 2 and the noise n d has variance ¾ 2 3h given by (3.13). From (3.14), the noise variance mean, denoted by ¹ ¾ 2 3h is ¾ 2 3h ¼ ¹ ¾ 2 3h =1+ ´ 2 P 2 · 2+ 4´ 1 P 1 ¸ (3.17) For the purpose of mathematical tractability, we will always approximate the noise variance of the received signaln d by its mean, an approach similar to that in [JH06]. ThisapproximationisequivalenttoanAWGNapproximationoftheoverall noise, which, as the simulation results will show, yields conservative estimates of the performance, but does not a®ect the diversity analysis. Starting from the model in (3.16) and noting that the received signal at the destination results from the concatenation of two Alamouti transmissions, we ¯rst average the PEP over the second hop channels h i;j ; i = 1;2 and j = 3;4. At high SNR ½, the Cherno® Bound, denoted by P cb (k;j;½), is a tight upper bound on P e (S k !S j ;½), which denotes the pairwise error probability (PEP) of detectingS j when S k is transmitted. Hence, P e (S k !S j ;½)·P cb (k;j;½)=E z det ¡1 £ I 4 +½M k;j diagfjz 1 j 2 ;:::;jz 4 j 2 g ¤ (3.18) where ½ , ° 3h 4¹ ¾ 2 3h is the SNR, M k;j , (S k ¡S j ) H (S k ¡S j ), and E z denotes the expectation over the vector z, · h 0;1 h 3;d h ¤ 0;2 h 3;d h 0;1 h 4;d h ¤ 0;2 h 4;d ¸ T . Since the 62 matrix (S k ¡S k ) has size 2£ 4, it has at most rank 2, a fact which also holds true for M k;j . Furthermore, the Alamouti structure of S k and S j ensures that this highest rank is achieved for all k and j. Hence, the processing of the second stage relays in the AAF-MH scheme, given in (3.9), is optimal (from a diversity point of view) for transmitted blocks of length 2. We can equivalently write the Cherno® Bound (3.18) as P cb (k;j;½)=E z 1 ;z 4 det ¡1 £ I 2 +2½M k;j diagfjz 1 j 2 ;jz 4 j 2 g ¤ (3.19) where M k;j , (S k ¡S j ) H (S k ¡S j ), z 1 = h 0;1 h 3;d and z 4 = h ¤ 0;2 h 4;d . From the orthogonal structure of and S k and S j , it turns out that M k;j = ¸ k;j I 2 , where ¸ k;j depends only on entries of S k and S j . Averaging over the channel gains jh 0;i j 2 , which are exponentially distributed, the Cherno® bound in (3.19) is given by P cb (½ ¸ )= 4 Y i=3 E jh i;d j 2 e 1 ½ ¸ jh i;d j 2 ½ ¸ jh i;d j 2 · ¡Ei µ ¡ 1 ½ ¸ jh i;d j 2 ¶¸ (3.20) where we dropped the indices k;j for clarity, ½ ¸ , 2½¸ k;j , and Ei(x) denotes the Exponential Integral function, de¯ned as Ei(x)=¡ Z 1 ¡x e ¡t t dt; x< 0: (3.21) 63 Carrying out the averaging over the channel gainsjh i;d j 2 in (3.20), yields P cb (½ ¸ )= 2 Y i=1 1 ½ ¸ Z 1 0 e 1 x i x i · ¡Ei µ ¡ 1 x i ¶¸ e ¡ x i ½ ¸ dx i (3.22) ThepresenceoftheExponentialIntegralintheintegrandoftheCherno®bound in (3.22) signi¯cantly complicates the analysis, since, to the best of our knowledge, no closed-form solution can be obtained for the resulting integral. We propose, therefore,toderiveupperandlowerboundson(3.22),withtheresultingexpressions yielding diversity gains (to be de¯ned subsequently) denoted by d u (½) and d l (½), respectively. Finally, we will observe that d u (½) = d l (½) , d(½) and the desired expression for the diversity will be obtained. 3.2.2.1 Lower Bound on (3.22) We ¯rst recall the following lower bound [AS64] ¡e u Ei(¡u)> 1 2 log µ 1+ 2 u ¶ ; u> 0 (3.23) Substituting x= 1 u in (3.23) we obtain ¡e 1 x Ei µ ¡ 1 x ¶ > 1 2 log(1+2x); x<1 (3.24) 64 Applying (3.24) to (3.22) and the change of variables u=1+2x i , yields P cb (½ ¸ ) > 2 Y i=1 1 ½ ¸ Z 1 0 log(1+2x i ) 2x i e ¡ x i ½ ¸ dx i (3.25) = Ã e 1 2½ ¸ 2½ ¸ Z 1 1 logu u¡1 e ¡ u 2½ ¸ du ! 2 (3.26) > Ã e 1 2½ ¸ 2½ ¸ Z 1 1 logu u+1 e ¡ u 2½ ¸ du ! 2 (3.27) Solvingtheright-handsideof(3.27),(see, e.g.,Equation4.351.3in[GR00]),results in the desired closed-form of the lower bound P cb (½ ¸ )> e 2 ½ ¸ 16½ 2 ¸ · Ei µ ¡ 1 2½ ¸ ¶¸ 4 (3.28) Now we recall the series expansion of Ei(x)[GR00] ¡Ei(¡x)=¡log(x)¡°¡ 1 X k=1 (¡1) k x k k¢k! =¡log(x)+O(1); x>0 (3.29) where ° =0:5772::: is Euler's constant. Substituting (3.29) in (3.28) with x= 1 2½ ¸ and applying the Taylor series expansion for e x , yields P cb (½ ¸ ) > e 2 ½ ¸ 16½ 2 ¸ " ¡°+log(2½ ¸ )¡ 1 X k=1 (¡1) k (2½ ¸ ) ¡k k¢k! # 4 (3.30) = ³ 1+ P 1 n=1 2 n n!½ n ¸ ´h log(½ ¸ )+log2¡°¡ P 1 k=1 (¡1) k (2½ ¸ ) ¡k k¢k! i 4 16½ 2 ¸ (3.31) 65 For high SNR ½ ¸ , the dominating term of the numerator in (3.31) is log 4 (½ ¸ ), since all the other ½ ¸ -dependent terms approach zero by increasing ½ ¸ . Therefore, we can write the lower bound in its ¯nal form P cb (½ ¸ )>P cb;l (½ ¸ ),K l log 4 (½ ¸ ) ½ 2 ¸ (3.32) where K l is a constant. 3.2.2.2 Upper Bound on (3.22) From the series expansion (3.29), we have the following upper bound -Ei µ ¡ 1 u ¶ ·log(u); u¸® (3.33) where ® ¼ 1:5 is a constant. The bound is shown in Figure 3.2. Performing the integration separately from 0 to ® and from ® to1, an upper bound to P cb (½ ¸ ) is obtained by applying (3.33) to (3.22), yielding P cb (½ ¸ )· µ 1 ½ ¸ ¶ 2 µZ ® 0 e 1=x x · ¡Ei µ ¡ 1 x ¶¸ e ¡ x ½ ¸ dx+ Z 1 ® e 1=x x log(x)e ¡ x ½ ¸ dx ¶ 2 (3.34) Noting that e ¡ x ½ ¸ ·1 when evaluating the ¯rst integral, yields P cb (½ ¸ )· µ 1 ½ ¸ ¶ 2 µZ ® 0 e 1=x x · ¡Ei µ ¡ 1 x ¶¸ dx+ Z 1 ® e 1=x x log(x)e ¡ x ½ ¸ dx ¶ 2 (3.35) 66 0 1 2 3 4 5 6 7 8 9 10 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Exponential integral function: −Ei(−1/x) x −Ei(−1/x) log(x) Figure 3.2: Exponential Integral function. Let K ® , R ® 0 e 1=x x £ ¡Ei ¡ ¡ 1 x ¢¤ dx, which does not depend on ½ ¸ . Since e 1=x ¸ e and log(x) x ¸0 for x¸1, we can further upper bound (3.35) as P cb (½ ¸ )· µ 1 ½ ¸ ¶ 2 µ K ® +e Z 1 1 log(x) x e ¡ x ½ ¸ dx ¶ 2 : (3.36) Performing integration by parts, we get P cb (½ ¸ ) = µ 1 ½ ¸ ¶ 2 µ K ® + e 2½ ¸ Z 1 1 log 2 (x)e ¡ x ½ ¸ dx ¶ 2 (3.37) · µ 1 ½ ¸ ¶ 2 µ K ® + e 2½ ¸ Z 1 0 log 2 (x)e ¡ x ½ ¸ dx ¶ 2 (3.38) 67 The integral in (3.38) can be solved in closed form[GR00], yielding the desired upper bound P cb (½ ¸ )· 0 @ K ® + e 2 h ¼ 2 6 +(°¡log(½ ¸ )) 2 i ½ ¸ 1 A 2 : (3.39) Forhigh½ ¸ , thenumeratorin(3.39)isdominatedbythelog 4 (½ ¸ )term, and, hence, we can write the ¯nal form of the upper bound as P cb (½ ¸ )·P cb;u (½ ¸ ),K u log 4 (½ ¸ ) ½ 2 ¸ (3.40) where K u is a constant. 3.2.2.3 Main Diversity Result The diversity gain at SNR ½ is de¯ned as[Nar06] d(½),¡ @logP e (½) @log(½) =¡ ½ P e (½) @P e (½) @½ (3.41) where P e (½) is the error probability for SNR ½. Hence, d(½) is given by the slope of the P e (½) versus ½ curve (on a log scale). Note that the above de¯nition is more general than the classical notion of diversity, denoted by d 1 , which does not incorporate the SNR dependence, and is given by d 1 ,¡ lim ½!1 @logP e (½) @log(½) : (3.42) 68 Applyingthede¯nition(3.41)totheboundsin(3.32)and(3.40), yields, respec- tively d l (½),¡ @logP cb;l (½) @log(½) =2¡ 4 log½ d u (½),¡ @logP cb;u (½) @log(½) =2¡ 4 log½ (3.43) Hence, since d u (½) = d l (½), we conclude that the diversity gain of the three-hop AAF scheme, denoted by d 3h (½), is d 3h (½)=2¡ 4 log½ (3.44) which shows that the diversity loss of the cooperative three hop AAF-MH scheme (compared to a single hop dual antenna system) is 4 log½ . It can be shown that the diversity gain of the AAF-2H scheme incurs half the loss of AAF-MH, and is d 2h (½)=2¡ 2 log½ (3.45) Figure 3.3 shows the simulated performance of the three-hop AAF-3H scheme em- ploying a BPSK rate 1.0 bit per channel use Alamouti DSTBC. Also shown are the performance under the AWGN assumption, and the asymptotic diversity curve predicted for the AAF-MH scheme. It is clear that, while the AWGN approxima- tion yields performance that is worse than the true non-Gaussian case, both curves match the predicted diversity in (3.44) for three hops. Furthermore, the diversity 69 −10 −5 0 5 10 15 20 25 30 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 ρ(dB) Block error rate Performance of AAF−MH for 3 hops AAF−MH, simulation, AWGN approx. AAF−MH, simulation Error prob. slope: AAF−MH, analytical Error prob. slope: AAF−2H, analytical Figure 3.3: AAF-MH scheme. performance is worse than that predicted in (3.45) for the two hop case (which is also shown for comparison). This loss can be attributed to the overall channel in AAF-MH being a \three-product" channel instead of a \two-product" channel as in AAF-2H. A natural conjecture is whether the results for AAF-2H and three-hop AAF-MH generalize to any number of hops. Extending the analysis in this section (details are given in Appendix 3.2), the diversity of the AAF-MH scheme over L hops is d Lh (½)=2¡ 2(L¡1) log½ : (3.46) Hence, the AAF-MH scheme incurs a diversity loss (due to the 2(L¡1) log½ factor in d Lh ) which increases with the number of hops L. As shown in the next section, 70 this loss can be mitigated by coding over longer blocks at the relays, yielding new distributed coding strategies. 3.3 Three-HopCommunicationviaRHandSME Codes In this section, two new coding strategies are proposed, for the speci¯c scenario of three-hopcommunication. Hence,thenetworktopologyisidenticaltothethree-hop AAF-MH topology. The new protocols are based on the structure of the rate-half (RH) and square-matrix embeddable (SME) space-time codes. The extension of the RH and SME schemes to more than 3 hops can also be done by repetition analogously to the AAF-MH scheme. 3.3.1 RH and SME Cooperation Protocols Throughout this section, we will use the superscripts 1 and 2 to denote signals for the RH and SME schemes, respectively. This will allow both schemes to be described simultaneously. The source transmission blocks - s (1) in RH and s (2) in SME - are s (1) = [s 1 s 2 s 3 s 4 s ¤ 1 s ¤ 2 s ¤ 3 s ¤ 4 ] T (3.47) s (2) = [s 1 s 2 s 3 s ¤ 3 ] T (3.48) 71 Thus, s (1) consists of a 8£ 1 block of 4 information symbols and their complex conjugates and s (2) has size 4£ 1 and contains 3 information symbols and the complex conjugate of s 3 . The received signals at the ¯rst stage of relays (denoted by s (1) i and s (2) i for RH and SME respectively, i=1;2) are given by s (l) i = p 2´ 0 s (l) h 0;i +w (l) i ; i=1;2 (3.49) where h 0;i »CN(0;1) and w (1) i »CN(0;I 8 ) and w (2) i »CN(0;I 4 ) are AWGN. We will use index i to denote the relays and l to denote the cooperation scheme , i.e, l = 1 for RH and l = 2 for SME. As in the AAF-MH case, the relays of the ¯rst stage normalize and amplify their received signals resulting in a transmit power of ´ 1 per relay. They can also perform time-shifting and conjugation operations over s (l) i , resulting in the transmitted signals from the ¯rst stagec (l) i . For relays 1 and 2, c (l) 1 = s ´ 1 P (l) 1 © (l) 1 s (l) 1 (3.50) c (1) 2 = s ´ 1 P (1) 1 © (1) 1 s (1) 2 (3.51) c (2) 2 = s ´ 1 P (2) 1 © (2) 1 s 0 (2) 2 (3.52) where s 0 (2) 2 denotes the component-wise complex conjugate of s (2) 2 , P (1) 1 = 1+2´ 0 and, for a reason that will become evident shortly, P (2) 1 = 3 4 (1+2´ 0 ). 72 The © (1) i are unitary permutation matrices with © (1) i = 2 6 6 4 F i 0 4 0 4 F i 3 7 7 5 ; F 1 = 2 6 6 6 6 6 6 6 6 6 6 4 1 0 0 0 0 ¡1 0 0 0 0 ¡1 0 0 0 0 ¡1 3 7 7 7 7 7 7 7 7 7 7 5 ; F 2 = 2 6 6 6 6 6 6 6 6 6 6 4 0 1 0 0 1 0 0 0 0 0 0 1 0 0 ¡1 0 3 7 7 7 7 7 7 7 7 7 7 5 (3.53) where 0 n denotes the n£n zero matrix. The matrices © (2) i ;i=1;2 are © (2) 1 = 2 6 6 6 6 6 6 6 6 6 6 4 1 0 0 0 0 ¡1 0 0 0 0 ¡1 0 0 0 0 0 3 7 7 7 7 7 7 7 7 7 7 5 ; © (2) 2 = 2 6 6 6 6 6 6 6 6 6 6 4 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 5 : (3.54) Note that in the SME processing, characterized by (3.54), the relays only transmit over three time slots. Since the transmission block is of length 4, a factor of 3=4 is present in P (2) i so that the average transmit power is maintained at ´ 1 by the ¯rst stagerelays. Relays3and4, atthesecondstage, receivetheresultofthecombined transmissions from relays 1 and 2. Therefore, we have s (l) i = 2 X j=1 c (l) j h j;i +w (l) i ; i=3;4 (3.55) 73 where h j;i denotes the fading channel between nodes j and i, as de¯ned in the previoussection,w (1) i »CN(0 8 ;I 8 )andw (2) i »CN(0 4 ;I 4 )areAWGN.Substituting (3.49) and (3.50-3.52) in (3.55), we have, for RH cooperation s (1) i = s 2´ 0 ´ 1 P (1) 1 S (1) 1;2 2 6 6 4 h 0;1 h 1;i h 0;2 h 2;i 3 7 7 5 + s ´ 1 P (1) 1 W (1) 1;2 2 6 6 4 h 1;i h 2;i 3 7 7 5 +w (1) i i=3;4 (3.56) where S (1) 1;2 , · © (1) 1 s (1) © (1) 2 s (1) ¸ is the equivalent distributed space-time code- word constructed by the ¯rst stage relays and W (1) 1;2 , · © (1) 1 w (1) 1 © (1) 2 w (1) 2 ¸ is the unitary-transformed noise of the ¯rst stage. Similarly, for SME cooperation, s (2) i = s 2´ 0 ´ 1 P (2) 1 S (2) 1;2 2 6 6 4 h 0;1 h 1;i h ¤ 0;2 h 2;i 3 7 7 5 + s ´ 1 P (2) 1 W (2) 1;2 2 6 6 4 h 1;i h 2;i 3 7 7 5 +w (2) i i=3;4 (3.57) where S (2) 1;2 , · © (2) 1 s (2) © (2) 2 s 0 (2) ¸ , W (2) 1;2 , · © (2) 1 w (2) 1 © (2) 2 w 0 (2) 2 ¸ and w 0 (2) 2 is the componentwise complex conjugate of w (2) 2 . The second stage relays perform pro- cessing of similar type to that of the ¯rst stage relays. For relays 3 and 4, c (l) 3 = r ´ 2 P 2 © (l) 3 s (l) 3 (3.58) c (1) 4 = r ´ 2 P 2 © (1) 4 s (1) 4 (3.59) c (2) 4 = r ´ 2 P 2 © (2) 4 s 0 (2) 4 (3.60) 74 where P 2 =1+2´ 1 is the average received power at each relay of the second stage (for both schemes), ´ 2 is the average transmitted power and, © (1) 3 =I 8 ;© (2) 3 =I 4 ;© (1) 4 = 2 6 6 4 F 4 0 4 0 4 F 4 3 7 7 5 ;© (2) 4 = 2 6 6 6 6 6 6 6 6 6 6 4 0 0 ¡1 0 0 0 0 ¡1 1 0 0 0 0 1 0 0 3 7 7 7 7 7 7 7 7 7 7 5 ; (3.61) where F 4 = 2 6 6 6 6 6 6 6 6 6 6 4 0 0 0 ¡1 0 0 ¡1 0 0 1 0 0 1 0 0 0 3 7 7 7 7 7 7 7 7 7 7 5 : (3.62) The received signal at the destination (after three hops) is, therefore y (l) 3h = 4 X j=3 c (l) j h j;d +w (l) d ; i=3;4 (3.63) where the destination noise w (1) d » CN(0 8 ;I 8 ) and w (2) d » CN(0 4 ;I 4 ). Note that w (l) d isthenoisecomponentthatwasaddedonlyinthelasthop. Substituting(3.56) and (3.57) in (3.58-3.58), and then into (3.63) we get y (l) 3h = p ° (l) S (l) f (l) 3h + q ® (l) 1 W (l) 1;2 v (l) + q ® (l) 2 W (l) 3;4 g d +w (l) d , p ° (l) S (l) f (l) 3h +n (l) d (3.64) 75 where n (l) d collects all the noise terms and, S (l) = · © (l) 3 S (l) 1;2 © (l) 4 S (l) 1;2 ¸ ; ° (l) = 2´ 0 ´ 1 ´ 2 P (l) 1 P 2 ; ® (l) 1 = ´ 1 ´ 2 P (l) 1 P 2 ; ® (l) 2 = ´ 2 P 2 (3.65) f (1) 3h = · h 0;1 h 1;3 h 3;d h 0;2 h 2;3 h 3;d h 0;1 h 1;4 h 4;d h 0;2 h 2;4 h 4;d ¸ T (3.66) f (2) 3h = · h 0;1 h 1;3 h 3;d h ¤ 0;2 h 2;3 h 3;d h ¤ 0;1 h ¤ 1;4 h 4;d h 0;2 h ¤ 2;4 h 4;d ¸ T (3.67) v (1) = · h 1;3 h 3;d h 2;3 h 3;d h 1;4 h 4;d h 2;4 h 4;d ¸ T (3.68) v (2) = · h 1;3 h 3;d h 2;3 h 3;d h ¤ 1;4 h 4;d h ¤ 2;4 h 4;d ¸ T (3.69) g d = · h 3;d h 4;d ¸ T (3.70) W (1) 1;2 = · © (1) 1 w (1) 1 © (1) 2 w (1) 2 © (1) 4 © (1) 1 w (1) 1 © (1) 4 © (1) 2 w (1) 2 ¸ (3.71) W (2) 1;2 = · © (2) 1 w (2) 1 © (2) 2 w 0 (2) 2 © (2) 4 © (2) 1 w 0 (2) 1 © (2) 4 © (2) 2 w (2) 2 ¸ (3.72) W (1) 3;4 = · w (1) 3 © (1) 4 w (1) 4 ¸ ; W (2) 3;4 = · w (2) 3 © (2) 4 w 0 (2) 4 ¸ : (3.73) Conditioned on all channels h i;j and from (3.64), the overall noise n (l) d is complex Gaussian with zero mean. Its covariance matrix, for the RH scheme, is given by K (1) n d =(1+® (1) 2 jjgjj 2 )I 8 +® (1) 1 2 X i=1 £ (jh i;3 j 2 jh 3;d j 2 +jh i;4 j 2 jh 4;d j 2 )I 8 +h i;3 h 3;d h ¤ i;4 h ¤ 4;d (© (1) 4 ) T +h ¤ i;3 h ¤ 3;d h i;4 h 4;d © (1) 4 i (3.74) 76 whereRf:g indicates the real part, and for SME is K (2) n d =(1+® (2) 2 jjgjj 2 )I 4 +® (2) 1 ³ I (¡4) 4 jh 1;3 j 2 jh 3;d j 2 +I (¡2) 4 jh 1;4 j 2 jh 4;d j 2 +I (¡3) 4 jh 2;3 j 2 jh 3;d j 2 +I (¡1) 4 jh 2;4 j 2 jh 4;d j 2 ´ (3.75) where we de¯ne I (¡k) n to be the n£n identity matrix with k-th diagonal element set to zero. Taking the expectation of (3.74) and (3.75) over h i;j , the average noise covariances are, respectively ¹ K (1) n d = ¹ ¾ 2 n d I 8 and ¹ K (2) n d = ¹ ¾ 2 n d I 4 , where ¹ ¾ 2 n d =1+2 ´ 2 1+2´ 1 +4 ´ 1 ´ 2 (1+2´ 0 )(1+2´ 1 ) = ¹ ¾ 2 3h : (3.76) From (3.56), (3.57) and (3.65), the RH and SME codewords, denoted by S (1) and S (2) respectively, are S (1) = 2 6 6 4 S (1) S 0(1) 3 7 7 5 (3.77) S (2) = 2 6 6 6 6 6 6 6 6 6 6 4 s 1 s ¤ 2 s ¤ 3 0 ¡s 2 s ¤ 1 0 ¡s ¤ 3 ¡s 3 0 s ¤ 1 s 2 0 s 3 ¡s ¤ 2 s 1 3 7 7 7 7 7 7 7 7 7 7 5 ; (3.78) 77 where S 0(1) denotes the complex conjugate of S (1) and S (1) = 2 6 6 6 6 6 6 6 6 6 6 4 s 1 s 2 s 4 s 3 ¡s 2 s 1 s 3 ¡s 4 ¡s 3 s 4 ¡s 2 s 1 ¡s 4 ¡s 3 s 1 s 2 3 7 7 7 7 7 7 7 7 7 7 5 : (3.79) TheresultingcodestructureofS (1) in(3.77)isthewellknownrate-halfcomplex orthogonal design for 4 antennas[TJC99]. Furthermore, if we place the constraint that s 2 is taken from a real valued constellation (such as BPSK or PAM), the code structure of S (2) is a square matrix embeddable STBC for 4 antennas[TH02]. Due to their orthogonal structure, the resulting codeword di®erence matrices have rank 4, which would yield a 4 th order diversity gain if the components of f (l) 3h were i.i.d complex Gaussian. We emphasize, however, that this is not the case, since the components ofh (l) , although uncorrelated, are not independent and are formed by products of complex Gaussian channels. In fact, from the cut-set bounds[CT91], the maximal possible diversity is two, since there are two independent channels departing from the source and two arriving at the destination. For detection, the standard ML decoder for orthogonal STBC's can be used and it can be shown that its complexity grows linearly with the codeword dimen- sion. This is a very desirable feature since, as can be inferred from the preceding 78 derivation, the number of columns of the distributed ST codeword doubles with each hop. NotethattheSMEcodeisobtainedbyinsertingtwoAlamouti2£2codestruc- turesinthemainblockdiagonalandathirdsymbolintheo®diagonal. Designsfor higher dimensions can be successively derived in this fashion, by embedding lower dimensional codes into larger block sizes[TH02]. The advantages of this embedded design over the RH design are its smaller decoding delay (since it has smaller block size) and higher symbol rate when the number of antennas is at most 4, as shown in [TH02, TJC99]. 