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Asymptotic performance analysis and design of wireless communication systems in generalized fading and… Nezampour Meymandi, Ali 2010

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ASYMPTOTIC PERFORMANCE ANALYSIS AND DESIGN OF WIRELESS COMMUNICATION SYSTEMS IN GENERALIZED FADING AND NOISE by ALI NEZAMPOUR MEYMANDI M.Sc., Sharif University of Technology, 2005 B.Sc., Amirkabir University of Technology, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2010 c Ali Nezampour Meymandi, 2010  Abstract One of the most important challenges for communication system designers is to combat the detrimental effects of fading and noise. Fading refers to random changes in the channel gain due to shadowing and multi–path propagation of the transmitted signal. Several statistical distributions have been proposed to model the fading phenomenon. On the other hand, various techniques, such as multi–antenna transmission and/or reception (e.g., space–time coding) have been proposed to improve the performance of wireless communication systems in fading channels. Clearly, the performance and consequently, the design of these systems depend on the considered fading model. Since the wireless environment is intrinsically dynamic and may also experience the keyhole effect, i.e., not fully correlated but rank deficient fading, it is important to have a comprehensive performance analysis framework which is applicable to a wide range of fading models. In this thesis, we provide such a framework for the asymptotic (hign signal–to–noise ratio) analysis of single–antenna transmission with receive diversity and space–time coding in generalized non–keyhole and keyhole fading channels including Rayleigh, Ricean, Nakagami–q, Nakagami–m, generalized–K, and Weibull fading. The presented analysis provides valuable new insights into system design and can be used to obtain tight asymptotic upper bounds for the bit, symbol, and frame error probabilities. The second subject studied in this thesis is non–Gaussian noise. Noise (which in our definition includes interference) in communication systems has been traditionally modelled as Gaussian. This is mainly motivated by the tractability of the design and analysis of ii  Abstract communication systems in Gaussian noise and is justified by the central limit theorem. However, wireless communication systems are often impaired by non–Gaussian noise and interference as well. Examples of non–Gaussian noise include co–channel interference, man–made or natural impulsive noise, and ultra–wideband interference. In this thesis, we analyze the performance of single-antenna transmission and space–time codes impaired by non–Gaussian noise and interference. Our general and easy–to–evaluate results reveal the effects of these types of noise and pave the way for designing robust detection techniques that perform close to optimum for a wide range of practical noise and interference environments. As an example, we propose an adaptive Lp –norm metric for robust detection in non–Gaussian noise which outperforms previously reported metrics.  iii  Preface The contributions of this thesis have been published or submitted in five journal and several conference papers as listed in Appendix F. Since all of these papers were co–authored by other colleagues, I would like to clarify my contributions to these publications, particularly those related to Chapter 5. In all publications, I contributed through surveying the literature, developing the research idea, formulating the problem, and performing simulations. Furthermore, I wrote the papers related to Chapters 2, 3, 4, and 6. I also wrote parts of the papers related to Chapter 5 and prepared four out of nine of the figures in those papers.  iv  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xix  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Motivation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Mathematical Modeling of Fading in Wireless Systems . . . . . . . . . . .  4  1.3  Non–Gaussian Noise in Wireless Systems  . . . . . . . . . . . . . . . . . .  11  1.4  Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  20  1.5  Literature Review  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  1.5.1  STCs in Non–keyhole Fading Channels with Gaussian Noise . . . .  23  1.5.2  STCs in Keyhole Fading Channels with Gaussian Noise  23  . . . . . .  v  Table of Contents 1.5.3  Diversity Combining Receivers in Non–Gaussian Noise . . . . . . .  24  1.5.4  STCs in Non–Gaussian Noise . . . . . . . . . . . . . . . . . . . . .  25  1.6  Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .  26  1.7  Organization of the Thesis  . . . . . . . . . . . . . . . . . . . . . . . . . .  28  2 Performance Analysis of STCs in Generalized Fading Channels . . . .  31  2.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  2.2  System Model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  2.3  Asymptotic Analysis  2.4  Numerical Results  2.5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  38  3 Performance Analysis of STCs in Generalized Keyhole Fading Channels 39 3.1  Introduction  3.2  System Model  3.3  Asymptotic PEP Analysis: Cases 1 and 2 (D = 0)  3.4  3.5  3.6  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  39  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  . . . . . . . . . . . . .  42  3.3.1  Asymptotic PEP Analysis: General Case  . . . . . . . . . . . . . .  42  3.3.2  Case 1 (D < 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  3.3.3  Case 2 (D > 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  Asymptotic PEP Analysis: Case 3 (D = 0)  . . . . . . . . . . . . . . . . .  3.4.1  Nakagami–m Fading at the Receiver Side  3.4.2  Ricean Fading at the Receiver Side  48  . . . . . . . . . . . . . .  48  . . . . . . . . . . . . . . . . .  50  . . . . . . . . . . . . . . . . . . . . . . . . .  52  3.5.1  Unified Asymptotic PEP Expression . . . . . . . . . . . . . . . . .  52  3.5.2  Implications for Code Design . . . . . . . . . . . . . . . . . . . . .  53  3.5.3  Keyhole vs. Non–keyhole Channels . . . . . . . . . . . . . . . . . .  54  Unification and Implications  Numerical Results  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  55 vi  Table of Contents 3.7  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4 Performance Analysis of Diversity Combining Receivers . . .  61  . . . . . .  62  4.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  4.2  Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  63  4.3  Asymptotic Performance Analysis  . . . . . . . . . . . . . . . . . . . . . .  64  4.3.1  Asymptotic Pairwise Error Probability (PEP) . . . . . . . . . . . .  65  4.3.2  Quadratic Diversity Combining Schemes . . . . . . . . . . . . . . .  67  4.3.3  Combining Gain and Comparison . . . . . . . . . . . . . . . . . . .  71  Calculation of Noise Moments . . . . . . . . . . . . . . . . . . . . . . . . .  72  4.4.1  Spatially Independent Noise . . . . . . . . . . . . . . . . . . . . . .  73  4.4.2  Spatially Dependent Noise  78  4.4.3  Calculation by Monte–Carlo Simulation  4.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  82  4.5  Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . .  83  4.6  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  89  5 Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  90  5.1  Introduction  90  5.2  System Model and Lp –Norm Metric  5.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91  5.2.1  Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91  5.2.2  Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  93  5.2.3  Lp –Norm Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  Asymptotic Analysis of Lp –Norm Combining  . . . . . . . . . . . . . . . .  96  . . . . . . . . . . . . . . . . . . . . . . . .  96  5.3.1  Asymptotic PEP of CC  5.3.2  Asymptotic PEPs of DC and NC  . . . . . . . . . . . . . . . . . .  97  5.3.3  Asymptotic BER . . . . . . . . . . . . . . . . . . . . . . . . . . . .  98 vii  Table of Contents 5.3.4 5.4  5.5  Combining and Diversity Gain  . . . . . . . . . . . . . . . . . . . .  98  Generalized Noise Moments . . . . . . . . . . . . . . . . . . . . . . . . . .  99  5.4.1  Exact Noise Moments for L = 2  99  5.4.2  Noise Moments for General L . . . . . . . . . . . . . . . . . . . . . 101  . . . . . . . . . . . . . . . . . . .  Metric Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.5.1  Off–line Optimization  . . . . . . . . . . . . . . . . . . . . . . . . . 103  5.5.2  On–line Optimization  . . . . . . . . . . . . . . . . . . . . . . . . . 105  5.6  Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . 112  5.7  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116  6 Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1  Introduction  6.2  System Model  6.3  6.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119  6.2.1  Channel and Noise Model . . . . . . . . . . . . . . . . . . . . . . . 120  6.2.2  Mahalonobis Distance (MD) Metric  . . . . . . . . . . . . . . . . . 121  Asymptotic Analysis of Space–Time Codes  . . . . . . . . . . . . . . . . . 122  6.3.1  Asymptotic PEP of Coherent STCs  . . . . . . . . . . . . . . . . . 122  6.3.2  Asymptotic PEP of Differential STBCs  6.3.3  Implications for STC and Metric Design  6.3.4  BEP, SEP, and FEP . . . . . . . . . . . . . . . . . . . . . . . . . . 129  Calculation of the Noise Moments  . . . . . . . . . . . . . . . 126 . . . . . . . . . . . . . . 127  . . . . . . . . . . . . . . . . . . . . . . 130  6.4.1  Additive White Gaussian Noise (AWGN)  . . . . . . . . . . . . . . 130  6.4.2  Additive Correlated Gaussian Noise (ACGN)  6.4.3  Spatially Independent Gaussian Mixture Noise (SIGMN) . . . . . . 134  6.4.4  Spatially Dependent Gaussian Mixture Noise (SDGMN) . . . . . . 135  . . . . . . . . . . . . 131  viii  Table of Contents 6.4.5  Asynchronous Co–channel Interference (CCI) . . . . . . . . . . . . 137  6.5  Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . 140  6.6  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148  7 Summary of Thesis and Future Research Topics . . . . . . . . . . . . . . 151 7.1  Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151  7.2  Future Work 7.2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153  Performance Analysis of STCs with Arbitrary Number of Transmit Antennas in Generalized Fading  . . . . . . . . . . . . . . . . . . . 153  7.2.2  Performance Analysis in α–stable Noise  . . . . . . . . . . . . . . . 154  7.2.3  Analyzing the Performance of Non–coherent STCs  7.2.4  Performance Analysis of Relay Networks in Non–Gaussian Noise  7.2.5  Optimizing Non–square STCs in Non-Gaussian Noise . . . . . . . . 156  7.2.6  More Robust Techniques  . . . . . . . . . 154 . 155  . . . . . . . . . . . . . . . . . . . . . . . 156  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157  Appendices A Calculation of I(θ) in (2.9)  . . . . . . . . . . . . . . . . . . . . . . . . . . . 167  B PDF of Y for NT = 1 and NT = 2  . . . . . . . . . . . . . . . . . . . . . . . 170  C Asymptotic Statistical Properties of X  . . . . . . . . . . . . . . . . . . . 173  D Evaluating a Function at Infinity Using its Power Series E Asymptotic PEP for CC  . . . . . . . . 174  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176  ix  Table of Contents F Publications Related to Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 178  x  List of Tables 2.1  Parameters α and β in (2.2) for common types of fading, cf. Chapter 1. . .  3.1  Parameters αk and βk in (3.3) and (3.4) for common types of fading, cf.  33  Chapter 1. Indices k, i, and j are non-negative integers. . . . . . . . . . . .  42  4.1  Parameters βM and dM for various signal constellations A. . . . . . . . . .  70  4.2  Moments Mn (N) = E{|n|2N } of scalar Gaussian RVs and the RVs discussed in Examples E1)-E3). Gaussian RV: Mean µn and variance σn2 . The parameters for the other RVs are defined in Section 4.4.1. . . . . . . . . . . . . .  4.3  74  Moments Mn (L) = E{|n|2L } of RVVs. Zero–mean i.i.d. Gaussian RVV: Variance of each element σn2 . Correlated Gaussian RVV: λl , 1 ≤ l ≤ L, are the eigenvalues of covariance matrix Rnn  E{nnH }. The parameters of  the Gaussian mixture RVV are defined in Section 4.4.2, Example E8). . . . 5.1  78  Generalized noise moments Mn (p) for L = 2 for various types of n.i.d. noise. In particular, we consider AWGN, i.i.d. GMN, n.i.d. Rayleigh–faded CCI– I (s  [s1 . . . sI ]T , si  k2 κ=k1  pi [κ]bi [κ], S contains all possible values of [b1,1 . . . b1,L ]T , MI contains  s), i.i.d. Rayleigh–faded CCI–II (I = 1, bI all possible values of bI , c1 i.i.d. GGN.  1,  c2  1−  1,  2 σ ¯g,1  2 2 σg,1 , σ ¯g,2  0), and  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100  xi  List of Tables 5.2  Approximations for the generalized noise moments Mn (p) for general L for the same types of n.i.d. noise considered in Table 5.1. Additionally, exact results for unfaded n.i.d. CCI–I (I = 1) and i.i.d. CCI–II (I = 1, ξ1 ξ2  6.1  1,  0) are provided. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103  Moments My (p)  E{(y H y)p } and My D (p)  p E{(y H D y D ) } relevant for  coherent (“C”) and differential (“D”) STCs, respectively. All variables in this table are defined in Section 6.4. . . . . . . . . . . . . . . . . . . . . . . 133 7.1  A comparison of the scenarios considered in Chapters 2–6. In this table, NT = 1 refers to systems with a single transmit antenna and quadratic diversity combining at the receiver; NT > 1 means space–time coding; non– Gaussian noise refers to any type of noise and interference with finite moments; Lp –norm includes the ED metric (based on L2 –norm) as a special case; and MD metric includes ED and ND metrics as special cases. The number of receiver antennas is arbitrary in all chapters. . . . . . . . . . . . 152  xii  List of Figures 1.1  The pdf of Ricean fading for Ω = 1 and Ω = 3 and different values of the Ricean factor, K[dB] = 10 log10 (K). . . . . . . . . . . . . . . . . . . . . . .  6  1.2  The pdf of Nakagami–m fading for Ω = 1 and different values of m. . . . .  7  1.3  The pdf of Nakagami–q fading for Ω = 1 and different values of q. . . . . .  8  1.4  The pdf of Weibull fading for Ω = 1 and different values of c. . . . . . . . .  9  1.5  The pdf of Generalized–K fading for Ω0 = 1 and different values m and k.  10  1.6  The keyhole phenomenon where there is rich scattering around the transmitter and the receiver but a barrier with a hole (e.g. a tunnel) separates the transmitter and the receiver [1]. . . . . . . . . . . . . . . . . . . . . . .  1.7  12  Another example of a keyhole channel. The reflections from the scatterers around the transmitter and receiver cause locally uncorrelated fading. However, since the distance between the scatterer rings is much larger than their size, there is a rank deficiency in the fading channel [2].  1.8  . . . . . . . . . .  The pdf of GGN with σ 2 = 1 and different values of β. The lower the value of β, the more impulsive the noise is. . . . . . . . . . . . . . . . . . . . . .  1.9  12  16  The diversity gain determines the slope of the asymptotic error probability, Pe , vs. SNR (dB) curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  1.10 A change in the coding gain shifts the asymptotic error probability, Pe , vs. SNR (dB) curve horizontally.  . . . . . . . . . . . . . . . . . . . . . . . . .  21  xiii  List of Figures 2.1  PEP of a length-2 error event of rate–2 4–state STTC in [3] for Nakagami–m fading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  2.2  FEP of TSC [3] and BBH [4] codes in Rayleigh and generalized–K fading. .  37  3.1  PEP of the length–3 error event in Example 1 over keyhole Nakagami–m fading channel. NT = 2 and NR = 1. Asymptotic PEP was obtained from (3.44) and (3.45a)–(3.45c). . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.2  56  PEP of the length–3 error event in Example 1 over keyhole Nakagami–m fading with parameter mT at the transmitter side and Ricean fading with K = 0 dB at the receiver side. NT = 2 and NR = 2. Asymptotic PEP was obtained from (3.44) and (3.45a)–(3.45c). . . . . . . . . . . . . . . . . . . .  3.3  56  BEP of a single Tx antenna system with NR = 1, 2 and the BBH STTC (NT = 2) with NR = 1 for Weibull–Rayleigh keyhole fading channel. βT = c/2 = 2 and βR = 1.  3.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  SEP of Alamouti STBC [5] with 4–PSK modulation over keyhole Nakagamim fading with mT = mR = 0.7, NT = 2, and NR = 1, 2, 3. . . . . . . . . . .  3.5  57  58  The simulated FEP of the TSC [3] and BBH [4] STTCs with 4–PSK modulation over keyhole and non–keyhole Rayeligh fading with NR = 1, 2, 3 receive antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.6  59  Simulation results and union bound for the BEP of the TSC [3] and SHN [1] STTCs with 4–PSK modulation over keyhole Nakagami-m fading with mT = mR = 2.0 and NR = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . .  60  xiv  List of Figures 4.1  BEP of BPSK with MRC and BFSK with NC vs. bit SNR per branch for i.i.d. Rayleigh fading and L = 3. Impairment by AWGN, i.i.d. –mixture noise ( = 0.25, κ = 10), and a Ricean faded Mi –PSK interferer with Ricean factor Ki = 6 dB (cf. E5). Markers: Simulated BEP. Solid lines: Asymptotic BEP [Eqs. (4.11), (4.14), and (4.15)]. . . . . . . . . . . . . . . . . . . . . .  4.2  84  BEP of 8–PSK vs. bit SNR per branch for differential EGC and MRC over an i.i.d. Ricean fading channel with Ricean factor K = 3 dB and spatially dependent –mixture noise ( = 0.25, κ = 10). Markers: Simulated BEP. Lines: Asymptotic BEP [Eqs. (4.11), (4.13), and (4.15)]. . . . . . . . . . .  4.3  84  SEP of 4–PSK vs. symbol SNR per branch for MRC over uncorrelated (α = 0.0) and correlated (α = 0.9) Rayleigh fading channels (L = 3) with uncorrelated (ρ = 0.0) and correlated (ρ = 0.9) Rayleigh faded asynchronous 4–PSK CCI. Markers: Simulated SEP. Solid lines: Asymptotic SEP [Eq. (4.11)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.4  86  BEP of 16–PSK vs. bit SNR per branch for MRC and differential EGC over correlated Ricean fading channels (K = 3 dB, α = 0.6) with correlated Rayleigh faded 16–PSK CCI (ρ = 0.6). Markers: Simulated BEP Lines: Asymptotic BEP [Eqs. (4.11), (4.13), and (4.15)]. . . . . . . . . . . . . . .  4.5  86  SEP of 4–PSK vs. symbol SNR per branch for MRC over i.i.d. Rayleigh fading channels with zero–mean correlated Gaussian interference. Markers: Simulated SEP. Lines: Asymptotic SEP [Eq. (4.11)].  4.6  . . . . . . . . . . . .  87  SEP of 16–QAM vs. symbol SNR per branch for MRC over i.i.d. Rayleigh fading channels (L = 2) with DS–UWB and MB–OFDM UWB interference. Markers: Simulated SEP. Lines: Asymptotic SEP [Eq. (4.11)]. . . . . . . .  88  xv  List of Figures 5.1  BEP vs. p for BPSK, i.i.d. Rayleigh fading, L = 2, SNR = 24 dB, and different types of n.i.d. noise. Noise parameters: I.i.d. –mixture noise ( = 0.1, κ = 10), n.i.d. Rayleigh–faded QPSK CCI–I (I = 1, τ1 = 0.25T with symbol duration T , raised cosine pulse shape with roll–off factor 0.22), i.i.d. Rayleigh–faded QPSK CCI–II (I = 1,  = 0.25), and i.i.d. GGN  1  (β = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2  BEP vs. p for BPSK, i.i.d. Rayleigh fading, L = 3, SNR = 20 dB, and different types of n.i.d. noise. Noise parameters: I.i.d. –mixture noise I ( = 0.1, κ = 10), i.i.d. –mixture noise II ( = 0.1, κ = 5), n.i.d. Rayleigh– faded and unfaded QPSK CCI–I (I = 1, τ1 = 0.25T , raised cosine pulse shape with roll–off factor 0.22), and i.i.d. Rayleigh–faded and unfaded QPSK CCI–II (I = 1,  5.3  1  = 0.41). . . . . . . . . . . . . . . . . . . . . . . . . . . . 106  Metric coefficients ql , 2 ≤ l ≤ 4, and pl , 1 ≤ l ≤ 4, vs. iteration t of FDSA algorithm. N1: I.i.d. Rayleigh–faded QPSK CCI–II (I = 1,  = 0.1)  and AWGN, where the CCI–II power is ten times larger than the AWGN variance; N2: I.n.d. Gaussian noise with variances σ12 = 1, σ22 = 0.5, σ32 = 0.5, σ42 = 2; N3: I.n.d. –mixture noise with  l  = 0.1, 1 ≤ l ≤ 4, and  κ1 = 20, κ2 = 40, κ3 = 50, κ4 = 100; N4: I.n.d. GGN with β1 = β2 = 3 and β3 = β4 = 1; N5: N.i.d. unfaded QPSK CCI–I (I = 1, τ1 = 0.3T , raised cosine pulse shape with roll–off factor 0.22). . . . . . . . . . . . . . . . . . 110 5.4  Metric coefficients ql , 2 ≤ l ≤ 4, and pl , 1 ≤ l ≤ 4, vs. iteration t of LRS algorithm. Noise types N1–N5 are specified in the caption of Fig. 5.3. . . . 110  5.5  BEP of BPSK with CC vs. iteration t for FDSA and LRS algorithms, respectively. For comparison BEP of L2 –norm combining is also shown. Noise types N1–N5 are specified in the caption of Fig. 5.3. . . . . . . . . . . . . . 111  xvi  List of Figures 5.6  BEP vs. SNR per bit per branch of 16–QAM with CC in i.i.d. Rayleigh fading (L = 2) and i.i.d. –mixture noise ( = 0.1, κ = 100). Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.1. . . . . . . . . . . . . . . . . . . . . 113  5.7  BEP vs. SNR per bit per branch of 16–QAM in i.i.d. Rayleigh fading (L = 2) and n.i.d. unfaded QPSK CCI–I (I = 1, τ1 = 0.3T , raised cosine pulse shape with roll–off factor 0.22). Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113  5.8  BEP vs. SNR per branch of BFSK with NC in i.i.d. Ricean fading (L = 3) and i.i.d. Rayleigh–faded QPSK CCI–II (I = 1,  1  = 0.25). Solid lines with  markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.2. . . . . . . . . . . . . . . . . . . . . 114 5.9  BEP vs. SNR per bit per branch of 4–PSK system with bandwidth B = 4 MHz and CC or DC in i.i.d. Rayleigh fading (L = 3) and MB–OFDM UWB [6] and IR–UWB (Nb = 32 bursts per symbol and Lc = 128 chips per burst) [7] interference. Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Monte– Carlo simulation of generalized noise moments. . . . . . . . . . . . . . . . . 115  6.1  SEP of orthogonal half–rate STBC (N = 8, NT = 4) from [8, Eq. (38)] with 4–PSK modulation vs. symbol SNR. NR = 1 and i.i.d. Rayleigh fading. Markers: Simulated SEP. Solid lines: Asymptotic SEP. . . . . . . . . . . . 142  6.2  SEP of orthogonal rate–3/4 STBC (N = 4, NT = 3) from [9, Eq. (4.104)] with 16–QAM modulation vs. symbol SNR. NR = 1 and i.i.d. Rayleigh fading. Markers: Simulated SEP. Lines: Asymptotic SEP. . . . . . . . . . . 142  xvii  List of Figures 6.3  FEP of rate–2, 4–state STTCs from [3] (TSC) and [4] (BBH) vs. symbol SNR. NR = 2 and i.i.d. Rayleigh fading. Markers: Simulated FEP. Lines: Asymptotic FEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144  6.4  SEP of coherent and differential Alamouti STBC (N = NT = 2) [5, 10] with 8–PSK modulation vs. symbol SNR. I.i.d. Ricean fading (K = 0 dB) for NR = 1 and NR = 2, –mixture noise with  = 0.25 and κ = 10. Markers:  Simulated SEP. Lines: Asymptotic SEP. . . . . . . . . . . . . . . . . . . . 145 6.5  SEP of Alamouti STBC (N = NT = 2) [5] with 4–PSK for NR = 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. symbol SNR. AWGN, ACGN (ρs = 0.9), and uncorrelated and correlated (ρs = 0.9) Rayleigh–faded synchronous 4–PSK CCI (τ1 = 0). Markers: Simulated SEP. Lines: Asymptotic SEP. . . . . . . . . . . . . . . . . . . . . . . . . . 147  6.6  SEP of Alamouti STBC (N = NT = 2) [5] with 4–PSK for NR = 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. estimated noise correlation ρˆs . ACGN (ρs = 0.9), correlated (ρs = 0.9) Rayleigh–faded synchronous 4–PSK CCI (τ1 = 0), and SNR = 20 dB. Markers: Simulated SEP. Lines: Asymptotic SEP. . . . . . . . . . . . . . . . . . . . . . . . . . 147  6.7  BEP of rate–2 diagonal STBC (N = NT = 2) from [11] for NR = 2 and i.i.d. Rayleigh fading vs. bit SNR. AWGN and spatially correlated (ρs = 0.7) Rayleigh–faded asynchronous 8–PSK CCI with I = 1 and τ1 = T /2. Markers: Simulated SEP. Lines: Asymptotic SEP. . . . . . . . . . . . . . . 149  6.8  BEP of TSC STTC code [3] and super–orthogonal STTC (JS) [12, Fig. 5] for NR = 1, 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. symbol SNR. Markers: Simulated SEP. Lines: Asymptotic SEP. . . . . 149  xviii  List of Abbreviations 3GPP  3rd Generation Partnership Project  4–PSK  4–ary Pulse Shift Keying  ACGN  Additive Correlated Gaussian Noise  AF  Amplify–and–Forward  AWGN  Additive White Gaussian Noise  BBH  The rate–2, 4–state STTC in [4]  BEP  Bit Error Probability  BFSK  Binary Frequency Shift Keying  BICM  Bit-Interleaved Coded Modulation  BPSK  Binary Phase Shift Keying  C  Coherent  CC  Coherent Combining  CCI  Co–Channel Interference  CDMA  Code–Division Multiple Access  CLT  Central Limit Theorem  CP  Channel Phase  CR  Combining Rule  D  Differential  DC  Differential Combining  DF  Decode–and–Forward xix  List of Abbreviations DS  Direct–Sequence  DSTC  Distributed Space–Time Coding  DS–UWB  Direct–Sequence Ultra–Wideband  ED  Euclidean Distance  EGC  Equal Gain Combining  FDSA  Finite Difference Stochastic Approximation  FEP  Frame Error Probability  FH  Frequency Hopping  GGN  Generalized Gaussian Noise  GMN  Gaussian Mixture Noise  IEEE  Institute of Electrical and Electronic Engineers  i.i.d.  Independent, Identically Distributed  i.n.d.  Independent, Non–Identically Distributed  IR  Impulse Radio  JS  The super–orthogonal STTC in [12]  LOS  Line–of–Sight  LRS  Localized Random Search  LTE  Long Term Evolution  MB  Multi–Band  MD  Mahalonobis Distance  MF  Matched Filter  M–FSK  M–ary Frequency Shift Keying  MGF  Moment Generating Function  MIMO  Multiple–Input Multiple–Output  ML  Maximum–Likelihood  xx  List of Abbreviations M–PAM  M–ary Pulse Amplitude Modulation  M–PSK  M–ary Phase Shift Keying  M–QAM  M–ary Quadrature Amplitude Modulation  MRC  Maximum–Ratio Combining  NB  Narrowband  NC  Noncoherent Combining  ND  Noise Decorrelating  OFDM  Orthogonal Frequency Division Multiplexing  OSTBC  Orthogonal Space Time Block Code  pdf  Probability Density Function  PEP  Pairwise Error Probability  PSK  Phase Shift Keying  QAM  Quadrature Amplitude Modulation  QPSK  Quaternary Phase Shift Keying  RHS  Right Hand Side  RV  Random Variable  RVV  Random Vector Vriable  SC  Selection Combining  SDGMN  Spatially Dependent Gaussian Mixture Noise  SEP  Symbol Error Probability  SHN  The rate–2, 4–state STTC in [1]  SIGMN  Spatially Independent Gaussian Mixture Noise  SNR  Signal–to–Noise Ratio  STBC  Space–Time Block Code  STC  Space–Time Code  xxi  List of Abbreviations STTC  Space–Time Trellis Code  TSC  The rate–2, 4–state STTC in [3]  UWB  Ultra–Wideband  xxii  Notation Throughout this thesis, bold upper case and lower case letters denote matrices and vectors, respectively. The remaining notation and operators used in this thesis are listed below: (2N − 1)!!  Double factorial of an odd number, (2N − 1)!!  (2N)!!  Double factorial of an even number, (2N)!!  L k1 ,...,kL  . A=B  Multinomial coefficient,  L k1 ,...,kL  1 · 3 · . . . · (2N − 1)  2 · 4 · . . . · 2N  L! k1 !...kL !  Asymptotic equivalence of A and B (in the regime of high SNR unless otherwise stated)  ·  A≤B  A is asymptotically (for high SNR) smaller than or equal to B  (·)∗  Complex conjugation  ∠(·)  Phase of a complex number  |·|  Absolute value of a complex number  {·}  Real part of a complex number  [X]ij  Element of matrix X in row i and column j  [·]T  Transposition  [·]H  Hermitian transposition  det(·)  Matrix determinant  tr{·}  Trace of a matrix  ⊗  Kronecker product of two matrices  vec{X}  Vectorizing operator, converts matrix X to a column vector by stacking the columns of X on top of each other xxiii  Notation diag{x1 , x2 , . . . , xX } X × X diagonal matrix with x1 , x2 . . . , xX on its main diagonal λi (X)  Eigenvalues of matrix X  IX  X × X Identity matrix  0X  X–dimensional all–zeros column vector  0X×Y  X × Y zero matrix  || · ||  L2 –norm of a vector or Frobenius norm of a matrix  E{·}  Statistical expectation of a random variable  Mx (p)  pth moment of the real RV |x|2 , Mx (p)  Mx (p, X)  pth moment of the quadratic form xH Xx, where X is a Hermitian  E{|x|2p } E{(xH Xx)p }  matrix and x is a complex RVV, Mx (p, X) Mx (p)  pth moment of ||x||2 for a complex RVV x, Mx (p)  Mx (p, I X ) = E{(xH x)p } = E{||x||2p }  Pr{·}  Probability of an event  I0 (·)  Zeroth order modified Bessel function of the first kind  Γ(·)  Gamma function, Γ(x)  ψ(·)  Digamma function, ψ(x)  γE  Euler’s constant, γE  Q(·)  Gaussian Q–function, Q(x)  2 F1 (·, ·; ·; ·)  Gaussian hypergeometric function  1 F1 (·, ·; ·)  Kummer confluent hypergeometric function  L{·}  Laplace transform of a function, Φ(s) = L{φ(x)}  o(·)  Order of a function, f (x) is o(x) if limx→0 f (x)/x = 0  O(·)  Order of complexity of an algorithm (the big O notation)  ∞ −t x−1 e t 0 d dx  dt  log Γ(x)  0.57722 √1 2π  ∞ x  2 /2  e−t  dt  ∞ −∞  φ(x)e−sx dx  xxiv  Acknowledgments First of all, I would like to express my sincere gratitude to my supervisor, Prof. Robert Schober for his whole–hearted support during the past four years. Not only he greatly helped me with the technical contributions of this thesis, but also he constantly encouraged and directed me toward the right direction in my research. His energy and determination combined with a friendly and understanding attitude provided the perfect environment for me to achieve my research goals. From the bottom of my heart, I appreciate the continuous and passionate love I have received from Yu (Vivian) Li since I met her in the second year of my PhD. My deep appreciation goes as well to my colleagues in the communications group who accompanied and encouraged me during the four years I was at the University of British Columbia. In particular, I thank Dr. Amir Nasri for his great help and his constructive comments. Also, I greatly thank the members of my doctoral committee, Dr. Vikram Krishnamurthy, Dr. Lutz Lampe, Dr. Victor Leung, and Dr. Vincent Wong, as well as my university and external examiners, Dr. Vijay Bhargava, Dr. Aria Nosratinia, and Dr. Mat´ıas Salibi´anBarrera for their invaluable feedback. Last but not the least, I thank my family for their love and inspiration from thousands of miles away and I dedicate my work to my dear father who passed away in the second year of my PhD but his memory and his words of wisdom have always driven me ahead. This work was supported by the University of British Columbia Four Year Fellowship.  xxv  Chapter 1 Introduction 1.1  Motivation  Since the advent of wireless communication systems in the 20th century, the main focus of research in this area has been on improving the quality, speed, and coverage of these systems. Particulary, achieving higher transmission speeds is a necessity since new applications such as web browsing and video streaming demand much higher data rates than existing systems can provide. In order to achieve this goal, a good understanding of the fading and noise phenomena in wireless environments is necessary. Fading refers to random changes in the channel gain due to shadowing (when there are obstacles between the transmitter and receiver) and/or multi–path propagation of the transmitted signal (when there are scatterers in the environment) [14]. Different distributions have been proposed to model fading effects and it has been shown that fading considerably degrades the performance of single antenna systems, cf. e.g. [15] and references therein. Therefore, single–input multiple–output (SIMO) and multiple–input multiple– output (MIMO) systems were proposed to improve the performance of communication systems in fading channels. SIMO systems employ a single transmit antenna and multiple receive antennas to introduce spatial diversity at the receiver side. Different diversity combining schemes [15] can be applied to combine the signals of the receive antennas. On the other hand, MIMO systems are equipped with multiple transmit antennas and employ space–time coding [3, 9] to encode the transmitted symbols. There are numerous papers 1  Chapter 1. Introduction in the literature that analyze the performance of SIMO and MIMO for different fading models. Based on these performance analyzes, different criteria for the design of receive combining techniques and space–time codes have been proposed [1, 16, 17, 18]. However, due to the mobility of most wireless devices and changes in the environment, the type of fading varies with time, speed, geographical position, and frequency. Fading channels may also be correlated or rank deficient [19, 20] which has to be considered in the analysis. An example is the keyhole phenomenon [2, 21, 22] which refers to a fading channel which is not fully correlated but is rank deficient. This type of fading occurs when the scatterer rings around the transmit and receive antennas are separated by a hard–to–penetrate barrier with a single hole or when they are located very far from each other so that there is a narrow virtual tunnel between them. In order to simplify the analysis, the vast majority of the literature considers specific scenarios for the fading channel and therefore the obtained results are not general. As a result, the designers of wireless systems cannot easily predict the performance of the system under design for different fading distributions or keyhole channels. Therefore, it is important to provide a generalized analysis which gives insight into the effects of different fading distributions and facilitates the derivation of general robust design criteria that are less sensitive to changes in the fading distribution. Our first goal in this work is to provide a unified performance analysis of systems with scalar modulation and receive diversity combining as well as general space–time codes in generalized non–keyhole and keyhole fading channels including Rayleigh, Ricean, Nakagami–q, Nakagami–m, generalized–K, and Weibull fading. Another unwanted but always present guest in wireless systems is noise. Communication systems have been traditionally designed with the assumption that the underlying noise is Gaussian distributed. This is mainly motivated by the tractability of the design  2  Chapter 1. Introduction and analysis of communication systems in Gaussian noise and is justified by the central limit theorem (CLT)1 . Nonetheless, wireless communication systems are often impaired by non–Gaussian noise and interference in practice.2 Examples of non-Gaussian noise include co–channel and adjacent channel interference [23]–[30], impulsive noise [31]–[33], and ultra–wideband (UWB) interference [34, 35]. It has been repeatedly acknowledged in the literature, cf. e.g. [36], that the performance of systems designed for Gaussian noise degrades drastically in non–Gaussian environments. Therefore, in order to achieve reliably high data rates in future communication systems, the inherent non–Gaussian distribution of the noise in many wireless environments has to be taken into account. Since the noise distribution usually changes rapidly and cannot be estimated easily in practical scenarios, most conventional communication systems are designed to operate optimally in Gaussian environments. Therefore, it is important to investigate the performance of these systems when they are impaired by non–Gaussian noise. However, the research in this area has mainly focused on only a few specific and simplified noise models. The lack of comprehensive analysis and design tools for communication systems impaired by non–Gaussian noise is the second motivation for our research. Our aim is to design robust detection techniques that perform close to optimum for a wide range of practical noise and interference environments. As an example, we propose an Lp –norm metric which adaptively optimizes the detection parameter p in order to achieve a better performance in comparison with the L2 –norm metric which is used for Gaussian noise. In the following, we review different mathematical models for fading and non–Gaussian noise, respectively. 1  Loosely speaking, the CLT states that when there is a sufficiently large number of small effects contributing to a single phenomenon, then that phenomenon has a Gaussian distribution [14]. 2 To simplify our notation, in the following, “noise” refers to any additive impairment of the received signal, i.e., our definition of noise also includes what is commonly referred to as “interference”.  