3.3.2 Diversity Analysis of RH and SME Protocols From (3.64), the received signal under the RH and SME schemes has similar struc- ture. Since RH and SME codes have the same rank, they have the same diversity performance. Therefore, we analyze the diversity of the RH scheme without loss of generality, and drop the corresponding superscript to simplify notation. Recalling (3.64), the received signal model is of the form y 3h = p °Sf 3h +n d (3.80) 79 wheref 3h = · h 0;1 h 1;3 h 3;d h 0;2 h 2;3 h 3;d h 0;1 h 1;4 h 4;d h 0;2 h 2;4 h 4;d ¸ T andSdenotesthe rate-half orthogonal STBC. The Cherno® Bound on the PEP becomes P e (S k !S j ;½)·P e;cb (S k !S j ;½)=E z det ¡1 £ I 4 +½M k;j diagfjz 1 j 2 ;:::;;jz 4 j 2 g ¤ (3.81) where ½ = ° 4¹ ¾ 2 3 and z, · h 0;1 h 3;d h 0;2 h 3;d h 0;1 h 4;d h 0;2 h 4;d ¸ . As before, we drop indicesk andj tosimplifynotationanddenotetheCherno®BoundP cb (S k !S j ;½) by P cb (½). Since M,M k;j =¸ rh I 4 is full rank due to orthogonality, we have P cb (½) = E z det ¡1 £ I 4 +½Mdiagfjz 1 j 2 ;:::;;jz 4 j 2 g ¤ (3.82) = E z ( 4 Y i=1 ¡ 1+½¸ rh diagfjz 1 j 2 ;:::;;jz 4 j 2 g ¢ ¡1 ) (3.83) Notethat, duetherankoftheRH/SMEcodesbeing4, theCherno®Boundexpres- sionin(3.82)hasadi®erentformwhencomparedtotheoneobtainedforAAF-MH, which uses a rank 2 Alamouti code. De¯ning ½ ¸ ,½¸ rh , averaging overjh 0;1 j 2 and jh 0;2 j 2 and denoting u,jh 3;d j 2 and z,jh 4;d j 2 , P cb (½ ¸ )=E u;z ·Z 1 0 (1+½ ¸ ux) ¡1 (1+½ ¸ zx) ¡1 e ¡x dx ¸ 2 : (3.84) Using the partial fraction expansion (1+½ ¸ ux) ¡1 (1+½ ¸ zx) ¡1 = A 1+½ ¸ ux + B 1+½ ¸ zx (3.85) 80 with A=u=(u¡z) and B =z=(z¡u), and substituting in (3.84) yields P cb (½ ¸ )= E u;z h u u¡z R 1 0 e ¡x 1+½ ¸ ux dx+ z z¡u R 1 0 e ¡z 1+½ ¸ zx dx i 2 (3.86) = E u;z ½ e 1 ½ ¸ u ½ ¸ (u¡z) h -Ei ³ ¡ 1 ½ ¸ z ´i + e 1 ½ ¸ z ½ ¸ (z¡u) h -Ei ³ ¡ 1 ½ ¸ z ´i ¾ 2 (3.87) Averaging over u and z yields the double-integral form P cb (½ ¸ )= 1 ½ 2 ¸ Z 1 0 Z 1 0 f(x;y)e ¡ x+y ½ ¸ dxdy: (3.88) where f(x;y)= 0 @ e 1 x £ -Ei(¡ 1 x ) ¤ ¡e 1 y h -Ei(¡ 1 y ) i x¡y 1 A 2 (3.89) We can split the integral in (3.88) as P cb (½ ¸ )= 1 ½ 2 ¸ ·Z ® 0 Z ® 0 f(x;y)e ¡ x+y ½ ¸ dxdy+2 Z 1 ® Z ® 0 f(x;y)e ¡ x+y ½ ¸ dxdy + Z 1 ® Z 1 ® f(x;y)e ¡ x+y ½ ¸ dxdy ¸ : (3.90) which for large ½ ¸ (high SNR) is dominated by the last term. This is because the exponential pd¯s approximately constant for high ½ ¸ (if z is not large), since the slope p 0 (z)¼¡ 1 ½ 2 ¸ is very small. Thus, a lower bound on P cb (½ ¸ ) can be written as P cb (½ ¸ )>P cb;l (½ ¸ ), 1 ½ 2 ¸ Z 1 ® Z 1 ® f(x;y)e ¡ x+y ½ ¸ dxdy (3.91) 81 while an upper bound is given as P cb (½ ¸ )<P cb;u (½ ¸ ), 4 ½ 2 ¸ Z 1 ® Z 1 ® f(x;y)e ¡ x+y ½ ¸ dxdy (3.92) which results from replacing the each of the three terms in (3.90) by the last term. Hence,theupperandlowerboundsonlydi®erbyaconstant,whichhasnoimpacton the diversity. Using the leading term from the series expansion of the Exponential Integral (3.29) in (3.89) yields f(x;y)= Ã e 1 x (logx+O(1))¡e 1 y (logy+O(1)) x¡y ! 2 (3.93) Retaining the most dominant term of the numerator in (3.93), the Cherno® Bound has the form P cb (½ ¸ )¼ K ½ 2 ¸ Z 1 ® Z 1 ® " (e 1 x logx¡e 1 y logy) x¡y # 2 e ¡ x+y ½ ¸ dxdy (3.94) for some constant K. Using the series expansion of e 1 x and e 1 y yields P cb (½ ¸ )¼ K ½ 2 ¸ Z 1 ® Z 1 ® " (logx¡logy)+ P 1 n=1 ( 1 n!x n logx¡ 1 n!y n logy) x¡y # 2 e ¡ x+y ½ ¸ dxdy (3.95) 82 Itisstraightforwardtoshowthat,forx¸®andy¸®,jlogx¡logyj¸j 1 n!x n logx¡ 1 n!y n logyj;n¸1. This allows the further simpli¯cation P cb (½ ¸ )¼ K ½ 2 ¸ Z 1 ® Z 1 ® " log( x y ) x¡y # 2 e ¡ x+y ½ ¸ dxdy (3.96) Recalling the Taylor series expansion of log(z) =2 P 1 k=1 1 2k¡1 ¡ z¡1 z+1 ¢ 2k¡1 ; for z >0, and replacing z =x=y, we have: log(x=y) x¡y = 2 x+y + 2 3 (x¡y) 2 (x+y) 3 + 2 5 (x¡y) 4 (x+y) 5 +:::= log(x=y) x¡y = 2 x+y +O ³ 1 x+y ´ p ;p > 1. By retaining the ¯rst term of the expansion we can thus obtain the approximation of (3.96) as P cb (½ ¸ )¼ 4K ½ 2 ¸ Z 1 ® Z 1 ® 1 (x+y) 2 e ¡ x+y ½ ¸ dxdy (3.97) By employing the substitutions z = x+y and w = x¡y the double integral in (3.97) can be reduced to a single integral yielding P cb (½ ¸ )¼ 4K ½ 2 ¸ Z 1 2® z¡2® z 2 e ¡ z ½ ¸ = 4K ½ 2 ¸ · ¡Ei µ ¡ 2® ½ ¸ ¶ ¡E 2 µ 2® ½ ¸ ¶¸ (3.98) where the function E 2 (z) is de¯ned as[AS64] E 2 (z)= Z 1 1 e ¡zt t 2 dt= ¡z (n¡1)! [¡log(z)¡°+0:5]¡ 1 X m=0;m6=n¡1 (¡z) m (m¡n+1)m! (3.99) 83 Substituting (3.99) and (3.29) in (3.97) yields that the leading term of P cb (½ ¸ ), denoted by P ¤ cb (½ ¸ ) is of the form P ¤ cb (½ ¸ )= K 0 ½ 2 ¸ log(½ ¸ ) (3.100) which yields a diversity slope of d RH;SME =2¡ 1 log(½) (3.101) To justify the approximations performed in this preceding derivations, Figure 3.4 compares the numerical evaluation of the exact Cherno® PEP bound in (3.88) with the error probability slope predicted by (3.101). At high SNR, it is clear that the analytical result becomes increasingly more accurate. Comparing (3.101) with (3.44),theRHandSMEschemesachievehigherdiversitythanthatoftheAAF-MH schemeforthreehops. Thesimulatedperformanceforrate1.0(bit/channeluse)RH andSMEschemesisshowninFigure3.5. Itcanalsobeveri¯edthat, athigh ½, the diversityperformanceshowsgoodagreementwiththeanalyticalresultin(3.101). A moregeneralcomparisonofthepotentialgainsachievedviathecooperationschemes analyzed in this chapter is given in Figure 3.6, which shows the block error rate versus the total average transmit power over all terminals. A path loss exponent of ·=4 is assumed, as well as a ¯xed S-D distance, hence, yielding shorter distances per hop as the number of hops increases. Clearly, all the cooperative schemes 84 0 5 10 15 20 25 30 35 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 PEP Chernoff bound for the RH−3 hop scheme PEP ρ λ (dB) Numerical Error prob. slope, analytical Figure 3.4: RH PEP and diversity slope achieve signi¯cant performance improvement over the noncooperative (single-hop) transmission, as expected. Comparing the three-hop AAF-MH and two-hop AAF- 2H schemes, we observe that AAF-MH has the best performance in the low and moderate power regimes (due to the shorter distance between hops) but worse for high power, which is explained by the larger diversity loss of the AAF-MH scheme,aspredictedbyouranalysis. Thethree-hopRHscheme,ontheotherhand, although o®ering worse performance than the AAF schemes at low and moderate power, achieves higher diversity and, hence, the best performance at high power. 85 −10 −5 0 5 10 15 20 25 30 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Performance of RH and SME for 3 hops ρ (dB) Block Error Rate SME, simulation RH, simulation RH, simulation, AWGN approx. Error prob. slope, analytical Figure 3.5: RH and SME schemes. 3.4 Practical Issues: Asynchronous Relays In this section, we examine the e®ect of synchronization errors on the performance of cooperative protocols. Thus far, all the cooperative schemes proposed in this chapter were analyzed assuming that all relays within each transmission stage were symbol-synchronized. While this assumption is quite realistic in the multi-antenna setup we are trying to emulate, it is not necessarily feasible for a distributed sce- nario. Furthermore, even under the symmetric relay topology considered in this chapter, such as in Figure 3.1, where signal travelling times are roughly the same, it is important to assess the performance loss incurred due to timing drifts that may occur between relays. In our analysis, we consider, for simplicity, the two-hop 86 25 30 35 40 45 50 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 Performance of cooperative protocols under path−loss attenuation, κ = 4 Total transmit power (dB) Block Error Rate RH−3 hop AAF−MH (3 hop) AAF−2H (2 hop) Non cooperative (1 hop) Figure 3.6: AAF-2H, AAF-MH and RH comparison in the presence of path-loss attenu- ation AAF-2Hscheme. Thedevelopmenthereincanbegeneralizedtoanyoftheproposed cooperative schemes. When symbol synchronization errors are present, the signal at a receiving node containscontributionsfromseveraltransmittedsymbolsforeachtransmittingnode. Hence each asynchronous communication link can be viewed as channel with inter- symbol interference (ISI). Due to the half-duplex constraint, the ISI present in every received block can be assumed to be constrained within its own symbols, i.e., there is no inter-block interference. Therefore, the AAF-2H received signal can be written as 87 y(k)=x T 1;2 (k) 2 6 6 4 h 1;d p 1 (¡1) h 1;d p 1 (0) h 1;d p 1 (1) h 2;d p 2 (¡1) h 2;d p 1 (0) h 2;d p 2 (1) 3 7 7 5 +w d (k),x T 1;2 (k)G d +w d (k) (3.102) where the ISI tap vector for relay i is denoted by p i , · p i (¡1) p i (0) p i (1) ¸ : (3.103) Note that (3.102) in fact generalizes the model (3.3) to account for the presence of ISI. Let ¿ =¿ 1 ¡¿ 2 denote the di®erence in arrival times (assumed to be perfectly known by the receiver) between the signals from the two relays. It turns out that the optimal sampling time at the receiver for maximum energy capture is halfway between ¿ 1 and ¿ 2 [MHSD05]. In this case, assuming without loss of generality that ¿ 1 >¿ 2 , we have p 1 = · R p (¡T + ¿ 2 ) R p ( ¿ 2 ) R p (T + ¿ 2 ) ¸ (3.104) p 2 = · R p (¡T ¡ ¿ 2 ) R p (¡ ¿ 2 ) R p (T ¡ ¿ 2 ) ¸ (3.105) where T is the symbol time and R p (t) is the pulse shape function. Since the pulse shapeissymmetric,wecandropthesubscriptinp i andde¯nep= · p ¡1 p 0 p 1 ¸ , p 1 . 88 From(3.102)andfollowingthederivationsinSection3.2.1, itisstraightforward to obtain the general model for the received signal under the AAF-2H scheme 2 as y 2h = p ° 2h · p 0 S S 0 ¸ 2 6 6 6 6 6 6 6 6 6 6 4 h 0;1 h 1;d h ¤ 0;2 h 2;d h 0;1 h 1;d h ¤ 0;2 h 2;d 3 7 7 7 7 7 7 7 7 7 7 5 +n d , p ° 2h S 0 2 6 6 6 6 6 6 6 6 6 6 4 h 0;1 h 1;d h ¤ 0;2 h 2;d h 0;1 h 1;d h ¤ 0;2 h 2;d 3 7 7 7 7 7 7 7 7 7 7 5 +n d (3.106) where S is the Alamouti codeword as in (3.7), and S 0 , 2 6 6 4 s(2)p ¡1 s ¤ (1)p 1 s(1)p 1 ¡s ¤ (2)p ¡1 3 7 7 5 (3.107) and the overall noise n d is n d =w d + r ´ 1 P 1 [Q 1 w 1 h 1;d +Q 2 ~ w 2 h 2;d ] (3.108) where ^ w 1 , · w 1 (2) w 1 (1) ¸ T ; Q 1 , 2 6 6 4 p 0 p ¡1 p 1 p 0 3 7 7 5 ; Q 2 , 2 6 6 4 p 0 p 1 p ¡1 p 0 3 7 7 5 (3.109) 2 The signal model can be generalized to AAF-MH and RH/SME schemes in an exactly anal- ogous fashion. 89 Hence, from the receiver point of view, ML detection under the asynchronous case consists of detecting the expanded version of the transmitted codeword S 0 from the received signal (3.106). It is straightforward to verify that the previous diversity analysis holds since S 0 retains the rank of S and, hence, the diversity performance at high SNR is the same. Figure 3.7 illustrates the performance of the AAF-2H scheme under several asynchronous scenarios. A pulse shape belonging to the raised-cosine family [Pro95] with rollo® factor ¯ = 0:5 is assumed. Note that, whileperformancedegradesas¿ increases,thediversityslopeoftheideal(perfectly synchronous) case is maintained. In fact, for ¿ = 0:1T the simulated performance was near identical to the the ideal case (and thus the ideal curve is omitted for clarity), which con¯rms the scheme's robustness to small synchronization errors between relays. 3.5 Conclusions In this chapter, cooperative protocols for distributed space-time multi-hop com- munications were proposed and analyzed. It was shown that an optimal power allocation strategy for the relays is to distribute power uniformly over all hops. Furthermore, by employing a SNR-dependent de¯nition of diversity, the achiev- able spatial diversity for each protocol was quanti¯ed for speci¯c schemes involving DSTBCs: the AAF-MH, the rate-half (RH) and the square matrix embeddable (SME) protocols. It was shown that the AAF-MH scheme has the drawback of an 90 −5 0 5 10 15 20 25 30 35 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 ρ (dB) Block Error Rate AAF−2H asynchronous performance, β=0.5 AAF−2H, simulation, τ=0.1T AAF−2H, simulation, τ=0.5T AAF−2H, simulation τ=0.8T Error prob. slope, AAF−2H, analytical Figure 3.7: Two-hop performance with asynchronous relays increasingdiversityloss,asthenumberofhopsincreases. Toalleviatethisproblem, new distributed coding strategies were proposed. Finally, the performance under asynchronous relays was also characterized and showed to be robust to even large synchronization errors. 3.6 Appendix A: Power Allocation In this section, we obtain the optimal power allocation for the AAF-MH, RH and SMEschemesbyoptimizingtheSNRatthedestination. Weuseinductiontoprove thattheequalpowerallocationstrategyisoptimalforanarbitrarynumberofhops. Without loss of generality, we assume an AAF-MH protocol, since from (3.9) and 91 (3.64)andthenoisevariances(3.14)and(3.76)theSNRforthethree-hopAAF-MH and RH schemes are the same and given by SNR= (2´ 0 )(2´ 1 )(2´ 2 ) P 1 P 2 h 1+2 ´ 2 P 2 ³ 1+2 ´ 1 P 1 ´i (3.110) and for the SME scheme it is 4 3 SNR. We denote the SNR of each relay of stage i by SNR i , S i N i , where S i and N i are, respectively, the received signal and noise powers at each relay of stage i. After L hops, the following recursive relation holds SNR ¡1 L (´ L¡1 )=SNR ¡1 L¡1 (´ L¡2 )+ F(´ L¡2 ) 2´ L¡1 (3.111) where´ k ,[´ 0 ;´ 1 ;:::;´ k ],F(´ L¡2 )= Q L¡2 k=0 ³ 1+ 1 2´ k ´ andthenotationSNR ¡1 i (´ i¡1 ) emphasizes that SNR ¡1 i depends only on the powers ´ 0 ;´ 1 ;:::;´ i¡1 . We wish to minimize SNR ¡1 L under the power constraint P = 2 P L¡1 i=0 ´ i . First we express the power constraint as a sum of two parts, which depend only on ´ L¡2 and ´ L¡1 respectively. Thus, P = L¡2 X i=0 2´ i +2´ L¡1 =®P +(1¡®)P; 0<®< 1; ® = P L¡2 i=0 2´ i P = ¾ L¡2 P (3.112) 92 The optimal power allocation vector for L hops can be found by solving (´ ¤ L¡2 ;® ¤ ) = arg min 0<®<1 min ¾ L¡2 =®P Ã SNR ¡1 L¡1 (´ L¡2 )+ F(´ L¡2 ) P(1¡®) ! (3.113) It is equivalent to carry out the minimization over in two parts: ¯rst over ´ L¡2 (¯xing®)andthenover®. ItcanbeshownthattheminimizationofSNR ¡1 i¡1 (´ L¡2 ) over ´ L¡2 is found by setting ´ ¤ L¡2 = µ ®P 2(L¡1) ;:::; ®P 2(L¡1) ¶ : (3.114) Substituting (3.114), the expression for the SNR ¡1 over L hops now becomes SNR ¡1 L = L¡1 ®P " L¡2 X i=0 µ 1+ L¡1 ®P ¶ L¡i¡2 # + 1 P(1¡®) µ 1+ L¡1 ®P ¶ L¡1 : (3.115) Di®erentiatingwithrespectto®wegetitsoptimumvalue® ¤ = L¡1 L ,fromwhich we get ´ ¤ i = P 2L i=0;:::;L¡1 (3.116) which is the ¯nal result. We have therefore shown that uniform power allocation is optimal for any number of hops. Finally, the optimal SNR value for a given total power P and number of hops L is found by substituting ® ¤ in (3.115) SNR L;opt = " µ L+P P ¶ L ¡1 # ¡1 : (3.117) 93 3.7 Appendix B: Diversity Analysis in the AAF- MH Scheme: General Case Under the general AAF-MH scheme, all the relays beyond the ¯rst stage perform identical operations of repetition and ampli¯cation. Following the argument in the previous section, this type of processing retains the maximum rank of the resulting DSTBC at the destination. Therefore, the Cherno® Bound on the PEP is given by P cb (½)=E z 1 ;z 2 L¡1 det ¡1 £ I 2 +2 L¡2 ½M k;j diagfjz 1 j 2 ;jz 2 L¡1j 2 g ¤ (3.118) where z 1 =h 0;1 Q L¡1 i=1 h 2i¡1;2i+1 , z 2 L¡1 =h ¤ 0;2 Q L¡1 i=1 Q L¡1 i=2 h 2i;2i+2 , and½= ° Lh 4¹ ¾ 2 Lh is the L-hop SNR. De¯ning ½ ¸ = ¸2 L¡2 ° Lh 4¹ ¾ 2 L and following similar steps as in the previous section, averaging over the channel gains, the Cherno® Bound generalizes to P cb (½ ¸ )= 2 Y i=1 1 ½ ¸ Z 1 0 ¢¢¢ Z 1 0 e 1 Q L¡2 k=1 x i;k Q L¡2 k=1 x i;k " ¡Ei Ã ¡ 1 Q L¡2 k=1 x i;k !