3  Chapter 1. Introduction  1.2  Mathematical Modeling of Fading in Wireless Systems  In this section, we briefly review different fading models that will be used throughout the thesis, particularly in Chapters 2 and 3. We assume the fading for the desired user is flat (also referred to as frequency non–selective), i.e., the variation of the channel gain over the transmission frequency band is neglibile3 . We also assume that the channel gain is constant during the transmission of a symbol (i.e., slow fading). Therefore, each transmitted symbol is simply multiplied by a complex gain α  aejφ while passing through the channel, where  a is the channel amplitude and φ is the channel phase.4 For coherent reception, as is common in the literature, we assume that the receiver knows the channel phase accurately and hence is able to compensate for it. Non–coherent receivers, however, do not require the knowledge of the channel phase. Therefore, the performance analysis of both coherent and non–coherent systems is independent of the distribution of the channel phase. For simplicity, it is common to assume that the channel phase is uniformly distributed in [−π, π) or [0, 2π). On the other hand, several models have been proposed for the distribution of the channel amplitude. In the following, we review the most common fading amplitude distributions in the literature. Rayleigh Fading The Rayleigh distribution is the most common model for fading channels in the literature. This model assumes that there is no line–of–sight (LOS) between the transmitter and the receiver and the environment is rich in scatterers. For instance, it has been shown that the 3  The fading for interfering users is assumed to be flat as well. The only exception is when the interferers are ultra–wideband (UWB) transmitters; cf. Chapters 4, 5, and 6. 4 Note that in SIMO and MIMO systems, the channel is modeled by a random vector and a random matrix, respectively, in which the elements represent the set of channel gains from each transmit antenna to each receive antenna.  4  Chapter 1. Introduction fading in densely built urban areas follows a Rayleigh distribution [37]. Due to the existence of several scatterers in such an environment and the lack of an LOS, the CLT predicts that the channel gain will be zero–mean complex Gaussian distributed. As a result, the fading amplitude follows a Rayleigh probability distribution function (pdf) given by  fa (a) =  where Ω  a2 2a exp(− ) Ω Ω  a ≥ 0,  (1.1)  E{a2 } denotes the mean–square (i.e., average energy) of the channel amplitude.  Ricean Fading In Ricean fading (also known as Nakagami-n fading), the fading gain is a non–zero–mean complex Gaussian random variable (RV) due to the existence of an LOS component in addition to the scattered component. Therefore, the fading amplitude has a Ricean pdf which is given by  fa (a) =  2(1 + K) exp(−K) a 1+K 2 exp − a Ω Ω  I0 2  K(1 + K) a Ω  a ≥ 0,  (1.2)  where I0 (·) is the zeroth–order modified Bessel function of the first kind, and K is the Ricean factor which is defined as the ratio of the power of the LOS component to the average power of the scattered component. Clearly, for K = 0, the Ricean pdf simplifies to a Rayleigh pdf. Therefore, Ricean fading is a more general model that includes Rayleigh fading as a special case. In Fig. 1.1, the Ricean pdf is plotted for Ω = 1 and Ω = 3, respectively, and different values of K[dB]  10 log10 (K). It can be seen from this figure  that increasing Ω shifts the pdf to the right and increasing the Ricean factor squeezes the pdf. Note that K[dB] = −∞ is equivalent to K = 0 and corresponds to Rayleigh fading.  5  Chapter 1. Introduction  1.4  K K K K  Probability distribution function, fa (a)  1.2  = = = =  −∞ dB −3 dB 0 dB 3 dB  1  0.8  0.6  Ω=3  0.4  0.2  0  Ω=1  0  0.5  1  1.5  2  2.5  Fading gain amplitude, a  Figure 1.1: The pdf of Ricean fading for Ω = 1 and Ω = 3 and different values of the Ricean factor, K[dB] = 10 log10 (K). Nakagami–m Fading For Nakagami–m fading, the pdf of the channel amplitude is given by  fa (a) =  where m ≥  1 2  2mm a2m−1 ma2 exp − Ωm Γ(m) Ω  a ≥ 0,  (1.3)  is the fading parameter and Γ(·) is the Gamma function. An interesting  property of this model is that the severity of fading can be controlled by the value of parameter m. In other words, m =  1 2  corresponds to the highest amount of fading and  m → ∞ represents a non–fading channel [15]. The pdf of Nakagami–m fading is plotted in Fig. 1.2 for Ω = 1 and different values of m. It is worth mentioning that the special case m = 1 corresponds to Rayleigh fading.  6  Chapter 1. Introduction 1.8  m m m m  1.6  = = = =  0.5 1.0 2.0 4.0  Probability distribution function, fa (a)  1.4  1.2  1  0.8  0.6  0.4  0.2  0  0  0.5  1  1.5  2  2.5  Fading gain amplitude, a  Figure 1.2: The pdf of Nakagami–m fading for Ω = 1 and different values of m. Nakagami–q Fading The Nakagami–q distribution, also known as Hoyt distribution, is given by 2  fa (a) =  (1 + q 2 )a (1 + q 2 ) 2 exp − a qΩ 4q 2 Ω  I0  (1 − q 4 )a2 4q 2 Ω  a ≥ 0,  (1.4)  where 0 < q ≤ 1 is the fading parameter. As shown in Fig. 1.3, small values of q indicate severe fading, i.e., higher likelihood of occurence of a low fading gain. Note that q = 1 represents Rayleigh fading. Weibull Fading Weibull fading has been used to model the radio channel in systems operating in the 800/900 MHz range of frequencies [15]. The pdf of the channel amplitude for Weibull 7  Chapter 1. Introduction 1  q q q q  0.9  = = = =  0.1 0.3 0.6 1.0  Probability distribution function, fa (a)  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  0  0  0.5  1  1.5  2  2.5  Fading gain amplitude, a  Figure 1.3: The pdf of Nakagami–q fading for Ω = 1 and different values of q. fading is controlled by parameter c as  fa (a) = c  where κ  Γ 1+  2 c  κ Ω  c/2  ac−1 exp −  κ Ω  c/2  ac  a ≥ 0,  (1.5)  . Special cases are c = 1 and c = 2 which correspond to exponential  and Rayleigh distributions, respectively. The Weibull pdf is plotted in Fig. 1.4 for Ω = 1 and different values of c. Generalized–K Fading As mentioned before, there are generally two different types of fading. Shadowing (or large–scale fading) refers to the fluctuations in the average power of the received signal which occur due to the existence of major obstacles between the transmitter and the  8  Chapter 1. Introduction 3.5  c c c c  Probability distribution function, fa (a)  3  = = = =  0.8 1.0 2.0 4.0  2.5  2  1.5  1  0.5  0  0  0.5  1  1.5  2  2.5  Fading gain amplitude, a  Figure 1.4: The pdf of Weibull fading for Ω = 1 and different values of c. receiver. On the other hand, in multi–path fading (also known as small–scale fading), the constructive or destructive addition of reflected signals from the scatterers in the vicinity of the transmitter and the receiver causes variations in the instantaneous power of the received signal. All of the aforementioned fading distributions model multi–path fading, rather than shadowing, i.e., the parameter Ω was assumed to be constant. In most wireless systems, this is a reasonable assumption since the effect of shadowing is partly compensated by power control. In these systems, multi–path fading is more detrimental and is usually the only type of fading which is considered in the analysis. However, once an efficient diversity system considerably reduces the effects of multi–path fading, the system performance will mainly depend on shadowing. Therefore, the effects of shadowing have to be considered as well. Generalized–K fading is a composite shadowing and multi–path fading model. In the following, we provide the pdf of the channel amplitude for this type of fading. 9  Chapter 1. Introduction 1  k k k k  0.9  = = = =  1, 1, 2, 2,  m m m m  = = = =  1 2 1 2  Probability distribution function, fa (a)  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  0  0  0.5  1  1.5  2  2.5  Fading gain amplitude, a  Figure 1.5: The pdf of Generalized–K fading for Ω0 = 1 and different values m and k. Empirical measurements confirm that in shadowed environments, Ω follows a log– normal distribution [15]. Nevertheless, the pdf of Ω can also be accurately modeled by a Gamma pdf given by [38]  fΩ (Ω) =  where Ω0  Ωk−1 Ω exp − k Ω0 Γ(k)Ω0  Ω ≥ 0,  (1.6)  E{Ω} and k is the shadowing parameter which depends on the physical  characteristics of the environment. Assuming a Nakagami–m distribution for multi–path fading, the conditional pdf of the channel amplitude can be found from (1.3). Combining Eqs. (1.3) and (1.6) results in  fa (a) =  4m(β+1)/2 aβ (β+1)/2 Γ(m)Γ(k)Ω0  Kk−m 2  m a Ω0  a ≥ 0,  (1.7)  10  Chapter 1. Introduction where β  k + m − 1 and Kk−m (·) is the modified Bessel function of order k − m. This  pdf is illustrated in Fig. 1.5 for different values of parameters k and m. Keyhole Fading Recently, it has been shown, both theoretically [21, 2] and experimentally [22], that MIMO systems can experience not fully correlated but rank–deficient fading. This phenomenon occurs when there is rich scattering around the transmitter and the receiver, but a barrier with a hole (e.g. a tunnel) between the transmitter and the receiver leads to a reduction of the rank of the channel matrix [22], see Fig. 1.6. It can also happen when there is a large distance between the scatterer rings of the transmitter and receiver [1, 2], as shown in Fig. 1.7. In these cases, the channel is mathematically modeled as the product of two independent fading components, one for the channel at the transmitter side and the other for the channel at the receiver side. This effect, which is known as keyhole or pinhole effect, reduces the achievable spectral efficiency of MIMO systems and will be discussed in detail in Chapter 3.  1.3  Non–Gaussian Noise in Wireless Systems  In order to analyze the performance of a communication system and optimize the receiver, the distribution of the noise that is added to the received signal has to be known. Traditionally, the noise in communication systems is assumed to be Gaussian distributed. Such an assumption mainly stems from the CLT and is further motivated by the mathematical tractability of analysis and design in Gaussian channels. The Gaussian model is reasonable for the so called background noise which consists of thermal noise, shot noise, and some other types of environmental noise. The background noise in communication systems is usually referred to as additive white Gaussian noise (AWGN). AWGN is predominantly assum– 11  Chapter 1. Introduction  Figure 1.6: The keyhole phenomenon where there is rich scattering around the transmitter and the receiver but a barrier with a hole (e.g. a tunnel) separates the transmitter and the receiver [1].  Narrow Pipe  Transmitter side scatterer ring  Receiver side scatterer ring  Figure 1.7: Another example of a keyhole channel. The reflections from the scatterers around the transmitter and receiver cause locally uncorrelated fading. However, since the distance between the scatterer rings is much larger than their size, there is a rank deficiency in the fading channel [2]. ed for the analysis and design of communication systems. However, there are many other types of noise in (terrestrial) communication systems, e.g., multi–user interference [39], co–channel [23]–[30] and ultra–wideband (UWB) interference [34, 35], radar clutter [40], noise from man–made devices such as vehicles, microwave ovens, etc. and other natural phenomena such as lightning or atmospheric changes [32]. These types of noise are intrinsically bursty and/or impulsive and are usually generated by a finite number of sources. As 12  Chapter 1. Introduction a result, the applicability of CLT for predicting the distribution of these types of noise is questionable. Generally, the aforementioned types of noise sources generate outliers (high amplitude noise) more often than Gaussian noise. As a result, the tail of the distribution for these types of noise is usually heavier than the Gaussian tail. The fact that the distribution of noise in real world communication systems is non–Gaussian has been confirmed by both measurement and analysis, e.g., cf. [33]–[35]. It has also been shown that the performance of systems designed to work optimally in Gaussian noise degrades dramatically in the presence of non–Gaussian noise. Nonetheless, communication systems are still designed with the assumption that the underlying noise is Gaussian. A reason for this is that the communication environments are so dynamic that the imaginary “always–optimal” receiver would require real–time estimation of the noise distribution. However, estimating the distribution of noise is a time and energy consuming task for a mobile device that is battery operated and has a limited processing power. Even if the exact distribution of the noise in an environment were known, the optimal receiver might be too complex to implement. So, two questions arise: First, how much is the performance of systems designed for Gaussian channels affected by different types of non– Gaussian noise? Second, are there low–complexity methods for improving the performance of communication systems in non–Gaussian noise without knowing its distribution? Before providing answers to these questions, we categorize and briefly review the most important non–Gaussian noise models in the following. Non–Gaussian noise models can be divided into two general categories, namely, noise models with exponentially–tailed pdfs and those with algebraically–tailed pdfs [45]. A random variable X is said to have an exponentially–tailed pdf, fX (x), if there exist c > 0 and α > 0 such that lim exp (xα ) fX (x) = c.  x→∞  (1.8)  13  Chapter 1. Introduction Similarly, X is said to have an algebraically–tailed pdf if there exist c > 0 and α > 0 such that lim xα fX (x) = c.  x→∞  (1.9)  Using (1.9), it can be easily shown that for algebraically–tailed noises, the qth absolute moment Mq (X)  E {|X|q } does not exist, i.e., is infinity, for any q ≥ α. This is in contrast  with the exponentially–tailed noises for which the moments for all values of q ≥ 0 are finite. Intuitively, this is a consequence of the fact that algebraically–tailed distributions generally have heavier tails than exponentially–tailed distributions. Examples of noises with exponentially–tailed distributions are Gaussian noise, generalized Gaussian noise, Middleton’s Class A noise, Gaussian mixture noise, and co–channel interference. On the other hand, Middleton’s Class B and Class C [32], Generalized–t [46], and alpha–stable noise [47] are algebraically–tailed. As we will see, the analysis presented in Chapters 4, 5, and 6 is valid only for exponentially–tailed types of noise since all noise moments need to be finite. Note that exponentially–tailed noises include the most common types of noise in communication systems. Therefore, here we only focus on these types of noise. The interested reader is referred to [45] for more information on algebraically–tailed distributions. Gaussian Noise The analysis that we will provide in Chapters 2 and 3 is valid only for Gaussian noise. Furthermore, Chapters 4, 5, and 6 compare the performance in Gaussian noise with that in various types of non–Gaussian noise. Therefore, we first briefly review some of the properties of Gaussian noise. As mentioned earlier, the Gaussianity of noise is usually justified by the CLT. Besides, other important reasons for the popularity of Gaussian noise in communication system analysis are its unique properties such as the preservation of the distribution under linear transformations and well–developed mathematical analysis tools  14  Chapter 1. Introduction such as the Gaussian Q–function [15] and Turin’s formula [48]. The Gaussian (also known as normal) pdf for a scalar (complex) random variable X with mean zero and variance σ 2 is given by fX (x) =  1 |x|2 exp − πσ 2 σ2  .  (1.10)  In multi–channel systems, the noise is a random vector or matrix in which each element has a pdf as in (1.10). It can be easily shown that the maximum–likelihood (ML) receiver in Gaussian noise minimizes the mean Euclidean distance (ED) between the decoded symbol and the received signal. In other words, the ML receiver decides in favor of the point in the signal constellation that is the closest to the received signal point. Unfortunately, the ED metric is not robust to variations in the noise distribution [45]. As a result, the performance of a system which is based on the ED metric drastically degrades in the presence of non–Gaussian noise. Laplacian Noise The first non–Gaussian noise model that is introduced here is known as Laplacian noise. The pdf of this type of noise is given by  fX (x) =  |x| 1 exp − 2σ σ  ,  (1.11)  where σ is a scale parameter analogous to the variance in the Gaussian pdf. It can be shown [49] that the ML receiver in Laplacian noise applies a median filter on the received signal. It is a well–known fact that median filters are more robust to the presence of outliers than the ED metric. However, we note that there is no strong evidence for the existence of Laplacian noise in practical communication systems [45].  15  Chapter 1. Introduction  0  10  −2  noise pdf, fX(x)  10  −4  10  −6  10  −8  β β β β  10  −10  10  = = = =  1.0 1.5 2.0 2.5  −2  −1  10  0  10  10  noise amplitude, |x|  Figure 1.8: The pdf of GGN with σ 2 = 1 and different values of β. The lower the value of β, the more impulsive the noise is. Generalized Gaussian Noise A generalization of both Gaussian and Laplacian noise models is known as Generalized Gaussian Noise (GGN) which is described by the pdf  fX (x) =  β Γ(4/β) |x|β exp − c 2π σ 2 (Γ(2/β))2  ,  (1.12)  where β > 0 is a parameter that controls the shape of the pdf, σ 2 is the variance of the noise, and c  σ 2 Γ(2/β) Γ(4/β)  β/2  . Note that β = 1 and β = 2 correspond to the Laplacian and  Gaussian pdf, respectively. Fig. 1.8 shows the pdf of GGN with σ 2 = 1 and different values of β. Note that in order to obtain a better resolution, both axes are plotted in logarithmic scale. It can be seen from Fig. 1.8 that lower values of β indicate more impulsive noise, i.e., a higher probability of occurrence of both low and high amplitude noise5 . 5  Roughly speaking, noise is impulsive if it is relatively quiet most of the time but once in a while generates high amplitude spikes (impulses).  16  Chapter 1. Introduction Middleton’s Class A Noise Model The most comprehensive characterization of (impulsive) non–Gaussian noise has been proposed by Middleton in [32] and further discussed in [50]–[53]. Based on the comparison between the coherence bandwidth of the noise and the receiver bandwidth, Middleton classified the noise into three categories. Class A noise identifies narrowband interference, i.e., when the coherence bandwidth of the noise is smaller than the receiver bandwidth. Class B refers to wideband interference and Class C is a combination of Classes A and B6 . Mathematically, the canonical form of Class A noise pdf is a sum of Poisson weighted zero–mean Gaussian terms as  fX (x) =  ∞ k=0  where ak  Ak exp(−A) k!  |x|2 ak exp − πσk2 σk2  ,  (1.13)  and A is referred to as the overlap (or impulsive) index and is defined  as the mean frequency of impulses times their mean length. The value of A is typically between 10−4 and 0.5 [50] with lower values indicating a more impulsive noise. The variance σk2 increases with k as σk2 =  k/A+Γ 2 σ , 1+Γ  where Γ, known as the Gaussian factor, is the ratio of  the power of the Gaussian component and the non–Gaussian component. Typical values of Γ are in the range of 10−5 to 0.1 with lower values indicating more powerful impulses. σ 2 denotes the variance of the noise. Class A noise accurately models the impulsive noise measurements in real communication environments. Interestingly, truncating the series in (1.13) to its first two or three terms still yields an accurate approximation of the noise distribution in practical systems [54]. 6  As mentioned before, since not all of the noise moments in Classes B and C are finite, the analysis that will be presented in this thesis is not applicable in these cases. Therefore, here we only focus on Middleton’s Class A noise model.  17  Chapter 1. Introduction Gaussian Mixture Noise Model Gaussian mixture noise (GMN) is a generalized version of Middleton’s Class–A noise model in which the number of interference sources is variable and their arrival time is not necessarily Poisson distributed. The pdf of this family of noise can be expressed as I  fX (x) = i=1  where ci > 0,  I i=1 ci  is given by σ 2 =  ci |x|2 exp − πσi2 σi2  ,  (1.14)  = 1, and σi2 , 1 ≤ i ≤ I, are noise parameters and the total variance  I 2 i=1 ci σi .  Eq. (1.14) indicates that at any time, a specific noise state  i ∈ {1, 2, . . . , I} is picked with probability ci and then a Gaussian random variable with variance σi2 is generated. Therefore, conditioned on the state of the noise (i.e., knowing the value of i which was picked), GMN is Gaussian distributed. We will exploit this conditional Gaussianity of GMN in Chapters 4, 5, and 6. GMN is a popular model for impulsive non–Gaussian noise in systems with receive antenna diversity [31], bit-interleaved coded modulation (BICM) [55, 56], and for partial band interference in frequency hopping (FH) systems [57] among others. Similar to Middleton’s Class A noise, the number of terms in (1.14) can be restricted with little effect on the accuracy of the model. In fact, for I = 2, GMN with variance σ 2 and c1 = 1 − , c2 = , σ22 = κσ12 =  κσ2 1− +κ  simplifies to the well–known –contamination or –mixture noise. In  this noise model, the pdf of the noise can be characterized by only two parameters, namely, 0<  < 1 and κ  1. The physical interpretation of these parameters is as follows. The  prevalent type of noise (i.e., the noise in (1 − ) fraction of time) in such an environment is a Gaussian background noise with variance σ12 = time (i.e., on average once every  1  σ2 1− +κ  . However, in  fraction of the the  seconds), a strong Gaussian interference with variance  σ22 = κσ12 appears. Therefore, the smaller the value of  and the larger the value of κ, the  18  Chapter 1. Introduction more impulsive the noise is. In multi–antenna systems, the GMN can be spatially independent (SIGMN) or spatially dependent (SDGMN). The SIGMN model can be used when the source of the noise affects the receiver antennas independently so that each antenna’s noise state is independent from that of the other antennas [31]. This is the case when there are several sources of noise or scatterers around the receiver and the spacing between the receiver antennas is sufficiently large. Otherwise, the SDGMN model should be used in which all antennas are in the same noise state at any given time [36]. Note that in both cases the noise is temporally (i.e., over time) independent. In other words, the set of noise states always changes independently from one symbol duration to the next one. A mathematical description of SIGMN and SDGMN will be provided in Chapters 4, 5, and 6. Co–channel Interference Co–channel interference (CCI) [23]–[30] is one of the most common types of interference, particularly in cellular systems due to frequency re–use by different cells. In fact, the capacity of cellular systems is mainly limited by CCI rather than background noise [58]. Therefore, it is very critical to have a good understanding of the effects of this type of non–Gaussian noise. Mathematically, CCI from a single source can be expressed as ku  x = a z,  g[k] i[k],  z  (1.15)  k=kl  where a represents the fading gain of the channel between the interferer and the desired user and kl , ku , g[k], and i[k] denote the lower limit, the upper limit, fixed (in general complex) coefficients, and the independent, identically distributed (i.i.d.) interference symbols taken from an Mi –ary alphabet. Eq. (1.15) can be used to model multiple synchronous, a single asynchronous, or multiple asynchronous co–channel interferers. For example, for I 19  Chapter 1. Introduction synchronous interferers we have kl = 1, ku = I, and g[k] and i[k] denote the gain and the transmitted symbol of the kth interferer. In contrast, for a single asynchronous interferer i[k] denotes symbols transmitted by the interferer in symbol intervals kl ≤ k ≤ ku and g[k]  g(kT + τ ), where g(t), T , and τ are the overall interference pulse shape, the symbol  duration, and the delay of the interferer compared to the desired user, respectively. kl and ku are appropriately chosen to ensure g(kT + τ ) ≈ 0 for k < kl and k > ku , respectively. Ultra–wide Band (UWB) Interference The interference from both multi–band orthogonal frequency division multiplexing (MB– OFDM) UWB and impulse radio (IR) (also referred to as direct–sequence (DS)) UWB is in general strongly non–Gaussian [35, 59]. The exact pdf of MB–OFDM and IR UWB interference were obtained in [35] and [60], respectively. In this thesis, we will consider the effect of this type of noise on the performance of SIMO and MIMO systems mainly through simulation.  1.4  Asymptotic Analysis  Throughout this thesis, we assume that the signal–to–noise ratio (SNR) is sufficiently high, i.e., our analysis is valid for the asymptotic regime of high SNR7 . In this case, the performance of the communication system can be characterized in a unified form as8 . Pe = (Gc γ)−Gd ,  (1.16)  7 Since, in this thesis, we refer to any additive impairment as “noise”, we use the term “SNR” even if the received signal is only impaired by what is traditionally referred to as “interference”. 8 We will use different performance benchmarks, namely, pairwise error probability (PEP), bit error probability (BEP), symbol error probability (SEP), and frame error probability (FEP). At high SNRs, all of these benchmarks are linearly related. Therefore, Pe in (1.16) can refer to any of these benchmarks.  20  Chapter 1. Introduction  0  10  −1  10  G =1  −2  10 P  e  d  −3  10  G =2 d  −4  10  Exact Asymptotic  −5  10  0  2  4  6  8  10 12 SNR (dB)  14  16  18  20  Figure 1.9: The diversity gain determines the slope of the asymptotic error probability, Pe , vs. SNR (dB) curve. 0  10  −1  Pe  10  G =1  G =4  c  c  −2  10  −3  10  0  Exact Asymptotic 2  4  6  8  10 12 SNR (dB)  14  16  18  20  Figure 1.10: A change in the coding gain shifts the asymptotic error probability, Pe , vs. SNR (dB) curve horizontally. 21  Chapter 1. Introduction where γ denotes the SNR, Gc is called coding gain9 , and Gd is the diversity gain. As shown in Figs. 1.9 and 1.10, the plot of Pe in (1.16) vs. γ[dB]  10 log10 (γ) on a double  logarithmic scale is a straight line which approaches the exact (i.e., non–asymptotic) curve for the error probability at high SNRs. It can be seen from these figures that the diversity gain determines the slope of the line and the coding gain shifts the line horizontally. Besides its generality, an asymptotic analysis reveals the effects of the system design, the modulation scheme, and the channel parameters on system performance.  1.5  Literature Review  In this section, we first introduce the SIMO and MIMO systems that will be considered in this thesis and briefly review some of the papers in the literature which consider the performance of these systems in different fading channels and non–Gaussian noise environments. As mentioned earlier, SIMO systems employ a single transmit antennas and multiple receive antennas. At the receiver of these systems, different schemes such as maximum–ratio combining (MRC), coherent or differential equal–gain combining (EGC), non–coherent combining (NC), and selection combining (SC) [15] can be applied to combine the signals from the receiver antennas before detecting the transmitted symbol.10 This process improves the system performance by introducing a diversity gain. MIMO systems which are equipped with multiple transmit and (possibly) multiple receive antennas can employ space–time codes (STCs) [3, 9] to create diversity (and coding) gain. There are different types of STCs such as space–time block codes (STBCs), space– time trellis codes (STTCs), super–orthogonal STTCs, etc. [9]. Note that STBCs can 9  In Chapters 4 and 5, we will use the term “combining gain” instead of “coding gain” as there is no coding in the system. 10 Note that MRC, EGC, and NC are also referred to as quadratic diversity combining techniques.  22  Chapter 1. Introduction be coherently or differentially decoded [9]. Often, the asymptotic regime of high SNR is considered for both design [3] and analysis [13] of STCs since this regime leads to analytically tractable results and reveals the effects of the STC structure and the channel properties on system performance. It is worth mentioning that SIMO systems with MRC detection can be considered as a special case of STCs for MIMO systems.  1.5.1  STCs in Non–keyhole Fading Channels with Gaussian Noise  Performance analysis of STCs in (non–keyhole) fading channels has been a popular research topic over the past decade. For instance, [61] analyzes the performance of STBCs and [62] provides bounds for the performance of STTCs in Rayleigh fading. Ricean fading is considered in [63]. There have been also attempts to analyze the performance of STCs in other types of fading. An upper bound for the performance of STTCs in Nakagami–m fading is given in [18]. In [64] and [65], the performance of orthogonal STBCs (OSTBCs) is analyzed in Nakagami–q (Hoyt) and Weibull fading, respectively. Wang and Giannakis introduced a generalized asymptotic analysis of OSTBCs which is applicable to different types of fading [16]. However, the results in [16] cannot be applied to non–orthogonal STCs. An analysis of STCs in generalized–K fading [66] does not seem to be available in the literature so far.  1.5.2  STCs in Keyhole Fading Channels with Gaussian Noise  Performance analysis and design of STCs in keyhole fading channels has mainly focused on OSTBCs over some specific types of fading channels. Peppas et al. [67] evaluated the performance of OSTBCs in Weibull fading. Zhao et al. [68] and Shin et al. [69] considered OSTBCs with M–ary phase shift keying (M–PSK) and square M–ary quadrature ampli23  Chapter 1. Introduction tude modulation (M–QAM) constellations over keyhole Rayleigh and Nakagami–m fading channels, respectively. Similarly, [70] provides lower and upper bounds on the bit error probability (BEP) and symbol error probability (SEP) of OSTBCs in keyhole Nakagami–m fading channels. The SEP of OSTBCs in uncorrelated and spatially correlated Rayleigh keyhole channels was calculated in [71] and [72], respectively. However, the final results in [69]–[72] are generally expressed as single definite integrals that have to be evaluated numerically. Besides, the above works are limited to one type of keyhole fading and are not applicable to space–time trellis codes (STTCs) which are generally non–orthogonal. In [73], the authors provide an upper bound for the pairwise error probability (PEP) of STCs (including STTCs). A code design criterion for STTCs with two transmit and one receive antenna was presented in [74]. Sanayei et al. provided a more general analysis in [1]. However, [1] concentrates on the design of STTCs and does not provide final expressions for the PEP. In addition, the results in [73], [74], and [1] are all valid for Rayleigh fading only and cannot be easily extended to other types of keyhole fading.  1.5.3  Diversity Combining Receivers in Non–Gaussian Noise  Clearly, if the noise distribution is known in parametric form, the distribution parameters can be estimated first, and optimal maximum–likelihood combining can be applied subsequently, cf. [75] and references therein. However, in many cases, such knowledge is not available at the receiver and the noise distribution may even change with time. Therefore, a handful of papers, e.g. [23, 26, 28, 30, 31] in the literature analyze the performance of conventional communication systems11 in non–Gaussian noise. All of these papers consider a specific noise model and therefore, their results are not applicable to other types of non–Gaussian noise. To the best of our knowledge, the only paper which considers general 11 By conventional communication systems, we mean systems that are designed with the assumption that the underlying noise is Gaussian distributed.  24  Chapter 1. Introduction non–Gaussian noise is [76] which provides a unified asymptotic SEP analysis of coherent equal gain combining (EGC) and coherent selection combining (SC) for i.i.d. diversity branches. Unfortunately, the approach used in [76] cannot be extended to the important class of quadratic diversity combining techniques which include coherent maximum ratio combining (MRC), differential EGC, and noncoherent combining (NC). An asymptotically optimum space–diversity detector for determining the presence or absence of a weak signal (rather than detecting a digitally modulated signal) in the low SNR regime was proposed in [77]. There have also been attemts to find robust combining schemes and metrics which perform well for a large class of noise distributions and possibly have a tunable parameter which can be adjusted to the underlying noise distribution. Prominent examples for such robust metrics include Huber’s M–metric [78], Myriad and Meridian metrics [79], metrics involving hard and soft limiters [57], and the Lp –norm metric [59, 80]. Thereby, the Lp –norm metric is particularly interesting since it performs well in both noise with heavy–tailed distributions (e.g. impulsive noise) and noise with exponentialally–tailed distributions (e.g., co–channel interference) if p is adjusted accordingly [80]. However, finding the optimum p for a particular type of noise is a formidable task, as appropriate optimization criteria are not known.  1.5.4  STCs in Non–Gaussian Noise  Compared to the large body of literature on STC design and analysis in Gaussian noise, relatively little is known about the performance of STCs in non–AWGN channels. In [81], the Chernoff bound for STCs in correlated Gaussian noise with noise decorrelating detection was derived. STC design for correlated Gaussian interference was considered in [82, 83]. Furthermore, STC design and analysis in non–Gaussian co–channel and multiuser interference was addressed in [84]–[87]. The effect of spatially dependent, temporally independent  25  Chapter 1. Introduction impulsive noise on STCs was studied in [36].  1.6  Contributions of the Thesis  As mentioned in Section 1.1, our goal in this thesis is to extend the previous works in the literature and provide a unified and generalized analysis for the performance of communication systems impaired by various types of fading channels and different types of (generally) non–Gaussian noise. In doing so, we try to be as general as possible. However, some categorization is inevitable since for example modeling of keyhole fading channels is essentially different from non–keyhole channels. In a nutshell, our analysis not only confirms the previously known results in the literature (mainly for Gaussian noise), but also leads to several new findings. In addition, we provide design criteria for different scenarios and propose new metrics for improving the performance of communication systems impaired by non–Gaussian noise. In the following, we provide a more detailed description of the contributions we have made in this thesis. • First, considering the practically important case of two transmit and an arbitrary number of receive antennas, we provide a simple closed–form expression for the asymptotic pairwise error probability (PEP) of STCs in Gaussian noise and generalized (non–keyhole) fading including Rayleigh, Ricean, Nakagami–q, Nakagami–m, Weibull, and generalized–K fading. The presented analysis provides new insights into STC design and can be used to obtain tight asymptotic upper bounds for the bit, symbol, and frame error probabilities of arbitrary STCs. • The analysis of STCs in Gaussian noise and generalized keyhole fading channels is our next contribution. We provide closed–form expressions for the asymptotic PEP of systems with a single transmit antenna as well as STCs with two transmit antennas 26  Chapter 1. Introduction and an arbitrary number of receive antennas. We show how the number of transmit and receive antennas as well as the fading parameters affect the diversity gain. We also derive the code design criteria and compare them with those in the non–keyhole fading channels. Similar to our analysis of non–keyhole channels, our results are insightful and can be used to obtain tight asymptotic upper bounds for the bit, symbol, and frame error probabilities of arbitrary STCs in keyhole fading. • After considering different types of fading, we develop a powerful and unified analysis for non–Gaussian noise environments. We consider quadratic diversity combining techniques in systems with a single transmit antenna as well as general coherent and differential STCs in MIMO systems with arbitrary numbers of transmit and receive antennas. The proposed analysis is applicable to Ricean and Rayleigh fading and any type of noise with finite moments and any quadratic combining scheme. Our analysis assumes a general Mahalonobis distance (MD) metric which includes Euclidean distance and noise decorrelating metrics as special cases. We show that the diversity gain is independent of the specific form of the MD metric and the type of noise. However, the coding gain is affected by both the MD metric and the noise moments. We also derive some expressions for the coding gain difference between different metrics as well as between differential and coherent transmission for different types of noise. Furthermore, we provide closed–form expressions for the noise moments for several practically relevant types of noise and interference. • Another contribution of this thesis is the introduction of an adaptive Lp –norm metric for robust coherent, differential, and noncoherent receive diversity combining in non– Gaussian noise and interference. We consider the general case where all diversity branches may use different combining weights and different Lp –norms. We derive a general closed–form expression for the asymptotic bit error probability (BEP) of 27  Chapter 1. Introduction Lp –norm combining in independent non–identically distributed Ricean fading and non–Gaussian noise and interference with finite moments. The asymptotic BEP expression reveals that the diversity gain of Lp –norm combining is independent of the type of noise and the metric parameters. In contrast, the combining gain depends on both the type of noise and the metric parameters. Thus, the asymptotic BEP can be minimized by optimizing the Lp –norm metric parameters for the underlying type of noise. For this purpose, finite difference stochastic approximation and localized random search algorithms are developed. Both adaptive algorithms do not require any a priori knowledge about the underlying noise and are able to track changes in the noise statistics. Simulation results confirm the validity of the derived asymptotic BEP expressions, the effectiveness of the proposed adaptive algorithms, and the excellent performance of the proposed adaptive Lp –norm metric compared to other popular metrics.  1.7  Organization of the Thesis  The rest of this thesis is organized as follows. In Chapter 2, we consider STCs with two transmit and an arbitrary number of receive antennas. Using the Taylor series expansion, we derive a unified representation for different fading distributions including Rayleigh, Ricean, Nakagami–q, Nakagami–m, Weibull, and generalized–K fading. Then, we analyze the performance of STCs in the aforementioned generalized fading channels with Gaussian noise. We derive a closed–form expression for the PEP which is applicable to both space– time block codes (STBCs) and space–time trellis codes (STTCs) and we provide the code design criteria. Chapter 3 considers the same problem as Chapter 2 but in generalized keyhole fading channels. Depending on the number of transmit and receive antennas and the fading 28  Chapter 1. Introduction parameters, we consider three different cases and derive closed–form expressions for the PEP in each case. We also compare the performance of keyhole fading with non–keyhole fading channels and derive the code design criteria. Several simulation results are provided to illustrate the applications of the proposed analysis. Unlike Chapters 2 and 3, Chapters 4, 5, and 6 consider only Ricean fading (which includes Rayleigh fading as a special case) but different types of non–Gaussian noise with finite moments. In Chapter 4, we study the asymptotic behavior of the BEP and symbol error probability (SEP) of quadratic diversity combining schemes such as coherent maximum–ratio combining (MRC), differential equal–gain combining (EGC), and noncoherent combining (NC) in correlated Ricean fading and non–Gaussian noise. We provide simple and easy–to–evaluate asymptotic BEP and SEP expressions which show that at high SNRs the performance of the considered combining schemes depends on certain moments of the noise and interference impairing the transmission. We derive general rules for calculation of these moments and provide closed–form expressions for the moments of several practically important types of noise such as spatially dependent and independent Gaussian–mixture noise, correlated synchronous and asynchronous co–channel interference, and correlated Gaussian interference. In Chapter 5, we propose an adaptive Lp –norm metric for robust coherent, differential, and noncoherent receive diversity combining in non–Gaussian noise and interference. For the asymptotic regime of high SNR, we derive closed–form expressions for the BEP which are valid for valid for independent non–identically distributed Ricean fading and non– Gaussian noise and interference with finite moments. For on–line metric optimization, we develop two efficient adaptive algorithms which do not require any a priori knowledge about the noise statistics and can also cope with non–stationary noise. Chapter 6 generalizes the analysis in Chapter 4 to general STCs (i.e., when the system  29  Chapter 1. Introduction uses both transmit and receive diversity). In addition, a general Mahalonobis distance metric is assumed which includes Euclidean distance and noise decorrelating metrics as special cases. Similar to Chapter 4, the noise moments are calculated for several practically important types of non–Gaussian noise and simulation results are provided to verify the derived expressions. Finally, Chapter 7 summarizes the findings of this thesis and presents some ideas for further research.  30  Chapter 2 Performance Analysis of STCs in Generalized Fading Channels12 2.1  Introduction  As mentioned in the previous chapter, most of the papers in the literature, e.g., [9] and references therein which analyze the performance of space–time codes (STCs) in fading channels consider a specific model for the fading (mainly Rayleigh and Ricean fading). In this chapter, we derive a simple closed–form asymptotic expression for the pairwise error probability (PEP) of space–time coded transmission with two transmit and an arbitrary number of receive antennas.13 This novel result reveals that STCs that achieve full diversity in Rayleigh fading also achieve full diversity in generalized fading. Furthermore, the derived PEP expression can be utilized for STC design and the comparison of existing STCs under various fading conditions. In the following, we first introduce the system model considered in this chapter. Then, we analyze the system performance and provide some numerical results. 12  Parts of this chapter were published in the IEEE Communication Letters, Vol. 13, No. 8, pp 561-563, Aug. 2009, cf. Appendix F. 13 In practice, the number of transmit antennas is usually limited due to the space constraints on the mobile devices. Therefore, STCs with two transmit antennas are the most practically important types of STCs. For instance, IEEE 802.11n [88], the IEEE 802.16e [89], and 3GPP Long Term Evolution (LTE) [90] standards support the use of two transmit antennas and multiple receive antennas for increasing the transmission rate and/or improving the performance.  31  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels  2.2  System Model  We consider a MIMO system with NT = 2 transmit and NR receive antennas. The space–time encoder at the transmitter generates an N × NT codeword matrix C, where E{tr{CC H }} = N and N denotes the codeword (frame) length. Assuming a quasi–static fading14 channel, the N × NR received signal matrix R can be modeled as R=  √  γ CH + Z,  (2.1)  where H and Z denote the NT × NR channel and noise matrices, respectively, and γ is the average SNR per receive antenna. The elements of Z are i.i.d. Gaussian random variables. The elements of H are modeled as hµν  [H]µν = aµν ejφµν , 1 ≤ µ ≤ 2, 1 ≤ ν ≤ NR ,  where aµν and φµν represent the amplitude and phase of the channel gain, respectively. As mentioned in Chapter 1, we assume that the phases φµν are uniformly distributed in [−π, π). Furthermore, we assume that the channel gains are i.i.d. and Ω = E{a2µν } = 1. It has been shown [16] that for high SNRs, the system performance depends on the behavior of the pdf of fading in the vicinity of zero, i.e., the probability of a deep fade. As aµν → 0+ , the Taylor series of the pdf of aµν can be written as f (a) = α a2β−1 + o(a2β−1 ),  (2.2)  where α and β are parameters that depend on the considered type of fading. In Table 2.1, α and β are listed for various common types of fading from Chapter 1. 14  In a quasi–static fading channel, the channel gain is constant for N symbols durations and changes independently in the next codeword length.  32  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels Table 2.1: Parameters α and β in (2.2) for common types of fading, cf. Chapter 1. Fading type  α  β  Rayleigh  2  1  Ricean  2(1 + K) exp(−K)  Nakagami-q  (1 + q )/q  1  Nakagami-m  2mm /Γ(m)  m  c/2  Weibull  (Γ(1 + 2/c)) t/2  2(m/π) Γ(|k−m|) Γ(m)Γ(k)  Generalized–K15  2.3  1  2  c/2 min{k, m}  Asymptotic Analysis  The PEP is defined as the probability of deciding in favor of a wrong codeword C 2 instead of the transmitted codeword C 1 . Defining the error matrix E enµ  C 1 − C 2 with entries  [E]nµ , 1 ≤ n ≤ N, 1 ≤ µ ≤ 2, the conditional PEP can be expressed as [3] Pe (E|H) = Q  where  NR  N  d (E)  (2.3)  2  2  2  γ d2 (E)/2 ,  hµν enµ  = tr{H H E H EH}.  (2.4)  ν=1 n=1 µ=1 H Exploiting tr{H H E H EH} = tr{E H EHH }, d2 (E) can be rewritten in terms of the   elements of matrix E H E   d1 d12   as  ∗ d12 d2 NR  d2 (E) = ν=1 NR  =  d1 |h1ν |2 + d2 |h2ν |2 + 2 {d12 h1ν h∗2ν } g(a1ν , a2ν , φν ),  (2.5)  ν=1 15  For generalized–K fading we assume that k = m.  33  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels √ d1 a21ν + d2 a22ν + 2ξ d1 d2 a1ν a2ν cos φν , ξ  where g(a1ν , a2ν , φν )  √ |d12 |/ d1 d2 , and φν  φ1ν − φ2ν + ∠d12 . Since E H E is a positive semi–definite matrix, det(E H E) ≥ 0 and ξ ≤ 1. Thus, the range of possible values for ξ is 0 ≤ ξ ≤ 1. Using the alternative representation 1 π  of the Q–function, Q(x)  π/2 −x2 /(2 sin2 θ) e dθ 0  obtain  π/2  1 Pe (E|H) = π  exp −  [15], and combining (2.3) and (2.5), we  NR ν=1 g(a1ν , a2ν , φν ) 2  γ  4 sin θ  0  dθ.  (2.6)  Exploiting that the channel gains are i.i.d. and that φν is uniformly distributed in [−π, π), the unconditional PEP can be expressed as π/2  1 Pe (E) = π  [I(θ)]NR dθ,  (2.7)  0  where I(θ)  1 2π  ∞ ∞ π  exp − 0  0 −π  γg(a1 , a2 , φ) 4 sin2 θ  f (a1 )f (a2 ) dφ da1 da2 .  (2.8)  It is shown in Appendix A that for high values of SNR and ξ < 1, I(θ) can be simplified to . α2 24β−2 [Γ(β)]2 2 F1 (β, β; 1; ξ 2) β sin θ . I(θ) = (γ 2 d1 d2 )β  (2.9)  Applying (2.9) in (2.7) and exploiting [91, Eqs. (3.621.3), (8.339.2)] results in the following closed–form expression for the PEP of space–time coded transmission with two transmit and NR receive antennas over generalized fading channels: 1 4βN −1 . 2 R Ξ Γ 2βNR + 2 √ Pe (E) = , π Γ(2βNR + 1) γ 2βNR  (2.10)  34  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels where Ξ  α2 [Γ(β)]2 2 F1 (β, β; 1; ξ 2) 4(d1 d2 )β  NR  ,  (2.11)  and for 0 ≤ ξ < 1 the Gaussian hypergeometric function can be efficiently computed from the fast converging series [91, Eq. (9.100)]  2 F1 (β, β; 1; ξ  2  )=  ∞ n=0  β+n−1 n ξ n  2  .  (2.12)  For general β > 0 and non–orthogonal STCs (0 < ξ < 1), (2.10) is a new result. For the special cases of Rayleigh, Ricean, and Nakagami–q fading, β = 1 (cf. Table 2.1) and 2 F1 (1, 1; 1; ξ  2  ) = 1/(1 − ξ 2 ) is valid. Thus, in these cases (2.10) can be simplified to16 α2NR Γ(2NR + 21 ) . . Pe (E) = √ NR 2 π det E H E (2NR )! γ 2NR  (2.13)  For the special case of orthogonal STBCs (ξ = 0), 2 F1 (β, β; 1; 0) = 1 holds and (2.10) is equivalent to the results in [16, Example 4]. . As mentioned in Chapter 1, the asymptotic PEP can be expressed as Pe (E) = (Gc γ)−Gd [16], where Gc and Gd denote the coding and the diversity gain, respectively. We observe from (2.10) that the maximum diversity gain is Gd = 2βNR . For a STC to achieve the full diversity gain in generalized i.i.d. fading channels ξ = 1 is a necessary and sufficient condition.17 This criterion is identical to the well–known determinant criterion, det(E H E) = d1 d2 − |d12 |2 = 0, for Rayleigh fading [3], i.e., a STC that achieves full diversity in Rayleigh fading also achieves full diversity in any other type of fading. However, the coding gain Gc depends on both the type of fading (via β) and the structure of the code (via ξ). Interestingly, this dependence is different for fading types with β = 1 and 16 17  Note that for Rayleigh fading, (2.13) is equivalent to the result in [13, Proposition 3]. For ξ = 1 the integral in (A.3) in Appendix A diverges and a loss in diversity results.  35  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels β = 1. For fading channels with β = 1, from (2.13) the coding gain can be expressed as √ 2 π(2NR )! × det E H E Gc = α2NR Γ(2NR + 21 )  NR  1 2NR  .  (2.14)  In (2.14), the first term depends only on the distribution of the channel while the second term depends only on the structure of the code. As a result, if a STC outperforms another STC for a fading distribution with β = 1, e.g. for Rayleigh fading, then it will also outperform in any other type of fading with the same value of β. e.g. for Nakagami–q fading. On the other hand, the difference between the coding gain of a specific STC in two different types of fading distribution with parameters (α1 , β1 = 1) and (α2 , β2 = 1) is simply given by ∆Gc =  α1 . α2  In general, for β = 1, the dependence of the coding gain on the code design and the fading parameters is not separable. Therefore, a comparison in these cases should be made using numerical results from (2.10). We also note that asymptotically accurate or tight upper bounds for the BEP, SEP, and frame error probability (FEP) of STCs can be obtained by combining the PEP in (2.10) with union bounds (cf. [15]).  2.4  Numerical Results  In Fig. 2.1, we consider Nakagami–m fading and compare the asymptotic PEP (2.10) of a length–2 error event of the rate–2, 4–state STTC in [3] (referred to as TSC in the thesis) with computer simulations, the exact PEP [9, Eq. (6.38)] for Rayleigh fading (m = 1), and the upper bound from [18]. The considered error event (e11 = e22 = 0 √ and e12 = e21 = (1 − j)/ 2) is the error event with the highest PEP. Fig. 2.1 confirms the asymptotic accuracy of (2.10) and reveals that the existing upper bound in [18] is somewhat loose.  36  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels  0  10  −2  10  Rayleigh  −4  PEP  10  −6  10  Nakagami-m (m = 2)  −8  10  Exact Asymptotic Upper bound Simulation  −10  10  5  10  Nakagami-m (m = 3) 15 20 SNR per symbol (dB)  25  30  Figure 2.1: PEP of a length-2 error event of rate–2 4–state STTC in [3] for Nakagami–m fading. 0  10  Generalized−K  (m = 2, k = 3) NR = 1  −1  10  −2  Generalized−K  FEP  10  (m = 1, k = 3) NR = 2  −3  10  −4  10  −5  10  TSC (Asymptotic) BBH (Asymptotic) TSC (Simulation) BBH (Simulation)  Rayleigh  NR = 2  −6  10  5  10  15 20 SNR per symbol (dB)  25  30  Figure 2.2: FEP of TSC [3] and BBH [4] codes in Rayleigh and generalized–K fading. 37  Chapter 2. Performance Analysis of STCs in Generalized Fading Channels In Fig. 2.2, we compare an asymptotic upper bound for the FEP with simulation results for the TSC code and the rate–2, 4–state STTC from [4] (referred to as BBH in the thesis). The asymptotic upper bound is obtained by combining (2.10) with a truncated union bound taking into account all error events up to length 3.18 We consider Rayleigh fading with NR = 2 and generalized–K fading with m = 1, k = 3, NR = 2 and m = 2, k = 3, NR = 1. As expected, both codes achieve a diversity gain of Gd = 2βNR = 4 for all three channels. However, interestingly, while the TSC code has a superior coding gain for the channel with NR = 1, the BBH code has a higher coding gain than the TSC code for the two channels with NR = 2. Thus, the best STC for a given application depends on the underlying type of fading and can be efficiently found based on the derived easy–to–compute PEP expression.  2.5  Conclusions  A simple closed–form expression for the PEP of STCs with two transmit antennas and an arbitrary number of receive antennas in generalized fading was derived. This result can be combined with union bounds to obtain tight asymptotic upper bounds for the bit, symbol, and frame error probabilities. The derived PEP is also useful for code search and design.  18  For these error events, ξ assumes values between 0 and 0.8165.  38  Chapter 3 Performance Analysis of STCs in Generalized Keyhole Fading Channels19 3.1  Introduction  As mentioned in Section 1.5.2, performance analysis and design of STCs in keyhole fading channels has mainly focused on orthogonal space–time block codes (OSTBCs) over some specific types of fading channels. In this chapter, we focus on general STCs with one or two transmit antenna(s)20 and an arbitrary number of receive antennas in a general uncorrelated keyhole fading channel. Due to the separation between the transmitter and receiver sides, we consider a general case in which the transmitter and receiver side can have different fading parameters and may be affected by different types of fading. We provide closed–form expressions for the asymptotic PEP. Our results are applicable to both STBCs and STTCs, are easy to evaluate, and can be used for code search and system design. We show how the number of transmit and receive antennas as well as the fading parameters affect the diversity gain. We 19  Parts of this chapter have been submitted to the IEEE Transactions on Wireless Communications, cf. Appendix F. 20 Note that systems with a single transmit antenna and maximum–ratio combining detection can be considered as a special case of STCs  39  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels also compare the performance of STCs in keyhole and non–keyhole fading channels from Chapter 2 and show that the performance gap between different codes is less sensitive to the number of receive antennas in keyhole fading than in non–keyhole fading. The remainder of this chapter is organized as follows. In Section 3.2, the system model is described. In Sections 3.3 and 3.4, we derive asymptotic expressions for the PEP of general STCs in generalized keyhole fading channels. The analytical results are discussed in Section 3.5 and verified through simulations in Section 3.6. Finally, the conclusions are drawn in Section 3.7.  3.2  System Model  We consider a MIMO system with NT ∈ {1, 2} transmit and NR receive antennas. The space–time encoder at the transmitter generates an N × NT codeword matrix C, where E{tr{CC H }} = N and N denotes the codeword (frame) length. Assuming a quasi–static fading channel, the N × NR received signal matrix R can be modeled as R=  √  γ CH + Z,  (3.1)  where H and Z denote the NT × NR channel and AWGN matrices, respectively, and γ is the average SNR per receive antenna. The elements of Z are independent, identically distributed (i.i.d.) Gaussian random variables. H models a keyhole channel matrix and is given by [1] H = h gH ,  (3.2)  where h and g are NT × 1 and NR × 1 channel vectors that correspond to the transmitter and receiver side, respectively. The elements of h and g are modeled as hµ = aµ ejφµ , 1 ≤ µ ≤ NT , and gν = bν ejϑν , 1 ≤ ν ≤ NR , respectively, with amplitudes aµ  |hµ | and 40  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels bν  |gν | and phases φµ  ∠hµ and ϑν  ∠gν . We assume that the phases φµ and ϑν are  uniformly distributed in [−π, π). Furthermore, the channel gains at both transmitter and receiver side are assumed to be i.i.d. with E{a2µ } = ΩT , E{b2ν } = ΩR , and ΩT ΩR = 1. We assume that the probability density functions (pdfs) of aµ and bν can be represented by their Taylor series expansions  faµ (a) = fbν (b) =  ∞ k=0 ∞ k=0  αT,k a2βT,k −1 ,  1 ≤ µ ≤ NT ,  (3.3)  αR,k b2βR,k −1 ,  1 ≤ ν ≤ NR ,  (3.4)  where αT,k and βT,k are parameters that depend on the considered type of fading for the transmitter side and αR,k and βR,k are the same parameters for the fading channel at the receiver side. In Table 3.1, we have dropped the indices T and R and listed general expressions for αk and βk for various common types of fading.21 From Table 3.1, we observe that, in general, βT,k and βR,k in (3.3) and (3.4) can be expressed as βT,k = ηT k + βT and βR,k = ηR k + βR , respectively, where parameters ηT , ηR , βT , and βR depend on the type of fading. In the next two sections, we will show that the asymptotic PEP analysis crucially depends on parameter D  NR βR − NT βT , which may be interpreted as the difference  between the receiver side diversity gain, NR βR , and the transmitter side diversity gain, NT βT . In particular, it will be convenient to distinguish between D < 0 (Case 1), D > 0 (Case 2), and D = 0 (Case 3). 21  Note that considering the first term (k = 0) of the entries in Table 3.1 and unit variance Ω = 1 results in Table 2.1.  41  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels Table 3.1: Parameters αk and βk in (3.3) and (3.4) for common types of fading, cf. Chapter 1. Indices k, i, and j are non-negative integers. Fading type  αk 2(−1) k! Ωk+1  Rayleigh Ricean  i+j=k  k+1  2(−1)i (1+K)i+j+1 exp(−K)K j i!(j!)2 Ωi+j+1 i  2 2i+1  4 2j  (−1) (1+q ) (1−q ) i+2j=k i!(j!)2 4i+3j q2i+4j+1 Ωi+2j+1  Nakagami–q  k  k+m  2(−1) m k! Γ(m)Ωk+m  Nakagami–m k  k+1 k+1 k+m  c(k+1)/2  (−1) (Γ(1+2/c)) k! Ωc(k+1)/2  Weibull  3.3  βk = ηk + β k  c  c 2 (k  + 1)  Asymptotic PEP Analysis: Cases 1 and 2 (D = 0)  In this section, we first consider the PEP of STCs in keyhole channels for the general case where D can be any real number, and subsequently develop closed–form asymptotic PEP expressions for D < 0 and D > 0, respectively.  3.3.1  Asymptotic PEP Analysis: General Case  The PEP is defined as the probability of deciding in favor of a wrong codeword C 2 instead of the transmitted codeword C 1 . Defining the error matrix E  C 1 − C 2 , the PEP can  be expressed as [1]  Pe (E)  E Q  γ  d2 (E)/2  1 = π  π/2  Ψd2 (E) 0  γ 4 sin2 θ  dθ,  (3.5)  where we exploited the alternative representation of the Q–function, Q(x) 1 π  π/2 −x2 /(2 sin2 θ) e 0  dθ [15], Ψd2 (E) (s)  2 (E)  E{e−sd  } is the moment generating function  42  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels (MGF) of d2 (E), and d2 (E) is defined as d2 (E)  with X  ||g||2 and Y  ||EH||2 = X Y  (3.6)  ||Eh||2 . With the help of [1, Lemma 1], Ψd2 (E) (s) can be  evaluated as ∞  Ψd2 (E) (s) = 0  =  ∞  fX (λ)ΨY (sλ) dλ  (3.7)  fY (λ)ΨX (sλ) dλ,  (3.8)  0  where fX (x) (fY (y)) and ΨX (s)  E{e−sX } (ΨY (s)  E{e−sY }) are the pdf and MGF of  X (Y ), respectively. Note that fX (x) depends only on the type of fading at the receiver side and can be easily calculated or estimated for different types of fading, cf. e.g. [92]. On the other hand, fY (y) depends on the fading distribution at the transmitter side and the elements of E H E. We note that E H E = |E|2 is a scalar for NT = 1 and, as  is  d1 d12  customary in the literature, is assumed to be a full rank matrix E H E   d∗12 d2 for NT = 2, since otherwise the STC cannot achieve a diversity gain compared to single– antenna transmission, cf. [3]. We show in Appendix B that for both NT = 1 and NT = 2, fY (y) can be expressed as fY (y) =  ∞  ck y ηT k+NT βT −1 ,  (3.9)  k=0  where ck is given by  ck        αT,k 2|E|  2βT,k  ,  αT,i αT,k−i Γ(βT,i )Γ(βT,k−i ) 2 F1 (βT,i ,βT,k−i k β β i=0 4 d1 T,i d2 T,k−i Γ(βT,i +βT,k−i )  NT = 1 ;1,ξ 2  )  (3.10)  , NT = 2  43  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels with ξ  √ |d12 |/ d1 d2 . For c0 , which will be of particular interest in the following, (3.10)  simplifies to     c0     αT,0 , 2|E|2βT (αT,0 Γ(βT ))2 2 F1 (βT ,βT ;1,ξ 2 ) 4 (d1 d2 )βT Γ(2βT )  NT = 1  .  (3.11)  , NT = 2  We note that for OSTBCs, ξ = 0 holds and the hypergeometric functions in (3.10) and (3.11) are equal to 1 which further simplifies the above results. It is clear from (3.5) that the asymptotic PEP is determined by lims→∞ Ψd2 (E) (s). According to the Bounded Convergence Theorem [93], the above lim operator can be moved inside the integrals in (3.7) and (3.8) if the corresponding integrand is integrable in the limit. In other words,  Ψ  d2 (E)  ∞  . (s) = 0  . = 0  fX (λ) lim ΨY (sλ) dλ  (3.12)  fY (λ) lim ΨX (sλ) dλ,  (3.13)  s→∞  ∞  s→∞  if the integrals converge. In order to determine the convergence of the above integrals, we note that from the definitions of ΨX (s) and ΨY (s), for λ not in the vicinity of 0, lims→∞ ΨX (sλ) = 0 and lims→∞ ΨY (sλ) = 0. Therefore, we break the limits of the integrals in (3.12) and (3.13) as  . Ψd2 (E) (s) =  fX (λ) lim ΨY (sλ) dλ + 0  . =  where  s→∞  fY (λ) lim ΨX (sλ) dλ + 0  ∞  s→∞  fX (λ) lim ΨY (sλ) dλ  (3.14)  fY (λ) lim ΨX (sλ) dλ,  (3.15)  s→∞  ∞  s→∞  → 0+ is a small positive value. The convergence of integrals (3.12) and (3.13) 44  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels depends on whether the first term in (3.14) and (3.15) is bounded or not, respecitvely. Therefore, we need to investigate the behavior of fX (0+ ) and fY (0+ ). It is shown in Appendix C that as x → 0+ , . fX (x) =  where AR  αR,0 Γ(βR ) 2  NR  AR xNR βR −1 , Γ(NR βR )  (3.16)  . Similarly, we observe from (3.9) that as y → 0+ , . fY (y) = c0 y NT βT −1 .  (3.17)  The asymptotic MGF of X and Y can be evaluated by applying the Laplace transform to (3.16) and (3.17), which leads to  where AT  . ΨX (s) = AR s−NR βR ,  (3.18)  . ΨY (s) = AT s−NT βT ,  (3.19)  c0 Γ(NT βT ). Eqs. (3.16)–(3.19) imply that as λ → 0+ and s → ∞ AT AR D−1 −NT βT λ s , Γ(NR βR ) AT AR −D−1 −NR βR . fY (λ)ΨX (sλ) = λ s . Γ(NT βT ) . fX (λ)ΨY (sλ) =  (3.20) (3.21)  Clearly, the first term of (3.15) is finite only if the exponent of λ in (3.21) is greater than −1, i.e., if D < 0. Similarly, the first term of (3.14) is finite when D > 0 in (3.20). Hence, we can use (3.13) and (3.12) if D < 0 and D > 0, respectively. In the following two subsections, we will consider these two cases, respectively. For D = 0 neither (3.13) nor (3.12) converge and a more elaborate analysis has to be applied, cf. Section 3.4.  45  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  3.3.2  Case 1 (D < 0)  It can be seen from (3.20) and (3.21) that for D < 0 the first terms of (3.14) and (3.15) are equal to infinity and zero, respectively. Therefore, (3.12) diverges while (3.13) remains finite. Combining (3.9) and (3.18), Ψd2 (E) (s) in (3.13) can be written as22 . Ψd2 (E) (s) = AR u(∞) s−NR βR ,  (3.22)  where u(λ)  ∞ k=0  ck ληT k−D . ηT k − D  (3.23)  We note that u(∞) required in (3.22) can be efficiently evaluated using the general method presented in Appendix D. Applying (3.22) in (3.5), the asymptotic PEP is obtained as 1 2N β −1 . 2 R R AR u(∞) Γ(NRβR + 2 ) √ , Pe (E) = π Γ(NR βR + 1) γ NR βR  (3.24)  where we have used the following identity [91, Eqs. (3.621.1), (8.335.1), and (8.384.1)] π/2 2x  sin θ dθ = 0  √  π Γ(x + 21 ) . 2 Γ(x + 1)  (3.25)  Example 1: Assume a single receive antenna (NR = 1) and Nakagami–m fading for both the transmitter and receiver side with βT = mT = 1.5, βR = mR = 1.0, and ΩT = ΩR = 1. Furthermore, consider the length 3 error event for the rate–2, 4–state STTC with NT = 2 in [3] for which C 1 and C 2 correspond to the all–zero and {0,1, 1, 0} state √ 1+j 1+j   0 sequences, respectively. For this error event E H = 22  . Employing 1+j 1+j 0  the appropriate parameters from Table 3.1 in (3.10) and (3.23), and with the help of 22  Note that since D ≤ −1, u(0) = 0 holds true.  46  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels the numerical method presented in Appendix D, we obtain u(∞)  0.448. Hence, the  asymptotic PEP can be readily computed from (3.24).  3.3.3  Case 2 (D > 0)  In this case, unlike Case 1, the first term in (3.15) becomes unbounded while (3.14) remains finite. Therefore, we use (3.12) and (3.19) to arrive at . Ψd2 (E) (s) = AT µX (−NT βT ) s−NT βT ,  where µX (−NT βT )  (3.26)  E{x−NT βT } is the NT βT th negative moment of X. Note that the  condition D = NR βR − NT βT > 0 guarantees the existence of this moment, cf. (3.16). Substituting (3.26) in (3.5) and using (3.25) leads to the asymptotic PEP expression 1 2N β −1 . 2 T T AT µX (−NT βT ) Γ(NT βT + 2 ) √ Pe (E) = . π Γ(NT βT + 1) γ NT βT  (3.27)  Example 2: For Nakagami–m fading with parameter mR at the receiver side, X is Gamma distributed with scale parameter ΩR /mR and shape parameter NR mR [69]. Therefore, µX (−NT βT ) can be evaluated as [91, Eq. (3.381.4)]  µX (−NT βT ) =  Γ(NR mR − NT βT ) . Γ(NR mR )(ΩR /mR )NT βT  (3.28)  The asymptotic PEP for Nakagami–m fading is obtained by combining (3.27) and (3.28). Note that for the special case of Rayleigh fading at both transmitter and receiver side, i.e., βT = mR = 1, substituting (3.28) in (3.26) results in the first case of [1, Theorem 1].  47  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  3.4  Asymptotic PEP Analysis: Case 3 (D = 0)  In this section, we consider the case D = 0, i.e., NT βT = NR βR for which using (3.20) and (3.21) in (3.14) and (3.15), respectively results in infinity in both cases. Thus, both (3.12) and (3.13) diverge and the Bounded Convergence Theorem can not be applied. As a result, a unified asymptotic PEP analysis for all considered types of fading in the spirit of Sections 3.3.2 and 3.3.3 is not possible. Instead, the exact pdfs and MGFs have to be used in (3.7) and (3.8) before the limit can be taken and a case–by–case analysis is necessary. Since in this case both (3.7) and (3.8) lead to the same result, we focus on (3.8) in the following. The exact pdf fY (y) for general fading distributions at the transmitter side is given in (3.9). However, the MGF ΦX (s) depends on the particular fading distribution at the receiver side. Therefore, a general asymptotic analysis for different types of fading distributions at the receiver side does not seem possible. Thus, in the following, we concentrate on the two most important types of receiver side fading in practice, namely Nakagami–m fading (βR = mR ) and Ricean fading (βR = 1). Note that both types of fading include Rayleigh fading as a special case.  3.4.1  Nakagami–m Fading at the Receiver Side  Recall from Example 2 that for Nakagami–m fading at the receiver side, X is Gamma distributed with scale parameter ΩR /βR and shape parameter NR βR . Therefore, we obtain for the MGF of X [94] ΨX (s) =  ΩR s+1 βR  −NR βR  .  (3.29)  48  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels For simplicity, we assume that ηT = 1, i.e., non–Weibull fading at the transmitter side, cf. Table 3.1. Thus, employing (3.9) and (3.29) in (3.8) yields ∞  Ψd2 (E) (s) = lim  z→∞  ck ∆k (s, z),  (3.30)  k=0  where z  ∆k (s, z) 0  =  λk+NT βT −1 ΩR sλ βR  +1  NR β R  dλ  z k+NT βT ΩR sz . 2 F1 k + NT βT , NR βR ; k + NT βT + 1, − k + NT βT βR  (3.31)  For derivation of the second line of (3.31), we have used [91, Eq. (3.194.1)]. We now employ the following asymptotic formulas for the Gauss hypergeometric function 2 F1 (a, b; c, x) as |x| → ∞ [95]: Γ(a − b)Γ(c) . Γ(b − a)Γ(c) = (−x)−a + (−x)−b ; a = b, (3.32) Γ(b)Γ(c − a) Γ(a)Γ(c − b) . Γ(c)(−x)−a (log(−x) − ψ(c − a) − ψ(a) − 2γE ) ; c − a ∈ Z.(3.33) 2 F1 (a, a; c, x) = Γ(a)Γ(c − a) 2 F1 (a, b; c, x)  Since NT βT = NR βR , assuming that s → ∞ and combining (3.31)–(3.33) yields  . ∆k (s, z) =  where Bk       Bk  ΩR s βR   ΩR    s βR  Γ(D−k)Γ(k+NT βT ) Γ(NR βR )  −(k+NT βT )  −(k+NT βT )  and Ck  + Ck z  log  1 . k−D  k−D  ΩR s βR  ΩR s βR  −NR βR  k ≥ 1,  (3.34a)  k = 0,  (3.34b)  It is easy to see that at s → ∞, the second  term of (3.34a) dominates the first term and is itself dominated by (3.34b). Therefore,  49  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels from (3.30) we obtain ΩR βR  . Ψd2 (E) (s) = c0  −NT βT  s−NT βT log  ΩR s . βR  (3.35)  Applying the above result in (3.5) and using (3.25) gives ΩR 2NT βT −1 γ c0 Γ(NT βT + 12 ) log 4β R . 2 Pe (E) = . √ Ω R NT β T N β Γ(NT βT + 1) γ T T π βR  (3.36)  Eq (3.36) fully characterizes the PEP for Case 3 for Nakagami–m fading at the receiver side.  3.4.2  Ricean Fading at the Receiver Side  In this section, we analyze the asymptotic PEP for Case 3 for Ricean fading at the receiver side. Recall that X = ||g||2 =  NR ν=1  b2ν . Since the fading gains bν ,  xν , where xν  0 ≤ ν ≤ NR , are i.i.d., the variables xν , 0 ≤ ν ≤ NR , are i.i.d. as well and the MGF of X is given by [15] MX (s) =  −NR  ΩR s+1 1+K  exp −  NR Ω R K s 1+K ΩR s+1 1+K  ,  (3.37)  where K is the Ricean factor. Using the Taylor series expansion of the exponential function, exp(x) =  ∞ n n=0 x /n!,  and using (3.9) and (3.37) in (3.8) results in  Ψd2 (E) (s) = lim  z→∞  ∞ k=0  ∞  ck  n=0  (−KNR )n ∆k,n (s, z), n!  (3.38)  where ∆k,n (s, z)  ΩR s 1+K  n  z 0  λk+NT βT +n−1 ΩR sλ 1+K  +1  NR +n  dλ.  (3.39)  50  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels Applying [91, Eq. (3.194.1)] in (3.39) yields  ∆k,n (s, z) =  ΩR s 1+K  n  z k+NT βT +n k + NT βT + n  × 2 F1 k + NT βT + n, NR + n; k + NT βT + n + 1, −  ΩR sz . (3.40) 1+K  We again exploit (3.32) and (3.33) to find the asymptotic form of ∆k,n (s, z) as s → ∞, which yields  . ∆k,n (s, z) =       Bk,n      where Bk,n  ΩR s 1+K  ΩR s 1+K  Γ(D−k)Γ(k+NT βT +n) . Γ(NR +n)  −(k+NT βT )  −(k+NT βT )  log  + Ck z  k−D  ΩR s 1+K  −NR  k ≥ 1, (3.41a)  ΩR s 1+K  k = 0,(3.41b)  Similar to the previous section, at s → ∞ the dominant  term in (3.38) is (3.41b) and we obtain . Ψd2 (E) (s) = c0  ΩR 1+K  −NT βT  s−NT βT exp(−KNR ) log  ΩR s . 1+K  (3.42)  Applying the above result in (3.5) and using (3.25) gives ΩR 2NT βT −1 γ c0 Γ(NT βT + 21 ) exp(−KNR ) log 4(1+K) . 2 . Pe (E) = √ Ω R NT β T π 1+K Γ(NT βT + 1) γ NT βT  (3.43)  It is worth mentioning that the log term in (3.35) and (3.43) has been reported before in [71] and [1] for the special case of Rayleigh fading. This term slightly reduces the slope of the PEP vs. SNR curve on a double–logarithmic scale. However, at high SNR, log γ is dominated by γ NT βT . We note that, as expected, for the special case of Rayleigh fading (3.43) with K = 0 and (3.36) with βR = 1 are identical. We also note that as long as an analytical expression 51  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels for ΨX (s) is available, a similar asymptotic analysis can be conducted for Case 3 for other types of fading at the receiver side as well.  3.5  Unification and Implications  In this section, we first unify the results derived in Sections 3.3 and 3.4 before we discuss their implications for system design and performance.  3.5.1  Unified Asymptotic PEP Expression  In order to arrive at a unified asymptotic PEP expression, we define a dummy parameter K = 0 for Nakagami–m fading. Consequently, the asymptotic PEP expressions in (3.24), (3.27), (3.36), and (3.43) can be expressed in a unified form as 1 2G −1 . 2 d Ξ Γ(Gd + 2 ) , Pe (E) = √ π Γ(Gd + 1) γ Gd  (3.44)  which is the same as Eq. (2.10) for the PEP of STCs in generalized non–keyhole fading channels. However, Ξ in (3.44) is different from (2.11) and is given by  Ξ   AR u(∞)      AT µX (−NT βT )   βR (1 + K)    c0 ΩR  Gd  exp(−KNR ) log  ΩR γ 4βR (1 + K)  D < 0,  (3.45a)  D > 0,  (3.45b)  D = 0.  (3.45c)  In the above Eqs., Gd is the diversity gain which can be expressed as  Gd = min{NT βT , NR βR }.  (3.46)  52  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels We note that (3.45a) and (3.45b) are valid for any type of fading while (3.45c) was shown to hold for Nakagami-m and Ricean fading at the receiver side, cf. Sections 3.3 and 3.4. Gd is the so–called diversity gain of the system and specifies the negative asymptotic slope of the PEP vs. SNR curves on a double–logarithmic scale. Eq. (3.46) generalizes the results in [71] and [1], which showed that the diversity gain for Rayleigh keyhole fading (βT = βR = 1) is Gd = min{NT , NR }.  3.5.2  Implications for Code Design  The PEP expression in (3.44) can be used to find optimal STCs in different scenarios. For D ≥ 0 (Cases 2 and 3), i.e., for the case when the diversity gain Gd is limited by the transmitter side, the PEP expression is influenced by the code error matrix, E, only via coefficient c0 (recall that AT = c0 Γ(NT βT ) in (3.45b), which is defined in (3.11). This leads to the STC design criterion  minimize max  2 F1 (βT , βT ; 1, ξ (d1 d2 )βT  2  )  ,  (3.47)  where the minimization and maximization are performed over all possible codes and all possible error events of a given code, respectively. Eq. (3.47) reveals that for D ≥ 0 the code design criterion is independent of the fading distribution at the receiver side. In the special case of Rayleigh fading at the transmitter side (βT = 1), 2 F1 (1, 1; 1, ξ 2) = 1/(1−ξ 2). Thus, in this case, the above design criterion reduces to 1/(d1d2 − d212 ) = 1/det{E H E}, which is the well–known determinant criterion for STCs in non–keyhole Rayleigh fading channels [3]. Here, we showed that the determinant criterion is also optimal for keyhole fading channels with transmitter side Rayleigh fading and any type of fading at the receiver side as long as D ≥ 0. For D < 0 (Case 1), a similar design criterion can be established, which now involves 53  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels minimizing the maximum value of u(∞). Since u(∞) is computationally more expensive to compute than the criterion in (3.47), the corresponding code search is more time consuming and finding simpler criteria may be an interesting topic for future research.  3.5.3  Keyhole vs. Non–keyhole Channels  It is interesting to compare the performance of STCs in keyhole and non–keyhole fading channels. As mentioned earlier, the PEP expression in generalized non–keyhole and keyhole fading channels (Eqs. (2.10) and (3.44), respectively) are essentially the same. However, the values of Ξ in these two equations are different and are given by (2.11 and (3.45) for non–keyhole and keyhole fading channels, respectively. Therefore, the asymptotic PEPs of generalized non–keyhole and keyhole (D ≥ 0) fading channels is proportional to 2 F1 (βT , βT ; 1, ξ (d1 d2 )βT  2  )  NR  and  2 F1 (βT , βT ; 1, ξ (d1 d2 )βT  2  )  ,  (3.48)  respectively. Eq. (3.48) suggests that if a given STC outperforms another STC in some type of non–keyhole fading, it will also perform better in keyhole fading provided that the transmitter side fading of the keyhole channel is identical to the non–keyhole fading and the receiver side fading is such that D ≥ 0. However, (3.48) also shows that while the performance gap between two STCs increases as the number of receive antennas NR increases in the non–keyhole fading channel, it is independent of NR in the keyhole fading channel. Finally recall that the asymptotic BEP, SEP, and FEP of different STBCs and STTCs can be easily approximated based on the derived PEP expression in (3.44) using standard approaches such as the truncated union bound, cf. e.g. [15].  54  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  3.6  Numerical Results  In this section, we verify our analytical results by computer simulations. In all figures we assume that fading distribution at the transmitter and receiver sides are equi–power, i.e., ΩT = ΩR = 1. Furthermore, the asymptotic PEP in Case 3 is evaluated only for medium and high SNRs at which the log term in (3.45c) is positive. In addition to the TSC and BBH STTCs introduced in Chapter 2, we denote the rate-2 4–state STTC in [1] by SHN. Fig. 3.1 shows the simulated and asymptotic PEP of the length–3 error event in Example 1 over a Nakagami–m keyhole fading channel with NT = 2, NR = 1, and different values of mT and mR , which correspond to Cases 1–3. Fig. 3.1 confirms the accuracy of our asymptotic PEP expressions. Fig. 3.2 shows the PEP of the same error event as in Fig. 3.1 when the fading distributions at the transmitter and receiver sides are Nakagami-m with different values of mT and Ricean with K = 0 dB, respectively and NT = NR = 2. It can be seen that our results are also accurate when the receiver side fading has a Ricean distribution and the diversity gain in each case can be verified by (3.46). Fig. 3.3 illustrates the effect of the number of transmit and receive antennas on the BEP. The channel is assumed to be a Weibull–Rayleigh keyhole fading channel, i.e., βT = c/2 = 2, and βR = 1. The modulation scheme is 4–PSK and the frame length of the channel is 130 symbol intervals. Three different scenarios are considered: 1) NT = NR = 1; 2) NT = 2 (BBH STTC) and NR = 1; and 3) NT = 1 and NR = 2 with maximum ratio combining (MRC) at the receiver. The asymptotic upper bound for Scenario 2 was obtained by combining (3.24) with a truncated union bound which takes into account all error events of lengths up to 3. This bound is somewhat looser than the asymptotic approximations for uncoded 4–PSK transmission with NT = 1, where only the dominant error event (corresponding to the minimum Euclidean distance of the 4–PSK constellation) 55  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels −1  10  −2  10  −3  10  −4  PEP  10  −5  10  −6  10  Case 1: mT = 1.5, mR = 1.0 (Simulation) Case 2: m = 1.0, m = 3.0 (Simulation) T  R  Case 1: m = 1.5, m = 2.5 (Simulation) T  −7  10  R  Case 2: mT = 1.0, mR = 2.5 (Simulation) Case 3: m = 1.0, m = 2.0 (Simulation) T  −8  10  10  R  Asymptotic 15  20  25  30  35  SNR per symbol (dB)  Figure 3.1: PEP of the length–3 error event in Example 1 over keyhole Nakagami–m fading channel. NT = 2 and NR = 1. Asymptotic PEP was obtained from (3.44) and (3.45a)–(3.45c). −1  10  −2  10  −3  10  −4  PEP  10  −5  10  Case 1: m = 1.5 (Simulation) T  −6  10  Case 2: m = 0.5 (Simulation) T  Case 1: m = 1.25 (Simulation) T  −7  10  Case 2: mT = 0.75 (Simulation) Case 3: mT = 1.00 (Simulation) Asymptotic  −8  10  10  15  20  25  30  35  SNR per symbol (dB)  Figure 3.2: PEP of the length–3 error event in Example 1 over keyhole Nakagami–m fading with parameter mT at the transmitter side and Ricean fading with K = 0 dB at the receiver side. NT = 2 and NR = 2. Asymptotic PEP was obtained from (3.44) and (3.45a)–(3.45c). 56  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels 0  10  −1  10  −2  BEP  10  −3  10  −4  10  Scenario 1: NT = 1, NR = 1 (Simulation) Scenario 2: NT = 2, NR = 1 (Simulation) Scenario 3: N = 1, N = 2 (Simulation) T  R  Asymptotic  −5  10  5  10  15  20  25  30  SNR per symbol (dB)  Figure 3.3: BEP of a single Tx antenna system with NR = 1, 2 and the BBH STTC (NT = 2) with NR = 1 for Weibull–Rayleigh keyhole fading channel. βT = c/2 = 2 and βR = 1. was considered. As expected from (3.46), we observe that the diversity gain in Scenarios 1, 2, and 3 is Gd,1 = 1, Gd,2 = 1, and Gd,3 = 2, respectively. A comparison of Scenarios 1 and 2 reveals that in this channel, increasing the number of transmit antennas does not improve the diversity gain and uncoded 4–PSK transmission with NT = 1 antenna outperforms the considered STTC using NT = 2 antennas. However, a similar comparison between Scenarios 1 and 3 indicates that increasing the number of receive antennas is beneficial. In general, (3.46) shows that antennas should be added to the side (transmitter or receiver) which limits the diversity gain. In this example, for Scenario 1 it is the receiver side as NR βR = 1 < NT βT = 2. It is worth noting that this phenomenon cannot be observed in Rayleigh keyhole fading channels [2, 71, 1] for which increasing NT or NR  57  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  0  10  −1  10  −2  10  SEP  NR = 1 −3  10  −4  10  NR = 2  −5  10  −6  10  NR = 3  Asymptotic Exact [6] Simulation 10  15  20  25  30  35  40  45  50  SNR per symbol (dB)  Figure 3.4: SEP of Alamouti STBC [5] with 4–PSK modulation over keyhole Nakagami-m fading with mT = mR = 0.7, NT = 2, and NR = 1, 2, 3. alone does not improve the diversity gain of Scenario 1. This underpins the importance of a careful modeling and analysis of keyhole fading channels and illustrates the relevance of the presented asymptotic results for system design. Fig. 3.4 depicts the SEP of Alamouti’s STBC [5] with 4–PSK modulation over a Nakagami-m fading channel with mT = mR = 0.7 and different number of receive antennas. For comparison, we have also plotted the PEP expression derived in [69]. Note that this expression involves a numerical integration of the hypergeometric function which has to be evaluated for each SNR separately. This figure illustrates that our asymptotic analysis can also be used to predict the performance of OSTBCs with low complexity and high accuracy. Note that increasing NR from two to three does not improve the diversity gain which is limited to Gd = mT NT = 1.4 by the transmitter side in both cases.  58  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  0  10  −1  10  FEP  NR = 1 −2  10  NR = 2  −3  10  TSC, keyhole fading BBH, keyhole fading TSC, non−keyhole fading BBH, non−keyhole fading  −4  10  0  5  10  NR = 3 15  20  25  30  SNR per symbol (dB)  Figure 3.5: The simulated FEP of the TSC [3] and BBH [4] STTCs with 4–PSK modulation over keyhole and non–keyhole Rayeligh fading with NR = 1, 2, 3 receive antennas. In Fig. 3.5, we compare the simulation results for FEP of the TSC and BBH STTCs over keyhole and non–keyhole Rayleigh fading channels with different numbers of receive antennas. Note that the asymptotic results are not plotted in this figure since otherwise the figure would become over–crowded. It can be observed that in the keyhole fading channel, regardless of the number of receive antennas, both codes have almost the same performance. In non–keyhole fading, however, while the two codes have almost the same FEP for NR = 1, there is an increasing performance gap between them for higher values of NR . The reason for this phenomenon, as explained in Section 3.5.3, is the presence of the exponent NR in the code design criterion in (3.48) for non–keyhole fading. This indicates that codes that perform better in non–keyhole fading do not necessarily perform better in keyhole fading channels and a new optimization should be performed to find the best codes in keyhole fading. 59  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  0  10  −1  10  −2  10  BEP  NR = 1 −3  10  −4  10  NR = 2  −5  10  TSC (Simulation) SHN (Simulation) TSC (Asymptotic bound) SHN (Asymptotic bound)  −6  10  6  8  10  12  14  16  18  20  22  24  SNR per symbol (dB)  Figure 3.6: Simulation results and union bound for the BEP of the TSC [3] and SHN [1] STTCs with 4–PSK modulation over keyhole Nakagami-m fading with mT = mR = 2.0 and NR = 1, 2. Fig. 3.6 compares the BEP of the TSC and SHN STTCs in Nakagami–m fading with mT = mR = 2.0 and NR = 1, 2. The asymptotic union bounds for both codes are obtained by considering all error events of length 2 and 3. We note that NR = 1 and NR = 2 correspond to Cases 1 and 3, respectively. The performance improvement of the SHN code compared to the TSC code in Rayleigh fading, which was reported in [1], is preserved in Nakagami–m fading with mT = mR = 2.0. We note that although the asymptotic upper bounds are not very tight for either code, they accurately predict the performance difference between both codes. As expected from Section 3.5.3, the performance gap between both codes does not increase with increasing number of receiver antennas.  60  Chapter 3. Performance Analysis of STCs in Generalized Keyhole Fading Channels  3.7  Conclusions  In this chapter, we presented asymptotic expressions for the PEP of single–antenna transmission and space–time coded transmission with two transmit antennas in generalized keyhole fading channels. Interestingly, the applicable asymptotic analysis techniques were shown to depend on the difference D between the receiver side diversity gain, NR βR , and the transmitter side diversity gain, NT βT . For D = 0, general PEP expressions valid for a wide range of types of fading could be obtained. In contrast, for D = 0, the effects of the receiver side fading had to be studied on a case by case basis and we obtained results for the practically important cases of Ricean and Nakagami–m fading. Nevertheless, for all values of D, the derived analytical results give valuable insight into the impact of the fading parameters, the numbers of transmit and receive antennas, and the properties of the STC on system performance. Thus, these results may be utilized to guide system design and STC optimization for keyhole fading channels. For example, our results revealed that the performance gap between different STCs is less sensitive to the number of receiver antennas in keyhole fading than it is in non–keyhole fading.  61  Chapter 4 Performance Analysis of Diversity Combining Receivers in Non–Gaussian Noise and Interference23 4.1  Introduction  We mentioned in Chapter 1 that practical wireless communication systems are often not only impaired by additive white Gaussian noise (AWGN) but also by non–Gaussian noise and interference. In this chapter, we present a novel powerful framework for analyzing quadratic diversity combining techniques in the high SNR regime when the received signal is impaired by correlated Ricean fading and general non–Gaussian noise. Since the only assumption that we make on the noise is that all of its moments exist, our results are applicable to a large number of practical scenarios. The resulting asymptotic BEP and SEP expressions are simple and easy to evaluate and only require the calculation of certain noise moments. We show how these noise moments can be efficiently obtained for several practically relevant types 23  Parts of this chapter were published in the IEEE Transactions on Communications, Vol. 57, pp. 1039-1049, Apr. 2009, cf. Appendix F.  62  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . of noise including spatially dependent and spatially independent gaussian–mixture noise (GMN), synchronous and asynchronous co–channel interference (CCI), spatially correlated Gaussian interference, and ultra-wide band (UWB) interference. The remainder of this chapter is organized as follows. In Section 4.2, some definitions and the considered signal model are introduced. In Section 4.3, general expressions for the asymptotic BEPs and SEPs of maximum–ratio combining (MRC), differential equal gain combining (EGC), and noncoherent combining (NC) are derived. Techniques for calculation of the moments of relevant types of noise are provided in Section 4.4. In Section 4.5, some examples are given to illustrate the application of the obtained analytical results, and conclusions are drawn in Section 4.6.  4.2  Signal Model  Assuming for the moment a linear modulation format24 , the signal rl [k] received in the lth diversity branch in the kth symbol interval can be modeled in equivalent complex baseband representation as rl [k] =  √  γ hl b[k] + nl [k],  1 ≤ l ≤ L,  (4.1)  where L, hl , b[k], and nl [k] denote the number of diversity branches25 , the fading gain of the lth branch, the transmitted symbol, and the noise in the lth diversity branch, respectively. Using vector notation, (4.1) can be rewritten as  r[k] =  √  γ h b[k] + n[k],  (4.2)  24  We will extend our signal model in Section 4.3.2 to binary frequency–shift keying (BFSK) modulation. Note that in systems with spatial diversity, L refers to the number of receiver antennas which is denoted by NR in Chapters 2, 3, and 6. Since a diversity branch can also refer to a frequency band or a time slot in frequency hopping (FH) and time hopping (TH) systems, respectively, we prefer the notation L rather than NR for the number of diversity branches in Chapters 4 and 5. 25  63  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . [r1 [k] r2 [k] . . . rL [k]]T , h  where r[k]  [h1 h2 . . . hL ]T , and n[k]  [n1 [k] n2 [k] . . . nL [k]]T .  To simplify our notation, in the following, we will drop the argument k whenever this is possible without loss of generality. The transmitted symbols b ∈ A are normalized to E{|b|2} = 1 and are taken from signal constellation A, cf. Section 4.3. ¯ The channel vector h is Gaussian distributed with mean h matrix Rhh  E{h} and covariance  ¯ ¯ H }. We assume that Rhh has full rank L and define the E{(h − h)(h − h)  Ricean factor of the lth branch as Kl  |µl |2 /σl2 , where µl and σl2 denote the lth element  ¯ and the lth main diagonal element of Rhh , respectively. For convenience we apply the of h ¯ = 0L and normalization Mh (1) = L, and we note that for Rayleigh and Ricean fading h ¯ = 0L , respectively. h The noise vector n is independent of h and normalized to Mn (1) = L. We note that the elements of n may be statistically dependent, non–circularly symmetric, and non–Gaussian. The only condition that we impose on n is that the pdfs of its elements are exponentially– tailed and therefore all of their joint moments exist, i.e., E{nκ1 1 (n∗1 )ν1 nκ2 2 (n∗2 )ν2 · · · nκLL (n∗L )νL } < ∞ for all κl ≥ 0, νl ≥ 0, 1 ≤ l ≤ L. Definition: We define the Nth moment of the real random variable (RV) |x|2 as Mx (N)  E{|x|2N }, where x is a complex RV. Similarly, for a complex random vector  variable (RVV) x we define the Nth moment of ||x||2 as Mx (N)  E{||x||2N }. We note  that Mx (0) = 1 and Mx (1) is the sum of the powers of the elements of x.  4.3  Asymptotic Performance Analysis  In this section, we develop asymptotic expressions for the BEPs and SEPs of coherent MRC, differential EGC, and NC for various modulation schemes. However, first we derive a general asymptotic result for the PEP of quadratic diversity combining receivers.  64  Chapter 4. Performance Analysis of Diversity Combining Receivers . . .  4.3.1  Asymptotic Pairwise Error Probability (PEP)  For quadratic diversity combining schemes the PEP can be generally expressed as √ Pe (d) = Pr{|| γ h e + n1 ||2 < ||n2 ||2 }, where n1 and n2 denote two noise vectors and e is a complex scalar with d2  (4.3)  |e|2 . n1 ,  n2 , and e will be specified for different combining schemes in Section 4.3.2. Based on (4.3) we can express the conditional PEP as ||n2 ||2  p∆ (x) dx,  Pe (d|n1 , n2 ) =  (4.4)  0  where p∆ (x) denotes the pdf of ∆  ||u||2 with u  ¯ a Gaussian random vector with mean u Ruu  √  γ h e + n1 . Conditioned on n1 , u is √ ¯ + n1 and covariance matrix E{u|n1 } = γ e h  ¯ )(u − u) ¯ H |n1 } = γ|e|2 Rhh . Therefore, the Laplace transform Φ∆ (s) of E{(u − u  p∆ (x) can be expressed as [96] √ ¯ √ ¯ exp −s[ γeh + n1 ]H (I L + sγ|e|2 Rhh )−1 [ γeh + n1 ] Φ∆ (s) = . 2 det(I L + sγ|e| Rhh )  (4.5)  Eq. (4.5) reveals that for full rank fading correlation matrices Rhh and γ → ∞ the Laplace transform Φ∆ (s) can be simplified to ¯ + n1 /(e√γ)] ¯ + n1 /(e√γ)]H R−1 [h . exp −[h hh Φ∆ (s) = . det(Rhh ) d2L γ L sL  (4.6)  An asymptotic expression for p∆ (x) can now be easily obtained by applying the inverse Laplace transform to (4.6). This result can then be used in (4.4) to obtain the asymptotic  65  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . conditional PEP −1 ¯ n n ¯ √1 H √1 . exp −[h + e γ ] Rhh [h + e γ ] Pe (d|n1 , n2 ) = ||n2 ||2L . L! det(Rhh ) d2L γ L  Using the expansion exp(x) =  ∞ k k=0 x /k!,  (4.7)  we can rewrite the exponential function in (4.7)  as n1 1 ¯+ n ¯ exp −[h √ ]H R−1 hh [h + √ ] e γ e γ  H  ¯ R−1 h ¯ (1 + f (n1 )/γ) , = exp −h hh  (4.8)  where f (n1 ) is implicitly defined in (4.8). Furthermore, f (n1 ) can be written as a sum of L products of the form Cκ1 ,ν1 ,··· ,κL ,νL nκ1,11 (n∗1,1 )ν1 nκ1,22 (n∗1,2 )ν2 · · · nκ1,L (n∗1,L )νL , where Cκ1 ,ν1 ,··· ,κL ,νL  are coefficients that are non–increasing in γ, n1,l , 1 ≤ l ≤ L, denote the elements of n1 , and κl ≥ 0 and νl ≥ 0 are integers. Assuming now that all individual and joint moments κ  κ  κ  κ  1,L 2,L of n1 and n2 exist (i.e., E{n1,11,1 (n∗1,1 )ν1,1 n2,12,1 (n∗2,1 )ν2,1 · · · n1,L (n∗1,L )ν1,L n2,L (n∗2,L )ν2,L } < ∞,  where n2,l are the elements of n2 , and ν1,l ≥ 0, ν2,l ≥ 0, κ1,l ≥ 0, and κ2,l ≥ 0, 1 ≤ l ≤ L), we obtain for γ → ∞ from Eqs. (4.7) and (4.8) for the asymptotic (unconditional) PEP the simple expression . ph Mn2 (L) −L γ Pe (d) = E{Pe (d|n1 , n2 )} = L! d2L with  (4.9)  H  ph  ¯ R−1 h ¯ exp −h hh det(Rhh )  .  (4.10)  From (4.9) we observe that n1 has no influence on the asymptotic PEP. We note, however, that although formally (4.9) can be obtained by simply neglecting n1 in (4.3), the provided rigorous derivation is necessary to prove this result and to understand its limitations. For example, since the above derivation shows that (4.9) is only valid if all individual and joint moments of n1 and n2 exist, the results of this chapter are not applicable if the elements 66  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . of n1 and/or n2 are e.g. samples of an α–stable process or any other noise with algebraic– tail pdf, cf. Chapter 1. However, for most types of noise and interference encountered in communication systems all individual and joint moments of n1 and n2 exist and (4.9) holds, cf. Section 4.4. We also observe from (4.9) that in this case n2 affects the asymptotic PEP via Mn2 (L), i.e., only the number of diversity branches L determines which moment ¯ and the correlation matrix Rhh of h have of ||n2 ||2 is relevant for the PEP but the mean h no influence in this regard. We note that the type of noise, the type of fading, and the number of diversity branches affect at what (finite) SNR value γ the asymptotic PEP in (4.9) becomes tight, cf. Figs. 4.14.6. We also emphasize that in general the asymptotic PEP expression in (4.9) is neither an upper nor a lower bound for the true PEP. Moreover, the above asymptotic analysis could be further refined by developing Φ∆ (s) into a power series with respect to γ −l , l ≥ L, and retaining not only the term with l = L [as was done in (4.6)] but also the terms with l > L. However, such a refined asymptotic analysis would result in significantly more complicated error rate expressions and is a topic for future research.  4.3.2  Quadratic Diversity Combining Schemes  In this section, we apply the general asymptotic PEP result from (4.9) to calculate the asymptotic SEP and BEP of coherent MRC, differential EGC, and NC. We note that diversity combining rules optimized for AWGN are in general suboptimum for non–Gaussian noise and optimum maximum–likelihood based combining rules may lead to large performance improvements [31]. However, in practice, it may be unrealistic to assume that the receiver can accurately estimate the noise and interference statistics which also may change with time. Therefore, as a pragmatic and popular choice [26, 30, 31], we will adopt here quadratic combining schemes optimized for AWGN also for non–Gaussian noise.  67  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Coherent Maximum Ratio Combining (MRC) √ The MRC decision rule can be expressed as ˆb = argmin {||r − γh˜b||2 }, where ˆb and ˜b ˜ b∈A  denote the estimated symbol and a hypothetical symbol, respectively. As a consequence, (4.3) is valid for MRC if we let n1 = n2 = n and e  b − ¯b, where b, ¯b ∈ A and b = ¯b. For  high SNR the SEP will be dominated by the PEP of the nearest–neighbor signal points of A. Therefore, exploiting (4.9) we obtain for the asymptotic SEP . . βM ph Mn (L) −L SEP = βM Pe (dM ) = γ , L! d2L M  (4.11)  where dM denotes the minimum Euclidean distance of A and βM is the average number of minimum–distance neighbors. For convenience, the values of βM and dM are listed in Table 4.1 for commonly used constellations A such as M–ary pulse amplitude modulation (M–PAM), M–ary quadrature amplitude modulation (M–QAM), and M–ary phase–shift keying (M–PSK). Differential Equal Gain Combining (EGC) Differential EGC26 is applicable to differential M–PSK transmission. In differential M– PSK the transmitted M–PSK symbols are obtained as b[k] = a[k]b[k − 1], where the differential symbols a[k] also belong to an M–PSK constellation A. The differential EGC decision rule can be expressed as a ˆ[k] = argmin {||r[k] − a ˜[k]r[k − 1]||2 }, where a ˆ[k] and a ˜[k]∈A  a ˜[k] are the estimated symbol and a hypothetical symbol, respectively. Therefore, (4.3) is applicable to differential EGC if we let e  (a[k] − a ¯[k])b[k − 1], where a[k], a ¯[k] ∈ A and  26  Differential EGC is also referred to as “differentially coherent” EGC and “post–detection” EGC in the literature, cf. e.g. [15].  68  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . a[k] = a ¯[k], n1  n[k] − a ¯[k]n[k − 1], and ˜ n2 = n  n[k] − a[k]n[k − 1].  (4.12)  If the marginal pdfs pnl (nl ), 1 ≤ l ≤ L, of all components of n[k − 1] are circular, i.e., pnl (nl ) = pnl (ejϕ nl ) for all ϕ ∈ [−π, π) [97], and n[k] and n[k − 1] are statistically in˜ and n ˜ dependent, a[k] has no influence on the distribution of n  n[k] − n[k − 1] may  be used instead of the definition in (4.12). Recalling again that the asymptotic SEP is dominated by the PEPs of the nearest–neighbor signal points of A, we obtain with (4.9) for differential EGC the expression . βM ph Mn˜ (L) −L SEP = γ , L! d2L M  (4.13)  with βM and dM as specified in Table 4.1 for M–PSK. Noncoherent Combining (NC) In this subsection, we consider NC (also referred to as “square law combining”) of BFSK. For BFSK the channel model in Section 4.2 has to be slightly modified since there are now two matched filters per receive antenna. We collect the outputs of these two matched √ ¯ √ ¯ γhb + n and r¯ γhb + n, filters in vectors r and r¯ which can be modeled as r ¯ are identically distributed where b, ¯b ∈ {0, 1} and b = ¯b [14]. We note that n and n and statistically independent if the channel noise is Gaussian [14]. For non–Gaussian ¯ are still identically distributed but not necessarily statistically independent. noise n and n ¯ does not affect the applicability of the Fortunately, the statistical dependence of n and n results of Section 4.3.1. For NC the magnitudes of ||r||2 and ||¯ r||2 are compared to arrive at a decision on the transmitted symbol and (4.3) is obviously applicable with e = 1.  69  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Table 4.1: Parameters βM and dM for various signal constellations A. Modulation Scheme M–PAM  βM 2 1−  BPSK (M = 2)  1  M–PSK (M ≥ 4)  2  M–QAM  4 1−  dM 1 M  2  3 M 2 −1  2  2 sin √1 M  π M  6 M −1  Therefore, the asymptotic SEP of BFSK can be expressed as . β2 ph Mn (L) −L SEP = γ , L! d2L 2  (4.14)  where β2 = 1 and d2 = |e| = 1. We note that unlike for linear modulations, d2 is not the Euclidean distance between the signal points of the BFSK constellation in the signal space. Bit Error Probability (BEP) For non–binary modulation with Gray mapping, the asymptotic BEPs can be obtained from the corresponding SEPs as [14] . BEP =  SEP . log2 (M)  (4.15)  Using Eqs. (4.11), (4.13)–(4.15) the SEPs and BEPs of MRC, differential EGC, and NC can be easily calculated as long as closed–form expressions for the moments Mn (L) and Mn˜ (L) are available. The calculation of these moments for practically relevant types of noise and interference is addressed in Section 4.4.  70  Chapter 4. Performance Analysis of Diversity Combining Receivers . . .  4.3.3  Combining Gain and Comparison  A comparison of Eqs. (4.11), (4.13), and (4.14) with (1.16) shows immediately that the diversity gain of all considered quadratic combining schemes is Gd = L independent of the type of noise. Furthermore, on a logarithmic scale, the combining gain27 and the diversity gain can be expressed in a unified manner as  Gc =  10 10 log10 (L!) + log10 L L  d2L M βM  −  10 10 log10 (ph ) − log10 (Mn2 (L)), L L  (4.16)  ˜ for MRC, NC and differential EGC, respectively. The second, where n2 = n and n2 = n the third, and the fourth term of (4.16) show the dependence of Gc on the modulation scheme, the channel statistics, and the noise statistics, respectively. Eq. (4.16) reveals ¯ and Rhh ), and that for a given L the modulation scheme, the channel statistics (i.e., h the noise statistics independently contribute to the combining gain. This new result is somewhat unexpected and means that increasing the correlation of the fading or changing the modulation scheme will horizontally shift the asymptotic SEP curves (on a log–log scale) for different types of noise by the same amount. On the other hand, the asymptotic performance difference of different types of noise is independent of the modulation scheme and the channel statistics. In the following, we will consider the combining gain for three special cases more in detail. 1) L = 1: Since Mn (1) = L = 1 is valid for all types of noise, (4.16) shows that for L = 1 the asymptotic error rate performance of MRC and NC is independent of the type of noise as long as all joint moments of n are finite. The same is true for EGC since in this case Mn˜ (1) = E{|n[k]−a[k]n[k −1]|2 } = 2Mn (1)−2 {E{a∗[k]}E{n[k]n∗ [k −1]}} = 2L = 2 27  In this chapter, combining gain is synonymous with “coding gain” in the literature, e.g. [16]. We prefer the term “combining gain” as channel coding is not applied here.  71  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . holds for all types of noise because E{a∗ [k]} = 0 for M–PSK. For L > 1 both Mn (L) and Mn˜ (L) depend on the type of noise and the same is true for the asymptotic error rate performance, of course. 2) MRC vs. NC: It is interesting to compare MRC and NC for binary modulation. Using β2 = 1 and the appropriate values for d2 in (4.16) shows that for any given number of diversity branches L the asymptotic SNR loss of BFSK with NC compared to BPSK with coherent MRC is 6 dB. While this is a well known result for channels impaired by i.i.d. AWGN [14], our derivation shows that the same loss results for any type of noise and interference as long as all of its moments exist. 3) MRC vs. differential EGC: Eq. (4.16) also reveals that for M–PSK the relative asymptotic performance loss of differential EGC compared to coherent MRC is  ∆GEM =  10 log10 L  Mn˜ (L) Mn (L)  .  (4.17)  Eq. (4.17) shows that ∆GEM only depends on the number of diversity branches L and the noise statistics. For L = 1 we obtain ∆GEM = 3 dB for all types of noise n with finite joint moments. Similarly, it will be shown in Section 4.4.1, that for i.i.d. AWGN ∆GEM = 3 dB holds for all L. However, for non–Gaussian noise and L > 1 the performance loss ∆GEM depends on L and may be smaller or larger than 3 dB, cf. Section 4.4.  4.4  Calculation of Noise Moments  The main difficulty in evaluating the asymptotic SEP expressions in Eqs. (4.11), (4.13), and (4.14) is the calculation of the moments Mn (L) and Mn˜ (L). In this section, we derive the moments of some basic types of noise and we introduce general rules that facilitate the calculation of the moments of more complicated, composite noises. For convenience we  72  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . discuss spatially independent and spatially dependent noise separately. We also present a Monte–Carlo based approach to moment calculation for cases where closed–form results cannot be obtained.  4.4.1  Spatially Independent Noise  For many practically relevant scenarios the noise components in different diversity branches are mutually independent. In this case, the multinomial expansion [98] can be used to simplify Mn (L) to  Mn (L) = k1 +...+kL =L  L Mn1 (k1 ) · . . . · MnL (kL ). k1 , . . . , kL  (4.18)  Therefore, for independent noise the calculation of Mn (L) reduces to finding the L(L + 1) scalar moments Mnl (kl ), 0 ≤ kl ≤ L, 1 ≤ l ≤ L. This motivates us to consider the moments of scalar RVs more in detail in the following subsections. Moments of Elementary Scalar RVs In Table 4.2, we provide the moments Mn (N) of elementary scalar RVs which are frequently encountered in the context of noise and interference. In particular, we consider Gaussian RVs with mean µn and variance σn2 , Gaussian mixture RVs, interference with a fixed channel, and Mi –PSK interference with a random channel phase. The non–Gaussian RVs are briefly discussed in the following. E1) Gaussian mixture RVs: Gaussian mixture RVs were introduced in Chapter 1, cf. (1.14). The moment for this type of noise is given in Table 4.2. E2) Interference with fixed channel: Interference with a fixed channel can be modeled as in (1.15) and its generalized moments are given in Table 4.2, where the set S includes  73  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Table 4.2: Moments Mn (N) = E{|n|2N } of scalar Gaussian RVs and the RVs discussed in Examples E1)-E3). Gaussian RV: Mean µn and variance σn2 . The parameters for the other RVs are defined in Section 4.4.1. Scalar Noise Model  Moments Mn (N)  Gaussian RV (µn = 0) Mn (N) = N! σn2N Gaussian RV (µn = 0) Mn (N) = N! 1 F1 (−N, 1; −|µn |2 /σn2 )σn2N Gaussian Mixture Mi –ary Interference with Fixed Channel Mi –PSK Interference with Random CP  I k=1 ck  Mn (N) = N! Mn (N) =  σk2N  1 k −kl +1  Mi u  n0 ∈S  |n0 |2N  Mn (N) = |g|2N  the Miku −kl +1 possible values of n corresponding to all Miku −kl +1 possible combinations of i[k], kl ≤ k ≤ ku . E3) Synchronous Mi –PSK interference with random channel phase (CP): In this case, the noise is given by n = g ejϕ i[k], where g, ϕ, and i[k] denote the channel gain, the random CP uniformly distributed in [−π, π), and the interfering Mi –PSK symbol, respectively. Note that i[k] has no influence on Mn (N) given in Table 4.2. Calculus for Moments of Scalar RVs In practice, the noise may consist of sums or/and products of different RVs. For example, CCI and Gaussian background noise may impair the received signal at the same time. For the case where the involved RVs are statistically independent, we establish some general combining rules for their scalar moments. The derivation of these combining rules (CRs) is relatively straightforward and, due to space limitation, we do not provide any proofs here. In the following, we assume that n1 and n2 are statistically independent RVs.  74  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . CR1) Product n = n1 n2 : It is easy to show that the Nth moment of |n|2 is given by Mn (N) = E{|n1 n2 |2N } = E{|n1|2N } E{|n2 |2N } = Mn1 (N) Mn2 (N).  (4.19)  CR2) Sum n = n1 + n2 : To arrive at a simple result, we have to assume that the pdfs of n1 and n2 are both circular [97]. For example, zero–mean Gaussian RVs and the RVs in E1) and E3) are circular, whereas non–zero mean Gaussian RVs and the RV in E2) are not circular. Assuming circularity, we obtain after some straightforward manipulations  Mn (N) = k1 +2k2 +k3 =N  N! Mn1 (k1 + k2 ) Mn2 (k2 + k3 ). k1 !(k2 !)2 k3 !  (4.20)  For example, for L = 1, 2, and 3 (4.20) yields Mn (1) = Mn1 (1)+Mn2 (1), Mn (2) = Mn1 (2)+ 4Mn1 (1)Mn2 (1)+Mn2 (2), and Mn (3) = Mn1 (3)+9Mn1 (1)Mn2 (2)+9Mn1 (2)Mn2 (1)+Mn2 (3). CR3) Scaling n = ξn1 , ξ constant: The Nth moment of |n|2 can be obtained as Mn (N) = |ξ|2N Mn1 (N).  (4.21)  Based on the scalar moments of the elementary RVs given in Table 4.2 and the combining rules established in this section, the moments of a large class of composite noises can be calculated. Moments of Composite Scalar RVs In order to illustrate the application of the moment combining rules CR1)–CR3) established in the previous subsection, we briefly discuss two relevant examples. E4) Differential EGC with i.i.d. noise: Assuming circular noise, the SEP of differential ˜ = n[k]−n[k −1], cf. Eqs. (4.12), (4.13). ThereEGC depends on the moments Mn˜ (L) of n  75  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . fore, if the noise nl [k], 1 ≤ l ≤ L, is spatially and temporally statistically independent, Mn˜ (L) can be calculated from the scalar moments by applying Eqs. (4.18) and (4.20). Examples where the noise is spatially and temporally statistically independent include the conventional AWGN model and GMN following “Model II” in [31]. SIGMN is an appropriate model for impulsive noise if the phenomenon causing the impulsive behavior affects the antennas independently, see [31] for a detailed discussion. We are now also in a position to shed some more light onto the performance loss of differential EGC compared to coherent MRC for L > 1. For example, assuming spatially and temporally i.i.d. circular noise (i.e., Mnl [k] (N) = Mn (N), 1 ≤ l ≤ L, ∀k) and L = 2 we obtain with Eqs. (4.17), (4.18), and (4.20)  ∆GEM = 5 log10 2  Mn (2) + 4Mn2 (1) Mn (2) + Mn2 (1)  ,  (4.22)  which can be simplified to  ∆GEM = 3 dB + 5 log10  (1 − + κ2 ) + 2(1 − + κ)2 2(1 − + κ2 ) + (1 − + κ)2  (4.23)  for i.i.d. –mixture noise. For = 0 the –mixture noise degenerates to Gaussian noise and ∆GEM = 3 dB follows from (4.23). On the other hand, for impulsive noise (i.e., > 0 and κ > 1) it can be shown that ∆GEM < 3 dB holds in general. E5) Ricean faded Mi –PSK interferer: The interference caused by a synchronous Ricean faded Mi –PSK co–channel interferer i[k] can be modeled as nl = (ejϕl gl +nl ) i[k], 1 ≤ l ≤ L, where ejϕl gl and nl denote the direct and the Rayleigh component, respectively. Assuming that ϕl is uniformly distributed in [−π, π), ejϕl gl i[k] can be modeled by Example E3), and nl i[k] is a zero mean Gaussian RV. Since the Mi –PSK symbol i[k] does not affect the distributions of ejϕl gl i[k] and nl i[k], we can drop it in the following. Furthermore,  76  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . since both ejϕl gl and nl are circular we can calculate the moments Mnl (N) using (4.20) and Table 4.2. Once these scalar moments are calculated, the moments of Mn (L) can be obtained from (4.18). I.I.D. Additive White Gaussian Noise (AWGN) Since the case of (i.i.d.) AWGN has been extensively studied in the literature, it is instruc˜ 2 tive to verify our novel asymptotic SEP results for this special case. Here, ||n||2 and ||n|| are both chi–square distributed RVs with 2L degrees of freedom and it is straightforward to derive Mn (L) as given in Table 4.3 and Mn˜ (L) = 2L Mn (L).  (4.24)  Therefore, according to (4.17), for AWGN the asymptotic performance loss ∆GEM of differential EGC compared to coherent MRC is 3 dB independent of the number of diversity branches and independent of the mean and the covariance matrix of h. For the special case of BPSK transmission over i.i.d. Rayleigh fading channels this 3 dB loss is a well–known result, cf. [14, Section 14.4.1]. For coherent MRC our results for the special case of AWGN can be compared with those given in [16]. On the other hand, an asymptotic analysis of differential EGC for general Ricean fading does not seem to be available in the literature even for AWGN. For NC the case of independent Ricean fading and AWGN was considered in [99], but our asymptotic results for correlated Ricean fading seem to be new even for AWGN. For i.i.d. Rayleigh fading and AWGN it can be verified that (4.11) for BPSK with MRC, (4.13) for BPSK with differential EGC, and (4.14) for BFSK with NC are equivalent to [14, Eq. (14.4-18)], [14, Eq. (14.4-28)], and [14, Eq. (14.4-33)], respectively.  77  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Table 4.3: Moments Mn (L) = E{|n|2L } of RVVs. Zero–mean i.i.d. Gaussian RVV: Variance of each element σn2 . Correlated Gaussian RVV: λl , 1 ≤ l ≤ L, are the eigenvalues of covariance matrix Rnn E{nnH }. The parameters of the Gaussian mixture RVV are defined in Section 4.4.2, Example E8). Vector Noise Model  Moments Mn (L)  I.I.D. Gaussian RVV  Mn (L) =  (2L−1)! (L−1)!  σn2L  Correlated Gaussian RVV  Mn (L) = L! k1 +...+kL =L  λk11 · . . . · λkLL  Spatially Dependent Gaussian Mixture  4.4.2  Mn (L) =  (2L−1)! (L−1)!  I k=1 ck  σk2L  Spatially Dependent Noise  In this subsection, we illustrate the calculation of the moments Mn (L) and Mn˜ (L) for spatially dependent noise for three different practically important types of noise and interference. E6) Correlated Gaussian noise/interference: We consider the case where n can be modeled as a zero–mean correlated Gaussian RVV with covariance matrix Rnn  E{nnH }.  Such a model applies for example if the received vector is impaired by I Rayleigh faded synchronous Mi –PSK co–channel interferers i[k] [23, 30] and AWGN I  g[k] i[k] + n0 ,  n=  (4.25)  k=1  where g[k] are mutually independent complex zero–mean Gaussian RVVs with covariance matrices C g[k]g[k]  E{g[k]g H [k]}, 1 ≤ k ≤ I, and n0 is the AWGN vector. Because of  the mutual independence of the fading vectors and the constant envelope of the Mi –PSK interference signals (|i[k]| = 1, 1 ≤ k ≤ I), n is a Gaussian vector with covariance matrix  78  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Rnn =  I k=1  C g[k]g[k] + σn2 0 I L , where σn2 0 denotes the variance of the elements of n0 . As  shown in Table 4.3, Mn (L) can be efficiently expressed in terms of the L eigenvalues λk , 1 ≤ k ≤ L, of Rnn . For the special case L = 2, Mn (L) can be simplified to Mn (2) = 2(4 − det(Rnn )),  (4.26)  where we have used the normalization Mn (1) = L. Interestingly, the moment assumes its minimum value Mn (L) = (2L − 1)!/(L − 1)! for the special case of uncorrelated Gaussian noise (λk = 1, 1 ≤ k ≤ L) and its maximum value Mn (L) = LL L! for fully correlated noise (λ1 = L and λk = 0, 2 ≤ k ≤ L). Therefore, using (4.11) we can bound the asymptotic SEP of MRC in correlated Gaussian noise as · βM ph 2L − 1 βM ph −L · γ ≤ SEP ≤ LL 2L γ −L . 2L dM dM L  (4.27)  A similar result can be derived for NC, cf. (4.14). The combining gain loss ∆Gc caused in MRC and NC by fully correlated Gaussian noise compared to i.i.d. AWGN is ∆Gc (L) = 10 log10 (L[L!(L−1)!/(2L−1)!]1/L ). For example, for L = 1, 2, and 3 we obtain ∆Gc (1) = 0 dB, ∆Gc (2) = 0.6 dB, and ∆Gc (3) = 1.4 dB, respectively. For large L, we can use Stir√ √ L 1) to obtain ∆Gc (L) ≈ 10 log10 ( 2 Lπ L/4) ≈ ling’s formula (x! ≈ 2πe−x xx+1/2 , x 10 log10 (L/4), L  1, which shows that the performance loss due to noise correlation  increases with L. It is interesting to note that not only correlation of the fading gains (elements of h) has an adverse effect on the performance of MRC and NC but also correlation of the noise in different diversity branches.  79  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . E7) Asynchronous CCI in correlated Rayleigh fading: A single Rayleigh faded asynchronous co–channel interferer can be modeled as  n = g · z,  (4.28)  where g is a correlated zero–mean Gaussian RVV and z can be modeled as in Example E2), i.e., z =  ku k=kl  g[k]i[k], cf. (1.15). Since g and z are statistically independent, the  moments of n can be calculated as  Mn (L)  E{||g z||2L } = E{||g||2L } E{|z|2L } = Mg (L)Mz (L),  (4.29)  where Mz (L) and Mg (L) can be obtained from Table 4.2 (interference with fixed channel) and Table 4.3 (correlated Gaussian RVV), respectively. Assuming time–invariant fading the relevant noise term for differential EGC is given ˜ = g˜ by n z , where z˜  z[k] − a[k]z[k − 1] and z[k] and z[k − 1] are the interference  contributions in two successive symbol intervals. z˜ can also be modeled as in (1.15) if g[k] and i[k] are replaced with appropriately defined effective coefficients g˜[k] and effective interference symbol ˜i[k], respectively. In order to illustrate this we consider the special case of a single synchronous co–channel interferer, i.e., z˜ = i[k] − a[k]i[k − 1]. Assuming that the desired user and the interferer use the same M–PSK constellation, i.e., Mi = M, we can simplify z˜ to z˜ = i[k] − i[k − 1], which is in the form of (1.15). The moment of z˜ can be calculated to Mz˜(L) =  1 L 2 M  M −1 L m=0 [1 − cos(2πm/M)] .  Therefore, since Mz (L) = 1  for M–PSK, the performance loss of differential EGC compared to MRC is given by  ∆GEM  10 = 3 dB + log10 L  1 M  M −1 m=0  2π m 1 − cos M  L  .  (4.30)  80  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Eq. (4.30) reveals that the performance loss suffered by differential EGC in correlated CCI depends only on L and the adopted M–PSK constellation. For example, for BPSK √ ∆GEM = 3(2−1/L) dB, whereas for 4–PSK ∆GEM = 3(1−1/L) dB+10 log10 ( L 2L−1 + 1). For both BPSK and 4–PSK we obtain ∆GEM = 3 dB for L = 1 and ∆GEM = 6 dB for L  1. For M  1 the sum in (4.30) can be approximated by an integral. This leads to  ∆GEM = 6 dB − 10 log10  L  (2L)!! (2L − 1)!!  ,  (4.31)  which yields ∆GEM = 3 dB, 3.9 dB, and 4.3 dB for L = 1, 2, and 3, respectively, and approaches 6 dB as L increases. From these considerations we conjecture that for a single synchronous M–PSK co–channel interferer and any constellation size M the asymptotic performance loss of differential EGC compared to MRC is between 3 dB and 6 dB, where the higher value is approached as L increases. Clearly, this is a very different behavior than that observed for AWGN, where we obtained ∆Gc = 3 dB for all L, cf. Section 4.4.1. We note that since correlation increases Mg (L), cf. E6, Eqs. (4.29), (4.11), (4.13), and (4.14) show that, similar to the Gaussian interference case, correlation in the interference channel has an adverse effect on the performance also for asynchronous CCI. E8) SDGMN: The pdf of spatially dependent GMN (“Model I” in [31]) is given by I  pn (n) = k=1  where ck > 0,  I k=1 ck  ck ||n||2 exp − 2 π L σk2L σk  ,  (4.32)  = 1, and σk2 , 1 ≤ k ≤ I, are constants. SDGMN may be spe-  cialized to a multi–dimensional version of Middelton’s Class–A noise (I → ∞) and multi– dimensional –mixture noise (I = 2). SDGMN is an appropriate model for impulsive noise if the physical process causing the impulsive behavior affects all antennas at the same time, see [31] for more discussion. The moments Mn (L) for spatially dependent GMN are given 81  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . in Table 4.3. ˜ Assuming temporally independent, spatially dependent GMN, it is easy to show that n is also a Gaussian mixture RVV whose pdf can be obtained from (4.32) by replacing I, ck , and σk2 by I˜  I(I + 1)/2, c˜k , and σ ˜k2 , respectively. The latter two parameters are defined  c2k and σ ˜k2  as c˜k  2σk2 for 1 ≤ k ≤ I, and c˜k  ˜ σi2 + σj2 for I + 1 ≤ k ≤ I,  2ci cj and σ ˜k2  1 ≤ i ≤ I, 1 ≤ j ≤ I, i = j. With these definitions, the combining gain loss of differential EGC compared to coherent MRC can be expressed as  ∆GEM  I˜ ˜k σ ˜k2L k=1 c I 2L k=1 ck σk  10 = log10 L  ,  (4.33)  which for –mixture noise can be simplified to  ∆GEM =  For  (1 − )2 2L + 2 (1 − )(κ + 1)L + 1 − + κL  10 log10 L  2  (2κ)L  .  (4.34)  = 0 (Gaussian case) (4.34) yields ∆GEM = 3 dB as expected, cf. Section 4.4.1. The  same is true for L = 1 and arbitrary L > 1,  > 0 and κ > 1, cf. Section 5.3.3. However, for  > 0, and κ > 1 differential EGC will cause a loss of less than 3 dB compared to  coherent MRC. For example, for  = 0.25 and κ = 10 we obtain from (4.34) ∆GEM = 3  dB, 2.25 dB, and 2.0 dB for L = 1, 2, and 3, respectively.  4.4.3  Calculation by Monte–Carlo Simulation  In some cases, the noise and interference statistics may be too complicated to obtain a closed–form expression for the moments Mn (L) or only measurements of n may be available. In those cases, a Monte–Carlo approach may be used to obtain moment estimates ˆ n (L) = 1 M Ne  Ne  k=1  ˆ k ||2L , ||n  (4.35)  82  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . ˆ k , 1 ≤ k ≤ Ne , are Ne realizations of n which may be obtained by simulation or where n ˆ n (L) and M ˆ n˜ (L) may then be used in Eqs. (4.11), (4.13), measurement. The estimates M and (4.14), respectively, instead of the true moments. This constitutes a semi–analytical approach to asymptotic SEP analysis. We note that this semi–analytical approach is ˆ n (L) and M ˆ n˜ (L) have to be much faster than Monte–Carlo simulation of the SEP since M calculated only once and are valid for all SNR values, whereas the SEP has to be simulated for each SNR value separately.  4.5  Numerical Results and Discussions  In this section, we verify the derived analytical expressions for the asymptotic BEP and SEP for several practically relevant cases with computer simulations. For calculation of the asymptotic BEP and SEP we used Eqs. (4.11), (4.13)–(4.15) and the required moments were obtained using the methods presented in Section 4.4. In all figures, we show the BEPs and SEPs as functions of the bit or symbol SNR per branch. BPSK vs. BFSK: Fig. 4.1 shows the BEPs of BPSK with MRC and BFSK with NC over an i.i.d. Rayleigh fading channel (L = 3) impaired by AWGN, i.i.d. –mixture noise ( = 0.25, κ = 10), and an i.i.d. Ricean faded Mi –PSK interferer with Ricean factor Ki = 6 dB (cf. E5), respectively. For all considered cases the simulation points quickly approach the asymptotic BEP curves as the SNR increases. Furthermore, Fig. 4.1 confirms that the 6 dB asymptotic performance loss of NC compared to MRC is independent of the type of noise, cf. Section 5.3.3. Spatially dependent –mixture noise (E8): In Fig. 4.2, we consider the BEP of 8–PSK with MRC and differential EGC for L = 1, 2, and 3 over an i.i.d. Ricean fading channel (K = 3 dB) impaired by spatially dependent –mixture noise ( = 0.25, κ = 10). The simulation results nicely confirm our asymptotic analysis also for spatially dependent noise. 83  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . 0  10  −1  10  −2  10  BFSK −3  10  −4  10  BEP  −5  10  BPSK  −6  10  −7  10  Ricean faded co−channel int. (simulation) AWGN (simulation) ε − mix. noise (simulation) theory  −8  10  −9  10  0  5  10  15  20  25  bit SNR per branch [dB]  Figure 4.1: BEP of BPSK with MRC and BFSK with NC vs. bit SNR per branch for i.i.d. Rayleigh fading and L = 3. Impairment by AWGN, i.i.d. –mixture noise ( = 0.25, κ = 10), and a Ricean faded Mi –PSK interferer with Ricean factor Ki = 6 dB (cf. E5). Markers: Simulated BEP. Solid lines: Asymptotic BEP [Eqs. (4.11), (4.14), and (4.15)]. 0  10  −1  10  −2  10  L=1  −3  10  −4  10  L=2  −5  BEP  10  L=3  −6  10  −7  10  −8  10  −9  10  MRC (simulation) EGC (simulation) MRC (theory) EGC (theory) 0  5  10  15  20  25  bit SNR per branch [dB]  Figure 4.2: BEP of 8–PSK vs. bit SNR per branch for differential EGC and MRC over an i.i.d. Ricean fading channel with Ricean factor K = 3 dB and spatially dependent –mixture noise ( = 0.25, κ = 10). Markers: Simulated BEP. Lines: Asymptotic BEP [Eqs. (4.11), (4.13), and (4.15)]. 84  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . Furthermore, Fig. 4.2 shows that the asymptotic performance loss of differential EGC compared to MRC is, respectively, 3 dB, 2.3 dB, and 2 dB for L = 1, 2, and 3, which is in perfect agreement with the results obtained from (4.34). Asynchronous CCI (E7): Fig. 4.3 shows the SEP of 4–PSK with MRC over a correlated Rayleigh fading channel (L = 3) impaired by a single asynchronous 4–PSK co–channel interferer which also experiences correlated Rayleigh fading. The correlation matrix Rhh of the desired user is a Toeplitz matrix with vector [1 α α2 ] as its first row, where α is the correlation coefficient. The correlation matrix C gg of the interferer has the same structure and its correlation coefficient is denoted by ρ. Both the desired user and the interferer employ square–root raised cosine filters with roll–off factor 0.22 for transmit pulse shaping and as receiver input filters. The timing offset between the desired user and the interferer is τ = T /4. Fig. 4.3 confirms that the performance of the desired user is not only negatively affected if its own channel is correlated but also if the interference channel is correlated, cf. Section 4.4.2, E7). Synchronous CCI (E7): In Fig. 4.4, we show the BEP of 16–PSK with MRC and differential EGC for correlated Ricean fading (K = 3 dB) and impairment by a correlated Rayleigh faded synchronous 16–PSK co–channel interferer. For L = 3 the correlation matrices Rhh and C gg of the desired user and the interferer, respectively, have the same structure as the corresponding matrices for Fig. 4.3 with α = ρ = 0.6. For L = 2 these matrices are also Toeplitz matrices with [1 α] and [1 ρ] as first rows, respectively, and the same values for α and ρ are valid as for the L = 3 case. From Fig. 4.4 we observe that for the considered type of interference the performance loss of differential EGC compared to MRC is 3 dB, 3.9 dB, and 4.3 dB for L = 1, 2, and 3, respectively, which is in perfect agreement with the values obtained from (4.31). Correlated Gaussian interference (E6): In Fig. 4.5, we consider the SEP of 4–PSK with  85  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . 0  10  −2  10  −4  10  SEP  α = 0.9  −6  10  α = 0.0  −8  10  ρ = 0.0 (simulation) ρ = 0.9 (simulation) theory  −10  10  0  5  10  15  20  25  30  symbol SNR per branch [dB]  Figure 4.3: SEP of 4–PSK vs. symbol SNR per branch for MRC over uncorrelated (α = 0.0) and correlated (α = 0.9) Rayleigh fading channels (L = 3) with uncorrelated (ρ = 0.0) and correlated (ρ = 0.9) Rayleigh faded asynchronous 4–PSK CCI. Markers: Simulated SEP. Solid lines: Asymptotic SEP [Eq. (4.11)]. 0  10  −1  10  −2  10  L=1 −3  10  L=2 −4  BEP  10  −5  10  −6  MRC (simulation) EGC (simulation) MRC (theory) EGC (theory)  10  0  5  L=3 10  15  20  25  bit SNR per branch [dB]  Figure 4.4: BEP of 16–PSK vs. bit SNR per branch for MRC and differential EGC over correlated Ricean fading channels (K = 3 dB, α = 0.6) with correlated Rayleigh faded 16–PSK CCI (ρ = 0.6). Markers: Simulated BEP Lines: Asymptotic BEP [Eqs. (4.11), (4.13), and (4.15)]. 86  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . 0  10  −1  10  L=1 −2  10  −3  10  L=2 −4  SEP  10  −5  10  L=3  −6  10  ρ ρ ρ ρ  −7  10  0  = = = =  0.5 0.0 0.5 1.0  (simulation) (theory) (theory) (theory) 5  10  15  20  25  symbol SNR per branch [dB]  Figure 4.5: SEP of 4–PSK vs. symbol SNR per branch for MRC over i.i.d. Rayleigh fading channels with zero–mean correlated Gaussian interference. Markers: Simulated SEP. Lines: Asymptotic SEP [Eq. (4.11)]. MRC over an i.i.d. Rayleigh fading channel with zero–mean correlated Gaussian interference. The interference correlation matrix Rnn has the same structure as C gg for Fig. 4.3 and asymptotic results for ρ = 0, 0.5, and 1.0 are shown. For clarity simulation results are only included for ρ = 0.5. As expected from the discussion in Section 4.4.2 E6), ρ = 0 and ρ = 1.0 result in lower and upper bounds for the SEP achievable for other values of ρ. We note that while fully correlated noise (ρ = 1.0) results in the worst performance, it does not cause a loss in diversity gain. A loss in diversity gain would result of course if the diversity branches of the desired user were fully correlated. UWB interference: In Fig. 4.6, we consider the SEP of a 16–QAM narrowband (NB) signal with MRC over i.i.d. Rayleigh fading (L = 2) and impairment by a single direct– sequence UWB (DS–UWB) [100] and multi–band orthogonal frequency multiplexing (MB– 87  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . 0  10  −1  10  −2  10  SEP  −3  10  −4  10  MB−OFDM, B = 1 MHz (simulation) MB−OFDM, B = 5 MHz (simulation) DS−UWB, B = 1 MHz (simulation) DS−UWB, B = 5 MHz (simulation) MB−OFDM (theory) DS−UWB (theory)  −5  10  0  5  10  15  20  25  30  35  symbol SNR per branch [dB]  Figure 4.6: SEP of 16–QAM vs. symbol SNR per branch for MRC over i.i.d. Rayleigh fading channels (L = 2) with DS–UWB and MB–OFDM UWB interference. Markers: Simulated SEP. Lines: Asymptotic SEP [Eq. (4.11)]. OFDM) UWB [101] interferer, respectively. The interference channels are modeled according to the CM1 UWB channel model [102] and are assumed to be spatially independent. The 16–QAM NB system uses square–root raised cosine filters with roll–off factor 0.35 for transmit pulse shaping and receive filtering. NB signal bandwidths of B = 1 MHz and B = 5 MHz are considered. Since derivation of a closed–form solution for Mn (L) for the relatively complicated UWB signal and channel models does not seem to be feasible, we ˆ n (L). used the Monte–Carlo method discussed in Section 4.4.3 to obtain an estimate M This estimate was used subsequently in (4.11) to calculate the asymptotic SEP. As can be observed from Fig. 4.6, the results obtained with this semi–analytical method are in excellent agreement with the simulation results at high SNR. Furthermore, Fig. 4.6 shows that for the considered scenario the SEP of the NB signal shows a stronger dependence on 88  Chapter 4. Performance Analysis of Diversity Combining Receivers . . . the NB signal bandwidth B for MB–OFDM UWB interference than for DS–UWB interference. However, for both considered values of B, DS–UWB interference is more harmful to the asymptotic SEP performance of the NB system than MB–OFDM UWB.  4.6  Conclusions  In this chapter, we presented simple, easy–to–evaluate, and insightful asymptotic BEP and SEP expressions for quadratic diversity combining receivers operating in correlated Ricean fading and non–Gaussian noise and interference. The only assumption necessary for the validity of the presented results is that all joint noise moments exist. We could show that while the diversity order of the considered combining schemes is independent of the type of noise, their combining gain is affected by the type of noise via certain noise moments. We provided general techniques for calculation of these noise moments and we tabulated the moments of several practically relevant types of noise. Our analytical results show that not only fading correlation but also noise correlation negatively affects the performance of diversity combiners. Furthermore, while BFSK with NC suffers from an asymptotic performance loss of 6 dB compared to BPSK with MRC regardless of the type of noise, the performance loss of differential EGC compared to MRC crucially depends on the type of noise if more than one diversity branches are available. Numerical results for various types of fading and noise confirmed the practical usefulness of the presented novel theoretical framework.  89  Chapter 5 Adaptive Lp–Norm Diversity Combining in Non–Gaussian Noise and Interference28 5.1  Introduction  In the previous chapter, we analyzed the performance of quadratic diversity combining schemes such as MRC, EGC, and NC in the presence of non–Gaussian noise. These combining schemes are based on the evaluation of an L2 –norm metric. Such metrics are not necessarily optimal for non–Gaussian noise. On the other hand, the noise distribution is usually unknown at the receiver and it may even change with time. As a result, maximum– likelihood (ML) detection in non–Gaussian noise is an impractical solution. In this chapter, we propose a robust Lp –norm metric for coherent, differential, and noncoherent combining techniques. This metric is tunable and performs well for a large class of noise distributions. We assume that different diversity branches may use different Lp –norms and different combining weights. We derive analytical expressions for the asymptotic BEP of the considered combining schemes with Lp –norm metric, which are valid for any type of noise with finite moments. This analysis is similar in spirit to the asymptotic analysis of L2 –norm metrics 28  Parts of this chapter were published in the IEEE Transactions on Wireless Communications, Vol. 8, pp. 4230-4240, Aug. 2009, cf. Appendix F.  90  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference for AWGN and non–Gaussian noise in [99, 16] and Chapter 4, respectively. The derived asymptotic BEP expressions show that the diversity gain is independent of the Lp –norm used and the type of noise. In contrast, the combining gain depends on a generalized moment of the noise samples at the diversity branches, which enables the development of simple metric optimization criteria that directly minimize the asymptotic BEP. We consider both off–line and on–line optimization of the metric parameters, and develop for the latter case adaptive multivariate finite difference stochastic approximation (FDSA) [103, 104] and localized random search (LRS) [104] algorithms. The remainder of this chapter is organized as follows. In Section 5.2, we introduce the system model and the Lp –norm metric. Asymptotic BEP expressions are derived in Section 5.3, and the calculation of generalized noise moments is discussed in Section 5.4. In Section 5.5, off–line and on–line optimization of the metric parameters is considered, and in Section 5.6, analytical and simulation results are presented. Conclusions are drawn in Section 5.7.  5.2  System Model and Lp–Norm Metric  In this chapter, we consider coherent combining, differential combining, and noncoherent combining for coherent linear modulation formats, e.g. M–QAM, M–PSK, differential M– PSK, and BFSK, respectively.  5.2.1  Signal Model  Assuming L diversity branches, for coherent linear modulation and differential M–PSK the received signal in the lth branch and in the kth symbol interval can be modeled in  91  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference equivalent complex baseband representation as  rl [k] =  √  γl hl b[k] + nl [k],  1 ≤ l ≤ L,  (5.1)  where γl , hl , and nl [k] denote the average SNR, the fading gain, and the noise in the lth diversity branch, respectively. The transmitted symbols b[k] ∈ A are normalized to E{|b[k]|2 } = 1 and taken from an M–ary alphabet A. In case of differential M–PSK, b[k] is obtained from a[k] ∈ A via differential encoding b[k] = a[k]b[k − 1]. The noise is assumed to be independent of the fading gains but the noise samples29 nl , 1 ≤ l ≤ L, may be statistically dependent and non–Gaussian. The noise variance is given by σl2  E{|nl |2 }, 1 ≤ l ≤ L. Similar to Chapter 4, the only restriction that  we impose on the noise is that all joint moments of the nl , 1 ≤ l ≤ L, exist, i.e., E{nκ1 1 (n∗1 )ν1 nκ2 2 (n∗2 )ν2 · · · nκLL (n∗L )νL } < ∞ for all κl ≥ 0, νl ≥ 0, 1 ≤ l ≤ L. Most practically relevant types of noise fulfill this condition (see next section). An exception is α–stable noise for which moments of order greater than α do not exist and which is sometimes used to model impulsive noise [105]. The fading gains hl are modeled as independent, non–identically distributed (i.n.d.) ¯l Gaussian random variables with mean h  E{hl } and variance σh2l  ¯ l |2 }, i.e., E{|hl − h  i.n.d. Ricean fading is assumed. Note that for γl in (5.1) to be the SNR, the power of the fading gains has to be normalized to E{|hl |2 } = σl2 , 1 ≤ l ≤ L. The Ricean factor is defined as Kl  ¯ l |2 /σ 2 and Rayleigh fading results as a special case for Kl = 0, 1 ≤ l ≤ L. |h hl  For BFSK the signal model in (5.1) has to be augmented since, in this case, in each diversity branch the outputs of two matched filters (MFs) are processed. The first MF 29  To simplify our notation, we drop the time index k in variables such as nl [k] whenever possible.  92  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference output is still given by (5.1) and the second MF output is modeled as  r¯l =  √  γl hl¯b + n ¯l ,  1 ≤ l ≤ L,  (5.2)  where b, ¯b ∈ {0, 1}, b = ¯b, and n ¯ l denotes the noise in the second MF output. While for AWGN nl and n ¯ l are statistically independent, this is not necessarily true for non–Gaussian noise. However, this does not affect the proposed asymptotic performance analysis and metric adaptation.  5.2.2  Noise Models  Since slight changes in the notation is necessary, in the following, we briefly discuss some important types of noise for which the analysis and metric optimization in this chapter is applicable. 1) Gaussian Mixture Noise (GMN): For i.n.d. GMN the probability density function (pdf) of the noise in the lth diversity branch is given by I  fn (nl ) = i=1  where ci,l > 0,  I i=1 ci,l  |nl |2 ci,l exp − 2 2 πσn,i,l σn,i,l  2 = 1, and σn,i,l ,  I 2 i=1 ci,l σn,i,l  ,  1 ≤ l ≤ L,  = σl2 , are constants. Special cases of  GMN include –mixture noise (I = 2, c1,l = 1 − l , c2,l = 2 2 σn,2,l = κl σn,1,l ,0≤  l  (5.3)  l,  2 σn,1,l = σl2 /(1 −  l  + κl l ),  < 1, and κl > 1) and Middleton’s Class A noise (I → ∞).  2) Co–Channel Interference I (CCI-I): The interference caused by I co–channel interferers in a system with receive antenna diversity can be modeled as [29] k2  I  nl [k] =  gi,l i=1  κ=k1  pi [κ]bi [k − κ],  1 ≤ l ≤ L,  (5.4)  93  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference where gi,l , pi [k], and bi [k] denote the fading gain at the lth receive antenna, the effective pulse shape, and the transmit symbols of the ith interferer, respectively. pi [k] depends on the transmit pulse shape of the interferer, the receiver input filter of the user, and the delay τi between the ith interferer and the user. The ith co–channel interferer is synchronous and asynchronous for τi = 0 and τi = 0, respectively. The limits k1 and k2 are chosen such that pi [k] ≈ 0 if k < k1 or k > k2 . Here, we model the interference channel gains gi,l 2 as (possibly correlated) Ricean fading gains with variances σg,i,l and Ricean factors Kg,i,l .  We note that CCI–I is spatially dependent even if the channel gains gi,l are independent because the term  k2 κ=k1  pi [κ]bi [k − κ] is common to all diversity branches.  3) CCI-II: The CCI model for FH systems with frequency diversity is slightly different from CCI-I. Assuming the synchronous case and that at hopping frequency l, 1 ≤ l ≤ L, co–channel interferer i, 1 ≤ i ≤ I, is present with probability  i,l ,  0≤  i,l  < 1, the resulting  interference can be modeled as I  nl =  Xi,l gi,l bi,l , i=1  1 ≤ l ≤ L,  (5.5)  where the Xi,l are mutually independent, and Xi,l = 1 and Xi,l = 0 with probabilities and 1 −  i,l ,  i,l  respectively. bi,l denotes the transmit symbols of the ith interferer at the lth  hopping frequency and the interference gains gi,l are modeled as i.n.d. Ricean fading with 2 variances σg,i,l and Ricean factors Kg,i,l . CCI–II can be used to model the interference in  systems that use FH for multiple access (e.g. Bluetooth). 4) Generalized Gaussian Noise (GGN): I.n.d. GGN is a popular model for non– Gaussian noise [59, 106]. The corresponding pdf for the lth diversity branch is given by  fn (nl ) =  |nl |βl βl Γ(4/βl ) exp − 2π(Γ(2/βl ))2 cl  ,  1 ≤ l ≤ L,  (5.6)  94  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference where cl  (Γ(2/βl )/Γ(4/βl ))βl /2 , and βl , 0 < βl < ∞, denotes the shape parameter. As  mentioned in Chapter 1, GGN contains Laplacian (βl = 1) and Gaussian (βl = 2) noise as special cases. We note that the Lp –norm metric with appropriately chosen parameters is the ML metric for i.n.d. GGN [80]. 5) UWB Interference: We will also test the theory and algorithms developed in this chapter for interference caused by the MB–OFDM UWB and IR–UWB signal formats standardized by ECMA [6] and IEEE 802.15.4a [7], respectively. We note that the proposed analysis is also applicable to any linear combination of the noises specified in 1)–5).  5.2.3  Lp–Norm Metric  In this subsection, we present the adopted Lp –norm metrics for the considered combining schemes. Coherent Combining (CC): The Lp –norm metric for CC is given by L  mc (˜b) = l=1  ql |rl −  √ γl hl˜b|pl ,  (5.7)  where ˜b ∈ A is a trial symbol, and ql > 0 and pl > 0, 1 ≤ l ≤ L, are metric parameters that can be optimized for performance maximization for the underlying type of noise.30 The decision ˆb is that ˜b which minimizes mc (˜b). For ql = 1 and pl = 2, 1 ≤ l ≤ L, the Lp –norm metric mc (˜b) is equivalent to MRC which is optimal in AWGN. For convenience we define the parameter vectors q  [q1 . . . qL ]T and p  [p1 . . . pL ]T .  Differential Combining (DC): DC is applied for differential M–PSK modulation We note that, strictly speaking, mc (˜b) is only a norm for pl ≥ 1, 1 ≤ l ≤ L. However, whether or not mc (˜b) is a norm is not important in our context. 30  95  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference and the corresponding Lp –norm metric is L  md (˜a) = l=1  ql |rl [k] − a ˜rl [k − 1]|pl ,  (5.8)  where a ˜ ∈ A is an M–PSK trial symbol. For the special case ql = 1 and pl = 2, 1 ≤ l ≤ L, the differential Lp –norm metric md (˜a) is equivalent to well–known differential EGC. The decision a ˆ is that a ˜ which minimizes md (˜a). Noncoherent Combining (NC): The considered NC metric for BFSK is L  ql (|rl |pl − |¯ rl | p l ) ,  mn = l=1  (5.9)  where we decide for b = 1 if mn ≥ 0 and for b = 0 otherwise. For ql = 1 and pl = 2, 1 ≤ l ≤ L, the Lp –norm metric in (5.9) is equivalent to conventional square–law combining for BFSK [15].  5.3  Asymptotic Analysis of Lp–Norm Combining  In this section, we develop asymptotic expressions for the PEP of the combining schemes described in Section 5.2.3 and relate these PEPs to the respective asymptotic BERs.  5.3.1  Asymptotic PEP of CC  We show in Appendix E that for any type of noise with finite moments, the asymptotic PEP of CC for γl → ∞, 1 ≤ l ≤ L, is given by . Pe (d) =  2L d2L  L l=1 L l=1  Γ  2 pl 2/pl  γl pl ql  1+Kl σl2  exp (−Kl )  Γ  L 2 l=1 pl  Mn (q, p),  (5.10)  +1  96  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference where Mn (q, p)  E  L pl l=1 ql |nl |  of the elements of noise vector n  L l=1  2/pl  can be interpreted as a generalized moment  [n1 . . . , nL ]T , and d denotes the Euclidean distance  between the alternative signal points considered for the PEP. The generalized noise moment Mn (q, p) in (5.10) can be calculated in closed form for special cases, cf. Section 5.4. Nevertheless, even if the generalized noise moment is not available in closed form, (5.10) can be used for fast evaluation of the asymptotic PEP since Mn (q, p) is independent of the SNR and has to be evaluated only once, which can be done e.g. by Monte–Carlo simulation. More importantly, (5.10) reveals how parameters ql and pl influence the asymptotic PEP, which will be exploited for metric optimization in Section 5.5. For complexity reasons it may be desirable for some applications to limit the number of metric parameters to be optimized. For this purpose we may set ql = q and pl = p, 1 ≤ l ≤ L, and simplify (5.10) to 2 L . 2 Γ p Pe (d) = d2L  where Mn (p)  E  L l=1  |nl |p  2L/p  L  L l=1  1+Kl σl2  L L l=1 (γl ) p Γ  exp (−Kl )  2L p  +1  Mn (p),  (5.11)  . Note that (5.11) depends on p but is independent  of q. For the special case p = 2, (5.11) is equivalent to (4.9) in Chapter 4 for i.i.d. fading.  5.3.2  Asymptotic PEPs of DC and NC  The asymptotic PEPs of DC and NC can be derived in the same way as those for CC. 1) DC: The asymptotic PEP of M–PSK with DC is also given by (5.10) and (5.11) if the respective generalized noise moments of n are replaced with the generalized noise moments of the effective noise vector  z = n[k] − a[k]n[k − 1].  (5.12) 97  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference If the nl [k] are rotational symmetric and n[k] and n[k − 1] are statistically independent, a[k] has no influence on the PEP and we may use z = n[k] − n[k − 1] instead of (5.12). 2) NC: It can be shown that we formally obtain the PEP of BFSK with NC by letting d = 1 in (5.10) and (5.11), respectively.  5.3.3  Asymptotic BER  As mentioned in Chapter 4, the asymptotic (average) BER can be obtained from the asymptotic PEP as . BER =  βM Pe (dM ), log2 (M)  (5.13)  where dM and βM denote the minimum Euclidean distance of signal constellation A and the average number of minimum–distance neighbors, respectively, and are listed in Table 4.1 for common modulation schemes.  5.3.4  Combining and Diversity Gain  . It is convenient to express the asymptotic BEP as BEP = (Gc γ)−Gd [16, 14], where Gc and Gd denote the combining and the diversity gain, respectively, and γ = ( i.e., γ [dB] =  1 L  L l=1  L l=1  γl )1/L ,  γl [dB]. From (5.10) we observe that the diversity gain is given by  Gd = L independent of metric parameters q and p, and independent of the type of noise. The combining gain for CC with Lp –norm metric can be expressed as  Gc [dB] = 10 log10 10 + L  L  l=1  d2min log2 (M)1/L 1/L 2ξmin    2/pl   pl ql log10   10 − L  L  l=1  L 2 i=1 pi  Γ Γ  2 pl  1 + Kl exp (−Kl ) σl2  1/L  log10 +1   10 log10 (Mn (q, p)) .