# £ e ¡ P L¡2 k=1 x i;k ½ ¸ dx i;1 ¢¢¢dx i;L¡2 (3.119) Once again, since the expression in (3.20) cannot be evaluated in closed-form (to our best knowledge), we resort to upper and lower bounds to provide insight on the performance of the general L-hop AAF-MH scheme. 94 3.7.1 Lower Bound From the bound (3.24), we can write a lower bound to (3.20) as P cb (½ ¸ )> 2 Y i=1 1 ½ ¸ Z 1 0 ¢¢¢ Z 1 0 log(1+2 Q L¡2 k=1 x i;k ) 2 Q L¡2 k=1 x i;k e ¡ P L¡2 k=1 x i;k ½ ¸ dx i;1 ¢¢¢dx i;L¡2 (3.120) which can be further bounded by restricting the integration interval as P cb (½ ¸ ) > 0 @ 1 2½ ¸ Z 1 1 ¢¢¢ Z 1 1 log ³ Q L¡2 k=1 x k ´ Q L¡2 k=1 (1+x k ) e ¡ P L¡2 k=1 x k ½ ¸ dx 1 ¢¢¢dx L¡2 1 A 2 (3.121) = Ã 1 2½ ¸ Z 1 1 ¢¢¢ Z 1 1 P L¡2 k=1 log(x k ) Q L¡2 k=1 (1+x k ) e ¡ P L¡2 k=1 x k ½ ¸ dx 1 ¢¢¢dx L¡2 ! 2 (3.122) = " L¡2 2½ ¸ Z 1 1 ¢¢¢ Z 1 1 log(x 1 ) L¡2 Y k=1 Ã e ¡ x k ½¡¸ x k +1 ! dx 1 ¢¢¢dx L¡2 # 2 (3.123) Performing the integration over x 1 in (3.123) yields P cb (½ ¸ ) > " L¡2 2½ ¸ Z 1 1 ¢¢¢ Z 1 1 · 1 2 e 1 ½ ¸ Ei 2 µ ¡ 1 ½ ¸ ¶¸ L¡2 Y k=2 Ã e ¡ x k ½ ¸ x k +1 ! dx 2 ¢¢¢dx L¡2 # 2 = 2 4 L¡2 4½ ¸ e 1 ½ ¸ Ei 2 µ ¡ 1 ½ ¸ ¶ Ã Z 1 1 e ¡ x ½ ¸ x+1 dx ! L¡3 3 5 2 (3.124) > 2 4 L¡2 4½ ¸ e 1 ½ ¸ Ei 2 µ ¡ 1 ½ ¸ ¶ Ã Z 1 1 e ¡ x+1 ½ ¸ x+1 dx ! L¡3 3 5 2 (3.125) From [GR00] we have Z 1 1 e ¡ x+1 ½ ¸ x+1 dx=¡Ei µ ¡ 2 ½ ¸ ¶ (3.126) 95 Hence, the desired expression for the lower bound is P cb (½ ¸ )> (L¡2) 2 16½ 2 ¸ e 2 ½ ¸ Ei 4 µ ¡ 1 ½ ¸ ¶· ¡Ei µ ¡ 2 ½ ¸ ¶¸ 2(L¡3) : (3.127) For large ½ ¸ , the series expansion of¡Ei( 1 ½ ¸ ) is dominated by log(½ ¸ ). Therefore, P cb (½ ¸ )>P cb;l (½ ¸ ),K 0 l (log 2 ½ ¸ ) (L¡1) ½ 2 ¸ : (3.128) 3.7.2 Upper Bound Starting from (3.119), we ¯rst break each integral into two integrals: the ¯rst from 0 to ® and the second from ® to1. We then use (3.33) to obtain the upper bound P cb (½ ¸ )· 0 @ 1 ½ ¸ Z ::: Z C e 1 Q L¡2 k=1 x k Q L¡2 k=1 x k " ¡Ei Ã ¡ 1 Q L¡2 k=1 x k !# e ¡ P L¡2 k=1 x k ½ ¸ dx 1 ¢¢¢dx L¡2 + 1 ½ ¸ Z 1 ® ¢¢¢ Z 1 ® e 1 Q L¡2 k=1 x k Q L¡2 k=1 x k " L¡2 X k=1 log(x k ) # e ¡ P L¡2 k=1 x k ½ ¸ dx 1 ¢¢¢dx L¡2 1 A 2 (3.129) whereC denotestheregioninR (L¡2) + inwhichoneormoreofthex i 'ssatisfyx i <®. Notethat the¯rstintegraloverC isin factasum of2 L¡2 ¡1integrals, eachin turn being less than the second integral in (3.129). 96 We can now write P cb (½ ¸ ) · " (L¡2)2 L¡2 ½ ¸ Z 1 ® ¢¢¢ Z 1 ® e 1 Q L¡2 k=1 x k log(x 1 ) L¡2 Y k=1 Ã e ¡ x k ½ x k ! dx 1 ¢¢¢dx L¡2 # 2 · " e(L¡2)2 L¡2 ½ ¸ Z 1 1 ¢¢¢ Z 1 1 log(x 1 ) L¡2 Y k=1 Ã e ¡ x k ½ x k ! dx 1 ¢¢¢dx L¡2 # 2 (3.130) Integration by parts of (3.130) over x 1 yields P cb (½ ¸ )· " e(L¡2)2 L¡2 2½ ¸ Z 1 1 ¢¢¢ Z 1 1 ³ log 2 (x 1 )e ¡ x 1 ½ ¸ dx 1 ´ L¡2 Y k=2 Ã e ¡ x k ½ x k ! dx 2 ¢¢¢dx L¡2 # 2 : (3.131) Upper bounding the integral over x 1 yields the desired expression P cb (½ ¸ )· " (L¡2)2 L¡2 e 2 µ ¼ 2 6 +(°¡log½ ¸ ) 2 ¶· ¡Ei µ ¡ 1 ½ ¸ ¶¸ L¡3 # 2 (3.132) Since the series expansion of¡Ei(¡ 1 ½ ¸ ) has leading term given by log(½ ¸ ) for high ½ ¸ , P cb (½ ¸ )·P cb;u (½ ¸ ),K 0 u (log 2 ½ ¸ ) (L¡1) ½ 2 ¸ : (3.133) which has the same asymptotic behavior as the lower bound (3.128). Applying the diversity de¯nition (3.41) we ¯nd that the diversity of the AAF-MH scheme over L hops is d Lh (½)=2¡ 2(L¡1) log½ : (3.134) 97 Chapter 4 Distributed Space-Time Cooperative Schemes for Underwater Acoustic Communications In resource limited, large scale underwater sensor networks, cooperative communi- cation over multiple hops o®ers opportunities to save power. Intermediate nodes between source and destination act as cooperative relays. Herein, protocols cou- pled with space-time block code (STBC) strategies are proposed and analyzed for distributed cooperative communication. Amplify-and-forward type protocols are considered, in which intermediate relays do not attempt to decode the information. The Alamouti-based cooperative scheme proposed by Hua et al (2003) for °at- fading channels is generalized in order to work in the presence of multipath, thus addressing a main characteristic of underwater acoustic channels. A time-reversal distributed space-time block code (TR-DSTBC) is proposed, which extends the dual-antennaTR-STBC(timereversalspace-timeblockcode)approachfromLind- skog and Paulraj (2000) to a cooperative communication scenario for signaling in 98 multipath. It is ¯rst shown that, just as in the dual-antenna STBC case, TR along withtheorthogonalityoftheDSTBCessentiallyallowsfordecouplingofthevector ISI detection problem into separate scalar problems, and thus yields strong perfor- mance (compared with single hop communication) and with substantially reduced complexity over non-orthogonal schemes. Furthermore, a performance analysis of the proposed scheme is carried out to provide insight on the performance gains, which are further con¯rmed via numerical results based on computer simulations and ¯eld data experiments. 4.1 Introduction Underwater sensor networks form an emerging technology paradigm that promises toenableorenhanceseveralkeyapplicationsinoceanicresearch, suchas: datacol- lection, pollution monitoring, tactical surveillance and disaster prevention[APM05, SSP00]. Exploiting sensor cooperation for terrestrial communications has attracted con- siderable recent attention, in order to increase reliability, coverage, throughput and capacity ([GB03] and references therein). A common feature among many of these networks is their multi-hop nature: communication is performed between a source and destination via intermediate terminals. This method provides several advan- tages over single hop schemes[GB03]: a) combating the severe signal attenuation over long distances, and therefore, saving transmission power; b) providing signal 99 pathsbetweenterminalswhichdonothaveadirectlineofsightbetweenthem; and c) providing multiple communication links for applications with a high data rate requirement which cannot be satis¯ed via a single link. Multihop networks can also provide additional gains through cooperation be- tween terminals. Recent information theoretic results show that cooperation can increasetheoverallcapacityofthesenetworksbytakingadvantageoftheirinherent increaseinspatialdiversity[SEA03a,SEA03c,LW03]. Anaturalwaytoexploitthis diversity is via Distributed Space-Time Block Coding (DSTBC) originally proposed in [LW03]. The goal of a DSTBC-based protocol is to allow the cooperating termi- nals to act, from the destination point of view, as a multi-antenna array employing a well designed Space-Time Block Code (STBC)[TJC99]. Several DSTBC schemes have been recently proposed [LW03, HMC03, ALK03, NBK04, JH06]. The idea that DSTBC schemes could be applied to underwater networks sug- gests itself naturally. The underwater acoustic channel, however, poses additional di±culties to the design of such communication protocols. The major challenges posed by underwater channels are[APM05]: severe range-dependent attenuation, extensive multipath propagation and highly variable propagation delays (due to slow sound propagation). In this chapter, we consider the problem of underwater communication between a single source and destination terminal. Data is relayed in a multi-hop fash- ion, through intermediate sensor nodes placed between the source and destination. 100 Communication protocols based on distributed space-time coding (DSTBC) are considered. The time reversal STBC (TR-STBC) approach proposed in [LP00] for dual co-located antennas and signaling in multipath, is extended to a dis- tributed communication scenario. The key feature of this approach is to allow remote terminals that not directly wired to the source to relay information to the destination, hence acting as virtual antennas. We show that, as in the STBC case [LP00, LSLL02], TR along with the orthogonality of the DSTBC essentially allows fordecouplingofavectorISIdetectionproblemintoseparatescalarproblems,with- out loss of optimality (neglecting "border e®ects" [LSLL02]) and, therefore, o®ers excellent performance with signi¯cant complexity reduction over non-orthogonal schemes. This chapter is organized as follows. Section 4.2 introduces the signal model and the TR-DSTBC scheme for 2 relays. Section 4.3 presents a diversity analysis of the proposed scheme employing a decision-feedback equalizer (DFE) at the re- ceiver, under standard simplifying assumptions. The robustness of the scheme to asynchronous relay operation is motivated in Section 4.4. A generalization of the scheme to more than two relays is presented in Section 4.5, based on the rate 1/2 orthogonal STBC. The ray-based underwater channel model used in our computer simulations, which is essentially based on [Sto05, ZYW95], is presented in Section 4.6. Numerical simulations and ¯eld test experimental results, which con¯rm the potential gains of the proposed approach, are presented in Section 4.7, which also 101 addresses some practical issues such as imperfect channel state information (CSI) at the receiver and the presence of channel time-variations. Section 4.8 presents the conclusions. Throughout the chapter, scalar quantities are denoted with lower or upper case normal font, vectors are denoted with bold faced lower case fonts, and matrices are denoted with bold faced upper upper case fonts. The symbol '*' denotes complex conjugate and the superscript 'H' denotes Hermitian (complex conjugate trans- pose). The distribution of a complex Gaussian random variable with mean ¹ and variance ¾ 2 will be denoted byCN(¹;¾ 2 ). 4.2 Signal Model We ¯rst consider the discrete-time signal model for a scenario with a single source terminal(S)communicatingtoadestinationterminal(D)viaastageoftwowireless relays as depicted in Figure 5.1. Since the channels between the multiple links contain ISI, we will employ a discrete-time model. To clarify the notation, for a generic input u(t) and channel ¯lter a(q ¡1 ) - with q ¡1 denoting the delay operator - the output v(t) is given by v(t)=a(q ¡1 )u(t)= Ã Lc X i=0 a i q ¡i ! u(t)= Lc X i=0 a i u(t¡i); t=1;:::;N (4.1) 102 S D R 1 R 2 g 1 (q -1 ) g 2 (q -1 ) f 2 (q -1 ) f 1 (q -1 ) Figure 4.1: Two-hop cooperative network with 2 relays. where L c +1 is the number of channel taps and N is the block size. It is useful to note that v(N¡t+1)=a(q ¡1 )u(N¡t+1)= Ã L a X i=0 a i q ¡i ! u(N¡t+1) t=1;:::;N (4.2) Denoting by time-reversed input and output by ¹ u(t), u(N ¡t+1) and ¹ v(t), v(N¡t+1) respectively, we have from (4.2) ¹ v(t)=v(N¡t+1)=a(q ¡1 )u(N¡t+1)=a(q)¹ u(t) (4.3) 103 Let h i (q ¡1 ) and g i (q ¡1 ) denote the S¡R i and R i ¡D channels respectively. Throughout this chapter, we shall assume that all channels are independent, with taps that are independently fading and quasi-static (time invariant for a duration 2N plus any required guard intervals, as explained next). Thesourcedividesitstransmissionsymbolstreams(t)intotwoblockss 1 (t)and s 2 (t), each of length N and transmits them separated by a guard band to avoid interblock interference. For the same reason, a preamble and a tail are inserted at the beginning and at the end of s(t) respectively [LSLL02]. The received signal at R i at sampling times t corresponding to the ¯rst transmission block is r i;1 (t)= p E s h i (q ¡1 )s 1 (t)+w i;1 (t); t=1;:::;N (4.4) and similarly for the second transmission block r i;2 (t)= p E s h i (q ¡1 )s 2 (t)+w i;2 (t); t=1;:::;N (4.5) Thus, r i;k (t) is the received signal at time t, at relay i and block k, where k =1;2. Thesignals k (t)isthet-thsourcetransmittedsymbolofblockk andistakenfroma PSKorQAMsymbolconstellation. TheAWGNsequence w i;k (t)hasunitvariance. Furthermore, w i;1 (t) and w i;2 (t) are assumed to be independent. The energy per transmit symbol is denoted by E s . 104 We assume, due to complexity and power limitations, that the relays can only perform amplify and forwarding type operations on their received signals (ampli- ¯cation, complex conjugation or time shift). No channel estimation or symbol detection is performed. In the case of °at fading, the work by Hua [HMC03] has shown how an Alamouti-type [Ala98] processing can be employed by the relays to achieve diversity gains. We now describe its extension to multipath channels, following an approach similar to [LSLL02, LP00] Both R 1 and R 2 transmit two blocks. Let u i;k (t) denote the signal transmitted by R i over block k and time t. In the ¯rst block R 1 and R 2 transmit, respectively, the following signals: u 1;1 (t) = r E r K 1 r 1;1 (t) (4.6) u 2;1 (t) = r E r K 2 ¹ r ¤ 2;2 (t) (4.7) where K i ;i = 1;2 is a normalizing factor applied to the received signal of relay i, to make it unit power, and E r is the transmit energy per symbol for each relay. In the second block, the transmitted signals are u 1;2 (t) = r E r K 1 r 1;2 (t) (4.8) u 2;2 (t) = ¡ r E r K 2 ¹ r ¤ 2;1 (t) (4.9) 105 We note that only R 2 conjugates and time-reverses during both of its blocks. Thisisdi®erentfromtheapproachin[LP00]wheretwoco-locatedantennasperform conjugationandtime-reversaloverthesecondblock. Althoughbothapproachesare equivalent for the STBC scenario, the latter would violate the quasi-static channel assumption of our DSTBC setup. The received signal at the destination for block k(k =1;2) is therefore given by y k (t)=g 1 (q ¡1 )u 1;k (t)+g 2 (q ¡1 )u 2;k (t)+n k (t) (4.10) Substituting the expressions for u i;k (t) from above we have y 1 (t)= r E s E r K 1 g 1 (q ¡1 )h 1 (q ¡1 )s 1 (t)+ r E s E r K 2 g 2 (q ¡1 )h ¤ 2 (q)¹ s ¤ 2 (t)+n 0 1 (t) (4.11) y 2 (t)= r E s E r K 1 g 1 (q ¡1 )h 1 (q ¡1 )s 2 (t)¡ r E s E r K 2 g 2 (q ¡1 )h ¤ 2 (q)¹ s ¤ 1 (t)+n 0 2 (t) (4.12) where n 0 1 (t) = n 1 (t)+ r E r K 1 g 1 (q ¡1 )w 1;1 (t)+ r E r K 2 g 2 (q ¡1 )¹ w ¤ 2;1 (t) (4.13) n 0 2 (t) = n 2 (t)+ r E r K 1 g 1 (q ¡1 )w 1;2 (t)¡ r E r K 2 g 2 (q ¡1 )¹ w ¤ 2;2 (t) (4.14) 106 andn k (t)isAWGNwithunitvariance. Itfollows,therefore,thatthepowerspectral density (PSD) of n 0 i (t), denoted by X(q ¡1 ;q) is given by X(q ¡1 ;q)=1+ E r K 1 g 1 (q ¡1 )g ¤ 1 (q)+ E r K 2 g 2 (q ¡1 )g ¤ 2 (q) (4.15) where we keep the PSD notation in the q domain instead of switching to the more conventional domain z, for notation simplicity. Computing the time reversal and conjugation of y 2 (t) we obtain ¹ y ¤ 2 (t)= r E s E r K 1 g ¤ 1 (q)h ¤ 1 (q)¹ s ¤ 2 (t)¡ r E s E r K 2 g ¤ 2 (q)h 2 (q ¡1 )s 1 (t)+~ n 2 (t) (4.16) where ~ n ¤ 2 = ¹ n 0 2 . De¯ning y(t)=[y 1 (t) ¹ y ¤ 2 (t)] T we have y(t)=H(q ¡1 ;q) 2 6 6 4 s 1 (t) ¹ s ¤ 2 (t) 3 7 7 5 + 2 6 6 4 n 0 1 (t) ~ n 2 (t) 3 7 7 5 (4.17) where the equivalent S¡D channel matrix is H(q ¡1 ;q), 2 6 6 4 q E s E r K 1 g 1 (q ¡1 )h 1 (q ¡1 ) q E s E r K 2 g 2 (q ¡1 )h ¤ 2 (q) ¡ q E s E r K 2 g ¤ 2 (q)h 2 (q ¡1 ) q E s E r K 1 g ¤ 1 (q)h ¤ 1 (q) 3 7 7 5 (4.18) 107 Note that the special case of °at fading S-R and R-D channels and K 1 = K 2 = K corresponds to the channel matrix H= r E s E r K 2 6 6 4 g 1 h 1 g 2 h ¤ 2 ¡g ¤ 2 h 2 g ¤ 1 h ¤ 1 3 7 7 5 (4.19) which has an Alamouti[Ala98] structure, as observed in [HMC03]. Assuming per- fect CSI, the receiver processes y(t) (given by (4.17)) with the space-time ¯lter H H (q ¡1 ;q) (as in [LP00]) followed by a whitening ¯lter. Due to the orthogonality of H(q ¡1 ;q), we have H H (q ¡1 ;q)H(q ¡1 ;q)= Ã 2 X i=1 E s E r K i g ¤ i (q)g i (q ¡1 )h ¤ i (q)h i (q ¡1 ) ! I,f(q ¡1 )f ¤ (q)I (4.20) where I is the identity matrix and we de¯ne f(q ¡1 ) as the result of the spectral factorization of the coe±cient of the identity matrix in (4.20). Hence, f(q ¡1 )f ¤ (q)= Ã 2 X i=1 E s E r K i g ¤ i (q)g i (q ¡1 )h ¤ i (q)h i (q ¡1 ) ! (4.21) The output vector is given by z(t),H H (q ¡1 ;q)y(t) (4.22) 108 whose components can be expressed as z 1 (t) = f(q ¡1 )f ¤ (q)s 1 (t)+v 1 (t) (4.23) z 2 (t) = f(q ¡1 )f ¤ (q)¹ s ¤ 2 (t)+v 2 (t) (4.24) and the output noise v(t),[v 1 (t) v 2 (t)] T has PSD X v (q ¡1 ;q) given by X v (q ¡1 ;q)=X(q ¡1 ;q)f(q ¡1 )f ¤ (q)I (4.25) Thus, v 1 (t) and v 2 (t) are independent and the problem of jointly detecting s 1 (t) and s 2 (t) from z(t) decouples: s 1 (t) is detected from z 1 (t) and s 2 (t) from z 2 (t). Comparing the ¯lter outputs given by (4.23) and (4.24) with the equivalent STBC relations for two antennas in [LP00] we note two key di®erences: ² ThenoisePSDgivenby(4.25)containsaterm X(q ¡1 ;q)6=1duetothenoise ampli¯cation at the relays. This noise is also colored (in time), since the R-D channel introduces ISI. ² The channel f(q ¡1 ) accounts for the overall e®ect of the \product" channels g i (q ¡1 )h i (q ¡1 ),whichincreasesISIandimpactsperformance,aswillbeshown in Section 4.3. 