(5.14) − L  98  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference Eq. (5.14) reveals that the combining gain consists of four terms. The first and the second term on the right hand side (RHS) of (5.14) depend on the signal constellation and the fading parameters, respectively, but are independent of the metric parameters q and p and the properties of the noise. The third term on the RHS of (5.14) is a function of q, p, and L but is also independent of the noise. Only the last term on the RHS of (5.14) depends on the properties of the noise via the generalized moment Mn (q, p) of the noise samples. Eq. (5.14) reveals that the optimal parameters q opt and popt , which maximize Gc , only depend on L and the type of noise (via Mn (q, p)) but are not influenced by the the modulation scheme and the Ricean factors Kl , 1 ≤ l ≤ L. For DC and NC similar observations as for CC can be made with respect to diversity gain and combining gain.  5.4  Generalized Noise Moments  In this section, we provide analytical results for the generalized noise moments defined in Section 5.3 for selected types of noise. To make the problem tractable, in this section, we consider not necessarily independent but identically distributed (n.i.d.) noise and Mn (p), which depends only on p, instead of Mn (q, p). To simplify our notation, in the following, we will drop subscript l in all noise parameters (e.g. in ci,l ,  l,  2 , Kg,i,l , etc.) if the κl , σg,i,l  noise is n.i.d. (which includes i.i.d. as a special case).  5.4.1  Exact Noise Moments for L = 2  First, we consider the special case L = 2. Furthermore, for independent complex Gaussian random variables (RVs) x1 and x2 having variances σx21 and σx22 we define MG (p; σx21 , σx22 ) E{(|x1 |p + |x2 |p )2L/p }. Using the substitutions |x1 | = r sin2/p ϕ and |x2 | = r cos2/p ϕ with  99  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference 0 ≤ r < ∞ and 0 ≤ ϕ ≤ π/2, we obtain MG (p; σx21 , σx22 ) = where κ  48κσx41 IG (p, κ), 24/p p  (5.15)  π/2 (sin(2ϕ))4/p−1 /(sin4/p 0  σx21 /σx22 , and the finite range integral IG (p, κ)  ϕ+  κ cos4/p ϕ)4 dϕ depends only on p and κ and can be easily evaluated numerically. Based on the result for MG (p; σx21 , σx22 ), we provide analytical expressions for the generalized moments of AWGN, n.i.d. Rayleigh–faded CCI–I (i.e., Kg,i = 0, 1 ≤ i ≤ I)31 , and i.i.d. Rayleigh–faded CCI–II (single interferer at each hopping frequency, i.e., I = 1) in Table 5.1. Furthermore, we also provide an expression for i.i.d. GGN in Table 5.1, which can be obtained in a similar fashion as the generalized moment in the Gaussian case and which contains the finite range integral IGG (p, β)  π/2 (sin(2ϕ))4/p−1 /(sin2β/p 0  ϕ+  cos2β/p ϕ)8/β dϕ. Table 5.1: Generalized noise moments Mn (p) for L = 2 for various types of n.i.d. noise. In particular, we consider AWGN, i.i.d. GMN, n.i.d. Rayleigh–faded CCI–I (s [s1 . . . sI ]T , k2 si κ=k1 pi [κ]bi [κ], S contains all possible values of s), i.i.d. Rayleigh–faded CCI–II (I = 1, bI [b1,1 . . . b1,L ]T , MI contains all possible values of bI , c1 1 − 1, 1 , c2 2 2 2 σ ¯g,1 σg,1 , σ ¯g,2 0), and i.i.d. GGN. Noise Model  Moments Mn (p)  AWGN  MG (p; 1, 1) I i=1  GMN CCI–I (Rayleigh) CCI–II (Rayleigh) GGN  1 |MI |  I 2 2 j=1 ci cj MG (p, σn,i , σn,j )  I I 1 2 2 2 2 s∈S MG (p, i=1 σg,i |si | , i=1 σg,i |si | ) |S| 2 2 2 2 ¯g,i |b1,1 |2 , σ ¯g,j |b1,2 |2 ) j=1 ci cj i=1 bI ∈MI MG (p, σ Γ(8/β)β I (p, β) (Γ(4/β))2 24/p−2 p GG  31  We note that the fading gains gl,i , 1 ≤ l ≤ L, of n.i.d. CCI–I are i.i.d. RVs. However, the resulting CCI-I is n.i.d. since each interferer affects all receive antennas simultaneously, cf. (5.4).  100  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference  5.4.2  Noise Moments for General L  For general L a closed–form expression for the generalized moment can be calculated for several special cases. In particular, we will provide accurate approximations for Mn (p) for n.i.d. noise distributions that are based on the Gaussian distribution (i.e., i.i.d. GMN, n.i.d. Rayleigh–faded CCI–I, i.i.d. Rayleigh–faded CCI–II), and exact results for unfaded n.i.d. CCI–I and i.i.d. CCI–II with I = 1 and Kg,1 → ∞, 1 ≤ l ≤ L. 1) Gaussian–based Noise Distributions: We first consider i.n.d. Gaussian RVs xl with variances σx2l , 1 ≤ l ≤ L, and our goal is to calculate MG (p; σx21 , . . . , σx2L ) E{(  L l=1  |xl |p )2L/p }. It can be shown that the pdf of yl = |xl |p is given by 2/p  fyl (yl ) =  2 2/p−1 y yl exp − l 2 2 pσxl σxl  ,  which is a Weibull pdf. We are interested in the pdf of z =  (5.16)  L l=1  yl . Unfortunately, a  closed–form expression for a sum of Weibull RVs is not known. However, an accurate approximation for the pdf of z is given by the α–µ pdf [107]  fz (z) =  µz α αµµ z αµ−1 exp − Ωµ Γ(µ) Ω  ,  (5.17)  where parameters α, µ, and Ω have to be optimized for the best possible agreement with the true pdf of z. For this purpose, the efficient moment–based method in [107, Eq. (5)–(9)] may be used. We note that in [107] only i.i.d. Weibull variables are considered, whereas we allow different variances σx2l . This small extension can be accommodated by replacing [107, Eq. (9)] by E{yln } = σxpnl Γ(1 + pn/2), n ∈ {0, 1, 2, . . .} (yl is referred to as Rl in [107]), and we found the corresponding approximation to be still very accurate. Using (5.17) we  101  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference obtain MG (p; σx21 , . . . ,  σx2L )  Γ(µ + 2L/(pα)) = Γ(µ)  Ω µ  2L/(pα)  .  (5.18)  Based on the approximation for MG (p; σx21 , . . . , σx2L ) in (5.18), we can find the generalized moments of AWGN, i.i.d. GMN, n.i.d. Rayleigh–faded CCI–I, and i.i.d. Rayleigh–faded CCI–II (I = 1) given in Table 5.2 for general L. 2) Unfaded CCI: We first consider n.i.d. CCI–I. Assuming a single, unfaded interferer (Kg,1 → ∞), (5.4) simplifies to k2 jΘ1,l  nl [k] = e  κ=k1  p1 [κ]b1 [k − κ],  1 ≤ l ≤ L,  (5.19)  with uniformly distributed, mutually independent phases Θ1,l ∈ (−π, π], 1 ≤ l ≤ L. Based on (5.19), the exact result for the generalized moment of unfaded CCI–I given in Table 5.2 can be obtained. Similarly, specializing (5.5) to I = 1 and Kg,1 → ∞, the exact expression for i.i.d. CCI–II in Table 5.2 can be derived. The asymptotic PEP for CC and NC can be (approximately or exactly) obtained by combining the generalized moments in Tables 5.1 and 5.2 with the PEP formula in (5.11). We note that since the effective noise for DC is more complicated than the noise for CC and NC, cf. (5.12), a closed–form evaluation of the generalized moments does not seem possible for DC in most cases.  5.5  Metric Optimization  In this section, we optimize the metric parameters p and q for minimization of the asymptotic BEP. In the following, we consider both off–line and on–line optimization.  102  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference Table 5.2: Approximations for the generalized noise moments Mn (p) for general L for the same types of n.i.d. noise considered in Table 5.1. Additionally, exact results for unfaded n.i.d. CCI–I (I = 1) and i.i.d. CCI–II (I = 1, ξ1 1, ξ2 0) are provided. Noise Model  Moments Mn (p)  AWGN  MG (p; 1, . . . , 1) I i1 =1  GMN CCI–I (Rayleigh) CCI–II (Rayleigh) CCI–I (Unfaded) CCI–II (Unfaded)  5.5.1  1 |MI |  2 i1 =1  1 |MI |  1 |S|  ···  I iL =1 ci1  ···  s∈S  MG (p,  2 iL =1 ci1  2 i1 =1  · · · c iL  2 2 · · · ciL MG (p, σn,i , . . . , σn,i ) 1 L  I i=1  2 σg,i |si |2 , . . . ,  bI ∈MI  1 L2L/p |S|  ···  2 iL =1 ci1  · · · c iL  I i=1  2 σg,i |si |2 )  2 2 MG (p, σ ¯g,i |b1,1 |2 , . . . , σ ¯g,i |b1,L |2 ) 1 L  s∈S  |s|2L  bI ∈MI  L p l=1 ξil |b1,l |  2L/p  Off–line Optimization  If the generalized noise moments are known, the metric parameters can be optimized off– line based on (5.10) or (5.11). If the underlying type of noise is a priori known, the generalized noise moments may be obtained in closed–form, cf. Tables 5.1 and 5.2, or, if this is not possible, from Monte–Carlo simulation using locally generated noise samples.32 Monte–Carlo simulation can also be applied to estimate the generalized noise moments from observed noise samples. To gain some insight and to make the problem tractable, we assume n.i.d. noise in this subsection. For n.i.d. noise we may set ql = q and pl = p, 1 ≤ l ≤ L, in metrics (5.7)–(5.9) without loss of optimality, i.e., we can base our off–line optimization on (5.11) and have to optimize only parameter p. Unfortunately, for most types of noise a closed–form optimization of p is not possible. An exception is n.i.d. unfaded CCI–I, where we can show based on (5.11) and Table 5.2 that the optimal p is given by 32 Note that if the underlying noise model is known a priori, ML combining can be applied, of course. However, even in this case the proposed Lp –norm metric may be preferable if the ML metric is computationally complex or causes numerical problems. For example, the GMN pdf consists of a sum of exponential functions which may cause numerical problems for high SNRs.  103  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference √ popt = ∞ corresponding to metric mc (˜b) = maxl∈{1,...,L} {|rl − γl hl˜b|}. Furthermore, exploiting (5.14) we obtain for the asymptotic SNR gain Gp of a metric using p > 2 over the L2 –norm metric  Gp [dB] = 10 log10  p 2Γ(2/p)  1−1/L  L1−2/p L!1/L  p→∞  = 10 log10  L L!1/L  .  (5.20)  For example, for L = 2 we obtain G20 = 1.3 dB and G∞ = 1.5 dB. Furthermore, using the Stirling formula [98] for L → ∞ we can show that G∞ = 10 log10 (e) = 4.3 dB. We note that it can be shown that popt = ∞ and (5.20) are also valid for DC and NC in n.i.d. unfaded CCI–I. If the optimal p cannot be obtained in closed form, numerical optimization is necessary. To illustrate this, we show in Figs. 5.1 and 5.2 the BEP of BPSK as a function of p for i.i.d. Rayleigh fading with L = 2 and L = 3, respectively. Details about the considered types of noise can be found in the captions of the figures. The solid lines represent analytical results generated based on (5.11), (5.13), and, respectively, Table 5.1 (Fig. 5.1) and Table 5.2 (Fig. 5.2). The markers indicate simulation results and the bold ”+” markers denote the minima of the analytical BEP. The agreement between analytical results and simulation results is excellent in both Figs. 5.1 and 5.2. As expected, Figs. 5.1 and 5.2 show that p = 2 is optimal for AWGN and also for Rayleigh–faded CCI–I. In constrast, for heavy–tailed types of noise such as –mixture noise and Rayleigh–faded CCI–II popt < 2 holds. For short–tailed noise such as unfaded CCI–I popt > 2 is valid. For i.i.d. GGN with β = 1 we obtain popt = 1 from Fig. 5.1 as expected. While all other BEP curves have a single minimum in the considered p range, the BEP for unfaded CCI–II in Fig. 5.2 has two local minima. Figs. 5.1 and 5.2 clearly illustrate the benefits of optimizing p and confirm our analysis.  104  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference  −5  Rayleigh−faded CCI−II  10  -mixture noise BER  GGN  Rayleigh−faded CCI−I  AWGN  0  0.5  1  1.5  2  2.5  3  3.5  4  p  Figure 5.1: BEP vs. p for BPSK, i.i.d. Rayleigh fading, L = 2, SNR = 24 dB, and different types of n.i.d. noise. Noise parameters: I.i.d. –mixture noise ( = 0.1, κ = 10), n.i.d. Rayleigh–faded QPSK CCI–I (I = 1, τ1 = 0.25T with symbol duration T , raised cosine pulse shape with roll–off factor 0.22), i.i.d. Rayleigh–faded QPSK CCI–II (I = 1, 1 = 0.25), and i.i.d. GGN (β = 1).  5.5.2  On–line Optimization  In practice, the statistical properties of the noise impairing a wireless communication system are often not known a priori and may change with time. Since multiplication of the Lp –norm metrics (5.7)–(5.9) with a positive constant does not change the decision, we can set q1 = 1 without loss of optimality and optimize only the 2L − 1 elements of vector x  [q2 . . . qL pT ]T . Since the metric coefficients may not be updated in every symbol interval, we introduce  a new time t = Nm k, where k is the symbol time and Nm > 1 can be used to specify how frequently the metric coefficients are updated. Furthermore, the proposed adaptive 105  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference  Rayleigh−faded CCI−II  −5  10  -mixture noise I  -mixture noise II −6  BER  10  Rayleigh−faded CCI−I AWGN  −7  Unfaded CCI−II  10  Unfaded CCI−I  1  2  3  4  5  p  6  7  8  9  10  Figure 5.2: BEP vs. p for BPSK, i.i.d. Rayleigh fading, L = 3, SNR = 20 dB, and different types of n.i.d. noise. Noise parameters: I.i.d. –mixture noise I ( = 0.1, κ = 10), i.i.d. – mixture noise II ( = 0.1, κ = 5), n.i.d. Rayleigh–faded and unfaded QPSK CCI–I (I = 1, τ1 = 0.25T , raised cosine pulse shape with roll–off factor 0.22), and i.i.d. Rayleigh–faded and unfaded QPSK CCI–II (I = 1, 1 = 0.41). algorithms require an estimate of the cost function to be minimized. For CC we obtain based on (5.10) the cost function estimate  Lt (x)  ˆ n (x, t) M  L l=1 L l=1  1 Ne  2/p pl ql l  Ne −1  L  ν=0  l=1  Γ Γ  2 pl L 2 l=1 pl  ˆ n (x, t) M  (5.21)  +1 L l=1  ql |ˆ nl [t − ν]|pl  2/pl  ,  (5.22)  where we have neglected all irrelevant terms and Ne denotes the number of time steps ˆ n (x, t) at time t. Furthermore, n used for estimation of the generalized moment M ˆ l [t]  106  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference rl −  √  γl hl b[t], where b[t] may be a training symbol or a previous decision. A similar  estimate for the cost function may be generated for DC and NC. In the following, two different algorithms for optimization of x are provided and compared. 1) Multivariate Stochastic Approximation: The first algorithm is based on the finite difference stochastic approximation (FDSA) framework with gradient estimation [104]. This framework is particularly well suited for the problem at hand since it employs a Kiefer– Wolfowitz type of gradient estimate gˆ t (xt ) avoiding cumbersome differentiation of Lt (x) [103]. In the tth iteration the FDSA algorithm generates the estimate xt for the optimal x as [104]  xt+1 = xt − at gˆ t (xt ), gˆ t (xt ) =  Lt (xt + ct e2L−1 ) − Lt (xt − ct e2L−1 ) Lt (xt + ct e1 ) − Lt (xt − ct e1 ) ... 2ct 2ct  (5.23) T  , (5.24)  where en is a column vector of length 2L − 1 with a 1 in position n and 0’s in all other positions. If n[k] is stationary and at and ct fulfill at > 0, ct > 0, at → 0, ct → 0, ∞ t=0  at = ∞, and  ∞ t=0  a2t /c2t < ∞, the above algorithm will find the global minimum if  the BEP has only one minimum and at least a local minimum otherwise [104] (as long as the BEP and Lt (x) meet the mild conditions required for convergence outlined in [104]). However, since, in practice, n[k] will be non–stationary, we may set at = a and ct = c, ∀t, where a and c are small positive constants to give the algorithm some tracking capability. Furthermore, since the pl may have a large dynamic range (e.g. popt = ∞ for unfaded CCI–I), the tracking ability of the algorithm can be improved by limiting the elements of xt to some finite value xmax at the expense of some loss in performance if the optimal element of x exceeds xmax . Note that for the problem at hand the FDSA algorithm may not find the global optimum as the cost function may have multiple local minima, cf. Fig. 5.2. 107  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference However, we did not find this to be a problem in practice as the BERs of most types of noise seem to have only a single minimum, and in case of multiple minima, all minima seem to result in similar performances. For initialization of the FDSA algorithm ql = 1 and pl = 2, 1 ≤ l ≤ L, is a good choice since this guarantees that the solution found by the algorithm in combination with CC, DC, and NC will not perform worse than conventional MRC, EGC, and NC, respectively. 2) Random Search Method: The second method that we consider is a localized random search (LRS) method. In contrast to FDSA algorithms, LRS algorithms do not get stuck in local minima and find the global minimum under mild conditions on the cost function [104]. Based on xt , the proposed LRS algorithm generates a new estimate [104]  ˆ t+1 = xt + dt , x  (5.25)  where dt is a random vector whose elements are i.i.d. Gaussian random variables with ˆ t+1 lies outside the predefined search space variance σd2 . If at least one of the elements of x ˆ t+1 are inside the search space. Subse[0, xmax ], (5.25) is repeated until all elements of x ˆ t+1 , otherwise xt+1 = xt . In a non–stationary quently, if Lt (ˆ xt+1 ) < Lt (xt ), we let xt+1 = x noise environment, the above algorithm is run continuously. The speed of convergence of the LRS algorithm depends crucially on the size of the search space (i.e., xmax ) and on σd2 [104]. For initialization, the same initial vector as for the FDSA is appropriate. Since LRS algorithms suffer from performance degradation if the cost function estimate is noisy [104], comparatively large Ne may be advisable. Note, however, that the tracking capabilities of the algorithm decrease as Ne increases. We found Ne = 100 to give a good compromise between estimation noise suppression and tracking capabilities for the application at hand. 3) Complexity: From a practical point of view, it is of interest to compare the complexity of the proposed adaptive algorithms assuming a fixed–point implementation 108  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference with s digits. Denoting the complexity of one multiplication by µ(s), the complexity of evaluating the Gamma and the power functions is O((log(s))2 µ(s)) [108], whereas that of a division is O(µ(s)). Taking this into account, neglecting the complexity of additions, and assuming that 2/pl , 1 ≤ l ≤ L, is obtained from a look–up table, the complexities of one iteration of the FDSA and LRS algorithms are given by CFDSA = O 2(2L − 1)(Ne (L + 1) + 2L + 1)(log(s))2 µ(s)  (5.26)  CLRS = O 2(Ne (L + 1) + 2L + 1)(log(s))2 µ(s) ,  (5.27)  and  respectively. A comparison of (5.26) and (5.27) shows that, since typically Ne has to be chosen much larger for the LRS algorithm (e.g. Ne = 100) than for the FDSA algorithm (e.g. Ne = 1), the complexity per iteration of the LRS algorithm is larger than that of the FDSA algorithm for typical values of L (e.g. L < 10). 4) Performance: In Figs. 5.3 and 5.4, we show metric coefficients ql , 2 ≤ l ≤ 4, and pl , 1 ≤ l ≤ 4, vs. iteration t of, respectively, the FDSA and the LRS algorithms for i.i.d. Rayleigh fading with L = 4 and SNR = 16 dB. The corresponding BERs of BPSK with CC are shown in Fig. 5.5. Five different types of noise are considered which are specified in the caption of Fig. 5.3 and at t = (ν − 1) · 106 we switch abruptly to a new noise Nν, 1 ≤ ν ≤ 5. For the FDSA algorithm we used at = a = 4 · 10−4 , ct = c = 10−5 , xmax = 10, Nm = 1, and Ne = 1. For the LRS algorithm we adopted σd2 = 0.1, xmax = 10, Nm = 1, and Ne = 100. For both algorithms xt was initialized with ql = 1, 2 ≤ l ≤ 4, and pl = 2, 1 ≤ l ≤ 4, and previous decisions ˆb[t] were used in the adaptation process. Figs. 5.3 and 5.4 show the results for one typical adaptation process and the corresponding BERs in Fig. 5.5 were calculated with (5.10) and (5.13), where the generalized noise moments were  109  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference q2, q3, q4  10  p ,p ,p ,p 1  2  3  4  9 8  ql , 2 ≤ l ≤ 4, pl , 1 ≤ l ≤ 4  7 6  N1  5  N2  N3  N4  N5  4 3 2 1 0  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5 6  t  x 10  Figure 5.3: Metric coefficients ql , 2 ≤ l ≤ 4, and pl , 1 ≤ l ≤ 4, vs. iteration t of FDSA algorithm. N1: I.i.d. Rayleigh–faded QPSK CCI–II (I = 1, = 0.1) and AWGN, where the CCI–II power is ten times larger than the AWGN variance; N2: I.n.d. Gaussian noise with variances σ12 = 1, σ22 = 0.5, σ32 = 0.5, σ42 = 2; N3: I.n.d. –mixture noise with l = 0.1, 1 ≤ l ≤ 4, and κ1 = 20, κ2 = 40, κ3 = 50, κ4 = 100; N4: I.n.d. GGN with β1 = β2 = 3 and β3 = β4 = 1; N5: N.i.d. unfaded QPSK CCI–I (I = 1, τ1 = 0.3T , raised cosine pulse shape with roll–off factor 0.22). q ,q ,q  10  2  3  4  p ,p ,p ,p 1  2  3  4  9 8  ql , 2 ≤ l ≤ 4, pl , 1 ≤ l ≤ 4  7 6  N1  5  N2  N3  N4  N5  4 3 2 1 0  0  0.5  1  1.5  2  2.5  t  3  3.5  4  4.5  5 6  x 10  Figure 5.4: Metric coefficients ql , 2 ≤ l ≤ 4, and pl , 1 ≤ l ≤ 4, vs. iteration t of LRS algorithm. Noise types N1–N5 are specified in the caption of Fig. 5.3. 110  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference −5  10  Adaptive Lp−norm metric (FDSA) Adaptive Lp−norm metric (LRS) L2−norm metric  −6  10  N2  BER  N1  N3  N4  N5  −7  10  −8  10  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  t  5 6  x 10  Figure 5.5: BEP of BPSK with CC vs. iteration t for FDSA and LRS algorithms, respectively. For comparison BEP of L2 –norm combining is also shown. Noise types N1–N5 are specified in the caption of Fig. 5.3. obtained by Monte–Carlo simulation. Figs. 5.3–5.5 show that both algorithms work well and that after each switching to a new type of noise, steady state operation is achieved quickly. Thereby, with the chosen settings, the steady state error of the LRS algorithm is larger than that of the FDSA algorithm, but the LRS algorithm converges faster to the new steady state after the type of noise has changed. Note, however, that the trade–off between tracking capabilities and residual error strongly depends on how the parameters of the algorithms (e.g. a, c, Ne , and σd2 ) are chosen [104]. Furthermore, as expected, Figs. 5.3 and 5.4 confirm that in steady state for the n.i.d. noises N1 and N5 all ql and pl are equal, respectively, whereas for the i.n.d. noises N2, N3, and N4 either the ql or/and the pl are not equal. For N5 pl = ∞, 1 ≤ l ≤ 4, is optimal and both algorithms yield pl = 10, 1 ≤ l ≤ 4, because we set xmax = 10. Fig. 5.5 shows that the Lp –norm metric with FDSA and LRS adaptation substantially outperforms the L2 –norm metric (i.e., MRC). 111  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference  5.6  Numerical Results and Discussions  In this section, we verify the analytical results derived in Sections 5.3 and 5.4 through simulations and compare the performance of the adaptive Lp –norm metric with that of other popular metrics. For convenience we consider n.i.d. noise throughout this section and drop subscript l in the noise parameters. The respective noise parameters are specified in the captions of the figures. The optimal metric parameter popt was obtained with the FDSA algorithm. In Figs. 5.6 and 5.7, we show the BEP of 16–QAM with CC in i.i.d. Rayleigh fading with L = 2 for the adaptive Lp –norm metric and several other popular robust metrics for, respectively, i.i.d. –mixture noise and n.i.d. unfaded QPSK CCI–I. To facilitate the √ definition of the various metrics, we introduce the notation ul |rl − γl hl˜b|. We consider the Huber metric m(˜b) =  L l=1  ml (˜b), ml (˜b) = u2l /2 if ul ≤ δ, and ml (˜b) = δul − δ 2 /2  if ul > δ [78], the Meridian metric m(˜b) = m(˜b) =  L l=1  L l=1  log(ul + δ) [79], and the Myriad metric  log(u2l + δ 2 ) [79]. Note that for all these robust metrics parameter δ has to  be optimized by exhaustive search, which is quite tedious, since, unlike for the Lp –norm metric, a systematic optimization framework is not available. For Figs. 5.6 and 5.7 the robust metrics were optimized by simulation for SNR = 30 dB. Fig. 5.6 shows that for the heavy–tailed –mixture noise the Lp –norm metric with popt = 0.4 outperforms the other robust metrics and the gap to the optimal ML metric is less than 1 dB. Fig. 5.7 shows that for short–tailed unfaded CCI–I the Huber and Myriad metrics are essentially equivalent to the L2 –norm metric and are outperformed by 1.3 dB by the Lp –norm metric with p = 20 (popt → ∞ holds in this case), as predicted in Section 5.5.1. We note that the ML metric does not seem tractable for unfaded CCI–I. Interestingly, while the Lp –norm metric was optimized based on the presented asymptotic analysis, Figs. 5.6 and 5.7 suggest that it also performs well for low SNRs. 112  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference −2  10  −3  BER  10  −4  10  L −norm (p p  opt  = 0.4, simulation)  L −norm (simulation) 2  Myriad (δ = 0.6, simulation) Meridian (δ = 1.1, simulation) Huber (δ = 0.01, simulation) ML (simulation) Lp−norm (popt = 0.4, theoretical asymptotic BER)  −5  10  L −norm (theoretical asymptotic BER) 2  6  8  10  12  14  16  18  20  22  24  26  SNR per bit [dB]  Figure 5.6: BEP vs. SNR per bit per branch of 16–QAM with CC in i.i.d. Rayleigh fading (L = 2) and i.i.d. –mixture noise ( = 0.1, κ = 100). Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.1. −1  10  −2  10  −3  BER  10  Lp−norm (p = 20, simulation) L2−norm (simulation)  −4  10  Myriad (δ = 100, simulation) Meridian (δ = 100, simulation) Huber (δ = 100, simulation) Lp−norm (p = 20, theoretical asymptotic BER) L −norm (theoretical asymptotic BER) 2  −5  10  0  5  10  15  20  25  SNR per bit [dB]  Figure 5.7: BEP vs. SNR per bit per branch of 16–QAM in i.i.d. Rayleigh fading (L = 2) and n.i.d. unfaded QPSK CCI–I (I = 1, τ1 = 0.3T , raised cosine pulse shape with roll–off factor 0.22). Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.2. 113  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference −1  10  −2  10  −3  10  −4  10  K = 0 dB  −5  10  −6  BER  10  −7  10  −8  10  Lp−norm (popt = 0.5, simulation) L2−norm (simulation)  −9  10  Lp−norm (popt = 0.5, theoretical asymptotic BER)  K = 6 dB  L2−norm (theoretical asymptotic BER) −10  10  0  5  10  15  20  25  SNR per bit [dB]  Figure 5.8: BEP vs. SNR per branch of BFSK with NC in i.i.d. Ricean fading (L = 3) and i.i.d. Rayleigh–faded QPSK CCI–II (I = 1, 1 = 0.25). Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Table 5.2. In Fig. 5.8, we show the BEP of BFSK with NC in i.i.d. Ricean fading with L = 3 and i.i.d. Rayleigh–faded QPSK CCI–II. Fig. 5.8 shows that the proposed Lp –norm combining also achieves considerable performance gains over L2 –norm combining for BFSK with NC and in Ricean fading. As expected from Section 5.3.4, the optimal value popt = 0.5 is independent of the Ricean factor K. Although for K = 6 dB the simulated BEP approaches the asymptotic BEP only for BEP < 10−10 , the Lp –norm metric optimized for the asymptotic BEP also results in large gains for higher BERs. For example, for BEP = 10−4 , the Lp –norm metric achieves a gain of 3.5 dB over the L2 –norm metric.  114  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference −2  10  −3  10  −4  10  −5  BER  10  IR−UWB  MB−OFDM UWB  −6  10  −7  CC with Lp−norm metric (p = 30, simulation)  10  CC with L2−norm metric (simulation) DC with L −norm metric (p = 30, simulation) p  −8  DC with L2−norm metric (simulation)  10  Lp−norm metric (p = 30, theoretical asymptotic BER) L2−norm metric (theoretical asymptotic BER) −9  10  5  10  15  20  25  30  35  SNR per bit [dB]  Figure 5.9: BEP vs. SNR per bit per branch of 4–PSK system with bandwidth B = 4 MHz and CC or DC in i.i.d. Rayleigh fading (L = 3) and MB–OFDM UWB [6] and IR–UWB (Nb = 32 bursts per symbol and Lc = 128 chips per burst) [7] interference. Solid lines with markers: Simulation results. Bold solid and dashed lines: Asymptotic BEP based on (5.11), (5.13), and Monte–Carlo simulation of generalized noise moments. Finally, in Fig. 5.9, we show the BEP of 4–PSK in i.i.d. Rayleigh fading with L = 3 and impairment by MB–OFDM UWB and IR–UWB interference following the ECMA [6] and IEEE 802.15.4a [7] standards, respectively. The bandwidth of the receiver input filter of the 4–PSK system is assumed to be B = 4 MHz. Results for both CC and DC are shown in Fig. 5.9. For both combining schemes and both types of UWB, p = 30 was close to optimal for the Lp –norm metric. Fig. 5.9 shows that Lp –norm combining also achieves substantial gains over L2 –norm combining in UWB interference. Thereby, the performance gains are larger for CC than for DC. This can be explained by the fact that the effective noise for DC is the sum of two independent noise samples, cf. (5.12), and thus, according to the Central Limit Theorem [109], is closer to a Gaussian distribution than the noise relevant for CC (and NC).  115  Chapter 5. Adaptive Lp –Norm Diversity Combining in Non–Gaussian Noise and Interference  5.7  Conclusions  In this chapter, we considered general Lp –norm coherent, differential, and noncoherent diversity combining in non–Gaussian noise and interference. For the asymptotic regime of high SNR we derived closed–form expressions for the BEP valid for i.n.d. Ricean fading and non–Gaussian noise and interference with finite moments. The asymptotic BEP expressions reveal that while the diversity gain of Lp –norm combining is independent of the type of noise and the metric parameters, the combining gain depends on generalized noise moments and on the metric parameters. For on–line metric optimization, we developed two efficient adaptive algorithms which do not require any a priori knowledge about the noise statistics and can also cope with non–stationary noise. Simulation results confirmed the analytical results presented in this chapter and showed that the proposed adaptive Lp –norm metric outperforms other robust metrics such as Huber’s metric, the Myriad metric, and the Meridian metric in both heavy–tailed and short–tailed noise.  116  Chapter 6 Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference33 6.1  Introduction  In Chapters 2 and 3, as is typical for the analysis and design of STCs, e.g. [3, 9, 10, 11, 13], we assumed that the only impairments of the received signal are fading and additive white Gaussian noise (AWGN). However, as mentioned in Chapter 1, practical wireless systems also often experience non–Gaussian noise and interference. Even if the noise is Gaussian, it may be correlated due to coupling effects in narrowly spaced receive antennas [110, 111]. In Chapters 4 and 5, we considered diversity combining techniques in systems with a single transmit antennas operating in non–Gaussian noise environments. In this chapter, we extend the analysis of Chapter 4 to STCs which are designed for systems with multiple transmit antennas. We present a novel, powerful, and unified framework for the calculation of the PEP of coherent and differential STCs in the high SNR regime when the received signal is impaired by correlated Ricean fading and (possibly) correlated non–Gaussian noise. We assume that the receiver has no or only limited knowledge about the statistical prop33  Parts of this chapter were published in the IEEE Transactions on Communications, Vol. 57, pp. 3353 - 3365, Nov. 2009, cf. Appendix F.  117  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference erties of the noise, and thus employs a Mahalonobis distance (MD) metric which includes the commonly used ED and noise ND metrics as special cases. The ED and ND metrics are optimum for AWGN and correlated Gaussian noise, respectively, but suboptimum for non–Gaussian noise. However, since the optimum ML metric for non–Gaussian noise requires knowledge of the complete noise distribution, which may be difficult to obtain in the dynamic and constantly changing environments of modern wireless systems, the simple ED and ND metrics are often applied even if the noise is not Gaussian. Similar to Chapters 4 and 5, the only restriction that we impose on the noise is that all of its moments exist, which makes our results applicable to a large number of practical scenarios. Unlike Chapter 4 in which the noise at the receiver was a random vector, the noise at the receiver of STCs is a matrix since the receiver processes the received signal after a complete block or frame is received. As a result, the PEP will depend on noise moments that are different than those derived in Chapter 4. Therefore, we will find new formulas for calculation of the noise moments. The novel PEP expressions that we will derive in this chapter can be combined with truncated union bounds to obtain tight approximations for the asymptotic BEP, SEP, FEP of arbitrary coherent and differential STBCs and coherent STTCs. As mentioned in Section 1.5.4, the current results on the performance of STCs in non– Gaussian noise, e.g., [36] and [84]–[87], are limited to the respective noise models. However, our results are more general and are applicable to many different types of noise including those in the aforementioned works. The rest of this chapter is organized as follows. In Section 6.2, the considered system model is introduced. In Section 6.3, the proposed asymptotic PEP expressions are developed and some implications for STC and metric design are discussed. The calculation of the moments of several relevant types of noise is addressed in Section 6.4. In Section 6.5, some examples are given to illustrate the application of the obtained analytical results,  118  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference and conclusions are drawn in Section 6.6. Definition: Here, we extend the definition of the moment of an RVV in Chapter 4. We define the pth moment of the quadratic form xH Xx as Mx (p, X)  E{(xH Xx)p },  where X and x are a Hermitian matrix and a complex random vector variable (RVV), respectively. If X is the X × X identity matrix, we obtain the moment defined in Chapter 4. Therefore, we use the short–hand notation Mx (p)  Mx (p, I X ) = E{(xH x)p }. We  recall that Mx (1) is the sum of the powers of the elements of x.  6.2  System Model  We consider a MIMO system with NT transmit34 and NR receive antennas. The transmitter is equipped with a space–time encoder which generates an N ×NT codeword matrix C[k] ∈ C, where N is the codeword (frame) length and C denotes the set of all codeword matrices. We assume that the N × NR received signal matrix at time k ∈ ZZ can be modeled as R[k] =  √  γ C[k]H T + Z T [k],  (6.1)  where H and Z[k] denote the NR × NT channel matrix and the NR × N noise matrix, respectively. We adopt the normalizations Mc (1) = N, Mh (1) = NT NR , and Mz (1) = NNR , where c[k]  vec{C[k]}, h  vec{H}, and z[k]  vec{Z[k]}, i.e., γ denotes the  average SNR per receive antenna. If a differential STBC is applied, N = NT holds and C[k] is obtained as C[k] = V [k]C[k − 1] from a unitary matrix V [k] ∈ V, where V is the set of all (differential) codeword matrices [11, 10]. For coherent and differential STCs we assume that H is constant over at least one and two consecutive codewords, respectively (quasi-static fading). 34  Note that in this chapter, unlike Chapters 2 and 3, the number of transmit antennas is arbitrary.  119  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference To simplify our notation, in the following, we will drop the time index k whenever possible. In the remainder of this section, we discuss the channel model, the noise model, and the adopted MD decision rule.  6.2.1  Channel and Noise Model  Element hnr nt  [H]nr nt of channel matrix H represents the channel gain from the nt th  ¯ + transmit antenna to the nr th receive antenna. We assume Ricean fading, i.e., H = H ˜ can be split into a line–of–sight (LOS) component H ¯ and a diffuse component H ˜ H ˜ nr nt having zero–mean Gaussian entries h  ˜ nr nt . Recall that the Ricean channel model [H]  ¯ = 0N ×N . For the diffuse contains the Rayleigh channel model as a special case with H R T ˜ we adopt the popular Kronecker correlation model [112]. Therefore, the component H covariance matrix of h is given by  Rhh ˜ where h  H  ˜h ˜ } = RT ⊗ R R , E{h  (6.2)  ˜ and RT and RR are the transmit–side and receive–side covariance vec{H},  matrices, respectively. As usual, we assume that noise matrix Z is independent from H and C. However, the entries of Z may be statistically dependent, non–circularly symmetric, and non–Gaussian. If the spatial and temporal correlation of the noise are independent, the autocorrelation matrix of z can be expressed as  Rzz  E{zz H } = Rzz,t ⊗ Rzz,s ,  (6.3)  where Rzz,t and Rzz,s denote the N × N temporal and the NR × NR spatial autocorrelation matrices, respectively, which are both assumed to have full rank. Furthermore, similar 120  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference to Chapters 4 and 5, throughout this chapter we assume that all joint moments of the elements of Z are finite. It is worth emphasizing that while most types of noise encountered in wireless systems meet this condition, cf. Section 6.4 for specific examples, there are types of noise with infinite moments (e.g. α–stable noise [105]) where the proposed analysis is not applicable, cf. Chapter 1.  