109 Applying the spectral factorization X(q ¡1 ;q) = x(q)x(q ¡1 ) and, hence, whiten the noise by applying the ¯lter W(q ¡1 ;q) = 1 x(q ¡1 )f ¤ (q) , resulting in the desired AWGN model ~ z 1 (t) = f(q ¡1 ) x(q ¡1 ) s 1 (t)+~ v 1 (t)¼ ~ c(q ¡1 )s 1 (t)+~ v 1 (t) (4.26) ~ z 2 (t) = f(q ¡1 ) x(q ¡1 ) ¹ s ¤ 2 (t)+~ v 2 (t)¼ ~ c(q ¡1 )¹ s ¤ 2 (t)+~ v 1 (t) (4.27) where ~ c(q ¡1 ) denotes the least square ¯nite impulse response (FIR) approximation of c(q ¡1 ), f(q ¡1 ) x(q ¡1 ) ; (4.28) whichhasanin¯niteimpulseresonse(IIR),ingeneral. DenotingrespectivelybyL f , L x andL ~ c thenumberoftapsinf(q ¡1 ),x(q ¡1 )and ~ c(q ¡1 ),wesetL ~ c =L f +L x ¡1. We shall assume that this approximation incurs negligible error compared to the desired¯lterresponse. FortheFIRmodelin(4.26)and(4.27),maximumlikelihood sequenceestimation(MLSE)canbecarriedoutviaaViterbi-typealgorithm[Pro95]. For channels experiencing extensive multipath however (such as the underwater channels), the MLSE detector is prohibitively complex, and a decision-feedback equalizer (DFE) has been shown to yield good results in practice[SFJ99]. 110 4.3 Performance Analysis In this section, we provide the performance analysis of the proposed cooperation scheme with decision-feedback equalization at the receiver. We employ the tech- nique described in [Mon77], and, for mathematical tractability, consider the par- ticular scenario in which the R¡D channels have only one multipath component. This case corresponds, for example, to the scenario where propagation occurs via is a single surface re°ection. Furthermore, we will also assume that all individual links S¡R i and R i ¡D are fading independently (maximum spatial diversity). First, we note that, due to the orthogonality of the channel matrix H(q ¡1 ;q), the dual-channel cooperative scheme can be analyzed as two independent single- channel branches: the ¯rst being S-R 1 -D and the second S-R 2 -D. For the same reason, the TR-STBC scheme in [LP00] was shown to be equivalent to a 2-antenna receive diversity system. Assuming, without loss of generality, communication through the S-R 1 -D chan- nel, the equivalent received signal at D and time t (after the space-time processing described in the previous section), denoted by r D (t), is given by r D (t)= r E s E r K 1 +1 X l=¡1 s(t¡l)h 1 (l)g 1 + r E r K 1 w 1 (t)g 1 + r E r K 2 w 2 (t)g 2 +n(t) (4.29) where g 1 and g 2 denote, respectively, the (°at) fading realization of the R 1 -D and R 2 -D channels and h 1 (l) the fading realization of the l-th path of the S-R 1 channel. 111 As in the previous section, the noise realizations at both relays are denoted by w 1 (t) and w 2 (t) and n(t) is the destination noise. In order to decode a desired symbol s(k), a DFE operates on a vector of N 1 +1 received symbols, each in turn given by (4.29), and N 2 past decoded symbols ^ s(k¡1);^ s(k¡2);:::;^ s(k¡N 2 ). Assuming, without loss of generality, that s(0) is the desired symbol to be detected, the output of the DFE is I 0 =e H Ã r E s E r K 1 s(0)g 1 h 1 (0)+ r E s E r K 1 N 1 X l=¡L c ;l6=0 s(l)g 1 h 1 (l)+ r E r K 1 w 1 (0)g 1 + + r E r K 2 w 2 (0)g 2 +n(0) ! + N 2 X u=1 b ¤ (u)^ s(¡u) (4.30) where L c +1 is the number of paths of the overall S-R 1 -D channel response, and N 1 +1 and N 2 denote the number of forward and feedback taps, respectively. For a general sequence a(t), we de¯ne a(k), · a(k) a(k+1) ::: a(k+N 1 ) ¸ T : (4.31) Thevectore,oflengthN 1 +1,istheforward¯lter. Thefeedbackcoe±cients,which operate on the previously decoded symbols, are denoted by b(1);:::;b(N 2 ). Asusualintheliterature,wemakethekeyassumptionthattheprevioussymbols are decoded perfectly [Mon77]. While this is not the case for low SNR, it is a good approximation in the high SNR regime and, furthermore, is the typical scenario 112 where diversity analysis is performed. Since the feedback ¯lter completely cancels the interference due to past symbols in (4.30), the DFE output becomes I 0 =e H Ã r E s E r K 1 s(0)g 1 h 1 (0)+ r E s E r K 1 N 1 X l=1 s(l)g 1 h 1 (l) + r E r K 1 w 1 (0)g 1 + r E r K 2 w 2 (0)g 2 +n(0) ! (4.32) which can be written as I 0 =e H Ã r E s E r K 1 s(0)g 1 h 1 (0)+n 1 +n 2 +n 0 ! (4.33) where the destination noise, the ISI due to future symbols, and the overall relay noise are given by, respectively n 0 = n(0) (4.34) n 1 = r E s E r K 1 N 1 X l=1 s(l)g 1 h 1 (l) (4.35) n 2 = r E r K 1 w 1 (0)g 1 + r E r K 2 w 2 (0)g 2 (4.36) Wenowmakefurthersimplifyingassumptions,alsoaccordingto[Mon77]. First, we replace the noise and interference covariance matrices by their ensemble aver- ages. Second, we assume that the interference symbols are complex Gaussian with variance 1. Furthermore, we assume Rayleigh fading, i.e, each tap k in h 1 (0) has 113 distribution CN(0;¾ 2 h k ) and g 1 has distribution CN(0;¾ 2 g ). Under these assump- tions, the noise and interference covariance matrices are K 0 = I (4.37) K 1 = E s E r K 1 ¾ 2 g diag " 0;¾ 2 h 0 ; 1 X i=0 ¾ 2 h i ;:::; L c X i=0 ¾ 2 h i ;:::; L c X i=0 ¾ 2 h i # (4.38) K 2 = 2¾ 2 g I (4.39) The noise plus interference covariance is K=K 0 +K 1 +K 2 (4.40) From (4.33) the signal power at the output of the equalizer is P 0 = E s E r K 1 e H h 1 (0)h 1 (0) H ejg 1 j 2 (4.41) and the noise plus interference power is W 0 =e H Ke (4.42) Following the same approach as in [Mon77], it can be shown that the optimal choice for the equalizer taps is e opt =K ¡1 h 1 (0)g 1 ; (4.43) 114 and the resulting optimal signal-to-noise ratio » of the equalizer output is » = E s E r K 1 h 1 (0) H K ¡1 h 1 (0)jg 1 j 2 (4.44) For BPSK modulation and given channel realizationsh 1 (0) and g 1 the bit error rate (BER) of the system can be estimated as Q( p 2»), which has to be averaged over the distributions of the channels. A simple upper bound approximation (for high SNR) of the resulting BER is given by the Cherno® Bound. Conditioned on g,g 1 =¾ g , where g»CN(0;1), the BER is BER 1 j g · 1 2 N 1 Y k=0 µ 1+ E s E r K 1 [K] k;k ¾ 2 h k ¾ 2 jgj 2 ¶ ¡1 · 1 2 µ 1+ E s E r K 1 [K] 0;0 ¾ 2 h 0 ¾ 2 g jgj 2 ¶ ¡1 (4.45) De¯ning ¹ » = E s E r K 1 [K] 0;0 ¾ 2 h 1;0 ¾ 2 g as an average SNR quantity, BER 1 j g · 1 2 ¡ 1+ ¹ »jgj 2 ¢ ¡1 (4.46) Recalling thatjgj 2 has an exponential distribution, the average BER is BER 1 · Z 1 0 1 2 ¡ 1+ ¹ »x ¢ ¡1 e ¡x dx (4.47) 115 After some manipulations, and noting that e 1= ¹ » ¼ 1 for high ¹ », we can write (4.47) as BER 1 · 1 2 ¹ » Z 1 1 ¹ » e ¡x x dx (4.48) Recalling that (e.g, see [GR00]) R 1 1 ¹ » e ¡x x dx = ¡Ei(¡ 1 ¹ » ), where Ei denotes the Euler exponential integral, the single-channel BER has the closed form approxima- tion BER 1 .¡ 1 2 ¹ » Ei µ ¡ 1 ¹ » ¶ (4.49) Thus, assuming perfect symmetry between the channel statistics of the two spatial diversity branches, the desired BER expression is BER 2 . 1 4 ¹ » 2 Ei 2 µ ¡ 1 ¹ » ¶ (4.50) Since Ei(¡1=x);x > 0 has an O(log(x)) behavior for large x [GR00], the BER decay with ¹ » is BER 2 .O µ log ¹ » ¹ » ¶ 2 =O µ 1 ¹ » ¶ 2¡2 loglog ¹ » log ¹ » : (4.51) We notice, therefore, that the e®ective diversity at ¯nite ¹ » is less than 2 by a factor of 2 loglog ¹ » log ¹ » . For ¹ » ! 1, the diversity is asymptotically 2 and, thus, full spatial diversity is achieved. The diversity loss at ¯nite SNR can be attributed to the overall S-R-D channel not being of Rayleigh statistics, but a \product" fading channel instead. 116 4.4 CopingwithImperfectRelaySynchronization The distributed nature of the cooperative communication strategy described in Section 4.2 naturally brings up the question of whether the relays need to operate under perfect synchronization. Indeed, this is a common assumption in several recently proposed distributed cooperation strategies[HMC03][SEA03a][NBK04]. In many practical situations, however, due to di®erent delays between the co- operative nodes and the destination, achieving perfect synchronization can be very di±cult[Li04]. The long sound propagation delays in underwater networks can, therefore, potentially exacerbate this problem. A direct consequence of imperfect synchronization between relays is the intro- duction of time dispersion in the channels; this can occur even in frequency °at channels, and is due to imperfect sampling times at the receiver[Li04][MHSD05]. Intuitively,therefore,theTR-DSTBCapproachforfrequencyselectivechannelscan also operate with asynchronous relays, if we assume a known upper bound on the relative transmission delays (to avoid interblock interference). Fromthereceivedsignalmodelattherelaysgivenby(4.4)and(4.5),thereceived signal for block k at D assuming a time delay of ² between transmissions from R 1 and R 2 is y 1 (t)= p E 1 g 1 (q ¡1 )h 1 (q ¡1 )s 1 (t)+ p E 2 g 2 (q ¡1 )h ¤ 2 (q)R p ² (q ¡1 ;q)¹ s ¤ 2 (t)+n 0 1 (t) y 2 (t)= p E 1 g 1 (q ¡1 )h 1 (q ¡1 )s 2 (t)¡ p E 2 g 2 (q ¡1 )h ¤ 2 (q)R p ² (q ¡1 ;q)¹ s ¤ 1 (t)+n 0 2 (t) 117 where E 1 = q EsEr K 1 and E 2 = q EsEr K 2 , the ¯lter R p² (q ¡1 ;q) denotes the transmit pulse correlation function for a delay ², with coe±cients R p² (k)=p(t+²)¤p(¡t)j t=kT ; ¡1<k <1 (4.52) where p(t) denotes the transmit pulse shape, usually a raised cosine pulse[SFJ99] of unit energy. Hence, in a perfectly synchronized scenario, R p 0 (q ¡1 ;q)=1. Clearly, from the expressions for y 1 (t) and y 2 (t), the imperfect synchronization increases the dispersion of the R 2 -D channel. However, de¯ning an equivalent channelh 0 2 (q ¡1 ),R p² (q ¡1 ;q)h 2 (q ¡1 )anidenticalformofthesignalmodelin(4.17) isobtained, withh 0 2 (q ¡1 )replacingh 2 (q ¡1 ). Thus, theTR-DSTBCformulationcan conveniently address the case of asynchronous relays by simple generalization of the channel model. In Section 4.7, experimental results for asynchronous relays (a consequence of the relays being deployed over di®erent depths) are reported. 4.5 IncreasingSpatialDiversitywithMoreRelays The key property that allows signal decoupling at receiver for the two-relay scheme presented in section 4.2 is the orthogonality of the Alamouti STBC. It allows two signal streams tobetransmitted via tworelaysover twoblocks oftime, and, hence, exhibits full rate of one. 118 R 1 R 4 S D R 2 R 3 g 1 (q -1 ) g 2 (q -1 ) g 3 (q -1 ) g 4 (q -1 ) f 1 (q -1 ) f 2 (q -1 ) f 3 (q -1 ) f 4 (q -1 ) Figure 4.2: Two-hop cooperative network with 4 relays Unfortunately, no full rate orthogonal STBC exists (for complex modulation) for more than two antennas[TJC99]. In this section we present a generalization of the two relay scheme for two hops presented earlier to arbitrary number of relays based on rate 1/2 orthogonal STBCs[TJC99], denoted by TR-1/2. For simplicity wedescribetheTR-1/2schemeforthecaseof4relaysdepictedinFigure4.2. Itcan be readily extended to more relays combining the distributed signaling approach described here with the rate 1/2 STBC structure in [TJC99]. The information stream d(t) is now divided into four equal length blocks d 1 (t);:::;d 4 (t). The source transmission occurs over eight blocks: in the ¯rst four blocks, s k (t)= p E s d k (t);k =1;:::;4 (4.53) 119 are transmitted; in the last four blocks, s k (t)= p E s ¹ d ¤ k¡4 (t);k =5;:::;8 (4.54) aresent. Notethattheblocksofthe secondhalftransmission are the time-reversed conjugates of the ¯rst half blocks. Hence the rate of the scheme is 1/2. The received signals at relay i;i=1;:::;4 and block k, k;i=1;:::;8 are r i;k (t)= p E s h i (q ¡1 )s k (t)+w i;k (t); t=1;:::;N (4.55) where, as before, the h i (q ¡1 ) denotes the channel coe±cients between source and relay i and w i;k (t) is AWGN. The processing performed by each relay over its received blocks is dictated by the underlying rate 1/2 STBC structure. In general for symbols x i ;i = 1;:::;4 from a complex alphabet, the rate 1/2 STBC is S= 2 6 6 4 S S ¤ 3 7 7 5 where S = 2 6 6 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 ¡x 2 x 1 ¡x 4 x 3 ¡x 3 x 4 x 1 ¡x 2 ¡x 4 ¡x 3 x 2 x 1 3 7 7 7 7 7 7 7 7 7 7 5 (4.56) 120 Thus, if each relay corresponds to a di®erent column of S, and each row to a di®erent time block, the relay transmissions can be represented by the matrix U, · u 1 (t) u 2 (t) u 3 (t) u 4 (t) ¸ , 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 q Er K 1 r 1;1 (t) q Er K 2 r 2;2 (t) q Er K 3 r 3;3 (t) q Er K 4 r 4;4 (t) ¡ q E r K 1 r 1;2 (t) q E r K 2 r 2;1 (t) ¡ q E r K 3 r 3;4 (t) q E r K 4 r 4;3 (t) ¡ q E r K 1 r 1;3 (t) q E r K 2 r 2;4 (t) q E r K 3 r 3;1 (t) ¡ q E r K 4 r 4;2 (t) ¡ q Er K 1 r 1;4 (t) ¡ q Er K 2 r 2;3 (t) q Er K 3 r 3;2 (t) q Er K 4 r 4;1 (t) q E r K 1 r 1;5 (t) q E r K 2 r 2;6 (t) q E r K 3 r 3;7 (t) q E r K 4 r 4;8 (t) ¡ q E r K 1 r 1;6 (t) q E r K 2 r 2;5 (t) ¡ q E r K 3 r 3;8 (t) q E r K 4 r 4;7 (t) ¡ q Er K 1 r 1;7 (t) q Er K 2 r 2;8 (t) q Er K 3 r 3;5 (t) ¡ q Er K 4 r 4;6 (t) ¡ q E r K 1 r 1;8 (t) ¡ q E r K 2 r 2;7 (t) q E r K 3 r 3;6 (t) q E r K 4 r 4;5 (t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (4.57) where each column u i (t) = [u i;1 (t);:::;u i;8 (t)] T consists of the transmission of 8 blocks, according to (4.57) and (4.55). Denoting by g i (q ¡1 ) the channels from relay i to destination, the received signal at destination (after time reversing and conjugating the second-half received blocks) is y(t)= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 f 1 (q ¡1 ) f 2 (q ¡1 ) f 3 (q ¡1 ) f 4 (q ¡1 ) f 2 (q ¡1 ) ¡f 1 (q ¡1 ) f 4 (q ¡1 ) ¡f 3 (q ¡1 ) ¡f 3 (q ¡1 ) f 4 (q ¡1 ) f 1 (q ¡1 ) ¡f 2 (q ¡1 ) ¡f 4 (q ¡1 ) ¡f 3 (q ¡1 ) f 2 (q ¡1 ) f 1 (q ¡1 ) f ¤ 1 (q) f ¤ 2 (q) f ¤ 3 (q) f ¤ 4 (q) f ¤ 2 (q) ¡f ¤ 1 (q) f ¤ 4 (q) ¡f ¤ 3 (q) ¡f ¤ 3 (q) f ¤ 4 (q) f ¤ 1 (q) ¡f ¤ 2 (q) ¡f ¤ 4 (q) ¡f ¤ 3 (q) f ¤ 2 (q) f ¤ 1 (q) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 4 d 1 (t) d 2 (t) d 3 (t) d 4 (t) 3 7 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 P 4 i=1 w i;1 (t)g i (q ¡1 )+´ 1 (t) . . . P 4 i=1 w i;4 (t)g i (q ¡1 )+´ 4 (t) P 4 i=1 ¹ w ¤ i;5 (t)g i (q ¡1 )+ ¹ ´ ¤ 5 (t) . . . P 4 i=1 ¹ w ¤ i;8 (t)g i (q ¡1 )+ ¹ ´ ¤ 8 (t) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (4.58) 121 where´ k (t)isadditivewhitenoiseatthereceiverandf i (q ¡1 )= q E s E r K i h i (q ¡1 )g i (q ¡1 ) is the overall channel S-R i -D. Just as in the 2-relay case, we can de¯ne the channel matrix H(q ¡1 ;q), 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 f 1 (q ¡1 ) f 2 (q ¡1 ) f 3 (q ¡1 ) f 4 (q ¡1 ) f 2 (q ¡1 ) ¡f 1 (q ¡1 ) f 4 (q ¡1 ) ¡f 3 (q ¡1 ) ¡f 3 (q ¡1 ) f 4 (q ¡1 ) f 1 (q ¡1 ) ¡f 2 (q ¡1 ) ¡f 4 (q ¡1 ) ¡f 3 (q ¡1 ) f 2 (q ¡1 ) f 1 (q ¡1 ) f ¤ 1 (q) f ¤ 2 (q) f ¤ 3 (q) f ¤ 4 (q) f ¤ 2 (q) ¡f ¤ 1 (q) f ¤ 4 (q) ¡f ¤ 3 (q) ¡f ¤ 3 (q) f ¤ 4 (q) f ¤ 1 (q) ¡f ¤ 2 (q) ¡f ¤ 4 (q) ¡f ¤ 3 (q) f ¤ 2 (q) f ¤ 1 (q) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (4.59) which satis¯es H H (q ¡1 ;q)H(q ¡1 ;q)= Ã 2 4 X i=1 r E s E r K i g i (q ¡1 )g ¤ i (q)h i (q ¡1 )h ¤ i (q) ! I,f(q ¡1 )f ¤ (q)I (4.60) and, hence, is orthogonal. The output vector is given by z(t),H H (q ¡1 ;q)y(t) (4.61) 122 consisting of 4 blocks, each one given by z i (t)=f(q ¡1 )f ¤ (q)d i (t)+v i (t);i=1;:::;4 (4.62) and the output noise v(t),[v 1 (t);:::;v 4 (t)] T has PSD X v (q ¡1 ;q)=X(q ¡1 ;q)f(q ¡1 )f ¤ (q)I (4.63) Hence,v 1 (t);:::;v 4 (t)areindependentandtheproblemofjointlydetectingd i (t);i= 1;:::;4 from z(t) again decouples, just like in the 2-relay case. It turns out that [TJC99]: i) a rate 1/2 complex orthogonal design exists for any size 2n£n where n is a power of two; ii) an orthogonal design of size l£n wherel <n, canbeobtainedbydeletingn¡l columnsofthe2n£ndesign. Hence, the TR-1/2 scheme can be generalized to any number of relays in straightforward fashion. 