6.2.2  Mahalonobis Distance (MD) Metric  We first consider the optimum ML decision rule for coherent detection which is given by ˆ = argmax C ˜ C  where r  log pz (r −  √  ˜ T }) γvec{CH  ,  (6.4)  vec{R}, pz (z) denotes the joint probability density function (pdf) of the ele-  ˆ ∈ C and C ˜ ∈ C denote the decoded and a hypothetical codeword ments of vector z, and C matrix, respectively. A similar ML decision rule can be derived for differential detection but is omitted here because of the space limitation. The ML decision rule has two main disadvantages. First, the ML decision rule requires knowledge of the noise pdf which may be difficult to obtain in practice. Second, depending on the type of noise, the computational complexity of the ML decision rule may be quite high. Both of these disadvantages are discussed more in detail for specific types of noise in Sections 6.4 and 6.5. In this chapter, to avoid the disadvantages of the optimum ML metric, we consider a simple, in general suboptimum MD metric [113]. For coherent and differential detection the MD decision rule is given by ˆ = argmin C ˜ C  ||Dt (R −  √ ˜ T T 2 γ CH )D s || ,  (6.5)  121  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference and Vˆ = argmin V˜  ||Dt (R[k] − V˜ R[k − 1])D Ts ||2 ,  (6.6)  respectively, where Vˆ ∈ V and V˜ ∈ V denote the decoded and a hypothetical differential STBC matrix, respectively. D t and D s are full–rank N ×N and NR ×NR matrices, respectively. The two most important special cases of the MD metric are (a) the conventional ED metric with D t = I N and D s = I NR and (b) the ND metric with Dt = (Rzz,t)−1/2 and Ds = (Rzz,s )−1/2 . It can be shown that the ED and ND metrics are equivalent to the optimum ML metric for AWGN [3, 11, 8] and correlated Gaussian noise [81], respectively. However, for non–Gaussian noise the MD metrics in (6.5) and (6.6) are suboptimum. The computational complexity of the MD decision rule is practically independent of D t and D s . Furthermore, the ED metric is independent of the statistical properties of the noise and the ND metric requires only knowledge of the noise correlation matrices Rzz,t and Rzz,s , which are much easier to estimate than the noise pdf. Therefore, ED and ND metrics are pragmatic choices for practical receivers regardless of the type of noise present. We note that the performance loss of these suboptimum metrics compared to the ML metric strongly depends on the type of noise.  6.3  Asymptotic Analysis of Space–Time Codes  In this section, we develop asymptotic expressions for the PEP of coherent and differential STCs, and discuss their relevance for STC and metric optimization.  6.3.1  Asymptotic PEP of Coherent STCs  The PEP Pe (E) is a function of the codeword difference matrix E  ˆ where C ∈ C C − C,  ˆ ∈ C denote the transmitted and the detected codewords, respectively. Conditioned and C 122  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference on Z, the PEP of coherent STCs can be calculated from (6.5) as √ Pe (E|Z) = Pr{||Dt ( γEH T + Z T )D Ts ||2 < ||Dt Z T D Ts ||2 |Z} ||˜ z||2  = Pr{∆ < ||˜ z ||2|Z} =  p∆ (x) dx,  (6.7)  0  √ √ where p∆ (x) is the pdf of ∆ ||Dt ( γEH T +Z T )DTs ||2 = ||v||2 , v vec{ γD s HE T DTt }+ √ ˜ is a Gaussian RVV, and z ˜ vec{D s ZDTt } = (D t ⊗ D s )z [114]. ˜ = γ(Dt E ⊗ D s )h + z z The Laplace transform Φ∆ (s)  E{e−s∆ |Z} of p∆ (x) can be obtained using Turin’s formula  [48] as Φ∆ (s) = ¯ where v  √  ¯ exp −s¯ v H (I N NR + sRvv )−1 v , det(I N NR + sRvv )  ¯ z with h ¯ γ(Dt E⊗D s ) h+˜  ¯ and Rvv vec{H}  γΓ with Γ  (6.8) D t ERT E H D H t ⊗  D s RR D H s denote the conditional mean and covariance matrix of v, respectively. Since both D t and D s are assumed to have full rank, Γ is a positive semi–definite matrix which has rt rr non–zero positive and NNR − rt rr zero eigenvalues, where rt and rr  rank{RR }. Consequently, I N NR + sRvv can be decomposed as I N NR + sRvv = U ΨU H   0rt rr ×(N NR −rt rr )   I rt rr + sγΛ Ψ  , I N NR −rt rr 0(N NR −rt rr )×rt rr  where Λ  rank{ERT E H }  (6.9) (6.10)  diag{λ1 , λ2 , . . . , λrt rr } and U contain the rt rr non–zero eigenvalues and the  eigenvectors of Γ, respectively. We decompose the unitary matrix U into U = [U 1 U 2 ], where U 1 and U 2 contain the rt rr eigenvectors corresponding to the non–zero eigenvalues and the NNR −rt rr eigenvectors corresponding to the zero eigenvalues, respectively. Based on the above considerations, using the Neumann series [115] we can express (I N NR +  123  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference sRvv )−1 as H −(1+δ) (I N NR + sRvv )−1 = (γs)−1 U 1 Λ−1 U H Ω, 1 + U 2 U 2 + o (sγ)  (6.11)  where Ω is a finite–norm matrix, which is independent of γ, and 0 < δ < 1. Similarly, we can express the inverse determinant of I N NR + sRvv as rt rr  det(I N NR + sRvv )  −1  (1 + sγλi )  =  −1  rt rr −1  =  i=1  λi  (sγ)−rt rr + o (sγ)−(rt rr +δ) . (6.12)  i=1  ¯E Furthermore, using (6.11) and the short–hand notation h  ¯ we can express (Dt E ⊗D s )h  the argument of the exponential function in (6.8) as −a − sb + o((sγ)−δ ) with √ ¯ ˜/ γ U 1 Λ−1 U H 1 hE + z  a  ¯ E + z˜ /√γ) h  H  b  √ ¯ ˜ γ hE + z  U 2U H 2  H  √ ¯ ˜ = z˜ H U 2 U H ˜, γ hE + z 2 z  (6.13) (6.14)  where we have exploited the identity (D t E ⊗ D s )H U 2 = 0N NR ×(N NR −rt rr ) in (6.14). Combining now (6.8) and (6.12)–(6.14) and considering the high SNR case, we obtain for Φ∆ (s) the asymptotic expression . Φ∆ (s) =  exp(−a − s b) . t rr λi ) (sγ)rt rr ( ri=1  (6.15)  Applying the inverse Laplace transform to Φ∆ (s) in (6.15) leads to . exp(−a)(x − b)rt rr −1 u(x − b) p∆ (x) = , t rr λi ) (rt rr − 1)!γ rt rr ( ri=1  (6.16)  where u(x) denotes the unit step function. Combining (6.7) and (6.16) yields r r  z ||2 − b) t r . exp(−a) (||˜ , Pe (E|Z) = t rr λi ) (rt rr )!γ rt rr ( ri=1  (6.17)  124  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference and the unconditional asymptotic PEP can be expressed as H . E exp(−a) z˜ X z˜ Pe (E) = t rr λi ) (rt rr )!γ rt rr ( ri=1  with X  (6.18)  H I N NR − U 2 U H 2 = U 1 U 1 . Using the expansion exp(x) =  ∞ k k=0 x /k!  and (6.13),  we can rewrite the exponential function in (6.18) as H  ¯ ¯ U 1 Λ−1 U H h exp(−a) = exp −h 1 E E  √ (1 + f (z)/ γ) ,  (6.19)  where f (z) can be written as a sum of products of the form Cκ1 ,ν1 ,··· ,κNNR ,νNNR z1κ1 (z1∗ )ν1 · · · κNN  νNNR ∗ , where Cκ1 ,ν1,··· ,κNNR ,νNNR are coefficients that are non–increasing in γ, zl , zN NRR (zN NR )  1 ≤ l ≤ NNR , denote the elements of z, and κl ≥ 0 and νl ≥ 0 are integers. Assuming now √ ˜ f (z)} < ∞), for γ → ∞, E{˜ ˜ f (z)}/ γ that all moments of z exist (i.e., E{˜ zH X z zH X z becomes negligible in (6.18) and we obtain for the PEP the simple asymptotic expression α Mz (rt rr , (Dt ⊗ Ds )H X(D t ⊗ D s )) α My (rt rr ) . α Mz˜ (rt rr , X) = = , Pe (E) = γ rt rr γ rt rr γ rt rr with α and y  ¯ H (D t E ⊗ D s )H U 1 Λ−1 U H (D t E ⊗ Ds )h ¯ exp −h 1 (rt rr )!  rt i=1  λi (D t ERT E H D H t )  rr  rr i=1  λi (Ds RR D H s )  rt  (6.20)  (6.21)  UH 1 (D t ⊗ D s )z. In (6.21) we exploited the fact that because of the special  H structure of Γ, λi = λµ (D t ERT E H DH t )λν (D s RR D s ), 1 ≤ i ≤ rt rr , 1 ≤ µ ≤ rt , 1 ≤ ν ≤  rr , holds. Eq. (6.20) can be viewed as a generalization of existing asymptotic results for AWGN and the ED metric in e.g. [13] to non–AWGN channels and the MD metric. We note from (6.20) that the type of noise impacts the asymptotic PEP only via the moment My (rt rr ). Furthermore, for rt and rr the inequalities 1 ≤ rt ≤ min{rank{E}, rank{RT }} ≤ min{N, 125  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference NT } and 1 ≤ rr ≤ NR hold. For the special case of full–rank square STBCs with N = NT and full–rank transmit and receive covariance matrices, we obtain rt = NT , rr = NR , and U = U 1 . Therefore, in this case, X = I NT NR and (6.20) and (6.21) can be simplified to H H α My (NT NR ) . α Mz (NT NR , Dt D t ⊗ D s D s ) Pe (E) = = , N N T R γ γ NT NR  (6.22)  with H  α  ¯ (R−1 ⊗ R−1 )h ¯ exp −h T R  NR [det(D D H )det(R )]NT (NT NR )![det(EE H )det(Dt D H s R s t )det(RT )]  .  (6.23)  Therefore, for full–rank square STBCs (such as Alamouti’s code) the code does not affect H the way in which the noise statistics impact the PEP because Mz (NT NR , DH t Dt ⊗ D s Ds)  is independent of the code. In contrast, for codes with N > NT (e.g. STTCs) this is not the case since My (rt rr ) also depends on the code via U 1 . If we specialize (6.22) and (6.23) to NT = 1 and Ds = I NR (ED metric), as expected, we obtain (4.9) and (4.10), which were derived for single–antenna transmission and MRC.  6.3.2  Asymptotic PEP of Differential STBCs  For differential STBCs we define the codeword error matrix as E  V − Vˆ , where V ∈ V  and Vˆ ∈ V denote the transmitted and the detected codeword matrix, respectively. Based on the MD decision rule in (6.6), we can express the conditional PEP Pe (E|Z[k], Z[k − 1]) √ γ(D t EC[k − 1] ⊗ for differential STBCs as in (6.7). However, now the definitions v ¯, z ¯ D s )h + z  T ˜ vec{Ds (Z[k] − Z[k − 1]Vˆ )D Tt }, and z  vec{Ds (Z[k] − Z[k − 1]V T )DTt }  are valid. For simplicity and practical relevance we assume that both RT and RR have full rank. Repeating the same steps as for coherent STCs in the previous section and exploiting 126  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference C[k − 1]C H [k − 1] = I NT we obtain for differential STBCs the asymptotic PEP H α MzD (NT NR , DH . α My D (NT NR ) t Dt ⊗ Ds Ds) Pe (E) = = , γ NT NR γ NT NR  where α is given by (6.23) and y D = z˜ = (Dt ⊗D s )z D with z D  (6.24)  z[k]−(V ⊗I NR )z[k −1].  We note that for the special case of NT = 1 transmit antenna and D s = I NR (ED metric), (6.24) gives the asymptotic PEP of differential PSK with EGC and is equivalent to (4.13) in Chapter 4.  6.3.3  Implications for STC and Metric Design  Comparing (6.20) and (6.21) with (1.16), we obtain the following expressions for the coding gain and diversity gain of coherent STCs  G d = rt rr  and  Gc [dB] =  10 10 log10 (α) + log10 (My (rt rr )). rt rr rt rr  (6.25)  Eq. (6.25) is also valid for differential STBCs if we set rt rr = NT NR and replace My (rt rr ) by My D (NT NR ). In the following, we will discuss the implications of (6.25) for STC and metric design. Coherent STC Design: Since on a double logarithmic scale Gd determines the slope of the PEP curve, the primary goal of most STC designs is to maximize the diversity gain by maximizing rt [3]. Since Gd is independent of the type of noise, this rank criterion is valid for all types of noise as long as the MD metric is used for detection35 . Therefore, our findings here generalize the results of [3] and [36] which showed the importance of the rank criterion for AWGN and impulsive noise, respectively. The secondary STC design goal is usually the maximization of the coding gain [3]. 35  We note that throughout this chapter “all types of noise” refers to all types of noise with finite moments, cf. Section 6.2.1.  127  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference Eq. (6.25) shows that the coding gain consists of two parts. The first term on the right hand side (RHS) of the coding gain in (6.25) depends on the STC design but is independent of the type of noise. In contrast, the second term depends on the type of noise via My (rt rr ) and potentially depends on the STC design via U 1 . An exception is the class of full–rank H square STBCs where My (rt rr ) = Mz (NT NR , DH t D t ⊗ D r D r ) holds for all types of noise  and the second term on the RHS of the coding gain can be ignored for STC optimization. This means that full–rank square coherent STBCs optimized for AWGN are also optimum for any other type of noise. In contrast, codes with N > NT optimized for AWGN are not necessarily optimum for other types of noise. If My (rt rr ) is known, the expression in (6.25) can be used for code search. We note, however, that the numerical evaluation of the asymptotic PEP has shown that the performance difference between different STCs in non–AWGN channels is practically identical to that in AWGN channels even if N > NT , cf. Figs. 6.3, 6.8. This suggests that STCs optimized for AWGN are also close to optimum for non–AWGN channels. Differential STBCs: Since in general y D depends on V , My D (NT NR ) generally depends on the STC design. Therefore, differential STBCs optimized for AWGN are not necessarily optimum for non–AWGN channels. However, similar to the coherent case, the numerical evaluation of the PEPs for different differential STBCs suggests that differential STBCs optimized for AWGN are also close to optimum for non–AWGN channels. It is interesting to compare the performance of coherent and differential STBCs. In particular, considering group codes where C = V is valid [11], the performance loss entailed by differential detection is given by  LD  My D (NT NR ) My (NT NR )  1/(NT NR )  ,  (6.26)  cf. (6.22), (6.24), which will be evaluated for specific types of noise in Section 6.4. 128  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference MD Metric Design: The diversity gain Gd in (6.25) is independent of matrices D t and D s . Therefore, any MD metric with full–rank matrices D t and Ds achieves the full diversity gain offered by the channel and the STC. However, the coding gain Gc does depend on the specific form of the MD metric and ideally D t and Ds should be optimized to maximize Gc for a given STC. For general types of noise, general Ricean fading, and general STCs this is a difficult task since both α and My (rt rr ) depend on D t and Ds in a relatively complicated manner. In this chapter, we are mainly interested in a comparison of the ND metric and the ED metric for full–rank square STBCs. For this purpose we define the coding gain advantage  GN D  Mz (NT NR ) −1 N R [det(Rzz,t )] [det(Rzz,s )]NT Mz (NT NR , R−1 zz,t ⊗ Rzz,s )  1/(NT NR )  (6.27)  that can be achieved with the ND metric compared to the ED metric. Due to its simplicity, in practice, the ED metric is often applied even if the noise is correlated. Therefore, it is interesting to investigate the performance loss  LCO  Mz (NT NR )|correlated Mz (NT NR )|uncorrelated  1/(NT NR )  (6.28)  suffered by the ED metric if the elements of z are correlated. We will evaluate GN D and LCO for various types of noise in Section 6.4.  6.3.4  BEP, SEP, and FEP  As mentioned in the previous chapters, the asymptotic PEP expressions in (6.20), (6.22), and (6.24) can be combined with truncated union bounds to arrive at tight asymptotic approximations for the BEP, SEP, and FEP. Here, we just note that we found the number  129  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference of dominant error events36 , which have to be included in the truncated union bound to obtain a tight approximation, to be independent of the type of noise.  6.4  Calculation of the Noise Moments  The general PEP expressions for coherent STCs (6.20) and differential STBCs (6.24) require the calculation of the moments My (rt rr ) and My D (NT NR ), respectively. For general highly unstructured types of noise such as UWB interference, the moments may be calculated using a Monte–Carlo approach. For example, we can estimate My (rt rr ) as 1 My (rt rr ) ≈ Ne  Ne  i=1  H H rt rr (z H , i (D t ⊗ D s ) U 1 U 1 (D t ⊗ D s )z i )  (6.29)  where z i denotes the ith realization of the noise process and Ne is the number of realizations. However, while (6.29) is easy to compute and allows for a quick and accurate asymptotic performance evaluation, it offers little insight into the problem. Furthermore, note that for the relevant case N > 1 the methods provided for moment calculation in Section 4.4 for scalar modulation are not applicable. Fortunately, for specific types of noise closed–form expressions for My (rt rr ) and My D (NT NR ) can be obtained. Some important examples are discussed in the following. Furthermore, we also briefly discuss ML detection for the considered non–Gaussian types of noise.  6.4.1  Additive White Gaussian Noise (AWGN)  It is instructive to consider the AWGN case. Since the ED metric is optimum for AWGN, we assume Dt = I N and D s = I NR . 36  The “dominant” error events are the error events whose corresponding PEPs are significantly larger than those of the remaining (non–dominant) error events.  130  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference Coherent STCs: If z is an AWGN vector, y = U H 1 z is also an AWGN vector of length rt rr since the columns of U 1 are orthogonal. The elements of y have variance σy2 = 1. Interestingly, for AWGN My (rt rr ) = (2rt rr − 1)!/(rt rr − 1)!, cf. Table 6.1, is independent of X and the provided PEP expressions can be simplified to expressions known from the literature. For example, it can be verified that for full–rank E, full–rank RT and RR , Rayleigh fading, and AWGN, (6.22) and (6.23) are equivalent to the asymptotic PEP expression in [13, Proposition 3]. Differential STBCs: Since V is a unitary matrix, z D is also an AWGN vector of length NT NR whose elements have variance σz2D = 2. Therefore, My D (NT NR ) in (6.24) is given by My D (NT NR ) = 2NT NR My (NT NR ). Consequently, for AWGN the asymptotic performance loss (6.26) entailed by differential detection is LD = 3 dB as expected, cf. e.g. [11]. Note that this result is valid for any differential space–time group code, any number of transmit and receive antennas, and arbitrarily correlated Ricean fading.  6.4.2  Additive Correlated Gaussian Noise (ACGN)  Zero–mean ACGN may be caused by coupling effects in narrowly spaced receive antennas [110, 111]. Coherent STCs: If z is ACGN with autocorrelation matrix Rzz , y is ACGN with auH tocorrelation matrix Ryy = U H 1 (D t ⊗ D s )Rzz (D t ⊗ D s ) U 1 . After tedious but straight-  forward calculations we obtain for the moment My (rt rr ) the expression given in Table 6.1, where ηi , 1 ≤ i ≤ rt rr denote the eigenvalues of Ryy . Assuming that spatial and temporal noise correlation are independent, cf. (6.3), the coding gain advantage (6.27) of the ND metric (which is equivalent to the optimum ML metric) for full–rank square STBCs can  131  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference be obtained as kN     2NT NR − 1 GN D =   NT NR  −1  k1 +...+kNT NR =NT NR  [det(Rzz,t  ξ1k1 · . . . · ξNTTNRR  )]NR [det(R  zz,s  N  )]NT  1/(NT NR )     ,  (6.30)  where ξi , 1 ≤ i ≤ NT NR , denotes the eigenvalues of Rzz . To gain more insight, we may consider the special case Rzz,s =  1 ρs ρs 1  and Rzz,t = I 2 , where ρs , −1 ≤ ρs ≤ 1, denotes  the spatial correlation coefficient. In this case, (6.30) simplifies to GN D (ρs ) = ([5(µ2 + 1/µ2) + 8(µ + 1/µ) + 9]/35)1/4 ,  where µ  (6.31)  (1 − ρs )/(1 + ρs ). From (6.31) we obtain GN D (0) = 0 dB (i.e., no gain since  ND metric simplifies to ED metric), GN D (0.9) = 4.4 dB, and limρs →±1 GN D (ρs ) → ∞ (i.e., perfect noise cancellation and error–free reception is possible for fully correlated noise). This shows that noise correlation is beneficial as long as it is exploited with the appropriate metric. On the other hand, if the ED metric is used, ACGN entails a performance loss of   LCO =   2NT NR − 1 NT NR  −1  kN  1/(NT NR )  ξ1k1 · . . . · ξNTTNRR   k1 +...+kNT NR =NT NR  N  (6.32)  compared to AWGN, cf. (6.28). Specializing (6.32) to spatially correlated, temporally uncorrelated ACGN with N = NT = NR = 2, we obtain LCO (ρs ) = ([35 + 42ρ2s + 3ρ4s ]/35)1/4 ,  (6.33)  which yields LCO (0) = 0 dB, LCO (0.9) = 0.77 dB, and LCO (±1) = 0.9 dB.  132  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference Differential STBCs: y D = (D t ⊗ D s )(z[k] − (V ⊗ I NR )z[k − 1]) is an ACGN vector. Consequently, the noise moment My D (NT NR ) can also be calculated using the result for ACGN from Table 6.1 once the corresponding autocorrelation matrix RyD yD  E{y D y H D}  has been determined. It can be shown that for general ACGN with temporal correlation RyD yD = 2Ryy holds. Consequently, for temporally correlated ACGN the performance loss LD caused by differential detection is different from 3 dB. In contrast, for temporally H uncorrelated ACGN (i.e., Rzz,t = I N ) RyD yD = 2Ryy = 2(D t D H t ⊗D s Rzz,s D s ) holds and  LD = 3 dB results from (6.26) and Table 6.1. This result is a generalization of a result in [11], which predicts this 3 dB loss for AWGN and the ED metric, to ACGN and arbitrary MD metrics. p Table 6.1: Moments My (p) E{(y H y)p } and My D (p) E{(y H D y D ) } relevant for coherent (“C”) and differential (“D”) STCs, respectively. All variables in this table are defined in Section 6.4.  Model for z  Moments My (p) and My D (p) (2p−1)! (p−1)!  AWGN ACGN  k  p! k1 +...+kp =p  η1k1 · . . . · ηp p  I NNR  SIGMN (C)  k  αi  p!  σy2p  i=1  k1 +...+kp =p  η1k1 [i] · . . . · ηp p [i]  I NNR I NNR  SIGMN (D)  k  αi αj  p! i=1  j=1  k1 +...+kp =p  IN  SDGMN (C)  p!  k  αi i=1  k1 +...+kp =p  IN IN  SDGMN (D) Asynch. CCI (C) Asynch. CCI (D)  p!  η1k1 [i, j] · . . . · ηp p [i, j]  αi αj  η1k1 [i] · . . . · ηp p [i] k  η1k1 [i, j] · . . . · ηp p [i, j]  i=1 j=1 k1 +...+kp =p k p! η1k1 (b) · . . . · ηp p (b) M ILb b∈BILb k1 +...+kp =p k p! η1k1 (bD ) · . . . · ηp p (bD ) M I(Lb +N) bD ∈BI(Lb +N) k1 +...+kp =p  133  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference  6.4.3  Spatially Independent Gaussian Mixture Noise (SIGMN)  SIGMN models impulsive or man–made noise for the case where the impulsive phenomenon impacts the antennas independently [31]. We note that SIGMN includes Middleton’s Class A noise and –mixture noise as special cases. For SIGMN the pdf of the noise znr [n] at receive antenna nr , 1 ≤ nr ≤ NR , at time n, 0 ≤ n < N, is given by I  pz (znr [n]) = i=1  where ci > 0,  I i=1 ci  ci |znr [n]|2 exp − πσi2 σi2  ,  (6.34)  = 1, and σi2 , 1 ≤ i ≤ I, are constants. For the important special  case of –mixture noise with variance σz2 , I = 2, c1 = 1 − , c2 = , σ12 = σz2 /(1 − + κ ), and σ22 = κσ12 with 0 <  < 1 and κ > 1. Conditioned on state i, 1 ≤ i ≤ I, znr [n] is a  Gaussian RV with variance σi2 . Since SIGMN is spatially and temporally uncorrelated, we only consider the ED metric in this section, i.e., Dt = I N and D s = I NR . Coherent STCs: Since vector z  [z1 [0] z2 [0] . . . zNR [0] z1 [1] . . . zNR [N − 1]]T has NNR  independent elements and each element can have I different variances, there are I N NR different possible states for z. Conditioned on state i, 1 ≤ i ≤ I N NR , z is a Gaussian RVV with diagonal covariance matrix Σi  E{zz H |i}. The NNR main diagonal elements  of the set of covariance matrices Σi , 1 ≤ i ≤ I N NR , contain all I N NR possible different combinations of variances σi2 , 1 ≤ i ≤ I. Consequently, conditioned on state i, vector y is also a Gaussian RVV with rt rr × rt rr covariance matrix Ri = U H 1 Σi U 1 , and the pdf of y can be written as I NNR  py (y) = i=1  where αi =  I km m=1 cm  αi exp −y H R−1 i y , π NT NR det(Ri )  2 and km is the number of times σm is contained in Σi , i.e.,  (6.35)  I m=1  km =  NNR holds. Based on (6.35) the moment My (rt rr ) can be calculated as given in Table 6.1, 134  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference where ηm [i], 1 ≤ m ≤ rt rr , denotes the eigenvalues of Ri . Differential STBCs: The pdf of y D = z[k] − (V ⊗ I NR )z[k − 1] can be expressed as I NNR I NNR  py D (y D ) = i=1  j=1  αi αj N N T R det(R π  ij )  −1 exp −y H D Rij y D ,  where i and j denote the states of z[k] and z[k−1], respectively, and Rij  (6.36)  E{y D y H D |i, j} =  Σi + (V ⊗ I NR )Σj (V ⊗ I NR )H . Thus, conditioned on i and j, y D is a Gaussian RVV and My D (rt rr ) can be calculated as shown in Table 6.1, where ηm [i, j], 1 ≤ m ≤ rt rr , denotes the eigenvalues of Rij . ML Detection: The ML metric required for coherent ML detection in (6.4) is given by log(pz (z)) =  NR nr =1  N −1 n=0  log(pz (znr [n])), where pz (znr [n]) is given in (6.34). Therefore,  assuming it is known that the underlying type of noise is SIGMN (which may not be the case in practice), the coherent ML metric still requires estimation of ci and σi2 , 1 ≤ i ≤ I. Furthermore, the coherent ML metric contains logarithms of sums of exponential functions which may give rise to numerical problems. Finally, if orthogonal STBCs with K M– ary scalar symbols per STBC matrix are used, unlike the ED metric [9], the ML metric cannot be decomposed into K independent metrics exploiting the orthogonality of the code. Therefore, ML detection requires M K /K metric computations per scalar symbol decision, whereas ED detection requires only M metric computations per scalar symbol decision [9]. Similar statements apply to the differential case.  6.4.4  Spatially Dependent Gaussian Mixture Noise (SDGMN)  SDGMN models impulsive or man–made noise for the case where the impulsive phe  135  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference nomenon impacts all antennas concurrently [31]. The corresponding pdf is given by [36] I  pz (z[n]) = i=1  ci ||z[n]||2 exp − σi2 π NR σi2NR  ,  (6.37)  [z1 [n] . . . zNR [n]]T contains the noise variables of all receive antennas at time  where z[n]  n, 0 ≤ n < N, and ci > 0,  I i=1 ci  = 1, and σi2 , 1 ≤ i ≤ I, are again constants. Eq. (6.37)  reveals that conditioned on the state i, 1 ≤ i ≤ I, all antennas are affected by i.i.d. Gaussian noise, where, in contrast to the SIGMN case, the noise variances at all antennas are identical. Thus, the unconditional SDGMN is spatially dependent but spatially and temporally uncorrelated. Therefore, we focus again on the ED metric and assume D t = I N and Ds = I NR . Coherent STCs: SDGMN can be treated in a similar way as SIGMN. The main difference is that now there are only I N different correlation matrices Σi , 1 ≤ i ≤ I N , for z since all znr [n] belonging to the same time instant n, 0 ≤ n < N, have the same variance. Therefore, the pdf of y is still given by (6.35) if we replace I N NR by I N . αi is still given by αi = i.e.,  I km m=1 cm I m=1  2 but now km is the number of times σm is contained in Σi divided by NR ,  km = N holds. The moment My (rt rr ) for SDGMN is also given in Table 6.1.  Differential STBCs: Using a similar approach as for SIGMN, we obtain for MyD (rt rr ) for SDGMN the expression given in Table 6.1, where ηm [i, j], 1 ≤ m ≤ rt rr , denotes the eigenvalues of Rij = Σi + (V ⊗ I NR )Σj (V ⊗ I NR )H , 1 ≤ i, j ≤ I, with Σi as specified in this section. ML Detection: In this case, the coherent ML metric in (6.4) is given by log(pz (z)) = N −1 n=0  log(pz (z[n])) with pz (z[n]) from (6.37). Regarding the computational complexity,  similar statements as in the SIGMN case apply.  136  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference  6.4.5  Asynchronous Co–channel Interference (CCI)  The impairment caused by I Rayleigh–faded asynchronous CCI signals can be modeled as ku  I  gnr [i]  znr [n] = i=1  k=−kl  fi [k]bi [n − k],  1 ≤ nr ≤ NR ,  (6.38)  where fi [k] (−kl ≤ k ≤ ku , kl , ku ≥ 0), bi [k] ∈ B, and gnr [i] denote the effective impulse response, i.i.d. symbols belonging to an M–ary symbol alphabet B, and the fading gain of the ith CCI signal, respectively. The fading gain vectors g i are modeled as i.i.d. Gaussian RVVs with Rgg given by fi [k]  [g1 [i] . . . gNR [i]]T , 1 ≤ i ≤ I,  E{gi g H i }. The CCI impulse responses are  f (kT + τi ), where f (t), T , and τi denote the convolution of the interferer  transmit pulse shape and the receiver input filter impulse response of the user, the symbol duration of the user, and the delay between the ith interferer and the user, respectively. Coherent STCs: Based on (6.38) the noise vector z can be represented as I  z= i=1  (F i bi ) ⊗ g i ,  (6.39)  where the jth row of the N×Lb matrix F i is given by [01×(j−1) fi [−kl ] fi [−kl +1] . . . fi [ku ] 01×(N −j) ], [bi [−ku ] bi [−ku + 1] . . . bi [N − 1 + kl ]]T has length  1 ≤ j ≤ N, and CCI vector bi Lb  N + ku + kl . Conditioned on b  [bT1 . . . bTI ]T , y = U H 1 (D t ⊗ D s )z is a zero–mean  Gaussian RVV with rt rr × rt rr covariance matrix I  Ryy (b)  H  E{yy |b} =  UH 1 i=1  H H H D t (F i bi bH i F i )D t ⊗ D s Rgg D s  U 1.  (6.40)  Consequently, the moment My (rt rr ) for asynchronous CCI can be obtained as given in Table 6.1, where ηm (b), 1 ≤ m ≤ rt rr , denotes the eigenvalues of Ryy (b). The outer sum in the expression for My (rt rr ) in Table 6.1 is over all M ILb possible vectors b. If the 137  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference received signal is impaired by both CCI and AWGN, My (rt rr ) can still be calculated using the formula provided in Table 6.1 after replacing ηm (b) by ηm (b) + σn2 , 1 ≤ m ≤ rt rr , where σn2 denotes the variance of the AWGN. The autocorrelation matrix of z is given by Rzz = ( the ND metric we obtain D t = (  I i=1  I i=1  F iF H i ) ⊗ Rgg . Therefore, for  −1/2 F iF H and D s = R−1/2 i ) gg . To gain more insight,  we consider the special case I = 1 and N = NT = NR = 2 in the following and assume Rgg =  1 ρs ρs 1  , where −1 ≤ ρs ≤ 1. In this case, for the ED metric the eigenvalues of  H H Ryy (b) are given by η1 (b) = (1 − ρs )bH F H 1 F 1 b, η2 (b) = (1 + ρs )b F 1 F 1 b, and η3 (b) = H −1 η4 (b) = 0, and for the ND metric we obtain η1 (b) = η2 (b) = bH F H 1 (F 1 F 1 ) F 1 b and  η3 (b) = η4 (b) = 0. Using this result for calculation of the appropriate moments, with the help of Table 6.1, the coding gain advantage (6.27) of ND detection compared to ED detection is    GN D (ρs ) =   (µ2 + 1/µ2 + µ + 1/µ + 1) 2 5[det(F 1 F H 1 )]  (b b∈BLb  H  4 (bH F H 1 F 1 b)  b∈BLb H −1 4 F 1 (F 1 F H 1 ) F 1 b)  1/4    ,  (6.41)  where µ = (1 − ρs )/(1 + ρs ). Interestingly, we obtain limρs →±1 GN D (ρs ) → ∞ for spatially fully correlated CCI independent of F 1 . Thus, (6.41) clearly shows that – though not optimum – the ND metric can achieve substantial performance gains compared to the simpler ED metric also for CCI. If the CCI is synchronous (τ1 = 0, kl = ku = 0), F 1 = I 2 holds, and (6.41) simplifies to GN D (ρs ) = [(µ2 + 1/µ2 + µ + 1/µ + 1)/5]1/4 ,  (6.42)  which yields GN D (0) = 0 dB (i.e., no gain for spatially uncorrelated synchronous CCI) and GN D (0.9) = 4.7 dB. Furthermore, under the same assumptions, the performance  138  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference degradation caused by the noise correlation if the ED metric is applied can be obtained as LCO (ρs ) = [(5 + 10ρ2s + ρ4s )/5]1/4 ,  (6.43)  which can be specialized to LCO (0) = 0 dB, LCO (0.9) = 1.1 dB, and LCO (±1) = 1.3 dB. A comparison of (6.33) and (6.43) shows that spatial noise correlation is even more harmful for CCI than for ACGN if it is not properly taken into account in the decoding metric. Differential STBCs: We obtain for zD the expression I  zD = i=0  where bi  (F i bi [k] − V F i bi [k − 1]) ⊗ g i ,  (6.44)  [bi [Nk − ku ] bi [Nk − ku + 1] . . . bi [Nk + N − 1 + kl ]]T . Conditioned on bD  [bi [N(k − 1) − ku ] bi [N(k − 1) − ku + 1] . . . bi [Nk + N − 1 + kl ]]T ∈ BI(Lb +N ) , y D is again a Gaussian RVV with autocorrelation matrix I  RyD yD (bD ) = i=1  H Dt (F i bi [k] − V F i bi [k − 1])(F i bi [k] − V F i bi [k − 1])H DH t ⊗ D s Rgg D s ,  (6.45) and the relevant moment My D (rt rr ) can be calculated as shown in Table 6.1, where ηm (bD ), 1 ≤ m ≤ NT NR , denotes the eigenvalues of RyD yD (bD ). To gain more insight into the performance loss entailed by differential detection, we consider again the case of one synchronous CCI signal and D t = I N . Assuming |b[k]| = 1 (i.e., M–ary PSK modulation) and exploiting the special structure of Ryy (b) and RyD yD (bD ) for I = 1, respectively, leads to LD =    1  1 N M 2N  bD ∈B2N  1/(NT NR )  ||b[k] − V b[k − 1]||2NT NR   ,  (6.46)  cf. (6.26) and Table 6.1, which holds for arbitrary NT = N, NR , full–rank Ds , and Rgg .  139  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference Since V is unitary, LD in (6.46) can be bounded as LD ≤ 4, i.e., the performance loss caused by differential detection does not exceed 6 dB. As an example, we consider diagonal STBCs, where V is a diagonal matrix with unit magnitude main diagonal elements [11], and assume that M  1. In this case, the sum in (6.46) can be approximated by a double  integral and we obtain for N = 2  LD =  LD =    1 1 2 (2π)2   1 1 2 (2π)2  2π 2π  0  0  [4 − 2(cos ϕ1 + cos ϕ2 )]2 dϕ1 dϕ2  =  2π 2π  0  0  1/2  √  1/4  [4 − 2(cos ϕ1 + cos ϕ2 )]4 dϕ1 dϕ2  =  5;  NR = 1, (6.47)  13/2; NR = 2, (6.48)  i.e., differential detection causes a performance loss of 3.5 dB and 4.1 dB for NR = 1 and NR = 2 receive antennas, respectively. ML Detection: For asynchronous CCI pz (z) is a sum of M ILb Gaussian pdfs with autocorrelation matrices  I i=1  H ILb F i bi bH different possible i F i ⊗ Rgg corresponding to the M  values of bi , cf. (6.39). Therefore, even if it is known that the underlying noise is asynchronous CCI, matrices F i , 1 ≤ i ≤ I, Rgg , and the M ILb different vectors bi have to be estimated for ML detection. In contrast, for the ND metric only the correlation matrix Rzz is required, which can be easily estimated without any prior knowledge about the underlying type of noise.  6.5  Numerical Results and Discussions  In this section, we verify the derived analytical expressions and show their versatility by comparing them with computer simulations for several practically relevant scenarios. The provided asymptotic BEP, SEP, and FEP approximations were obtained by combining the asymptotic PEP expressions with truncated union bounds, cf. Section 6.3.4. The 140  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference noise moments required for evaluation of the asymptotic PEPs were computed with the expressions given in Table 6.1 and the error probabilities are shown as functions of the symbol SNR per receive antenna unless stated otherwise. For all results involving CCI, we assume that both transmitter and receiver use square–root raised cosine filters with roll–off factor 0.22. For most STCs and types of noise considered in this section, we found the ML metric to be computationally too complex and/or numerically unstable. Therefore, except for Fig. 6.4, we only show results for the ED and ND metrics. Unless stated otherwise, we assume that the noise correlation matrix is perfectly known if the ND metric is applied. Coherent STCs with ED Metric: First, we consider in Fig. 6.1 the SEP of the orthogonal half–rate STBC (N = 8, NT = 4) from [8, Eq. (38)] with 4–PSK modulation for NR = 1 receive antenna and i.i.d. Rayleigh fading. We consider impairment by AWGN, –mixture noise (a special case of SIGMN, cf. Section 6.4.3), and asynchronous 4–PSK CCI with I = 1 and τ1 = T /4. Fig. 6.1 validates the presented asymptotic analysis and shows that all types of noise lead to a diversity gain of Gd = 4 but different coding gains as expected from Section 6.3.3. Furthermore, we observe that the impulsive –mixture noise is more detrimental to system performance than AWGN and CCI. Thereby, the performance is worse for  = 0.1 than for  = 0.25 since –mixture noise is more impulsive for smaller  . We note that for the non–Gaussian types of noise considered in Fig. 6.1, ML detection (not shown) would require 44 /4 = 64 (computationally expensive) metric evaluations per 4–PSK symbol decision, whereas ED detection requires only 4 (simple) metric evaluations per symbol decision, cf. Section 6.4.3, [9]. Fig. 6.2 shows the SEP of the orthogonal rate–3/4 STBC (N = 4, NT = 3) from [9, Eq. (4.104)] with 16–ary quadrature amplitude modulation (16–QAM) modulation for NR = 1 receive antenna, i.i.d. Rayleigh fading, and impairment by Rayleigh–faded multi– band orthogonal frequency division multiplexing (MB–OFDM) [101] or direct–sequence  141  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference 0  10  −1  10  −2  10  −3  10  −4  SEP  10  −5  10  −6  10  AWGN (simulation) asynch. co−channel interference (simulation) ε − mixture noise, ε = 0.25, κ = 10 (simulation) ε − mixture noise, ε = 0.10, κ = 10 (simulation) asymptotic  −7  10  −8  10  0  5  10  15  20  SNR [dB]  Figure 6.1: SEP of orthogonal half–rate STBC (N = 8, NT = 4) from [8, Eq. (38)] with 4–PSK modulation vs. symbol SNR. NR = 1 and i.i.d. Rayleigh fading. Markers: Simulated SEP. Solid lines: Asymptotic SEP. 0  10  −1  10  −2  10  −3  10  −4  SEP  10  MB−OFDM, B = 1 MHz (simulation) MB−OFDM, B = 5 MHz (simulation)  −5  10  DS−UWB, B = 1 MHz (simulation) DS−UWB, B = 5 MHz (simulation) MB−OFDM, B = 1 MHz (asymptotic)  −6  10  MB−OFDM, B = 5 MHz (asymptotic) DS−UWB, B = 1 MHz (asymptotic) DS−UWB, B = 5MHz (asymptotic)  −7  10  10  15  20  25  30  SNR [dB]  Figure 6.2: SEP of orthogonal rate–3/4 STBC (N = 4, NT = 3) from [9, Eq. (4.104)] with 16–QAM modulation vs. symbol SNR. NR = 1 and i.i.d. Rayleigh fading. Markers: Simulated SEP. Lines: Asymptotic SEP. 142  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference (DS) [100] UWB interference. The space–time coded victim system employs square–root raised cosine transmit and receive filters with roll–off factor 0.22 and bandwidth B. The UWB interference was generated based on the IEEE 802.15.3a MB–OFDM and DS–UWB standard proposals [101] and [100], respectively. After receive filtering, the received signal is sampled and the resulting discrete–time signal is processed using the ED metric.37 For this complex interference model the moments required for the proposed asymptotic analysis were obtained with the Monte–Carlo approach (6.29) using Ne = 103 . Fig. 6.2 illustrates the accuracy of this technique for high SNRs and shows that while the harmfulness of MB–OFDM strongly depends on the bandwidth B of the victim receiver, the effect of DS–UWB on the victim system is practically independent of B. This can be explained as follows. Since for both considered bandwidths the symbol duration of the victim system is much larger than the chip duration, Tc = 0.762 ns [100], of the DS–UWB system, the effective noise z1 [n] after receive filtering and sampling is the sum of several RVs. Consequently, for both considered bandwidths z1 [n] is approximately Gaussian and a similar performance results in both cases. On the other hand, MB–OFDM UWB has a sub–carrier spacing of fs = 4.125 MHz and the receiver filter filters out a fraction of a sub–carrier and approximately one sub–carrier for B = 1 MHz and B = 5 MHz, respectively. Therefore, in this case, the statistics of the resulting effective noise are rather different for the two considered bandwidths causing the considerable difference in error performance. In Fig. 6.3, we compare the FEPs of two rate–2, 4–state STTCs (TSC [3] and BBH [4]) for different types of noise assuming NT = 2, NR = 2, and i.i.d. Rayleigh fading. The frame length is 130 symbols and for both codes and all considered types of noise only simple error events [116] of length two were considered in the truncated union bound for the FEP. We observe from Fig. 6.3 that the known performance advantage of the BBH code over the TSC code for AWGN [4] is preserved for i.i.d. –mixture noise. As expected, for both codes 37  More details regarding the considered UWB interference model can be found in [35].  143  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference 0  10  −1  10  −2  10  −3  FEP  10  −4  10  −5  10  BBH, AWGN (simulation) TSC, AWGN (simulation) BBH, i.i.d. ε − mix.noise, ε = 0.25, κ = 10 (simulation) TSC, i.i.d. ε − mix. noise, ε = 0.25, κ = 10 (simulation) BBH, i.i.d. ε − mix.noise, ε = 0.25, κ = 100 (simulation) TSC, i.i.d. ε − mix. noise, ε = 0.25, κ = 100 (simulation) BBH (asymptotic) TSC (asymptotic)  −6  10  10  12  14  16  18  20  22  24  SNR [dB]  Figure 6.3: FEP of rate–2, 4–state STTCs from [3] (TSC) and [4] (BBH) vs. symbol SNR. NR = 2 and i.i.d. Rayleigh fading. Markers: Simulated FEP. Lines: Asymptotic FEP. the more impulsive –mixture noise (κ = 100) causes an additional performance loss of approximately 0.9 dB compared to the less impulsive noise (κ = 10). Furthermore, for both codes the asymptotic tightness of the truncated union bound is practically identical for all considered types of noise. Differential STCs with ED Metric: In Fig. 6.4, we investigate the impact of –mixture noise ( = 0.25, κ = 10) on the coherent and differential versions of Alamouti’s STBC (N = NT = 2) with 8–PSK modulation [5, 10]. I.i.d. Ricean fading (Ricean factor K = 0 dB) for NR = 1 and NR = 2 receive antennas is adopted. For NR = 2 both spatially dependent and i.i.d. –mixture noise, which are special cases of SDGMN and SIGMN, respectively, are considered. As expected from our analysis in Section 6.3.3, both the coh– 144  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference 0  10  −1  10  NR = 1  −2  10  −3  10  −4  SEP  10  −5  10  NR = 2  −6  10  i.i.d. ε − mixture noise (simulation) spatially dep. ε − mixture noise (simulation) coherent STBC (asymptotic) differential STBC (asymptotic) i.i.d. ε − mixture noise, coherent STBC, ML detection (simulation)  −7  10  0  5  10  15  20  25  30  SNR [dB]  Figure 6.4: SEP of coherent and differential Alamouti STBC (N = NT = 2) [5, 10] with 8–PSK modulation vs. symbol SNR. I.i.d. Ricean fading (K = 0 dB) for NR = 1 and NR = 2, –mixture noise with = 0.25 and κ = 10. Markers: Simulated SEP. Lines: Asymptotic SEP. erent and the differential STBCs achieve diversity gains of Gd = 2 and Gd = 4 for NR = 1 and NR = 2, respectively, also for –mixture noise. For NR = 2 the coding gain of the spatially dependent noise is smaller (worse performance) since the spatial dependencies reduce the benefits of diversity combining at the receiver. For comparison, for NR = 2 and i.i.d. –mixture noise Fig. 6.4 also contains simulation results for coherent ML detection. As can be observed, for this type of noise the ED metric entails a performance loss of less than 1.5 dB. The performance losses of the ED metric compared to the ML metric for the spatially dependent –mixture noise and for differential detection are even smaller. For clarity, the corresponding SEP curves are not included in Fig. 6.4. Note that for the considered case ML detection and ED detection require 82 /2 = 32 and 8 metric computations per 8–PSK symbol decision, respectively, cf. Section 6.4.3, [9]. 145  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference ND Metric: In Fig. 6.5, we show the SEP of Alamouti’s STBC (N = NT = 2) [5] with 4–PSK and ED and ND detection for NR = 2 and a Rayleigh fading channel with a receiver antenna correlation of 0.7. We consider impairment by AWGN, ACGN (ρs = 0.9), and uncorrelated and correlated (ρs = 0.9) Rayleigh–faded synchronous 4–PSK CCI (τ1 = 0). Note that all considered types of noise are temporally uncorrelated. Fig. 6.5 nicely confirms (6.31), (6.33), (6.42), and (6.43). In particular, the performance loss suffered by the ED metric due to correlation is LCO (0.9) = 0.8 dB and 1.1 dB for ACGN and CCI, respectively. Furthermore, the gain achievable with the ND metric compared to the ED metric is GN D (0.9) = 4.4 dB and 4.7 dB for ACGN and CCI, respectively. Fig. 6.5 clearly shows that, though suboptimum, the ND metric can achieve large performance gains if the CCI is correlated. Adopting the same STBC, ACGN parameters, and CCI parameters as in Fig. 6.5, we investigate the robustness of the ND metric in Fig. 6.6. For this purpose, for SNR = 20 dB we show the SEP as a function of the estimated noise correlation ρˆs which is used in the ND metric for calculation of matrix D s . For ρˆs = 0, Ds = I 2 results and the ND metric using the mismatched correlation is identical to the ED metric. As expected, the best performance is achieved for ρˆs = ρs = 0.9. However, Fig. 6.6 shows that the ND metric results in significant performance gains even if the noise correlation cannot be estimated accurately. Fig. 6.7 shows the BEP of the rate–2 diagonal STBC (N = NT = 2) from [11] for NR = 2 and i.i.d. Rayleigh fading vs. bit SNR. Results for coherent detection (without differential encoding) and differential detection are shown for impairment by AWGN (ED metric) and spatially correlated Rayleigh–faded asynchronous 8–PSK CCI (I = 1, τ1 = T /2, ρs = 0.7, ED and ND metric), respectively. In the ND metric only spatial decorrelation was performed since the temporal correlation caused by the asynchronous CCI was small  146  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference 0  10  −1  10  −2  10  −3  10  ED metric  −4  SEP  10  −5  10  AWGN, ED metric (simulation) ACGN, ED metric (simulation) uncorrelated synch. CCI, ED metric (simulation) correlated synch. CCI, ED metric (simulation) ACGN, ND metric (simulation) correlated synch. CCI, ND metric (simulation) ED metric (asymptotic) ND metric (asymptotic)  −6  10  −7  10  ND metric  −8  10  0  5  10  15  20  SNR [dB]  Figure 6.5: SEP of Alamouti STBC (N = NT = 2) [5] with 4–PSK for NR = 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. symbol SNR. AWGN, ACGN (ρs = 0.9), and uncorrelated and correlated (ρs = 0.9) Rayleigh–faded synchronous 4–PSK CCI (τ1 = 0). Markers: Simulated SEP. Lines: Asymptotic SEP. ACGN (simulation) correlated synch. CCI (simulation) ACGN (asymptotic) correlated synch. CCI (asymptotic) −5  SEP  10  −6  10  −7  10  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  ρˆs  Figure 6.6: SEP of Alamouti STBC (N = NT = 2) [5] with 4–PSK for NR = 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. estimated noise correlation ρˆs . ACGN (ρs = 0.9), correlated (ρs = 0.9) Rayleigh–faded synchronous 4–PSK CCI (τ1 = 0), and SNR = 20 dB. Markers: Simulated SEP. Lines: Asymptotic SEP. 147  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference (less than 0.2). We observe from Fig. 6.7 that the performance difference predictions for synchronous CCI in (6.42) and (6.48) are also approximately valid for the asynchronous case. In particular, for both coherent and differential detection the performance gain achievable with the ND metric is GN D (0.7) = 2.2 dB which is in perfect agreement with (6.42). The asymptotic performance loss entailed by differential detection compared to coherent detection is LD = 3.8 dB for both ED and ND metric, which agrees well with the 4.1 dB loss predicted for synchronous CCI in (6.48). In Fig. 6.8, we compare the BEPs of the rate–2, 4–state TSC STTC [3] and the rate–2, 4–state super–orthogonal STTC in [12, Fig. 5] (JS) for NT = 2, NR = 1, 2, and Rayleigh fading with a receiver antenna correlation of 0.7. For NR = 2 we consider AWGN with ED metric and ACGN (ρs = 0.7) with ED and ND metric. For both codes we consider only simple error events [116] of lengths up to two in the truncated union bound for the BEP. For NR = 2 the performance advantage of the JS code compared to the TSC code is approximately 1.2 dB independent of the type of noise and independent of the metric used for detection. Furthermore, since short error events dominate the overall performance of both STTCs, the performance difference predictions in Section 6.4.2 for full–rank STBCs are also accurate in this case. In particular, for both codes the performance loss suffered by the ED metric because of the noise correlation is LCO (0.7) = 0.5 dB, cf. (6.33), and the performance gain achieved by the ND metric is GN D (0.7) = 2.0 dB, cf. (6.31).  6.6  Conclusions  In this chapter, we presented asymptotic PEP expressions for coherent and differential STCs operating in correlated Ricean fading and non–Gaussian noise and interference. The presented expressions are much more general than the existing PEP results in the literature as they are valid for arbitrary coherent STTCs, coherent and differential STBCs, arbitrary 148  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference 0  10  −1  10  −2  10  −3  10  −4  BEP  10  −5  10  −6  10  AWGN (simulation) asynch. CCI, ED metric (simulation) asynch. CCI, spatial ND metric (simulation) coherent code (asymptotic) differential code (asymptotic)  −7  10  −8  10  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  bit SNR [dB]  Figure 6.7: BEP of rate–2 diagonal STBC (N = NT = 2) from [11] for NR = 2 and i.i.d. Rayleigh fading vs. bit SNR. AWGN and spatially correlated (ρs = 0.7) Rayleigh– faded asynchronous 8–PSK CCI with I = 1 and τ1 = T /2. Markers: Simulated SEP. Lines: Asymptotic SEP. 0  10  −1  10  NR = 1 −2  10  −3  10  NR = 2  −4  BEP  10  −5  10  TSC, AWGN (simulation) TSC, ACGN, ρs = 0.7, ED metric (simulation)  TSC, ACGN, ρs = 0.7, ND metric (simulation)  −6  10  JS, AWGN (simulation) JS, ACGN, ρs = 0.7, ED metric (simulation)  JS, ACGN, ρ = 0.7, ND metric (simulation)  −7  10  s  TSC (asymptotic) JS (asymptotic) −8  10  0  5  10  15  20  SNR [dB]  Figure 6.8: BEP of TSC STTC code [3] and super–orthogonal STTC (JS) [12, Fig. 5] for NR = 1, 2 and Rayleigh fading with a receiver antenna correlation of 0.7 vs. symbol SNR. Markers: Simulated SEP. Lines: Asymptotic SEP. 149  Chapter 6. Performance Analysis of Space-Time Codes in Non–Gaussian Noise and Interference MD metrics including ED and ND metrics, and any type of noise and interference with finite moments. Based on the presented analysis we showed that the diversity gain of an STC is independent of the specific form of the MD metric and the type of noise. In contrast, the coding gain is affected by both the MD metric and the noise properties. In particular, for correlated noise such as ACGN or correlated CCI large performance gains are possible if the ND metric is used instead of the simpler ED metric. On the other hand, if the ED metric is adopted, spatial and temporal noise dependencies adversely affect the performance. Furthermore, our results show that while full–rank square coherent STBCs optimized for AWGN are also strictly optimum for non–AWGN noise, non–square coherent STBCs, differential STBCs, and coherent STTCs designed for AWGN are close to optimum for non–AWGN noise.  150  Chapter 7 Summary of Thesis and Future Research Topics In this chapter, we summarize the main results obtained in this thesis and propose ideas for future related research.  7.1  Summary of Results  The main goal of this thesis was to provide generalized analysis and design techniques for wireless communication systems impaired by various types of fading and noise. Table 7.1 lists the scenarios considered in Chapters 2–6. In the following, we briefly review the main results of each chapter. In Chapter 2, we obtained a simple closed–form expression for the PEP of STCs with two transmit antennas in Gaussian noise and generalized (non–keyhole) fading channels including Rayleigh, Ricean, Nakagami–q, Nakagami–m, Weibull, and generalized–K fading. Unlike the existing results in the literature, our analysis is valid for both STBCs and STTCs. In Chapter 3, this analysis was generalized to keyhole fading channels. In addition to STCs with two transmit antennas, systems with a single transmit antenna and MRC reception were also considered. Similar to Chapter 2, we showed how the diversity gain is related to the number of transmit and receive antennas and the fading parameters. We 151  Chapter 7. Summary of Thesis and Future Research Topics Table 7.1: A comparison of the scenarios considered in Chapters 2–6. In this table, NT = 1 refers to systems with a single transmit antenna and quadratic diversity combining at the receiver; NT > 1 means space–time coding; non–Gaussian noise refers to any type of noise and interference with finite moments; Lp –norm includes the ED metric (based on L2 –norm) as a special case; and MD metric includes ED and ND metrics as special cases. The number of receiver antennas is arbitrary in all chapters. Chapter  NT  Fading  Noise  Metric  2  2  generalized i.i.d. non–keyhole  Gaussian  ED  3  1 and 2  generalized i.i.d. keyhole  Gaussian  ED  4  1  correlated Ricean  non–Gaussian  ED  5  1  correlated Ricean  non–Gaussian  Lp  6  ≥1  correlated Ricean  non–Gaussian  MD  also derived code design criteria which are useful for predicting the performance of different codes and finding optimal codes in different scenarios. In Chapter 4, we studied the asymptotic performance of quadratic diversity combining schemes such as MRC, differential EGC, and NC in correlated Ricean fading and non– Gaussian noise. We compared the performance of these combining schemes for various types of noise distributions. In Chapter 5, we proposed a robust adaptive Lp –norm metric to improve the performance of the coherent, differential, and non–coherent combining schemes in the presence of non–Gaussian noise and interference. The proposed metric does not need any a priori knowledge about the distribution of the noise and uses stochastic approximation methods to optimize the performance of the system. We showed that the proposed Lp –norm metric outperforms other robust metrics such as Huber’s metric, the Myriad metric, and the Meridian metric in both heavy–tailed and short–tailed noise. In Chapter 6, we extended the analysis of Chapter 4 to STCs. In addition, we considered a general MD metric which includes the ED and ND metrics as special cases. We predicted  152  Chapter 7. Summary of Thesis and Future Research Topics the performance loss caused by fading and/or noise spatial correlation if the ED metric is used and illustrated that the performance can be improved if the ND metric is applied in the case of correlated noise. In Chapters 4, 5, and 6, we showed that the diversity gain is independent of the type of noise and the considered metrics. However, the combining (coding) gain in general depends on the code structure (in case of space–time coding), the adopted metric, and the noise moments which were derived in closed form for several practically important types of noise. Based on this dependency, we drew some interesting conclusions about the optimality of different metrics and the code design criteria. In summary, the presented analytical framework provides simple and insightful closed– form expressions that can be employed to predict or optimize the performance of different diversity combining techniques and general STCs in various types of fading and noise.  7.2  Future Work  In the following, we propose some ideas for further research that are similar to or can be based on the results of this thesis.  7.2.1  Performance Analysis of STCs with Arbitrary Number of Transmit Antennas in Generalized Fading  Table 7.1 shows that our analyses in Chapters 2 and 3 are valid for systems with a single transmit antenna or STCs with two transmit antennas. While this is an important subclass of MIMO systems, there are STCs with more than two transmit antennas, cf. e.g. [9]. As the demand for high data–rate systems increases and the technology improves, these STCs become practically more important. For example, the 3GPP LTE standard also supports MIMO systems with four transmit antennas [90]. Therefore, it would be interesting to  153  Chapter 7. Summary of Thesis and Future Research Topics extend our analyses in Chapters 2 and 3 to include STCs with more than two transmit antennas in both non–keyhole and keyhole fading channels.  7.2.2  Performance Analysis in α–stable Noise  A broad and increasingly important class of non-Gaussian phenomena encountered in practice can be characterized by α–stable distributions [105] which have thicker tails compared to the Gaussian distribution. Underwater acoustic signals, low-frequency atmospheric noise, and some types of man-made impulsive noise have all been found to follow α–stable distributions. The α–stable distribution is very flexible as a modeling tool in that it has a parameter 0 ≤ α ≤ 2, called the characteristic exponent, that controls the heaviness of its tails. A small positive value of α indicates severe impulsiveness, while a value of α close to 2 indicates a more Gaussian type of behavior. The α–stable distribution includes the Gaussian distribution as a special case for α = 2. This noise model is in fact justified by the generalized central limit theorem which suggests that if an observed signal or noise can be thought of as the sum of a large number of i.i.d. random variables, possibly with infinite moments, then an α–stable model may be appropriate. As we mentioned earlier, our analyses in Chapters 4, 5, and 6 are not applicable to α–stable noise since this type of noise has infinite higher order moments for 0 ≤ α < 2, c.f. [105]. Therefore, new analysis techniques have to be found to deal with this class of non-Gaussian noise.  7.2.3  Analyzing the Performance of Non–coherent STCs  In Chapter 6, we compared the performance of coherent and differential STCs in nonGaussian noise. In coherent STCs, the channel gains are assumed to be known by the receiver. Differential transmission embeds the user information in two consecutive trans154  Chapter 7. Summary of Thesis and Future Research Topics mitted symbols or blocks and therefore does not require knowledge of the channel gains. However, differential transmission assumes that the channel gains are constant at least for two consecutive time intervals. In contrast, for non–coherent STCs (e.g. unitary STCs [117]) the channel may change independently from codeword to codeword. Note that decoding of non–coherent STCs is based on non–MD metrics and therefore these codes cannot be analyzed with the method presented in Chapter 6 for coherent and differential STCs. Accordingly, an interesting topic for future research is the analysis and design of non–coherent STCs in non–Gaussian noise and interference.  7.2.4  Performance Analysis of Relay Networks in Non–Gaussian Noise  STCs exploit the independent paths between the transmit and receive antennas of a single user in a point–to–point wireless system to provide spatial diversity. Similarly, in cooperative diversity systems, multiple users of a wireless network serve as relays for each other to provide spatial diversity. Among the most widely used cooperative strategies are amplify–and–forward (AF) [118] and decode–and–forward (DF) [119] techniques. Distributed space–time coding (DSTC) [120, 121] is another new technique which may or may not require decoding at the relays. Note that DSTC can be implemented in many different ways, c.f. e.g. [121, 122] and references therein. Interestingly, in AF relaying the effective noise at the receiver involves products of noise and fading. Therefore, this effective noise is non–Gaussian even in AWGN channels. It would be interesting to apply the proposed asymptotic analysis to this AF relaying and study its performance in general non-Gaussian noise and interference.  155  Chapter 7. Summary of Thesis and Future Research Topics  7.2.5  Optimizing Non–square STCs in Non-Gaussian Noise  As we mentioned in Chapter 6, full–rank square coherent STBCs optimized for AWGN are also optimum for any other type of noise with finite moments as long as the MD metric is considered; however, non–square codes (codes for which N > NT ) which are optimized for AWGN are not necessarily optimum for other types of noise. Therefore, assuming that the MD metric is employed at the receiver, searching for the optimal non–AWGN tolerant non–square STCs is an interesting topic for future researsch.  7.2.6  More Robust Techniques  The adaptive Lp –norm metric proposed in Chapter 5 for receive diversity combining systems performs well when the noise distribution can be approximated by a generalized Gaussian distribution. However, more improvement is still possible in environments with highly impulsive noise. 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Nunan, “Evaluation of a Function at Infinity Using its Power Series,” Physical Review Letters, vol. 46, pp. 1255–1256, May 1981.  166  Appendix A Calculation of I(θ) in (2.9) Changing variables a1 and a2 to polar form as ρ cos ϕ =  √  γd1 a1 and ρ sin ϕ =  √  γd2 a2 ,  (2.8) can be rewritten as  I=  1 √  2πγ d1 d2  ∞ π/2 π  0  0 −π  exp −  ρ2 (1 + 2ξ cos ϕ sin ϕ cos φ) 4 sin2 θ ×f  ρ cos ϕ √ γd1  ρ sin ϕ √ γd2  f  ρ dφ dϕ dρ, (A.1)  where we have dropped the argument of I(θ) for simplicity of notation. As shown in [16], for high SNR, decision errors only occur when the channel gains approach zero and f (a)  . I=  αat−1 from (2.2) can be applied in (A.1). This leads to  α2 2π(γ 2 d1 d2 )β  ∞ π/2 π  exp − 0  0 −π  ρ2 (1 + ξ sin 2ϕ cos φ) 4 sin2 θ β−1  ×ρ  sin 2ϕ 2  2β−1  dφ dϕ dρ. (A.2)  Changing the order of integration in (A.2) and using [91, Eq. (3.381.4)] results in β . α2 Γ(2β) 22β sin θ I= 2π(γ 2 d1 d2 )β  π/2 π  (sin 2ϕ)2β−1 dφ dϕ. (1 + ξ sin 2ϕ cos φ)2β  (A.3)  0 −π  167  Appendix A. Calculation of I(θ) in (2.9) Using the change of variables z = (1/ sin(2ϕ) − 1)/2, [91, Eqs. (8.384.1), (3.197.9)], and assuming ξ < 1, (A.3) can be simplified to α2 [Γ(2β)]2 sinβ θ . J, I=√ π(γ 2 d1 d2 )β Γ(2β + 12 )  (A.4)  where π  J 0  2 F1  1 2β, 2β; 2β + ; (1 − ξ cos ψ)/2 dψ 2  π/2  1 2β, 2β; 2β + ; (1 − ξ cos ψ)/2 dψ 2  2 F1  = 0  π/2 2 F1  +  1 2β, 2β; 2β + ; (1 + ξ cos ψ)/2 dψ, 2  (A.5)  0  Each of the integrals in (A.5) can be written as [91, Eq. (9.136)] π/2  0  2 F1 (2β, 2β; 2β  1 1 − ξ cos ψ + ; )dψ = AX + BY, 2 2  (A.6)  1 1 + ξ cos ψ )dψ = AX − BY, + ; 2 2  (A.7)  π/2 2 F1 (2β, 2β; 2β 0  where A  √ π Γ(2β+ 12 ) [Γ(β+ 21 )]2  and B are constants and π/2 2 F1 (β, β; 2β  X  1 + ; ξ 2 cos2 ψ)dψ, 2  (A.8)  0  168  Appendix A. Calculation of I(θ) in (2.9) π/2 2 F1 (β  Y  1 3 1 + , β + ; ; ξ 2 cos2 ψ) cos ψ dψ. 2 2 2  (A.9)  0  From (A.5)–(A.9), we obtain π/2 2 F1  J = 2A X = 2A  1 β, β; 2β + ; ξ 2 cos2 ψ dψ. 2  (A.10)  0  Changing variable ψ to u  cos2 ψ results in 1  J =A 0  1 2 2 F1 (β, β; 2 ; ξ u)  u(1 − u)  du.  (A.11)  Using [91, Eq. (7.512.11)] and replacing A in (A.11) yields 3  π 2 Γ(2β + 12 ) 2 J= . 2 F1 β, β; 1; ξ [Γ(β + 12 )]2  (A.12)  Substituting (A.12) in (A.4) we obtain . πα2 [Γ(2β)]2 2 F1 (β, β; 1; ξ 2) β I(θ) = sin θ . [Γ(β + 12 )]2 (γ 2 d1 d2 )β Finally, using the identity Γ(2β) =  22β−1 √ π  (A.13)  Γ(β) Γ(β + 12 ) [91, Eq. (8.335.1)] in (A.13)  results in (2.9).  169  Appendix B PDF of Y for NT = 1 and NT = 2 In this appendix, we derive (3.9)–(3.11) for NT = 1 and NT = 2 transmit antennas, respectively. For the case of a single transmit antenna, E and h = a1 ejφ1 are scalar values. Therefore, the pdf of Y = ||Eh||2 = |E|2 a21 as given in (3.9)–(3.11) follows directly from the pdf of a1 in (3.3). H H 2 For two transmit antennas,  Y can be expressed as Y = ||Eh|| = tr{h E Eh}. Thus,   using E H E   d1 d12  , we obtain  d∗12 d2  Y = d1 a21 + d2 a22 + 2|d12 |a1 a2 cos φ,  (B.1)  φ1 − φ2 + ∠d12 is uniformly distributed in [−π, π). Changing variables a1 and √ √ a2 to polar form as ρ cos ϕ = d1 a1 and ρ sin ϕ = d2 a2 , (B.1) can be written as  where φ  Y = ρ2 (1 + 2ξ cos ϕ sin ϕ cos φ) ,  where ξ  (B.2)  √ |d12 |/ d1 d2 , ρ ≥ 0, and 0 ≤ ϕ ≤ π/2.38 Therefore, the cumulative distribution  function (cdf) of Y is given by  FY (y) 38  1 Pr{Y < y} = 2π  π −π  √  π/2 0  y 1+2ξ cos ϕ sin ϕ cos φ  fρ,ϕ (ρ, ϕ) dρ dϕ dφ,  (B.3)  0  Note that 0 ≤ ξ < 1 for full–rank STCs with det{E H E} > 0.  170  Appendix B. PDF of Y for NT = 1 and NT = 2 where fρ,ϕ (ρ, ϕ) is the joint pdf of ρ and ϕ. Using (3.3) and noting that a1 and a2 are i.i.d., fρ,ϕ (ρ, ϕ) can be written as39 [94]  fρ,ϕ (ρ, ϕ) =  ∞  ∞  αi αj  βi βj i=0 j=0 d1 d2  ρ2βi +2βj −1 cos2βi −1 ϕ sin2βj −1 ϕ.  (B.4)  Using (B.4) in (B.3) and evaluating the innermost integral leads to  FY (y) =  1 4π  ∞  ∞  αi αj Λij  βi βj i=0 j=0 d1 d2 (βi  + βj )  y βi+βj ,  (B.5)  where π  π/2  Λij −π  cos2βi −1 ϕ sin2βj −1 ϕ (1 + 2ξ cos ϕ sin ϕ cos φ)βi +βj  0  dϕ dφ.  (B.6)  The inner integral in (B.6) can be evaluated by the change of variable z = cot ϕ and using [91, Eqs. (3.252.11) and (8.771)] which results in  Λij =  Γ(2βi )Γ(2βj ) Γ(2βi + 2βj )  π −π  2 F1  1 1 − ξ cos φ 2βi , 2βj ; βi + βj + , 2 2  dφ.  (B.7)  We use [91, Eqs. (8.384.1) and (9.136)] to arrive at  Λij = √  4 A Γ(2βi )Γ(2βj ) Γ(2βi + 2βj )  πΓ(βi +βj +1/2) . Γ(βi +1/2)Γ(βj +1/2)  where A  π/2 0  2 F1  1 βi , βj ; , ξ 2 cos2 φ 2  dφ,  (B.8)  Applying the change of variable u = cos2 φ, using [91, Eq.  7.512.11], and inserting A results in 3  2π 2 Γ(2βi )Γ(2βj )Γ(βi + βj + 21 ) πΓ(βi )Γ(βj ) 2 2 = Λij = 2 F1 βi , βj ; 1, ξ , 1 2 F1 βi , βj ; 1, ξ 1 Γ(βi + βj ) Γ(2βi + 2βj )Γ(βi + 2 )Γ(βj + 2 ) (B.9) 39  For simplicity of notation, in this section, we drop the index T from αT,j and βT,j  171  Appendix B. PDF of Y for NT = 1 and NT = 2 where we have used [91, Eq. (8.335.1)]. Exploiting (B.9) in (B.5) yields  FY (y) =  ∞  ∞  i=0 j=0  αi αj Γ(βi )Γ(βj ) 2 F1 (βi , βj ; 1, ξ 2) β 4 dβ1 i d2 j (βi  + βj )Γ(βi + βj )  y βi+βj .  (B.10)  The pdf of Y can be obtained by differentiating (B.10), which results in  fY (y) =  ∞  ∞  χij y βi+βj −1 ,  (B.11)  i=0 j=0  where χij  β  αi αj Γ(βi )Γ(βj ) 2 F1 (βi , βj ; 1, ξ 2)/(4 dβ1 i d2 j Γ(βi + βj )). Since βi = ηT i + βT  and βj = ηT j + βT , fY (y) can be represented in the form of a single sum as shown in (3.9)–(3.11).  172  Appendix C Asymptotic Statistical Properties of X In this appendix, we derive the asymptotic pdf and MGF of X = ||g||2 = xν  NR ν=1  xν , where  b2ν . From (3.4), we know that the pdf of bν as bν → 0+ is given by . R −1 , fbν (bν ) = αR b2β ν  1 ≤ ν ≤ NR .  (C.1)  Therefore, the asymptotic pdf of xν is given by [94] √ 2βR −1 αR βR −1 . αR xν fxν (xν ) = = x , √ 2 xν 2 ν  (C.2)  and the asymptotic MGF of xν is obtained as  Ψxν (s)  . αR Γ(βR ) −βR s . E{e−xν } = 2  (C.3)  Since bν and therefore xν are i.i.d., the MGF of X is given by . ΨX (s) =  NR  Ψxν (s) = AR s−NR βR ,  (C.4)  ν=1  where AR  αR Γ(βR ) 2  NR  . Applying the inverse Laplace transform to (C.4) yields the  asymptotic pdf of X in (3.16). 173  Appendix D Evaluating a Function at Infinity Using its Power Series In this appendix, we generalize the method presented in [124] for evaluating a function at infinity using its power series. Assume that the function f (z) can be represented by the series ∞  f (z) = z 1−  an z n ,  (D.1)  n=I−1  where 0 ≤  < 1 is a rational number and I − 1 ≥ 0 is the index of the first non-zero  coefficient of the series, i.e., aI−1 = 0. We note that a special case of (D.1) with = 0 and I = 1 was considered in [124]. We assume in the following that f (∞) exists and is finite. By changing index n to k  n − I + 1 we can rewrite (D.1) as f (z) = z  I−  ∞  ak+I−1 z k .  (D.2)  k=0  The series in (D.2) can be written as  f (z) = z  I−  ∞  bk z  k  −1  ,  (D.3)  k=0  174  Appendix D. Evaluating a Function at Infinity Using its Power Series where the coefficients bk , k ≥ 0, can be calculated from the polynomial division 1/ Thus, we can write N  (f (z)) = z  ∞  N (I− )  (N ) ck z k  ∞ k k=0 ak+I−1 z .  −1  ,  (D.4)  k=0 (N )  where ck  is found by convolving sequence bk N − 1 times with itself. Since  is a rational  number, one can choose an integer number N such that N(I − ) is an integer. We approximate the above expression by considering only the first N(I − ) terms of the series in (D.4) to arrive at   −1  N (I− )  (f (z))N  z N (I− )   (N )  k=0  The above series has a limit at ∞, i.e.,  (f (∞))N  In fact, it can be shown that if N and  ck z k   (N )  cN (I− )  −1  .  .  (D.5)  (D.6) (N )  are chosen such that cN (I− ) > 0 [124]  f (∞) = lim  N →∞  (N )  cN (I− )  −1/N  .  (D.7)  Our simulation results show that truncating the series in (D.1) and choosing a moderate value for N (e.g. N = 20) in (D.7) still leads to accurate results. Therefore, the above algorithm can be used to accurately evaluate u(∞) in (3.24) and (3.45a) with low complexity.  175  Appendix E Asymptotic PEP for CC Assuming that b was transmitted and ˆb = b was detected, the corresponding PEP can be expressed as Pe (d) = Pr{mc (b) > mc (ˆb)}, where d vector n  |e| and e  (E.1)  b − ˆb. In a first step, we calculate the PEP conditioned on the noise  [n1 . . . nL ]T . With (5.7) and (E.1) this conditional PEP can be obtained as mc (b)  fc (z) dz,  Pe (d|n) =  (E.2)  0  where we have used the fact that due to the conditioning on n, mc (b) = Ll=1 ql |nl |pl is √ L pl a constant, and fc (z) is the pdf of mc (ˆb) = l=1 ql | γl hl e + nl | , which we calculate step–by–step in the following. √ The conditional pdf of xl = | γl hl e + nl | is a Ricean pdf given by √ x2l + | γl ¯hl e + nl |2 2xl fxl (xl ) = 2 2 exp − d γl σhl d2 γl σh2l  I0  √ ¯ xl | γl h l e + nl | 2 2 d γl σh2l  The pdf of the transformed variable yl = xpl l is given by fyl (yl ) = the scaling with ql leads to zl = ql yl with pdf fzl (zl ) =  .  1 1/pl −1 y fxl pl l  1 f (z /q ). q l yl l l  (E.3)  1/pl  yl  and  Taking into account  176  Appendix E. Asymptotic PEP for CC √ these identities the pdf of zl = ql | γl hl e + nl |pl is given by fzl (zl ) =  2/pl  2/pl −1  2zl  2/pl  d2 γl σh2l pl ql  exp −  zl  2/pl  + ql  √ ¯ 2 | γl h l e + nl | 2/pl  d2 γl σh2l ql  1/pl  I0  2  zl  √ ¯ | γl h l e + nl | 1/pl  d2 γl σh2l ql  . (E.4)  Considering the asymptotic case γl → ∞ and exploiting the Taylor series expansions of exp(·) and I0 (·), fzl (zl ) can be written as Cl 2/pl −1 zl + o(γl−1 ), γl  fzl (zl ) =  where Cl  2/pl  ¯ l |2 /σ 2 /(d2 σ 2 pl q 2 exp −|h hl hl l  ).  (E.5)  Thus, the moment generating function  E{e−szl } = Cl Γ(2/pl )γl−1 s−2/pl + o(γl−1 ). Since  (MGF) of zl can be expanded as Φzl (s)  conditioned on n the zl are statistically independent, the MGF of mc (ˆb) is given by Φc (s) =  L l=1 Φzl (s),  and the asymptotic expansion of the corresponding pdf is given  by L l=1  fc (z) =  Cl Γ  L 2 l=1 pl  Γ  2 pl L l=1  z  L 2 l=1 pl −1  L  γl−1 .  +o  γl  (E.6)  l=1  Using this result in (E.2) leads to  Pe (d|n) =  L l=1  Γ  Cl Γ  L 2 l=1 pl  +1  2 pl  L 2 l=1 pl  L L l=1  γl  l=1  ql |nl |  pl  L  γl−1  +o  .  (E.7)  l=1  If all joint moments of the elements of n are finite, averaging Pe (d|n) in (E.7) with respect to n yields (5.10). The assumption of finite joint noise moments is necessary, since the terms absorbed into o(  L −1 l=1 γl )  in (E.7) involve sums of products of the elements of n  which have to remain finite after expectation.  177  Appendix F Publications Related to Thesis The papers that have been published/submitted from the work presented in this thesis are listed in the following.  Chapter 2 • A. Nezampour, and R. Schober, “Asymptotic Performance of Space–Time Codes in Generalized Fading Channels”, IEEE Communication Letters, Vol. 13, No. 8, pp 561-563, Aug. 2009.  Chapter 3 • A. Nezampour, A. Nasri, and R. Schober, “Asymptotic Analysis of Space–Time Codes in Generalized Keyhole Fading Channels”, Submitted to IEEE Transactions on Wireless Communications, Apr. 2010. • A. Nezampour, A. Nasri, and R. Schober, “Asymptotic Analysis of Space–Time Codes in Generalized Keyhole Fading Channels”, in Proceedings of IEEE Wireless Communication & Networking Conference (WCNC), Sydney, Australia, Apr. 2010.  Chapter 4 • A. Nezampour, A. Nasri, R. Schober, and Y. Ma, “Asymptotic BEP and SEP of Quadratic Diversity Combining Receivers in Correlated Ricean Fading, Non-Gaussian Noise, and Interference”, IEEE Transactions on Communications, Vol. 57, pp. 10391049, Apr. 2009. 178  Appendix F. Publications Related to Thesis • A. Nezampour, A. Nasri, R. Schober, and Y. Ma, “Asymptotic BEP and SEP of Differential EGC in Correlated Ricean Fading, and Non–Gaussian Noise”, in Proceedings of IEEE Global Telecommunications Conference (GLOBECOM), Washington, D.C., USA, Nov. 2007. • A. Nezampour, A. Nasri, R. Schober, and Y. Ma, “Asymptotic BEP and SEP of MRC in Correlated Ricean Fading, and Non–Gaussian Noise”, in Proceedings of IEEE Vehicular Technology Conference (VTC), Baltimore, USA, Oct. 2007  Chapter 5 • A. Nasri, A. Nezampour, and R. Schober, “Adaptive Lp –Norm Diversity Combining in Non-Gaussian Noise and Interference”, IEEE Transactions on Wireless Communications, Vol. 8, pp. 4230-4240, Aug. 2009. • A. Nasri, A. Nezampour, and R. Schober, “Adaptive Coherent Lp –Norm Combining”, in Proceedings of IEEE International Conference on Communications (ICC), Dresden, Jun. 2009  Chapter 6 • A. Nezampour, R. Schober, and Y. Ma, “Asymptotic Analysis of Coherent and Differential Space–Time Codes in Non–Gaussian Noise and Interference”, IEEE Transactions on Communications, Vol. 57, pp. 3353 - 3365, Nov. 2009. • A. Nezampour, R. Schober, and Y. Ma, “Asymptotic Analysis of Space–Time Codes with Mahalonobis Distance Decoding in Non–Gaussian Noise and Interference”, in Proceedings of European Wireless Conf., Invited Paper, Prague, June 2008. • A. Nezampour, R. Schober, and Y. Ma, “Asymptotic Analysis of Space–Time Codes 179  Appendix F. Publications Related to Thesis in Non–Gaussian Noise and Interference”, in Proceedings of IEEE Vehicular Technology Conference (VTC), Singapore, May 2008  180  

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