4.6 Underwater Channel Model We adopt a ray-based model similar to [ZYW95, Sto05] to model the multipath sound propagation. The main di®erence in our model is that we assume multipath componentsarefading. Weconsideraquasi-staticfadingmodel,inwhichthechan- nel is constant within a ¯xed duration and changes to an independent realization 123 over the next time frame. We analyze the e®ects of a slowly time varying channel within the frame as well as non-perfect CSI via simulations in the next section. A given multipath arrival p is characterized by its mean magnitude gain ® p and delay t p . These quantities are dependent on the path length l p , which in turn is a function of the given range R. The path magnitude gain is given by ® p = ¡ p p A(l p ) (4.64) where ¡ p ;¡ p ·1 is the amount of loss due to re°ection at the bottom and surface. The acoustic propagation loss, represented by A(l p ) is given by Thorp's formula A(l p )=l k p [a(f c )] l p (4.65) where k = 1:5 for practical spreading, f c is the carrier frequency and absorbtion coe±cient a(f c ) (in dB/km) given by Thorp's formula 10loga(l p )= 0:11f 2 c 1+f 2 c + 44f 2 c 4100+f 2 c +2:75:10 ¡4 f 2 c +0:003 (4.66) Finally, the path delay is given by t p = l p =c, where c = 1500 m/s is the speed of sound. Figure 4.3 shows the channel path delays and magnitudes for a distance of 3 km between transmitter and receiver, f c =15 kHz and depth of 75 m. 124 0 5 10 15 20 25 30 35 40 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 delay (ms) magnitude 2−hop channel path properties Figure 4.3: Two-hop channel pro¯le: d = 3 km (range per hop), f c = 15 kHz, depth = 75m 4.7 Simulation Results 4.7.1 Computer Channel Simulations For simulation purposes, we consider an underwater network with S-D distance of 6km, carrierfrequencyf c =15kHzandchanneldepthof75m. Threecommunica- tion strategies are compared: single hop, where the source communicates directly tothedestination; andcooperative(two-hop)witheither2or4relays, asdescribed in sections 4.2 and 4.5. Our ¯rst simulation scenario compares performance of the MLSE and DFE to justifytheuseofthelatterforsignaldetectioninourmulti-hopcooperativesetting. The modulation is BPSK with a symbol duration of T = 2:5 ms and, therefore, a data rate of 400 b/s. 125 This results in a channel with 2 taps per hop in the cooperative strategy, and 3 taps for the direct communication approach. In the latter case, multipath arrivals can occur within a fraction of a symbol time. In this case these arrivals can add up constructively [ZYW95], and thus result in a stronger path, or destructively [CPH04], weakening the resulting overall multipath component. We consider both scenarios separately here. Figure 4.4 shows the performance of the MLSD and DFE approaches for the cooperativeandsinglehopstrategies. FortheDFE,thenumberoffeedforwardand feedback ¯lter taps is N 1 +1 = 15 and N 2 = L c respectively, where L c +1 is the number of taps of the overall S-D channel. It is clear for all cases considered that theDFEperformanceisveryclosetoMLSD.Furthermore,theresultshighlightthe signi¯cant performance improvement introduced by cooperation. The two main reasons are: a) multi-hoping gain, since the overall attenuation su®ered by the signal is less severe; b) spatial diversity gain, which is due to the spatial diversity inherently available in the distributed network, which is exploited via distributed space-time processing. Note that, although both the single-hop and two-hop cooperative communica- tion approaches yield diversity gain through multipath combining, only the latter provides extra gains through spatial diversity. This explains the larger decay slope of the error probability curves for the cooperative schemes. 126 0 5 10 15 20 25 30 35 40 10 −4 10 −3 10 −2 10 −1 10 0 total energy/symbol (dB) BER Performance in UWA channels, range = 6 km, f c = 15kHz, depth = 75m, rate = 400 b/s 1−hop constructive − MLSD 1−hop constructive − DFE 1−hop destructive − MLSD 1−hop destructive − DFE 2−relay cooperative, MLSD 2−relay cooperative, DFE 4−relay cooperative − MLSD 4−relay cooperative − DFE Figure 4.4: Two-hop cooperation performance Finally, we note that increasing the number of cooperating relays further in- creases the available spatial diversity in the system and, hence, further improves the performance. This can also be veri¯ed for the 4-relay case in Figure 4.4, which employs the TR-1/2 scheme. Figure 4.5 shows the Cherno® Bound in (4.50) with ° = 1, compared with the simulation performance. Better approximations can be obtained by ¯ne-tuning ° via simulations[Mon77]. It can be observed that, at high SNR, the bound indeed con¯rms the decay order predicted by (4.51). 4.7.1.1 Sensitivity to Imperfect CSI Inapracticalscenario, thereceiverdoesnothaveperfectknowledgeofthechannel. In our current framework, CSI at the receiver can be obtained via known training 127 20 25 30 35 40 45 10 −4 10 −3 10 −2 10 −1 10 0 Total transmit energy/symbol (dB) BER DFE performance and Chernoff Bound Simulation Chernoff Bound Approximation Figure 4.5: Cherno® Bound for two-hop cooperation symbols sent from the relays (to estimate the R i -D channels) and from the source (to estimate the S-R i -D channels, such as in [HMC03]). Due to the presence of channel estimation errors, a degradation in the idealized performanceresultsisnaturallyexpected. Inthissection,weassess,viasimulations, the performance of the receiver employing a DFE under di®erent imperfect CSI conditions. We model the channel estimation error as a random error matrix that is added to the true channel matrix H(q ¡1 ;q). The estimated channel matrix ^ H(q ¡1 ;q), is given by ^ H(q ¡1 ;q)=H(q ¡1 ;q)+E(q ¡1 ;q) (4.67) 128 with E(q ¡1 ;q) expressed as E(q ¡1 ;q)= 2 6 6 4 E 11 (q ¡1 ;q) E 12 (q ¡1 ;q) E 21 (q ¡1 ;q) E 22 (q ¡1 ;q) 3 7 7 5 (4.68) In terms of the actual channel estimates ^ g i (q ¡1 ;q) and ^ h i (q ¡1 ;q), we can also write ^ H(q ¡1 ;q), 2 6 6 4 ^ g 1 (q ¡1 ) ^ h 1 (q ¡1 ) ^ g 2 (q ¡1 ) ^ h ¤ 2 (q) ¡^ g ¤ 2 (q) ^ h 2 (q ¡1 ) ^ g ¤ 1 (q) ^ h ¤ 1 (q) 3 7 7 5 (4.69) where the constant terms q EsEr K i are now incorporated into the channels, for nota- tional simplicity. The channel estimates are given by ^ h i (q ¡1 ) = h i (q ¡1 )+e h i (q ¡1 ) (4.70) ^ g i (q ¡1 ) = g i (q ¡1 )+e g i (q ¡1 ) (4.71) Hence, we can write E 11 (q ¡1 ;q) = e h 1 (q ¡1 ) ¡ e g 1 (q ¡1 )+g 1 (q ¡1 ) ¢ +e g 1 (q ¡1 )h 1 (q ¡1 ) (4.72) E 12 (q ¡1 ;q) = e ¤ h 2 (q) ¡ e g 2 (q ¡1 )+g 2 (q ¡1 ) ¢ +e g 2 (q ¡1 )h ¤ 2 (q) (4.73) E 21 (q ¡1 ;q) = ¡e h 2 (q ¡1 ) ¡ e ¤ g 2 (q)+g ¤ 2 (q) ¢ ¡e ¤ g 2 (q)h 2 (q ¡1 ) (4.74) E 22 (q ¡1 ;q) = e ¤ h 1 (q) ¡ e g 2 (q ¡1 )+g ¤ 1 (q) ¢ +e ¤ g 1 (q)h ¤ 1 (q) (4.75) 129 The receiver processing in (4.22) is now performed using the channel estimate ^ H(q ¡1 ;q). From (4.17) and (4.22), ^ z(t)= ^ H H (q ¡1 ;q)y(t)=z(t)+ E H (q ¡1 ;q)H(q ¡1 ;q) 2 6 6 4 s 1 (t) ¹ s ¤ 2 (t) 3 7 7 5 +E H (q ¡1 ;q) 2 6 6 4 ´ 0 1 (t) ~ ´ 2 (t) 3 7 7 5 (4.76) Assumings 1 (t)istobedetected,wenoticefrom(4.76)thattheimperfectCSIincurs not only in noise enhancement of the decision statistic, but also self and inter block interference from s 1 (t) and s 2 (t) respectively. Figure 4.6 shows the performance sensitivity of the 2-relay scheme for imperfect CSI. The estimation error in each path p for all channels is assumed to be complex Gaussian with variance ¾ 2 ep = ¾ 2 e ¾ 2 p , where ¾ 2 p is the average square magnitude of path p. For ¾ 2 e =¡20 dB, the degradationfromperfectCSIisverysmallanditdeterioratesrathergracefullywith increasing ¾ 2 e . As expected, for large estimation errors the system cannot take full advantageofthespatialdiversity, andsigni¯cantperformancedeteriorationoccurs. 130 0 5 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 Total transmit energy/symbol (dB) BER Performance with imperfect CSI σ 2 e = −5dB σ 2 e =−10dB σ 2 e =−15dB σ 2 e =−20dB Figure 4.6: Two-hop cooperation performance: sensitivity to channel estimation errors 4.7.1.2 Sensitivity to Channel Time Variations To investigate the performance of the system with channel time variations, we assume a ¯rst order time-varying model for each path of the overall channel f(q ¡1 ) in the model derived in (4.23) and (4.24). Hence the receiver ¯lter outputs are z 1 (t) = f(q ¡1 ;t)f ¤ (q;t)s 1 (t)+v 1 (t) (4.77) z 2 (t) = f(q ¡1 ;t)f ¤ (q;t)¹ s ¤ 2 (t)+v 2 (t) (4.78) A ¯rst-order autoregressive (AR) model is assumed for each of the tap coe±- cients of the ¯lter f(q ¡1 ;t). A similar approach to model channel time variation 131 characteristics was also employed in [EBP00]. Each time-varying tap f i (t) is mod- elled as f i (t)=¾f i (t¡1)+ p 1¡¾ 2 ²(t) (4.79) where ²(t) is a complex white Gaussian innovation process of unit variance. For a symbolratef s ,wecanrelatetheDopplerspreadandtheARparameter¾as[EBP00] ¾ =2¡cos(w d =2)¡ p cos 2 (w d =2)¡4cos(w d =2)+3 (4.80) Figure 4.7 shows the performance of the scheme under several time varying conditions, controlled by the parameter ¾. Two types of frame length were used: 50 and 30 symbols. For N = 50 the time variations accumulated across the frame incurs a slightly more severe degradation compared to a N = 30 frame. In both cases, mostoftheperformancegainsarestillobservedforchannelvariationsonthe range of ¾ =¡40dB to¡30dB. 4.7.2 Experimental Simulations To further verify the potential gains achieved via cooperation, the proposed 2-relay protocol was compared with a direct S-D communication approach with measured channeldata. ThedataacquisitionexperimentswereperformedattheWoodsHole Oceanographic Institution, MA, USA, in June 2006. 132 0 5 10 15 20 25 30 35 40 10 −4 10 −3 10 −2 10 −1 10 0 Total transmit energy/symbol (dB) BER Sensitivity analysis of channel time variation σ 2 = −20dB, N=30 σ 2 =−30 dB, N=30 σ 2 =−20dB, N=50 σ 2 =−30 dB, N=50 σ 2 =−40 dB, N=50 Figure 4.7: Two-hop cooperation performance: sensitivity to channel time variations The experimental setup is depicted in Figure 4.8. The carrier frequency was 12 kHz. The source transmitter and destination hydrophone were deployed in about 5 mdeepwater, andthetwohydrophonesattherelayswereplacedin10mand20m deepwater. Allhydrophoneswereomnidirectional. Theprobedataweremaximum- length shift pseudonoise sequences of unshaped (rectangular) pulses. Figures 4.9, 4.10and4.11showtheaveragedchannelresponseoveranensembleofmeasurements for the ¯rst hop, second hop and direct links respectively. Note that, compared to simulated ray-based model given in Figure 4.3, the experimental data has a signi¯cantly larger delay spread. Typical delay spreads of 80 to 100 symbols were observedatadatarateof1.2kb/s. Furthermore,theR-Dchannelsarenotperfectly synchronized, as can be seen in Figure 4.12. 133 Dock Transmitter (depth = 5m) Relays (depths: 10m and 20m) Destination (depth = 5m) 100m 26m Figure 4.8: Experimental setup Inallcasesastrongmainarrivalisperceived,alongwithclustersofarrivalswith delays of more than 100ms. Furthermore, it can be observed that the direct S-D channel was signi¯cantly weaker than the S-R and R-D channels. This constitutes a favorable scenario for testing the multi-hop cooperation strategy. The simulated performance with experimental channel data is shown in Figure 4.13. The data rate is 1.2 kb/s and the BPSK modulation scheme was employed. The data blocks length was set to N =1000 symbols, and a DFE with 50 forward taps was employed. Since perfect CSI was assumed, the number of feedback taps was set to be the the delay spread (in symbols intervals) of the overall channel. Comparing the cooperative scheme performance with direct communication, both 134 0 50 100 150 200 250 300 350 400 450 −45 −40 −35 −30 −25 −20 −15 −10 time(ms) Normalized average channel response: S−R link average power(dB) Figure 4.9: Source-relay average channel response multi-hopping as well as diversity gains are observed. For the direct hop case a diversity (slope of the error probability curve) of around 1.2 was observed, whereas itisaround2forthecooperativecase. Theadditionalgaindiversityobservedinthe directhopcasecanbefullyattributedtomultipathcombining. Forthecooperative strategy, in section 4.3 a diversity gain slightly smaller than 2 was predicted when no multipath gain is present. Clearly, therefore, the cooperative strategy is able to take advantage of diversity in both levels: space and the frequency selectivity of the channel. 4.8 Conclusions In this chapter, cooperative protocols for distributed space-time communications in underwater networks were proposed and analyzed. The proposed protocols, 135 0 50 100 150 200 250 300 350 400 450 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 time(ms) averagge power (dB) Normalized average channel response: R−D link Figure 4.10: Relay-destination average channel response inspired by the time-reversal STBC[LP00], were shown to yield signi¯cant perfor- mance improvement over single hop communication in underwater channels and can be generalized to networks with any number of relays. It was also shown, both numerically and experimentally, that the TR-DSTBC approach has the added ad- vantageofnotrequiringacomputationallyexpensivemultidimensionalequalization at the receiver, as well as being robust to asynchronism between the relays. On- going research focuses on the development of new, high rate, distributed protocols and space-time codes which can be systematically applied to any number of relays. 136 0 50 100 150 200 250 300 350 400 450 −55 −50 −45 −40 −35 −30 −25 −20 −15 time(ms) average power (dB) Normalized average channel response:S−D link Figure 4.11: Source-destination average channel response 16 17 18 19 20 21 22 −50 −45 −40 −35 −30 −25 −20 −15 −10 time(ms) average power (dB) Normalized average channel response Relay 1: depth = 10m Relay 2: depth = 20m Figure 4.12: Channel responses for asynchronous relays 137 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 Total energy/symbol (dB) BER Experimental performance, rate = 1.2 Kb/s 1−hop 2−relay cooperative Figure 4.13: Error rate performance of single hop and cooperative transmission 138 Chapter 5 Achievable Rates, Outage Probability and Throughput of Cooperative DSTBC Protocols In this chapter, which is part of ongoing work, several communication protocols for two-hop communication are analyzed from an achievable rate (outage mutual information-OMI),outageprobabilityandthroughputperspective. First, theper- formance of the the Alamouti Amplify & Forward (AAF) protocol for two relays is analytically characterized: exact expressions and closed form bounds are derived for its outage probability, outage mutual information (OMI) and throughput. Fur- thermore, it is shown that the best choice of transmission scheme (between AAF and no-cooperation) is highly dependent on the total energy available for trans- mission. Extensions of the AAF scheme to protocols for three and four relays are also considered and their performance is assessed via simulations. In particular, a new protocol for four relays is proposed, which is inspired by the quasi-orthogonal STBCs for co-located antennas (Papadias and Foschini, 2003). It is shown that the 139 new protocol can achieve both higher outage capacity and lower outage probability than the AAF protocol. 5.1 Introduction Satisfying the ever-increasing demand for higher data rates by user applications is undoubtedlyamajorchallengeinthedesignofcurrentandfuturewirelessnetworks. Cost-e®ectivesolutionsthatmakethemostoftheavailableresources,suchaspower and bandwidth, need to be sought. In this context, multi-hop communication appears to be a natural way to increase service coverage over larger areas while simultaneously saving transmit power (and, hence, battery life) by reducing the amount of signal attenuation due to path-loss per multi-hop link. Besides power savings, multi-hopping in distributed networks can also pro- videincreasedcapacityviacooperativecommunicationstrategies[SEA03b,SEA03d, LTW04, NBK04]. Due to the inherent spatial diversity available in such networks (due to nodes typically being located signi¯cantly far apart), typical gains ob- served in classical multi-antenna MIMO systems can be obtained via Distributed Space-Time Coding [LW03] to signi¯cantly improve transmission reliability and/or capacity. An additional requirement is that proposed solutions be feasible. The cur- rent state-of-the-art transceivers are not capable of simultaneous transmission and 140 reception and, thus, operate in half-duplex mode[NBK04]. Hence, cooperation be- tween terminals in a multi-hop fashion clearly incurs some loss in achievable rate, and thus, might erode some (or even most) of the potential gains from power savings[SLH + 04]. The work in [SLH + 04] investigated the tradeo® between power e±ciency and achievable rate for a linear network where the links were AWGN channels and intermediate nodes fully decode the source information. By using the standard formula for the capacity for point-to-point channels, the optimum number of hops for such networks was characterized as a function of the total power and target rate. Not surprisingly, it was shown that cooperation is mostly bene¯cial in the power limited (low SNR) regime, whereas no cooperation is preferred at high SNR. Forfadingchannels, theworkin [AGS05]analyzed theperformanceofAmplify- and-Forward (AF) and Decode-and-Forward (DF) two-hop cooperative strategies in terms of the achievable diversity vs. multiplexing gain (DMG) tradeo®[ZT03]. The two-hop strategies considered therein are in fact a mix of single and two-hop, since the source is able to transmit to the destination directly in [AGS05]. While the DMG tradeo® is certainly an important metric for asymptotic characterization of MIMO-based schemes, it relies on a high SNR assumption, which, in general, is not the most common scenario in multi-hop networks. In this chapter, we consider networks with more than a single relay in each hop (compared to [SLH + 04]) and more than two hops (compared to [AGS05]). We 141 obtain general expressions, for arbitrary SNR, to characterize the rates achievable by DSTBC-inspired protocols in fading channels, via multi-hop cooperative proto- cols and a half-duplex constraint. We focus on AF-type protocols employing an underlying DSTBC scheme and with perfect CSI at the decoding nodes. Towards this goal, we emphasize that most of the standard expressions for achievable rate-such as the Shannon Capacity, Ergodic Mutual Information and Outage Mutual Information - do not yield practical rate metrics for the proposed analysis; unliketheAWGNcaseinvestigatedin[SLH + 04], wecannotusethepoint- to-point Shannon capacity formula in block-fading channels. Since the channels betweentheterminalsarefadingingeneral,asweshallsee, themutualinformation between source and destination, I SD , turns out to be dependent on the random fadingrealizations. Achannel-independentcharacterizationisalternativelygivenin twoways,byanalyzingeithertheaverage(mean)ortailbehaviorofthedistribution of I SD . Thus, the ergodic mutual information (EMI) is de¯ned as C e =EfI SD g (5.1) where the operatorEf:g denotes the ensemble average over the statistics of all the fading channels. If the transmitted codewords are allowed to span an in¯nite number of inde- pendently fading blocks, the EMI is achieved via a circularly symmetric isotropic complex Gaussian codebook[PGNB04]. More importantly, in practical systems, 142 codewords have a ¯nite duration and span a single block, which yields a zero Shannon capacity. The reason is because, regardless of the information rate at the transmitter, there is always a nonzero probability that the fading channel re- alization will not support it. Therefore, in non-ergodic and/or delay-constrained scenarios, thep¡outage mutual information (p¡OMI),(oroutagerate)isthevalue C o;p that satis¯es P(I SD ·C o;p )=p (5.2) Thus,theoutagemutualinformationC o;p istheinformationratethatisguaranteed for 100p% of the channel realizations. The EMI and p-OMI are not practical performance metrics for the purpose of providing a general and useful performance index; the EMI assumes coding over in¯nitenumberofblockswhilethep¡OMIisonlyvalidforaspeci¯cvalueofp. On the other hand, for a given information rate R, the outage probability P o (R) is the probability that the mutual information I SD falls below R, and can be expressed as 1 . P o (R)=P(I SD ·R) (5.3) ForMIMOsystemsingeneral, (5.2)and(5.3)giverisetoaninterestingtradeo® between SNR, rate and PER. Speci¯cally, for a ¯xed rate R, one can trade an increase in the SNR for a decrease in PER with slope given by the maximum 1 At this point, we recall that the outage probability can be viewed as the minimum achievable packet error rate (PER). This is particularly useful when analyzing the achievable throughput in distributed networks [PGNB04, ZV05] 143 diversity order; conversely, for a ¯xed target PER, one can trade an increase in SNR for a linear increase in achievable rate[ZT03, PGNB04]. Our objective is to investigate a similar tradeo® for multi-hop/multi-relay sys- tems. We quantify the achievable rate in terms of the Zero-outage Throughput (or simply throughput in this work), de¯ned in[AB03]. For a general scheme with probability of outage P o (R), the throughput ¹ R is given as ¹ R,max R fR[1¡P o (R)]g (5.4) Note that the function maximization can be performed over R since the function R t ,R[1¡P o (R)] is concave. Speci¯cally, R t (R)!0 as R!0 or R!1. Our main result is in the form of approximate performance bounds on the p- OMI, PER and throughput for several types of DSTBC protocols. They show that, for the protocols under investigation, most of the gains of cooperation are present in the power-limited regime, while at the bandwidth-limited regime single- hop communication is preferable. This chapter is organized as follows. Section 5.2 develops the signal models for thecooperativeprotocolsunderconsideration,whicharebasedonorthogonal[TJC99] orquasi-orthogonal[PF03]distributedSTBCs. Section5.3presentstheoutageanal- ysis(OMI,outageprobabilityandthroughput)forthesingle-hop(non-cooperative) and the two-relay AAF scheme. Section 5.4 presents numerical results for compar- ison with the other protocols, and conclusions are outlined in Section 5.5 144 5.2 ReceivedSignalModelsforCooperativeSpace- Time Protocols 5.2.1 Single-Hop (Non-Cooperative) Communication The received signal in single-hop scheme is given by y = p P 1 ¯xh+w (5.5) where P is the source transmit power per symbol, ¯ is the (single-hop) path-loss attenuation factor from source to destination, x is the transmitted symbol, h » CN(0;1) is the random fading realization and w»CN(0;1) is AWGN. 5.2.2 The Two-Hop AAF Scheme We consider the two-hop setup depicted in Figure 5.1. Each transmission block s from the source consists of 2 complex and independent information symbols of zero mean unit variance. Hence the scheme has a (full) rate of 1 symbol/s/Hz during the source transmission. Each relay receives a noisy and distorted version s i of the source block, s i (k)= p P s °h i s(k)+w i (k);k =1;2 and i=1;2 (5.6) 145 Figure 5.1: 2 hop cooperative network with 2 relays. where the channel and noise assumptions remain identical, P s is the source-relay average transmit power per block and ° is the power attenuation due to path-loss between the source and the relays. After the relays perform AAF processing, the signal at destination is y = s P s P r ®° 2(P s °+1) 2 6 6 4 s(1) ¡s ¤ (2) s(2) s ¤ (1) 3 7 7 5 2 6 6 4 h 1 g 1 h ¤ 2 g 2 3 7 7 5 +w d (5.7) where[g 1 ;g 2 ],®andP r denote, respectively, therelay-destinationchannels, attenu- ation, and average transmit power per block. The overall noise w d has distribution CN(0;¾ 2 w I 2 ), with ¾ 2 w = Pr® 2(Ps°+1) P 2 i=1 jg i j 2 +1. By taking the complex conjugate of y(2) in (5.7) , the received signal can be represented as y 0 =GH 2 6 6 4 s(1) s ¤ (2) 3 7 7 5 +w 0 d (5.8) 146 where G= q P s P r ®° 2(Ps°+1) , and the equivalent channel matrix is H= 2 6 6 4 ~ h 1 ~ h 2 ¡ ~ h ¤ 2 ~ h ¤ 1 3 7 7 5 (5.9) where ~ h 1 =h 1 g 1 and ~ h 2 =h 2 g ¤ 2 . After channel matched-¯ltering at the receiver, we obtain y mf =H H y 0 =G¢ 2 2 6 6 4 s(1) s ¤ (2) 3 7 7 5 +w mf =G¢ 2 ~ s+w mf (5.10) where, due to the orthogonality of H, the post-matched ¯ltering noise w mf has distributionCN(0;¢ 2 ¾ 2 w ) with ¢ 2 = 2 6 6 4 ¸ 0 0 ¸ 3 7 7 5 ; where ¸=j ~ h 1 j 2 +j ~ h 2 j 2 : (5.11) If more spatial degrees are available in the network (i.e., more than two relays), it is desirable to extend the simple AAF scheme to further improve performance. This, however, comes at the price of either an upfront rate penalty to maintain the orthogonality of the overall channel H, or sacri¯cing the orthogonality of H to maintain the rate. Wenextdescribe3protocolsforcommunicationviamorethan2relays: the¯rst two,denotedbyrate1/2andrate3/5respectively,areorthogonalDSTBCschemes, 147 while the third, denoted by D-QSTBC (distributed quasi-orthogonal STBC) main- tains full rate. We emphasize the distinction between the rate 1/2 and rate 3/5 protocols for 2 hops presented in this chapter and the RH and SME protocols for 3 hops in our multi-hopping work [VM06]. 5.2.3 The Two-hop Rate 1/2 Scheme WewillconsiderthescenarioinFigure5.1withastageoffourrelaysinsteadoftwo. This scheme can also be extended to three or more than four relays as suggested in [HMC03]. Each transmission block s from the source consists of 4 complex and independent information symbols of zero mean unit variance, followed by their complex conjugates. Hence the scheme has a rate of 0.5 symbol/s/Hz during the source transmission. The relay processing and received signal at destination can be obtained in straightforward fashion. The resulting signal at destination is y 0 =GH 2 6 6 6 6 6 6 6 6 6 6 4 s(1) s(2) s(3) s(4) 3 7 7 7 7 7 7 7 7 7 7 5 +w 0 d (5.12) 148 where G= q P s P r ®° 2(Ps°+1) , and the equivalent channel matrix is H= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ~ h 1 ~ h 2 ~ h 3 ~ h 4 ~ h 2 ¡ ~ h 1 ~ h 4 ¡ ~ h 3 ¡ ~ h 3 ~ h 4 ~ h 1 ¡ ~ h 2 ¡ ~ h 4 ¡ ~ h 3 ~ h 2 ~ h 1 ~ h ¤ 1 ~ h ¤ 2 ~ h ¤ 3 ~ h ¤ 4 ~ h ¤ 2 ¡ ~ h ¤ 1 ~ h ¤ 4 ¡ ~ h ¤ 3 ¡ ~ h ¤ 3 ~ h ¤ 4 ~ h ¤ 1 ¡ ~ h ¤ 2 ¡ ~ h ¤ 4 ¡ ~ h ¤ 3 ~ h ¤ 2 ~ h ¤ 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.13) where ~ h i =h i g i . After channel matched-¯ltering at the receiver, we obtain y mf =H H y 0 =2G¢ 4 2 6 6 6 6 6 6 6 6 6 6 4 s(1) s(2) s(3) s(4) 3 7 7 7 7 7 7 7 7 7 7 5 +w mf =2G¢ 4 ~ s+w mf (5.14) 149 where, again due to the orthogonality of H, the post-matched ¯ltering noise w mf has distributionCN(0;2¢ 4 ¾ 2 w ) with ¾ 2 w = Er® 4(E s °+1) P 4 i=1 jg i j 2 +1 and ¢ 4 = 2 6 6 6 6 6 6 6 6 6 6 4 ¸ 0 0 0 0 ¸ 0 0 0 0 ¸ 0 0 0 0 ¸ 3 7 7 7 7 7 7 7 7 7 7 5 ; where ¸= 4 X i=1 j ~ h i j 2 : (5.15) 5.2.4 The Two-Hop Rate 3/5 Scheme For the speci¯c case of 3 relays, a two-hop scheme with source transmission rate of 0.6 symbol/s/Hz is possible, while also retaining the orthogonality of channel. In this case, each source transmission block s consists of 3 complex and independent informationsymbolsofzeromeanunitvariance,followedbythecomplexconjugates of s(2) and s(3). Denoting by r i;j the received signal at relay i, and index j (1 · i · 3 and 1·j·5), the relay transmission blocks v i are given by v 1 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r ¤ 1;1 ¡r ¤ 1;2 r ¤ 1;3 0 3 7 7 7 7 7 7 7 7 7 7 5 ; v 2 =G 1 ; 2 6 6 6 6 6 6 6 6 6 6 4 r 2;2 r 2;1 0 ¡r 2;5 3 7 7 7 7 7 7 7 7 7 7 5 ; v 3 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r 3;3 0 ¡r 3;1 r 3;4 3 7 7 7 7 7 7 7 7 7 7 5 (5.16) 150 where G 1 = q P s 3(P s °+1) 4 3 . The joint transmission of v i by the relays results in the received signal y at the destination. By complex conjugating y 2 ;y 3 and y 4 , the destination obtains y 0 =GH 2 6 6 6 6 6 6 4 s(1) s ¤ (2) s ¤ (3) 3 7 7 7 7 7 7 5 +w 0 d (5.17) where G= p P r °G 1 and the overall noise w 0 d has distributionCN(0;§ w ) with § w =I 4 +G 2 1 2 6 6 6 6 6 6 6 6 6 6 4 jg 1 j 2 +jg 2 j 2 +jg 3 j 2 0 0 0 0 jg 1 j 2 +jg 2 j 2 0 0 0 0 jg 1 j 2 +jg 3 j 2 0 0 0 0 jg 2 j 2 +jg 3 j 2 3 7 7 7 7 7 7 7 7 7 7 5 : (5.18) The equivalent channel matrix H is H= 2 6 6 6 6 6 6 6 6 6 6 4 ~ h 1 ~ h 2 ~ h 3 ~ h ¤ 2 ¡ ~ h ¤ 1 0 ¡ ~ h ¤ 3 0 ~ h ¤ 1 0 ~ h ¤ 3 ¡ ~ h ¤ 2 3 7 7 7 7 7 7 7 7 7 7 5 (5.19) 151 where ~ h 1 = h ¤ 1 g 1 , ~ h 2 = h 2 g 2 and ~ h 3 = h 3 g 3 . After channel matched-¯ltering at the receiver, we obtain y mf =H H y 0 =G¸I 3 2 6 6 6 6 6 6 4 s ¤ (1) s(2) s(3) 3 7 7 7 7 7 7 5 +w mf (5.20) where ¸ = P 3 i=1 j ~ h i j 2 and the post-matched ¯ltering noise w mf has distribution CN(0;H H § w H). 5.2.5 The Two-Hop Rate 1 D-QSTBC Scheme We now present a 4-relay scheme with full (1 symbol/sec/Hz) source transmis- sion rate per block. Hence, in the proposed scheme, s consists of 4 complex and independent information symbols of zero mean unit variance. Denoting by r i;j the received signal at relay i, and index j (1 · i · 4 and 1·j·4), the relay transmission blocks v i are v 1 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r 1;1 r 1;2 r 1;3 r 1;4 3 7 7 7 7 7 7 7 7 7 7 5 ; v 2 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r ¤ 2;2 ¡r ¤ 2;1 r ¤ 2;4 ¡r ¤ 2;3 3 7 7 7 7 7 7 7 7 7 7 5 ; v 3 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r 3;3 ¡r 3;4 ¡r 3;1 r 3;2 3 7 7 7 7 7 7 7 7 7 7 5 ; v 4 =G 1 2 6 6 6 6 6 6 6 6 6 6 4 r ¤ 4;2 r ¤ 4;1 ¡r ¤ 4;4 ¡r ¤ 4;3 3 7 7 7 7 7 7 7 7 7 7 5 (5.21) 152 whereG 1 = q P r 4(P s °+1) . Aftercomplexconjugationofthesecondandforthelements of the received signal y at the destination, the resulting signal is y 0 =GH 2 6 6 6 6 6 6 6 6 6 6 4 s(1) s ¤ (2) s(3) s ¤ (4) 3 7 7 7 7 7 7 7 7 7 7 5 +w 0 d (5.22) where G = p P s °G 1 and the overall noise w 0 d has distribution CN(0;¾ 2 w I 4 ) with ¾ 2 w =G 2 1 P 4 i=1 jg i j 2 +1. The channel matrix H is given by H= 2 6 6 6 6 6 6 6 6 6 6 4 ~ h 1 ~ h 2 ~ h 3 ~ h 4 ¡ ~ h ¤ 2 ~ h ¤ 1 ~ h ¤ 4 ¡ ~ h ¤ 3 ¡ ~ h 3 ¡ ~ h 4 ~ h 1 ~ h 2 ¡ ~ h ¤ 4 ~ h ¤ 3 ¡ ~ h ¤ 2 ~ h ¤ 1 3 7 7 7 7 7 7 7 7 7 7 5 (5.23) where ~ h i = h i g i , i = 1;3 and ~ h i = h ¤ i g i , i = 2;4. The channel matched-¯ltering at the receiver results in y mf =G¢ 4 2 6 6 6 6 6 6 6 6 6 6 4 s(1) s ¤ (2) s(3) s ¤ (4) 3 7 7 7 7 7 7 7 7 7 7 5 +w mf (5.24) 153 where the matrix ¢ 4 is ¢ 4 = 2 6 6 6 6 6 6 6 6 6 6 4 ¸ 0 ² 0 0 ¸ 0 ¡² ¡² 0 ¸ 0 0 ² 0 ¸ 3 7 7 7 7 7 7 7 7 7 7 5 ; where ¸= 4 X i=1 j ~ h i j 2 ; ²=2j=( ~ h ¤ 1 ~ h 3 + ~ h ¤ 4 ~ h 2 ): (5.25) Note that, due to the non-orthogonality of the scheme, the matrix ¢ 4 contains the interference term ². However, the particular structure of ¢ 4 still makes it full rank in general (since det(¢ 4 )=(° 2 +² 2 ) 2 ). Furthermore, a closer look at the structure of¢ 4 revealsthatthematched-¯lteredsystemin(5.10)canbe completely decoupled into two subsystems: 2 6 6 4 y mf (1) y mf (3) 3 7 7 5 = ¢ 2 2 6 6 4 s(1) s(3) 3 7 7 5 + 2 6 6 4 w mf (1) w mf (3) 3 7 7 5 (5.26) 2 6 6 4 y mf (2) y mf (4) 3 7 7 5 = ¢ 2 2 6 6 4 s ¤ (2) s ¤ (4) 3 7 7 5 + 2 6 6 4 w mf (2) w mf (4) 3 7 7 5 (5.27) where ¢ 2 = 2 6 6 4 ¸ ² ¡² ¸ 3 7 7 5 (5.28) 154 Noticethat(5.26)and(5.27)sharethesamechannelmatrix¢ 2 ,withidentically distributed, but statistically independent noise vectors. In addition, the nonzero term ² represents the interference within the "even" and "odd" substreams. Since the decoupled systems (5.26) and (5.27) are independent, the subsequent processing can be described, without loss in generality, for (5.26) only, with the understanding that it will be repeated in identical fashion for (5.27). From (5.24), (5.26) and (5.28), we observe that the covariance matrix of the vector [w mf (1) w mf (3)] T is § 2 = ¾ 2 w ¢ 2 . Since ¢ 2 is in general full rank, it can be decomposed as ¢ 2 =¡ 2 ¡ H 2 = 2 6 6 4 ¹ º ¡º ¹ 3 7 7 5 2 6 6 4 ¹ º ¡º ¹ 3 7 7 5 H (5.29) where it can be readily shown that ¹ = µ 1 p 2 ¶q ¸+ p ¸ 2 +² 2 (5.30) º = µ 1 p 2 ¶ jsign µ ² j ¶q ¸¡ p ¸ 2 +² 2 (5.31) Premultiplying both sides of (5.26) by ¡ ¡1 2 , we obtain 2 6 6 4 y 0 mf (1) y 0 mf (3) 3 7 7 5 =¤ 2 6 6 4 s(1) s(3) 3 7 7 5 + 2 6 6 4 w 0 mf (1) w 0 mf (3) 3 7 7 5 (5.32) 155 where ¤ = ¡ ¡1 2 ¢ 2 , and the noise terms w 0 mf (1) and w 0 mf (3) are independent and Gaussian with variance ¾ 2 w . 5.3 Outage Analysis of Cooperative Protocols We now turn our attention towards obtaining expressions for the OMI and proba- bility of outage of the cooperative protocols discussed in the previous sections. We start by discussing the single-hop (non-cooperative) scheme, which will serve as a baseline for comparison with the other protocols. 5.3.1 Single-Hop Communication From (5.5), the mutual information I D achieved by the direct communication scheme is I D =log(1+P 1 ¯jhj 2 ) (5.33) The outage probability for a rate R is, therefore, P o (R)=P(I D <R)=P ¡ log(1+P 1 ¯jhj 2 )<R ¢ =P µ jhj 2 < 2 R ¡1 P 1 ¯ ¶ (5.34) Since z =jhj 2 has exponential distribution with CDF F(z) = 1¡e z , the prob- ability of outage is P o (R)=1¡e ¡ 2 R ¡1 P 1 ¯ (5.35) 156 The p¡OMI, denoted by R p , is the rate that is achievable for 100(1¡p)% of the channel realizations. Thus, from (5.35), we have p=1¡e ¡ 2 R p ¡1 P 1 ¯ (5.36) which yields R p =log 2 (¡¯P 1 log(1¡p)+1) (5.37) Substituting (5.35) in (5.4), and setting the derivative of R t (R) to zero, we obtain the throughput ¹ R D for the single-hop scheme is ¹ R D = W(P 1 ¯) log(2) e ¡ 2 W(P 1 ¯) log(2) ¡1 P 1 ¯ (5.38) where W(x) denotes the Lambert W function, which is de¯ned as the inverse func- tion of f(z) = ze z . The corresponding source rate ¹ R i that achieves ¹ R D is given by ¹ R i = W(P 1 ¯) log(2) : (5.39) 157 5.3.2 AAF Protocol From (5.10) and (5.11), and the overall system rate, the mutual information of the AAF scheme, conditioned on the fading channel realizations, is I A = 1 4 log 2 det µ I 2 + G 2 ¾ 2 w ¢ 2 ¶ = 1 2 log 2 Ã 1+ G 2 ¾ 2 w 2 X i=1 j ~ h i j 2 ! = 1 2 log 2 (1+½¸)b/s/Hz (5.40) where ½= 2G 2 (Ps°+1) P r ®(jg 1 j 2 +jg 2 j 2 )+2(P s °+1) is an SNR factor and ¸=jh 1 j 2 jg 1 j 2 +jh 2 j 2 jg 2 j 2 . The outage probability of the AAF scheme is given by Po(R)=P(I A <R) (5.41) Unfortunately, an exact analysis of the outage probability of the AAF scheme is not available, since the overall noise variance now depends on the relay-destination channel realizations. We will, therefore approximate the noise variance by its mean (AWGN approximation). From Jensen's inequality, I A ¸ ~ I A = 1 2 log 2 (1+ ¹ ½¸) (5.42) where ¹ ½= G 2 (P s °+1) Pr®+Ps°+1 . An upper bound on the probability of outage is given by P o (R)·P( ~ I A <R)=P µ ¸< 2 2R ¡1 ¹ ½ ¶ ,P (u) o (R) (5.43) 158 5.3.2.1 Exact Expression for P (u) o (R) Recalling that, conditioned on g 1 and g 2 , ¸ is a sum of independent Chi-square random variables, and de¯ning r = 2 2R ¡1 ¹ ½ , we have P (u) o (R)=P (¸<rjx 1 ;x 2 )= 1 2jx 1 ¡x 2 j ½ 1 µ 1 (1¡e ¡µ 1 r )¡ 1 µ 2 (1¡e ¡µ 2 r ¾ (5.44) where x i =jg i j 2 =2, and µ 1 = x 1 +x 2 ¡jx 1 ¡x 2 j 4x 1 x 2 µ 2 = x 1 +x 2 +jx 1 ¡x 2 j 4x 1 x 2 (5.45) Thus, the exact expression for P (u) o (R) is P (u) o (R)= Z 1 0 Z 1 0 P (¸<rjx 1 ;x 2 )e ¡x 1 ¡x 2 dx 1 dx 2 (5.46) 5.3.2.2 Approximate Bounds on P (u) o (R) Due to the somewhat involved nature of the integrand, the exact bound in (5.46) canonlybecomputednumerically. Toobtainsome furtherinsight, wenowdevelop approximate upper and lower bounds on P (u) o (R). These bounds are approximate in the sense that they rely on an approximation of the PDF of the product of independent exponential random variables, developed in the Appendix (Section 5.6). 159 From the expression for P (u) o (R) given in (5.43), we can obtain the upper bound P (u) o (R)=P µ ¸< 2 2R ¡1 ¹ ½ ¶ · P µ jh 1 j 2 jg 1 j 2 < 2 2R ¡1 ¹ ½ ¶ P µ jh 2 j 2 jg 2 j 2 < 2 2R ¡1 ¹ ½ ¶ = · P µ Z < 2 2R ¡1 ¹ ½ ¶¸ 2 (5.47) where Z has PDF given by (see Appendix, Section 5.6) f(z)=2K 0 (2 p z) (5.48) A lower bound on P (u) o (R) is given by P (u) o (R)=P µ ¸< 2 2R ¡1 ¹ ½ ¶ ¸ P µ jh 1 j 2 jg 1 j 2 < 2 2R ¡1 2¹ ½ ¶ P µ jh 2 j 2 jg 2 j 2 < 2 2R ¡1 2¹ ½ ¶ = · P µ Z < 2 2R ¡1 ¹ ½ ¶¸ 2 (5.49) UsingtheapproximateCDFofZ obtainedinSection5.6yieldstheapproximate bounds P (u) o (R). " 2 p ¼ ° Ã 3 2 ;2 s 2 2R ¡1 ¹ ½ !#2 (5.50) P (u) o (R)& " 2 p ¼ ° Ã 3 2 ;2 s 2 2R ¡1 2¹ ½ !#2 (5.51) 160 Figure 5.2 compares the simulated AAF outage probability (for R= 1 bps/Hz) withtheboundsobtainedhere: thenumericallycomputedGaussianapproximation boundin(5.46),itsapproximateupperboundgivenby(5.50)andlowerboundgiven by (5.51). As expected, the Gaussian approximation bound indeed yields an upper bound on the true performance. The approximate upper bound further bounds the performance, yielding a looser bound, whereas the approximate lower bound yields a much tighter approximation on the true performance. Due to the PDF approximation, however, it is not a strict lower bound for high transmit energy. Also shown in Figure 5.2 are the outage probabilities for single-hop communication underpathlossexponentsof® =2and® =4. Asexpectedagain,duetoitslackof cooperative diversity, the single-hop scheme performs poorly at moderate to high energy scenarios. Finally, the performance when the relay-destination channels are static (non-fading) is also shown. Clearly, the exponential-product nature of the overall channel has a signi¯cant impact on the performance. 5.3.2.3 Approximate Bounds on the OMI Theexpressionsin(5.50)and(5.51)canalsobeusedtoobtainapproximatebounds on the p¡OMI of the AAF protocol. From the upper bound on P (u) o (R) an approx- imate lower bound on the p¡OMI, denoted by R (l) p satis¯es 2 4 2 p ¼ ° 0 @ 3 2 ;2 s 2 2R (l) p ¡1 ¹ ½ 1 A 3 5 2 =p (5.52) 161 0 5 10 15 20 25 30 35 40 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Outage probability for AAF protocol (R = 1 b/s/Hz) P out Total energy [dB] AAF simulation AAF Gaussian approximation bound AAF lower bound(pdf approximation) AAF upper bound(pdf approximation) single hop (α =2 ) single hop (α = 4) AAF static 2 nd hop Figure 5.2: Outage probability of the AAF protocol which yields R (l) p = 1 2 log 2 Ã 1 4 · ° ¡1 µ 3 2 ; p p¼ 2 ¶¸ 2 ¹ ½+1 ! (5.53) where ° ¡1 (a;x) denotes the inverse of the Incomplete Gamma function. Similarly, an approximate upper bound on the p- OMI is given by R (u) p = 1 2 log 2 Ã 1 2 · ° ¡1 µ 3 2 ; p p¼ 2 ¶¸ 2 ¹ ½+1 ! (5.54) Figure 5.3 compares the 10% OMI simulation results with the bounds obtained in this section. As in the outage probability case, the Gaussian approximation naturallyincursalowerpredictedperformance. Furthermore,whilethelowerbound 162 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Total energy [dB] bps/Hz 10% OMI for the AAF protocol AAF OMI − simulation Gaussian approx − simulation Upper bound on Gaussian approx. Lower bound on Gaussian approx. static 2 nd hop (Rayleigh fading) Figure 5.3: 10% OMI of the AAF protocol (5.53) yields a conservative predicted OMI, the upper bound (5.54) apparently predictstheactualsimulationresultsquitewell. Figure5.4comparesthe10%OMI of the AAF and single-hop schemes. As expected, the AAF scheme su®ers, at high transmitenergy, ofaratepenaltyduetothepracticalhalf-duplexconstraintonthe transmissions. In the single-hop scheme, on the other hand, the source transmits newsymbolsateverytimeslot. Nevertheless, itsclearthatthebestchoicebetween both schemes is dependent on the available transmit energy. 163 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 10% OMI for AAF and single−hop Total transmit energy [dB] bps/Hz AAF − simulation Single hop (α= 2) Single hop (α =4) Figure 5.4: 10% OMI of AAF and single-hop protocols 5.3.2.4 Throughput Approximate values for the throughput of the AAF scheme can be found via the probability of outage expressions given in (5.50) and (5.51). Speci¯cally, an upper bound approximation is ¹ R (u) =max R 8 < : R 2 4 1¡ " 2 p ¼ ° Ã 3 2 ;2 s 2 2R ¡1 2¹ ½ !#2 3 5 9 = ; (5.55) and a lower bound is ¹ R (l) =max R 8 < : R 2 4 1¡ " 2 p ¼ ° Ã 3 2 ;2 s 2 2R ¡1 ¹ ½ !#2 3 5 9 = ; (5.56) 164 Althoughnoclosedformexiststoourbestknowledge,thesolutionsto(5.55)and (5.56) can be computed numerically in straightforward fashion. Shown in Figures 5.5 and 5.6 are the simulation results on the e®ective throughput achieved by the sourceasafunctionofthesourcerate,fortotaltransmitenergyof30dBand10dB respectively. The upper and lower bounds in (5.55) and (5.56) are also shown for comparison. In both cases, it appears that the optimum source transmission rate canbereasonablywellpredictedbyaveragingtheoptimumratesresultingfromthe upper and lower bounds. Furthermore, at high transmit energy, the upper bound yields a better prediction of the e®ective throughput. The throughput comparison between AAF and single-hop communication is showninFiguresand. Athightransmitenergyandlowattenuationfactor(® =2), single-hop is clearly preferable due to the half-duplex constraint of cooperation. However, at low transmit energy and attenuation factor (® = 4), the throughput of cooperative transmission is signi¯cantly superior. 5.3.3 Rate 1/2, Rate 3/5 and D-QSTBC Protocols From (5.14) and (5.15), the mutual information and the p¡OMI for the rate 1/2 scheme with four relays are, respectively I1 2 = 1 4 log 2 Ã 1+ 2G 2 ¾ 2 w 4 X i=1 j ~ h i j 2 ! b/s/Hz (5.57) R1 2 ;p = arg C¸0 n P(I1 2 ·C)=p o (5.58) 165 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 source rate [bps/Hz] effective throughput [bps/Hz] Effective throughput for the AAF scheme, E = 30 dB upper bound simulation lower bound Figure 5.5: AAF throughput, for total energy of 30 dB and, for the rate 3/5 scheme, I3 5 = 1 9 log 2 det Ã I 3 +G 2 3 X i=1 j ~ h i j 2 (H H § w H) ¡1 ! b/s/Hz (5.59) R3 5 ;p = arg C¸0 n P(I3 5 ·C)=p o (5.60) From (5.28) and (5.32), the mutual information for D-QSTBC is I Q = 1 4 log 2 det µ I 2 + G 2 ¾ 2 w ¢ 2 ¶ b/s/Hz (5.61) and the q% OMI is R Q;p =arg C¸0 fP(I Q ·R)=pg (5.62) 166 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 source rate [bps/Hz] effective throughput [bps/Hz] Effective thoughput for the AAF scheme, E = 10dB lower bound simulation upper bound Figure 5.6: AAF throughput, for total energy of 10 dB The penalty (in terms of achievable rate) incurred by the D-QSTBC scheme can be assessed by comparing (5.61) with the ideal four-relay mutual information, denoted by I 0 . It is found by setting the interference term ² in (5.28) to zero, yielding I 0 = 1 2 log 2 Ã 1+ G 2 ¾ 2 w 4 X i=0 j ~ h i j 2 ! b/s/Hz (5.63) Finally, the outage probability P o (R) of a given protocol with mutual informa- tion I x and for a given rate R can be found by P o (R)=P(I x ·R) (5.64) 167 1.5 2 2.5 3 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5 4 Effective throughput, E = 30 dB source rate [bps/Hz] effective throughput [bps/Hz] AAF single hop (α = 2) single hop (α = 4) Figure 5.7: Single-hop and AAF throughput, for total energy of 30 dB where I x corresponds to one of the expressions for the mutual information found in this section. 5.4 Numerical Results In this section, we provide some numerical results on the OMI and outage prob- abilities for the several schemes discussed in this chapter. Figure 5.9 shows the 10% OMI for orthogonal and proposed D-QSTBC scheme, for ° = ® = 1 (no at- tenuation). Note that, among the schemes analyzed in the previous section, only the four relay D-QSTBC approach e®ectively improves the OMI over AAF. The reason is because that both the rate 1/2 and rate 3/5 protocols sacri¯ce rate in 168 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Effective throughput, E=10 dB source rate [bps/Hz] effective throughput [bps/Hz] AAF single hop (α = 2) single hop (α =4) Figure 5.8: Single-hop and AAF throughput, for total energy of 10 dB order to preserve orthogonality, while AAF is the only full rate orthogonal scheme. Furthermore, we also notice from Figure 5.9 that the OMI achieved by D-QSTBC is very close to the optimal (ideal) OMI for four relays. Theoutageprobability,orachievablepacketerrorrate(PER)for R =1bps/Hz is shown in Figure 5.10. The PER for direct source-destination communication is also included for comparison with the cooperative schemes. The path-loss attenu- ation at distance d was computed according to the expression[ZV05] ° =® =K µ d d 0 ¶ 2 (5.65) 169 0 5 10 15 20 25 30 35 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Average power per symbol [dB] bps/Hz 10% Outage capacities for different protocols rate 1, orthogonal (AAF), 2 relays ideal (no interference), 4 relays rate 1, proposed, 4 relays rate 1, proposed, static 2 nd hop rate 3/5, orthogonal, 3 relays rate 1/2, orthogonal, 4 relays Figure 5.9: 10% OMI for di®erent cooperation protocols where K is the attenuation at the reference distance d 0 . In our simulations, we employed d 0 =10m, K =1, and d=50m for two-hop and d=100m for single-hop (direct) communication. Note that, at su±ciently high transmit power, all coop- erative protocols exhibit superior performance compared to direct communication, withtheD-QSTBCschemeagainexhibitingthebestperformance. Furthermore,in the same scenario, increasing the number of relays improves performance in terms of PER. Figure 5.10 also shows that, even though the rate 1/2 scheme has lower achievable rate than AAF (see Figure 5.9), it can o®er superior PER performance. 170 5 10 15 20 25 30 35 40 45 50 10 −4 10 −3 10 −2 10 −1 10 0 Average power per symbol [dB] PER Outage Probabilities, R = 1 b/s/Hz direct transmission AAF, 2 relays rate 1/2, 4 relays D−QSTBC, 4 relays Figure 5.10: Achievable packet error rate for di®erent cooperation protocols 5.5 Conclusions In this chapter, the achievable rates, outage probabilities and throughput of coop- erative protocols based on orthogonal and quasi-orthogonal distributed STBCs are investigated. Forthetwo-relayAAFprotocol,approximateupperandlowerbounds ontheoutagemutualinformation, outageprobabilityandthroughputwerederived in analytical form. Compared to single-hop transmission, it was shown that, al- thoughpractical(half-duplex)cooperationgenerallyhassuperiorperformancefrom an outage probability perspective, this does not always hold true for the OMI and throughput. Inparticular, theidealmodeofoperation(cooperationvs. singlehop) tomaximizethethroughputishighlydependentonthepowerregime: inthepower 171 limited scenario, cooperation is highly bene¯cial, whereas in the bandwidth limited case, single-hop is preferable. 5.6 Appendix: Exponential-ProductDistribution Let X and Y be two independent random variables with standard exponential densities, i.e., f(x) = e ¡x and f(y) = e ¡y , where x;y ¸ 0. Our goal is to ¯nd the PDF f(z) and CDF F(z) of Z =XY. De¯ning the auxiliary variable W = X, the joint PDF of (Z;W) can be ex- pressed in terms of the joint PDF of (X;Y) as f(z;w)= f(x;y)j x=w;y= z w jJ(z;w)j = e ¡w e ¡ z w w (5.66) where J(z;w) is the Jacobian determinant de¯ned as J(z;w)= ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ y x 1 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ x=w;y= z w =¡w (5.67) The PDF of Z is therefore, f(z)= Z 1 0 e ¡(w+ z w ) w dw =2K 0 (2 p z) (5.68) 172 where K 0 (x) is a Modi¯ed Bessel function of the second kind. A useful approxima- tion of K 0 (x) is K 0 (x)¼ r ¼ 2x e ¡x (5.69) It is straightforward to show that R 1 0 p ¼ 2x e ¡x dx = ¼ 2 p 2 , and, thus the PDF of Z can be approximated as f(z)¼2 r 2 ¼ z ¡ 1 4 e ¡2 p z (5.70) The CDF can be approximated using (5.70), yielding F(z)¼ Z z 0 2 r 2 ¼ t ¡ 1 4 e ¡2 p t dt= 2 p ¼ ° µ 3 2 ;2 p z ¶ (5.71) where °(®;x) is the Incomplete Gamma function, de¯ned as °(®;x)= Z x 0 t a¡1 e ¡t dt (5.72) Figure 5.11 compares the exact PDF in (5.68) with the approximation given by (5.70). 173 0 0.2 0.4 0.6 0.8 1 10 −1 10 0 10 1 10 2 z f(z) Exponential−product pdf true pdf approx pdf Figure 5.11: Exponential-product PDF 174 Chapter 6 Conclusions In this thesis, space-time block coded schemes for single-hop and multi-hop com- munication for quasi-static wireless fading channels were proposed and analyzed. In the ¯rst part, several performance criteria for single-hop transmission based on space-time block codes were developed based on generalization of the union bound approach. The new performance metrics, such as the indecomposable union bound and the progressive union bound, were shown to be tighter than the classic Cherno® Bound technique, especially for high code cardinality and at low SNR. Furthermore, key properties of the new criteria were developed which found appli- cation in the construction of new improved codes. In the second part, distributed multi-hop communication protocols based co- operative transmissions between terminals were investigated. In all the proposed schemes, cooperation was achieved by employing a distributed space-time block code (DSTBC), and suitably designing the transmitted signals for individual ter- minals. Unlike non-DSTBC based schemes, the proposed approach do not incur 175 loss in spectral e±ciency, since signals transmitted from separate terminals are not transmitted orthogonally (such as in TDMA or FDMA schemes). The proposed schemes were also shown to be fairly robust to relay asynchronism. The third part this thesis investigated multi-hop communication protocols in the presence of ISI. The proposed TR-DSTBC protocols were shown to exhibit low complexity decoding, without requiring a vector Viterbi-type receiver for optimal decoding. Furthermore,theperformanceofadecision-feedbackequalizerwasinves- tigated and shown to be quite close to that of the optimal receiver. The potential gains of the proposed multi-hop protocols were con¯rmed by experimental results performed on underwater acoustic channels. The last part of this thesis focused on achievable rates of distributed space- time cooperative schemes. 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Abstract (if available)

Abstract The presence of random fading in wireless channels constitutes one of the most challenging problems for achieving reliable digital communications. At the same time, fading also offers opportunities for improved performance via multi-dimensional communications, also known as multiple-input/multiple-output (MIMO) signaling. Given the ever increasing demand for high data rate in real-time applications involving transmission of voice, data and multimedia content, modern communication networks need be properly designed in order to fully extract the potential gains offered by MIMO schemes. In point-to-point communications, the use of multiple antennas at the transmitter and/or receiver constitutes one of the standard approaches to MIMO signaling, through the use of Space-Time Codes. Such codes enable dramatic performance improvement over single-antenna systems.

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Asset Metadata

Creator Vajapeyam, Madhavan S. (author)

Core Title Space-time codes and protocols for point-to-point and multi-hop wireless communications

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 02/22/2007

Defense Date 01/22/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cooperative communications,OAI-PMH Harvest,space-time coding,wireless communications

Language English

Advisor Mitra, Urbashi (committee chair), Baxendale, Peter H. (committee member), Caire, Giuseppe (committee member), Weber, Charles L. (committee member)

Creator Email vajapeya@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m286

Unique identifier UC1159788

Identifier etd-Vajapeyam-20070222 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-162871 (legacy record id),usctheses-m286 (legacy record id)

Legacy Identifier etd-Vajapeyam-20070222.pdf

Dmrecord 162871

Document Type Thesis

Rights Vajapeyam, Madhavan S.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu