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Asymptotic performance analysis for EGC and SC over arbitrarily correlated Nakagami-m fading channels Li, Xianchang 2011

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A SYMPTOTIC P ERFORMANCE A NALYSIS FOR EGC AND SC OVER A RBITRARILY C ORRELATED N AKAGAMI -m FADING C HANNELS by  X IANCHANG L I B.Eng., Northwestern Polytechnical University, China, 2004  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Electrical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (O KANAGAN) April 2011 c Xianchang Li, 2011  Abstract The multi-branch diversity combining technique has been widely used in wireless communications systems to overcome the adverse effect of multipath fading. The three most popular diversity combining schemes are selection combining (SC), equal gain combining (EGC), and maximal ratio combining (MRC). In the performance analysis of multi-branch diversity combining, the asymptotic technique is an attractive approach to obtain compact and accurate error rate and outage probability in large signal-to-noise ratio (SNR) regions. Asymptotic performance can be obtained either in the time domain by finding the probability density function (PDF) of the instantaneous output SNR, or in the frequency domain by finding the moment generating function (MGF) of the square root of the instantaneous output SNR. In this thesis, the PDF of the instantaneous SNR at the output of selection combiner and the MGF of the square root of the instantaneous SNR at the output of equal gain combiner over arbitrarily correlated Nakagami-m fading channels are derived and used to obtain asymptotic error rate and outage probability expressions of SC and EGC, respectively. These expressions provide accurate and rapid estimation of error rates and outage probabilities. The accuracy of our analytical results is verified by computer simulation. More importantly, our analytical results provide physical insights into the transmission behavior of EGC and SC over arbitrarily correlated Nakagami-m fading channels. For instance, it is shown that the asymptotic error rates over correlated branches can be obtained by scaling the asymptotic error rates over independent branches with a factor, detm (M), where det(M) is the determinant of matrix M whose elements are the square root of corresponding elements in the branch power covariance correlation matrix R. A similar relationship can also be found for the outage probabilities. ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  List of Acronyms  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  List of Symbols  1.1  Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.3  Thesis Outline  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  2 Multipath Fading and Diversity Combining Techniques . . . . . . . . . . . .  8  2.1  2.2  Multipath Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.1.1  Rayleigh Distribution  . . . . . . . . . . . . . . . . . . . . . . . .  9  2.1.2  Rician Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .  9  2.1.3  Nakagami-m Distribution  . . . . . . . . . . . . . . . . . . . . . .  10  Diversity Combining Techniques  . . . . . . . . . . . . . . . . . . . . . .  11  iii  Table of Contents 2.2.1  Maximal Ratio Combining  . . . . . . . . . . . . . . . . . . . . .  11  2.2.2  Equal Gain Combining  . . . . . . . . . . . . . . . . . . . . . . .  13  2.2.3  Selection Combining . . . . . . . . . . . . . . . . . . . . . . . . .  13  2.3  Asymptotic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  2.4  Construction of Multiple Correlated Nakagami-m RVs . . . . . . . . . . .  16  2.4.1  Construction of Multiple Correlated Nakagami-m RVs . . . . . . .  17  2.4.2  Joint PDF of Multiple Correlated Nakagami-m RVs  18  . . . . . . . .  3 Asymptotic Performance Analysis of SC over Arbitrarily Correlated Nakagamim Fading Channels  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  20  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  20  3.2  A Series Representation of Qm (α , β )  . . . . . . . . . . . . . . . . . . . .  22  3.3  PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels . .  23  3.3.1  System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  3.3.2  PDF of SC Output SNR . . . . . . . . . . . . . . . . . . . . . . .  25  3.4  3.5  Asymptotic Performance Analysis of SC  . . . . . . . . . . . . . . . . . .  27  3.4.1  Asymptotic Error Rate . . . . . . . . . . . . . . . . . . . . . . . .  28  3.4.2  Asymptotic Outage Probabilities  . . . . . . . . . . . . . . . . . .  28  . . . . . . . . . . . . . . . . . . . . .  29  Discussions and Numerical Results 3.5.1  Discussions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  3.5.2  Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . .  30  4 Asymptotic Performance Analysis of EGC over Arbitrarily Correlated Nakagamim Fading Channels  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  35  4.1  System Model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  35  4.2  MGF of Square Root of Instantaneous SNR at the Output of EGC . . . . .  36  4.3  Asymptotic Performance Analysis of EGC  . . . . . . . . . . . . . . . . .  37  Asymptotic Error Rate . . . . . . . . . . . . . . . . . . . . . . . .  38  4.3.1  iv  Table of Contents 4.3.2 4.4  Asymptotic Outage Probabilities  Discussions and Numerical Results  . . . . . . . . . . . . . . . . . .  38  . . . . . . . . . . . . . . . . . . . . .  39  4.4.1  Discussions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  39  4.4.2  Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . .  39  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  5.1  Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . .  45  5.2  Future Work  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  5 Conclusions  Appendices A Derivation of Error Rate for Noncoherent Modulation  . . . . . . . . . . . .  55  B Derivation of (2.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  58  C Derivation of (2.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  v  List of Tables 2.1  Parameters p and q for different coherent and noncoherent modulation schemes 16  3.1  The exact and approximate values of Qm (α , β ) for different values of α and β 24  vi  List of Figures 1.1  Q  β γ¯ as functions of β for γ¯ = 0, 5, and 15 dB. . . . . . . . . . . . . .  6  2.1  Multi-branch diversity combining receiver. . . . . . . . . . . . . . . . . . .  12  3.1  The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3branch correlated Nakagami-m channels with matrices M1 and M2 when m = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.2  31  The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3branch correlated Nakagami-m channels with matrices M1 and M2 when m = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3  The asymptotic and simulated outage probabilities of SC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 0.5. . . .  3.4  42  The asymptotic and simulated outage probabilities of EGC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 0.5. .  4.4  41  The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 2. . . . .  4.3  34  The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 0.5. . . .  4.2  33  The asymptotic and simulated outage probabilities of SC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 2. . . . .  4.1  32  43  The asymptotic and simulated outage probabilities of EGC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 2. . .  44 vii  List of Acronyms Acronyms  Definitions  AWGN  Additive White Gaussian Noise  BER  Bit Error Rate  BDPSK  Binary Differential Phase Shift Keying  BNCFSK  Binary Noncoherent Frequency Shift Keying  BPSK  Binary Phase Shift Keying  CDF  Cumulative Distribution Function  EGC  Equal Gain Combining  LOS  Line-of-Sight  MGF  Moment Generating Function  M-PAM  M-ary Pulse Amplitude Modulation  M-PSK  M-ary Pulse Shift Keying  M-QAM  M-ary Quadrature Amplitude Modulation  MRC  Maximal Ratio Combining  PDF  Probability Density Function  PSD  Power Spectral Density  RV  Random Variable  SC  Selection Combining  SER  Symbol Error Rate  SNR  Signal-to-Noise Ratio  viii  List of Symbols Symbols  Definitions  [·]T  Matrix transpose operator  d  Transmitted data symbol  det(M)  Determinant of matrix M  exp(·)  Exponential function  E [·]  The expectation of a RV  ES  Average symbol energy of the transmitted data symbol  f (β )  PDF of RV β  f (γ )  PDF of instantaneous SNR  f (x)  PDF of RV X  fγSC (γ )  PDF of γSC  f (r1 , r2 , · · · , rL )  Joint PDF of R1 , R2 , · · · , RL  F(a, b; c; z)  Hypergeometric function defined as F(a, b; c; z) =  Γ(c) ∞ Γ(a+n)Γ(b+n) zn n! Γ(a)Γ(b) ∑n=0 Γ(c+n)  F(r|z)  Marginal CDF of R depending on Z  F (r1 , r2 , · · · , rL |z)  Conditional CDF of R1 , R2 , · · · , RL depending on Z  F (r1 , r2 , · · · , rL )  Joint CDF of R1 , R2 , · · · , RL  FγSC (γ )  CDF of γSC  Gc  Coding gain  Gd  Diversity gain  Gil  Zero-mean complex Gaussian RV  ix  List of Symbols hi  Channel fading amplitude in the i-th branch  Hi  Nakagami-m RV  Iv (·)  vth-order modified Bessel function of the first kind defined as v+2k  (x/2) Iv (x) = ∑∞ k=0 k!Γ(v+k+1)  Mh (s)  MGF of RV h  max{γi }Li=1  Maximum of γ1 , γ2 , · · · , γL  ni  Complex AWGN in the i-th branch  N0  Power spectral density of complex AWGN  o ()  Small o notation  p  Parameter in the conditional SER  pe (β )  Conditional SER in terms of β  pe (γ )  Conditional SER in terms of γ  Pe  Average SER for coherent or noncoherent modulation  PeE  Asymptotic SER of EGC reception with coherent modulation  PeS,C  Asymptotic SER of SC reception with coherent modulation  PeS,N  Asymptotic SER of SC reception with noncoherent modulation  (PeEGC )asym  Asymptotic SER of EGC over arbitrarily correlated Nakagami-m fading channels  EGC ) (Pe,i asym  Asymptotic SER of EGC over independent Nakagami-m fading channels  (PeSC )asym  Asymptotic SER of SC over arbitrarily correlated Nakagami-m fading channels  SC ) (Pe,i asym  Asymptotic SER of SC over independent Nakagami-m fading channels  E (γ ) Pout th  Asymptotic outage probability of EGC reception  S (γ ) Pout th  Asymptotic outage probability of SC reception  q  Parameter in the conditional SER x  List of Symbols ∞ √1 x 2π  Q(·)  Gaussian Q-function defined as Q(x) =  Qv (α , β )  Generalized Marcum Q-function defined as Qv (α , β ) =  exp −y2 /2 dy  2 2 ∞ x v−1 − x +2α e Iv−1 (α x) dx β x α  ri  Received signal in the i-th branch  rEGC  Output signal of equal gain combiner  rMRC  Output signal of maximal ratio combiner  S−1  Inverse of matrix S  Var [·]  Variance of a RV  wi  Weighting factor in the i-th branch  β  A RV depending on the channel statistics  γ  Instantaneous SNR at the output of the diversity combiner  γEGC  Instantaneous SNR at the output of equal gain combiner  γMRC  Instantaneous SNR at the output of maximal ratio combiner  γSC  Instantaneous SNR at the output of selection combiner  γi  Instantaneous SNR of the i-th branch  γ¯  The average SNR at the combiner output  Γi  Average SNR of the i-th branch  Γ(·)  Gamma function defined as Γ(x) =  γ (m, z)  Incomplete Gamma function defined as  γ (m, z) =  ∞ x−1 −t e dt 0 t  z m−1 −t e dt 0t  δu,v  Kronecker delta function  ρi  Correlation factor  ϕi  Channel fading phase in the i-th branch  N µ, σ 2  Gaussian distributed with mean µ and variance σ 2  χn 0, σi2  Central chi-square distribution with n degrees of freedom and common Gaussian variance σi2  xi  List of Symbols χn s, σi2  Noncentral chi-square distribution with n degrees of freedom, noncentrality parameter s2 and common Gaussian variance σi2  xii  Acknowledgments I am deeply grateful to my supervisor Dr. Julian Cheng for his enthusiasm, guidance, advice, encouragement, and support. I will continue to be influenced by his rigorous scholarship, clarity in thinking, and professional integrity. I would like to express my thanks to Dr. Heinz Bauschke for his willingness to serve as my external examiner. I would also like to thank Dr. Richard Klukas and Dr. Jonathan Holzman for serving on the committee. I really appreciate their valuable time and constructive comments on my thesis. I am also grateful to my dear colleagues Xian Jin, Maggie Luyan Mei, Mingbo Niu, Nathan Nianxin Tang, Andr´e Johnson, Vicki Feng Wei, Sapphire Yu Lan, Yeyuan Xiao, Yuefeng Yao, and Chiun-Shen Liao for sharing their academic experiences and constructive viewpoints generously with me during our discussions. Special thanks are given to Xuegui Song for his long-lasting help in research and daily life since 2003. I would also like to thank my dear friends Kefeng Xu, Xiaosa Zhang, Xiangkui Song, Er’tao Lv, Pengfei Zhou, and Shuaibin Li for their help. Finally, I would like to thank my parents, sister, and brothers for their patience, understanding and support over all these years. I also want to thank my girlfriend and her parents for their encouragement and support. All my achievements would not have been possible without their constant encouragement and support.  xiii  Chapter 1 Introduction 1.1 Background and Motivation In the 1860s, James Clerk Maxwell proposed Maxwell’s equations, which form the foundation of electromagnetics and make wireless communications possible. However, it was not until 1895 that Guglielmo Marconi opened the way for modern wireless communications by transmitting the Morse code for the letter ‘S’ over a distance of approximately one and a half kilometers utilizing electromagnetic waves [1]. Since then, wireless communications have become a significant element of modern society. From satellite broadcasting to wireless Internet to the now ubiquitous cellular telephones, wireless communications have revolutionized the way societies function. On the one hand, wireless communications have gained wide popularity around the world. For instance, it is reported that the number of mobile phone subscribers is on the order of 4 billion currently and will rise to 5.6 billion in 2013 worldwide. On the other hand, wireless communications are faced with a number of challenges, among which multipath fading is the main cause leading to system degredation. Multipath fading is mainly caused by the scattering, reflection, refraction, and diffraction of radiowaves when wireless signals go through complex physical mediums. Multipath fading leads to a loss of signal power without reducing noise power, and hence causing poor performance in wireless communication systems. The diversity concept was introduced to overcome the adverse effects of multipath fading on the performance of wireless communication systems by providing multiple faded replicas of the same signal at the receiver [2]. The intuition behind this concept is to exploit the low probability of concurrence of deep  1  1.1. Background and Motivation fades in all the replicas. In the spatial domain, diversity can be realized through multiplereceiver antennas, i.e., antenna diversity. Multi-branch diversity combining is an effective antenna diversity technique to combat the detrimental effects of multipath fading in wireless communications [3]. The three most popular diversity combining schemes are maximal ratio combining (MRC), selection combining (SC), and equal gain combining (EGC). Among them, MRC is the optimal diversity combiner but with high implementation complexity; SC and EGC provide comparable performance to MRC with relatively lower complexity. In the study of diversity combining over correlated fading channels, analytical methods have been used to obtain accurate expressions for the average error rate and outage probability. While exact, the results are often complicated and difficult to apply since they generally demand one or two fold numerical integration and the exactness for the estimation of the average error rate and outage probability may be constrained by a selected numerical integration routine. In addition, these analytical results usually cannot provide useful physical insights into the transmission characteristics of correlated fading channels [4]. However, the asymptotic technique is a powerful analytical tool that usually provides compact and accurate estimation for error rate and outage probability in large signal-to-noise ratio (SNR) regions. Besides the lower computational complexity, another significant advantage of asymptotic analysis is that the resulting solutions can often reveal some important insights that can not be easily obtained otherwise. For instance, Liu et al. observed that the asymptotic error rates of MRC, EGC, and SC over arbitrarily correlated Rayleigh channels can be obtained by scaling the asymptotic error rates over independent branches with the determinant of the normalized channel correlation matrix [5].  2  1.2. Literature Review  1.2 Literature Review In the study of multi-branch diversity combining, performance analyses over independent fading channels have been widely published (see references in [2]). However, correlated fading among several diversity branches exists for many real systems [6], [7] due to the insufficient separation between the antennas. Therefore, quantitative analysis of the degradation of diversity systems due to correlated fading is important from both theoretical and practical standpoints. While a comprehensive theoretical performance analysis for MRC over various correlated fading channels is available, it is challenging for SC and EGC reception. This is because, with the exception of the dual-branch case, the cumulative distribution function (CDF) or probability density function (PDF) at the output of SC and EGC combiners operating over correlated branches is generally unknown. A number of researchers have studied the performance of SC and EGC over correlated fading channels. For SC, most of the existing performance analysis over correlated fading channels has primarily focused on two or three branches [8–12]. In [13], Zhang and Lu proposed a general approach for studying L-branch SC in arbitrarily correlated fading. However, their method requires an L-dimensional integration, and the computation complexity increases exponentially with L [14]. In [15], Karagiannidis et al. obtained an expression for the joint CDF of multivariate Nakagami-m random variables (RVs) with exponential correlation, and this result was subsequently applied to the performance analysis of SC over correlated Nakagami-m fading channels. In another related work, Karagiannidis et al. proposed an efficient approach for evaluating the PDF and CDF of multiple Nakagami-m RVs with arbitrary correlation by approximating the correlation matrix with a Green’s matrix [16]. However, their results were expressed in terms of an infinite series whose convergence can be slow when L is large and when the correlation coefficient is close to one. Recently, Chen and Tellambura derived the CDF of L-branch SC output SNR in equally correlated Nakagami-m fading by noting that a set of equally correlated complex Gaussian RVs can be obtained by linearly combining a set of independent Gaussian RVs [17]. Their technique 3  1.2. Literature Review was further applied by Zhang and Beaulieu for the performance analysis of generalized SC in arbitrarily correlated Nakagami-m fading [14]. Analytical solutions presented in [14] and [17], while exact, are complex and can be difficult to apply because they require double integration involving the m-th order generalized Marcum Q-function. The exact performance analysis of EGC over correlated fading channels with arbitrary diversity order is equally challenging. Chen and Tellambura studied the performance of EGC in equally correlated Nakagami-m fading channels in [18], where the equally correlated Nakagami-m fading channels are transformed into a set of conditionally independent chisquare RVs and the moments of the EGC output SNR are expressed in terms of the Appell hypergeometric function. This solution, while exact, is complex and can be difficult to apply because it requires multiple numerical integrations. Karagiannidis analyzed the performance of EGC by approximating the moment generating function (MGF) of the output SNR, where the moments are determined exactly only for exponentially correlated Nakagami-m channels in terms of a multi-fold infinite series [19]. More recently, various simple and accurate approximations to the PDF of the sum of an arbitrary number of correlated Nakagami-m RVs are proposed in [20–23], which then are used for analytical EGC performance evaluation. All of the approaches proposed in [19–23] are complex because they require approximations for the MGF of the output SNR or the PDF of a sum of arbitrary number of Nakagami-m RVs. The asymptotic technique described in [24], [25] is a powerful analytical tool to obtain accurate error rate and outage probability in large SNR regions. Let γ = β γ¯ be the instantaneous SNR at the output of the diversity combiner, where β is a RV depending on the channel statistics and γ¯ is the average SNR at the combiner output. For coherent modu√ lation, the conditional symbol error rate (SER) in terms of γ is pe (γ ) = pQ qγ , where Q(·) is one-dimensional Gaussian Q-function defined as Q(x) =  ∞ √1 x 2π  exp −y2 /2 dy [2,  eq. (4.1)], and p and q are constants related to the modulation formats. In [25], Wang and Giannakis observed that the conditional SER curve for coherent modulation behaves like an impulse function when instantaneous SNR is close to zero and the conditional SER is  4  1.3. Thesis Outline nearly zero for large instantaneous SNR (see Fig. 1.1). For correlated fading channels, the expression of f (γ ) can be difficult to obtain. Since the average SER is defined as  Pe =  ∞ 0  pe (γ ) f (γ )d γ  (1.1)  where f (γ ) is the PDF of instantaneous SNR, based on the observation of Wang and Giannakis, one can approximate the unknown PDF of the instantaneous SNR near its origin to compute the average SER for large values of average SNR. The PDF near zero can be derived from a Taylor series expansion. Finally, one can obtain the asymptotic average SER by (1.1). This is the fundamental idea of asymptotic techniques for performance analysis over correlated fading channels. Another approach, which can also yield the same asymptotic solution, is to calculate the MGF of the instantaneous SNR [25]. The asymptotic technique is suitable for providing highly accurate estimation for error rate and outage probability in large SNR regions, and for revealing physical insights into the transmission characteristics of multi-branch diversity combining over correlated fading channels.  1.3 Thesis Outline This thesis consists of five chapters. Chapter 1 presents background knowledge of wireless communication developments and technologies. In modern wireless communication, mitigating the detrimental effects of multipath fading is important for any mature wireless communication system. Multi-branch space diversity is commonly used to improve the error rate performance of wireless communication systems. However, the analytical expressions for the error rate of diversity combining systems are often complex and unfeasible for correlated diversity branches. This motivates researchers to apply asymptotic techniques to study the error rate and outage probability of diversity combining systems in large SNR regions. Chapter 2 provides detailed technical background for the entire thesis. Firstly, multipath fading is presented and the three basic and most widely used fading models, namely Raleigh, 5  1.3. Thesis Outline  0.6 0.5 0.4 0.3  γ¯ = 0 dB Q  0.2  √  β¯ γ  γ¯ = 5 dB 0.1  γ¯ = 15 dB 0 −0.1 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2  β Figure 1.1: Q  β γ¯ as functions of β for γ¯ = 0, 5, and 15 dB.  6  1.3. Thesis Outline Rician and Nakagami-m fadings, are introduced. Secondly, the three most popular diversity schemes, namely MRC, EGC, and SC, are explained. Thirdly, the asymptotic technique is reviewed to show how the approximate MGFs or PDFs can be used to calculate the asymptotic error rate. Finally, a single integral representation is presented for the joint PDF of multiple correlated Nakagami-m RVs with specified correlation matrix. In Chapter 3, we derive a new series representation of the generalized Marcum Q-function and give a simple accurate approximation for Marcum Q-function Qm (α , β ) when β → 0+ . Utilizing this approximation, we obtain the PDF of the instantaneous SNR at the output of SC when SNR is near its origin and then derive closed form expressions for asymptotic SER and outage probability. In Chapter 4, using the series expansion of the modified Bessel function of the first kind, we obtain the MGF of the square root of instantaneous SNR at the output of the equal gain combiner, and then derive the asymptotic average SER and outage probability of EGC over arbitrarily correlated Nakagami-m channels. Chapter 5 summarizes the entire thesis and lists our contributions in this thesis. In addition, future work related to our current research is suggested.  7  Chapter 2 Multipath Fading and Diversity Combining Techniques In this chapter, we will present some background knowledge concerning multipath fading, diversity combining techniques, the asymptotic technique, as well as construction of multiple Nakagami-m random variables with specified correlation matrix.  2.1 Multipath Fading Radiowave propagation through wireless channels is a complicated process characterized by various effects such as multipath fading and shadowing. Multipath fading is due to the constructive and destructive combination of randomly delayed, reflected, scattered, and diffracted signal components. This type of fading is relatively fast and therefore is responsible for short-term signal variations, where both the signal envelope and signal phase fluctuate over time. Depending on the relative relation between the symbol period of the transmitted signal and the coherence time of fading channels, fading can be classified into slow fading and fast fading [2], [3]. Coherence time is defined as the time period over which we can consider the fading process to be correlated. Slow fading occurs when the symbol duration is less than the channel coherence time, and fast fading is the opposite. Similarly, according to the relative relation between the transmitted signal bandwidth and the channel coherence bandwidth, fading can also be classified into frequency-nonselective fading and frequency-selective fad-  8  2.1. Multipath Fading ing. Coherence bandwidth is defined as the frequency range over which the fading process is correlated. If the transmitted signal bandwidth is much smaller than the channel coherence bandwidth, the fading is considered to be flat, and otherwise it is frequency selective. In this thesis, we only focus on slow and frequency-nonselective flat fading channels. When the multipath fading process possesses these properties, it is common to use statistical distributions to describe the random behavior of the received signal amplitude. Most widely used statistical models include the Rayleigh, Rician, and Nakagami-m distributions.  2.1.1 Rayleigh Distribution The Rayleigh distribution is frequently used to model the time varying characteristics of the received signal amplitude in a wireless channel where there is no direct line-of-sight (LOS) path between the transmitter and the receiver. It is well known that the envelope of the sum of two independent and identically distributed (i.i.d.) Gaussian signals with zero mean and variance σ 2 obeys a Rayleigh distribution. The PDF of the Rayleigh distribution is given by x x2 exp − σ2 2σ 2 2x x2 = exp − , x≥0 Ω Ω  f (x) =  (2.1)  where Ω = 2σ 2 , is the mean square value of the received signal amplitude.  2.1.2 Rician Distribution When a strong LOS path exists between the transmitter and receiver in addition to many weaker random multipath signal components, the randomness of the received signal ampli-  9  2.1. Multipath Fading tude is modeled using the Rician distribution whose PDF is given by  f (x) =  2x(K + 1) (K + 1)x2 exp −K − I0 2x Ω Ω  K(K + 1) Ω  , x≥0  (2.2)  where K = A2 /(2σ 2 ) is the Rician factor defined as the ratio of the LOS power A2 to the scattered power 2σ 2 , the average amplitude power is denoted by Ω = E[X 2 ] = A2 + 2σ 2 where E [·] denotes the expectation of a RV, and Iv (·) is the vth-order modified Bessel function of v+2k  (x/2) the first kind defined as Iv (x) = ∑∞ k=0 k!Γ(v+k+1) where Γ(·) is the Gamma function defined as  Γ(x) =  ∞ x−1 −t e dt 0 t  [26, eq. (8.310.1)]. As the strength of the dominant signal diminishes,  i.e. K = 0, the Rician distribution specializes to a Rayleigh distribution. As the value of K becomes large, the fading effect tends to vanish.  2.1.3 Nakagami-m Distribution Introduced by Nakagami in the early 1940’s [27], the Nakagami-m distribution is a versatile distribution used to model multipath fading in wireless channels. Empirical data show that the Nakagami-m fading model often gives the best fit to land-mobile and indoor-mobile multipath propagation [2]. The PDF of the Nakagami-m distribution is given by  f (x) =  m 2 Γ(m) Ω  m  x2m−1 exp −  mx2 1 , x ≥ 0, m ≥ Ω 2  (2.3)  where Ω is the mean square value of the amplitude. The fading severity parameter m is defined as Ω2 /E[(X 2 − Ω)2 ]. Beyond its empirical justification, the Nakagami-m distribution is often used for the following reasons. First, the Nakagami-m distribution can be used to model fading conditions more or less severe than Rayleigh fading. When m = 1, the Nakagami-m distribution becomes the Rayleigh distribution. When m = 0.5, it becomes a one-sided Gaussian distribution. As the value of the parameter m increases, the fading severity decreases. Second, the Rician distribution can be approximated by the Nakagami  10  2.2. Diversity Combining Techniques distribution with K =  √  √ m2 − m/(m − m2 − m) and m = (K + 1)2 /(2K + 1) for m > 1 [3].  The corresponding squared Nakagami-m fading amplitude has a Gamma PDF as  f (x) =  1 m Γ(m) Ω  m  xm−1 exp −  mx 1 , x ≥ 0, m ≥ . Ω 2  (2.4)  2.2 Diversity Combining Techniques Diversity techniques were introduced to overcome the detrimental effects of multipath fading on wireless communication systems. The principle of diversity techniques is that, if several copies of the same information bearing signal are available and they all experience independent fading, then the probability that all copies are in deep fading simultaneously is small. If signal copies are appropriately combined at the receiver end, one can reduce the effect of multipath fading and improve the performance of wireless communication systems. There are several known methods to obtain the independent copies of the signal: space diversity, frequency diversity and time diversity. Among them, space diversity is widely used because it is simple to implement and requires no additional bandwidth. In this thesis, we focus only on diversity in the spatial domain with multi-branch reception. The structure of multi-branch diversity combining is shown in Fig. 2.1 where ri denotes the received signal in the i-th branch, hi is the channel fading amplitude, ϕi is the corresponding phase, d is the transmitted data symbol, ni is the complex additive white Gaussian noise (AWGN) in the i-th branch, and wi is the weighting factor. The three commonly used multi-branch reception schemes are MRC, SC, and EGC. They are briefly described in the following subsections.  2.2.1 Maximal Ratio Combining In the maximal ratio combining mechanism, the weighting factors are the complex conjugation of the channel gains, i.e., wi = hi e jϕi . Therefore, the combiner eliminates the influence  11  2.2. Diversity Combining Techniques  Figure 2.1: Multi-branch diversity combining receiver.  of random phase, which is induced by multipath fading, on the output signal. The combiner amplifies the strong signals and attenuates the weak ones to maximize the instantaneous output SNR at the combiner. Assuming the noise components of the input branches are mutually independent, the output signal of maximal ratio combiner is L  L  i=1  i=1  rMRC = ∑ wi ri = ∑ h2i d + hi e jϕi ni  (2.5)  and the instantaneous output SNR γMRC is given by  γMRC =  L 2 L hi ES (∑Li=1 h2i )2 ES = = ∑ ∑ γi (∑Li=1 h2i )N0 i=1 N0 i=1  (2.6)  where ES is the average symbol energy of the transmitted data symbol, N0 is the power spectral density (PSD) of complex AWGN in the i-th branch, and γi =  h2i ES N0  denotes the  instantaneous SNR of the i-th branch. MRC is the optimal diversity combiner in the sense of maximizing the combiner output SNR, in the absence of other interfering sources [28]. However, MRC is also known for having the highest complexity to implement due to the fact that phase-lock and amplitude  12  2.2. Diversity Combining Techniques weighting must be performed.  2.2.2 Equal Gain Combining In equal gain combining, the weighting factors have constant amplitude value, but have opposite phase to the channel gains, i.e., wi = e jϕi , which is known as the co-phasing operation. Therefore, the combiner only eliminates the influence of random phase with equal weights. Assuming equal noise powers in all branches, the instantaneous output signal of EGC is given by L  L  i=1  i=1  rEGC = ∑ wi ri = ∑ hi d + e jϕi ni  (2.7)  and the instantaneous output SNR γEGC is given by (∑Li=1 hi )2 ES γEGC = . LN0  (2.8)  In practice, exact estimation of the correct weighting factors for MRC can be difficult. Hence, EGC becomes a practical combining scheme with lower implementation complexity than MRC.  2.2.3 Selection Combining In selection combining, the combiner only picks one best branch out of the L noisy received signals ri (i = 1, · · · , L). The weighting factor for the selected branch is unity and the other weighting factors are zero. Suppose all branches have the same noise power spectral density N0 . Then the output of SC can be expressed as rSC = rindex(max{hi }L  i=1 )  (2.9)  13  2.3. Asymptotic Technique where index(yi ) denotes the index i corresponding to yi . Then the instantaneous output SNR  γSC is given by γSC = max{γi }Li=1 where γi =  h2i ES N0  (2.10)  denotes the instantaneous SNR of the i-th branch. Since SC processes only a  single branch, it has a much lower complexity compared to MRC and EGC. However, since SC ignores information provided by other diversity branches, its performance is poorer than EGC and MRC. SC can be used with coherent modulations, noncoherent modulations, and differentially coherent modulations.  2.3 Asymptotic Technique Let γ = β γ¯ be the instantaneous SNR at the output of the diversity combiner, where β is a RV depending on the channel statistics and γ¯ is the average SNR at the combiner output. Suppose the PDF of β can be approximated by a single polynomial term for β → 0+ as f (β ) = cβ t + o (β t ).1 For coherent modulation with conditional SER pe (β ) = pQ  qβ γ¯ , the average SER  for large SNR is given by [25]  Pe = √  2t cΓ t + 32 p  1 + o . γ¯t+1 π (t + 1) (qγ¯)t+1  (2.11)  Many coherent modulation schemes, such as binary phase shift keying (BPSK), M-ary phase shift keying (M-PSK), M-ary pulse amplitude modulation (M-PAM) and M-ary quadrature amplitude modulation (M-QAM), have conditional SERs of the form pe (β ) = pQ  qβ γ¯ ,  and the corresponding values of p and q are tabulated in Table 2.1. To the author’s best knowledge, an asymptotic error rate expression for noncoherent modulation in large SNR regions has not been derived. We present this result in the following 1 We  write a function p(x) as o(x) if limx→0+ p(x)/x = 0.  14  2.3. Asymptotic Technique proposition. Proposition 1: For noncoherent modulation with conditional symbol error rate of the form pe (β ) = p exp(−qβ γ¯), the error rate at large SNR can be expressed as  Pe =  cΓ (t + 2) p 1 + o t+1 t+1 γ¯ (t + 1) (qγ¯)  (2.12)  where p and q are constants related to specific modulation formats. The proof of (2.12) is given in Appendix A. The values of p and q for noncoherent modulations, like binary noncoherent frequency shift keying (BNCFSK) and binary differential phase shift keying (BDPSK), are tabulated in Table 2.1. Therefore, to compute the asymptotic SER at large SNR, one needs to determine the parameters c and t from the PDF of β , or, from the MGF of β when s → ∞. Another approach is to consider the square root of γ at the output of the diversity combiner. Let √ h = γ = β γ¯. From the PDF of β , with a change of variable, it is straightforward to obtain the PDF of γ as f (γ ) = c  γt γt + o . γ¯t+1 γ¯t+1  (2.13)  The PDF of h is h2t+1 h2t+1 f (h) = 2c t+1 + o γ¯ γ¯t+1  (2.14)  and the corresponding MGF of h can be expressed as  Mh (s) =  1 2cΓ(2t + 2) + o t+1 2t+2 . t+1 2t+2 γ¯ s γ¯ s  (2.15)  Therefore, one can simply obtain the asymptotic SER by extracting the parameters c and t from (2.13) to (2.15) in the frequency domain. On the other hand, the asymptotic SER can also be approximated by (see, e.g., [25], [29])  Pe = (Gc · γ¯)−Gd + o  1 γ¯Gd  (2.16) 15  2.4. Construction of Multiple Correlated Nakagami-m RVs Table 2.1: Parameters p and q for different coherent and noncoherent modulation schemes Modulation Scheme Conditional SER p q √ BPSK Q ( 2γ ) 1 2 √ π π M-PSK M ≥ 4 ≈ 2Q 2γ sin M 2 2 sin2 M 6γ M 2 −1  M-PAM  2(1 − M1 )Q  M-QAM  4(1 − √1M )Q  BNCFSK BDPSK  3γ M−1  2(1 − M1 )  6 M 2 −1  4(1 − √1M )  3 M−1  1 2 1 2  1 2  1 1 2 exp − 2 γ 1 2 exp (−γ )  1  where Gc is the coding gain, and Gd is the diversity order. The diversity order Gd determines the slope of the SER versus average SNR curve, at high SNR, in a log-log scale. Gc (in decibels) determines the shift of the curve in SNR relative to a benchmark SER curve of γ¯−Gd . Comparing (2.16) with (2.11) and (2.12) respectively, we observe that Gd = t +1 for both coherent and noncoherent modulation, Gc = q Gc = q  1 cpΓ(t+2) − t+1 t+1  2t cpΓ(t+ 23 ) √ π (t+1)  1 − t+1  for coherent modulation, and  for noncoherent modulation.  2.4 Construction of Multiple Correlated Nakagami-m RVs In this section, we present a framework to obtain a single integral representation for multiple Nakagami-m distributions with a specified correlation matrix. The joint PDF of correlated Nakagami-m RVs is expressed explicitly in terms of single integral solutions. A remarkable feature of these expressions is that the computational complexity reduces to a single integral computation for an arbitrary number of dimensions. The basic idea for the construction of multiple correlated Nakagami-m RVs is that a set of equally correlated complex Gaussian RVs can be obtained by linearly combining a set of independent Gaussian RVs [30]. The original construction approach was proposed in [17] to obtain the CDF of Lbranch SC output SNR in equally correlated Nakagami-m fading. In [14], this approach was used to evaluate the performance of diversity combiners with positively correlated branches. 16  2.4. Construction of Multiple Correlated Nakagami-m RVs  2.4.1 Construction of Multiple Correlated Nakagami-m RVs Similar to [17, eq. (5)], [30] for integer m we express the L correlated Nakagami-m RVs through Lm zero-mean complex Gaussian RVs by √ 1 − ρi Xil + ρi X0l + jσi  Gil = σi  √ 1 − ρiYil + ρiY0l  (2.17)  for i = 1, · · · , L and l = 1, · · · , m, where j2 = −1, 0 ≤ ρi ≤ 1, and X0l , Xil , Y0l and Yil are independent Gaussian RVs with distribution N (0, 1/2). That is, for any u, v ∈ {0, 1, · · · , L}, and l, n ∈ {1, · · · , m}, E[Xul Yvn ] = 0, E[Xul Xvn ] = E[Yul Yvn ] = δu,v δl,n /2, where δu,v is the Kronecker delta function. 2 Let Ri denote the summation of the absolute square of Gil , i.e., Ri = ∑m l=1 |Gil | . Then,  it can be shown Ri is the sum of squares of m independent Rayleigh envelopes with central chi-square distribution χ2m (0, σi2 /2) [14]. The cross-correlation between Ri and Rk can be shown to be  ρRi ,Rk =  E [Ri Rk ] − E [Ri ] E [Rk ] Var [Ri ] Var [Rk ]  = ρi ρk , i = k  (2.18)  where Var [·] denotes the variance of a RV. The proof of (2.18) is given in Appendix B. √ Let Hi = Ri , then it can be shown H1 , H2 , · · · , HL are L correlated Nakagami-m RVs with identical fading parameter m and mean-square value mσi2 . The relationship between the correlation of Hi and Hk and the correlation of Ri and Rk , denoted by ρHi ,Hk and ρRi ,Rk respectively, is [27]  ρHi ,Hk =  F − 12 , − 12 ; m; ρRi ,Rk − 1 ψ (m) − 1  (2.19)  where ψ (m) = Γ(m)Γ(m + 1)/Γ2 (m + 1/2) and F(·) is the hypergeometric function defined as F(a, b; c; z) =  Γ(c) ∞ Γ(a+n)Γ(b+n) zn n! Γ(a)Γ(b) ∑n=0 Γ(c+n)  [31, eq. (15.1.1)].  17  2.4. Construction of Multiple Correlated Nakagami-m RVs  2.4.2 Joint PDF of Multiple Correlated Nakagami-m RVs 2 2 Let Z = ∑m l=1 X0l +Y0l . When X0l = x0l and Y0l = y0l (l = 1, · · · , m) are fixed, the real √ and imaginary parts of Gil have equal variance of σi2 (1 − ρi )/2 and means σi ρi x0l and √ σi ρi y0l , respectively. Therefore, R1 , R2 , · · · , RL are independent noncentral chi-square RVs 2 2 2 with distribution χ2m ( ρi ∑m l=1 x0l + y0l , σi (1 − ρi )/2) and whose marginal CDF is given  by [29, eq. (2-1-124)]  F (ri |z) = Pr (Ri ≤ ri |z) = 1 − Qm  2zρi , 1 − ρi  2ri 2 σi (1 − ρi )  (2.20)  where Qm (·, ·) denotes the m-th order Marcum Q-function defined as Qm (α , β ) =  2 2 ∞ x m−1 − x +2α e Im−1 (α x) dx x β α  [32]. The joint marginal CDF of R1 , · · · , RL is  given by L  F (r1 , r2 , · · · , rL |z) = ∏ 1 − Qm i=1  2zρi , 1 − ρi  2ri 2 σi (1 − ρi )  .  (2.21)  2 2 Z = ∑m l=1 X0l +Y0l follows a central chi-square distribution χ2m (0, 1/2) and its PDF is  given by f (z) =  zm−1 e−z , z ≥ 0. Γ (m)  (2.22)  Averaging the joint marginal CDF in (2.21) with respect to the PDF of Z, the joint CDF of R1 , R2 , · · · , RL becomes  F (r1 , r2 , · · · , rL ) =  1 Γ (m)  ∞ L  ∏  0 i=1  1 − Qm  2zρi , 1 − ρi  2ri σi2 (1 − ρi )  zm−1 e−z dz. (2.23)  Taking partial derivatives of (2.23) with respect to r1 , r2 , · · · , rL , one can obtain the joint PDF  18  2.4. Construction of Multiple Correlated Nakagami-m RVs of R1 , R2 , · · · , RL as follows 1 f (r1 , r2 , · · · , rL ) = Γ (m)  ∞ L 0  1 ∏ σ 2(1 − ρi) i=1 i  ri zρi σi2  2 1 − ρi  zρi ri σi2  × Im−1  m−1 2  exp −  ri 2 σi (1 − ρi ) ρ  z  L i m−1 − 1+∑i=1 1−ρi z  e  (2.24)  dz.  Eqns. (2.23) and (2.24) essentially are the joint CDF and PDF of L correlated Gamma RVs respectively, and will be useful in studying the asymptotic performance of SC. Using variable transformation, we can obtain the joint PDF of H1 , H2 , · · · , HL as 1 f (h1 , h2 , · · · , hL ) = Γ (m)  ∞ L 0  2hi ∏ σ 2(1 − ρi) i=1 i  h2i zρi σi2  2 1 − ρi  zρi h2i σi2  × Im−1  m−1 2  exp −  h2i σi2 (1 − ρi )  (2.25)  ρ  z  L i m−1 − 1+∑i=1 1−ρi z  e  dz.  The proof of (2.25) is given in Appendix C. Eqn. (2.25) is the joint the PDF of L correlated Nakagami-m RVs and will be useful in studying the asymptotic performance of EGC.  19  Chapter 3 Asymptotic Performance Analysis of SC over Arbitrarily Correlated Nakagami-m Fading Channels 3.1 Introduction The generalized Marcum Q-function has a long history in signal processing literature, especially in the analysis of target detection by pulsed radars with single or multiple observations [32]. This special function is also frequently used in the performance analysis involving noncoherent and differential detection of multichannel narrowband signals over fading channels [2], [29]. The generalized Marcum Q-function is given by [32]  Qv (α , β ) =  ∞  β  x  x α  v−1  e−(x  2 +α 2  )/2 I  v−1 (α x) dx  (3.1)  where α and β are non-negative real numbers, and v is a positive real number. Although v can take any positive real number, when v is a positive integer m, Qm (α , β ) is the complementary cumulative distribution function of a noncentral chi-square RV with 2m degrees of freedom [33]. Because of its importance, the generalized Marcum Q-function has been the subject of considerable research over the past several decades. The precise computation of Qv (α , β ) can be difficult especially for large α and v values, because the integral in (3.1) involves the  20  3.1. Introduction modified Bessel function of the first kind [34]. In order to calculate the generalized Marcum Q-function, power series expansion methods were used in [35–37] and the Neuman series expansion method was proposed in [38]. Since direct computation of (3.1) is difficult, various alternative representations of Qv (α , β ) were proposed. For example, several integral forms of Qv (α , β ) were developed in [39–45] and the references therein. During the past decade, several upper and lower bounds were proposed for the generalized Marcum Q-function [46– 55]. Most of these bounds were obtained by utilizing the bounds of the integrand in (3.1) or by changing the integral region via a geometric interpretation of the functions. In this thesis, using a series expansion of the modified Bessel function of the first kind Im−1 (x) we present a new series representation of Qm (α , β ) in terms of the incomplete gamma function γ (m, x) and obtain a simple accurate approximation of Qm (α , β ) when  β → 0+ . Utilizing this approximation of Qm (α , β ), we study the asymptotic performance of selection combining over arbitrarily correlated Nakagami-m channels. Of practical value, we derive new compact analytical results that can be used to provide rapid and accurate error rate and outage probability estimation. Of theoretical interest, we reveal some physical insights into the transmission characteristics of SC over correlated Nakagami-m channels, and in particular, how the branch power covariance coefficient matrix can influence the average error rate and outage probability in large SNR regions. The rest of this chapter is organized as follows. In Section 3.2, an alternative series representation and a new approximation of Qm (α , β ) when β → 0+ are presented. In Section 3.3, the PDF of L-branch SC output SNR near its origin over arbitrarily correlated Nakagami-m fading channels is derived. Section 3.4 derives the asymptotic error rate and outage probability of SC. Section 3.5 discusses some important insights and provides some numerical results.  21  3.2. A Series Representation of Qm (α , β )  3.2 A Series Representation of Qm(α , β ) We first present a new series representation of Qm (α , β ) as follows. Proposition 2: The generalized Marcum Q-function can be written as Qm (α , β ) = 1 − e−  α2 2  ∞  α 2k β2 γ m + k, ∑ k 2 k=0 Γ (m + k) k! · 2  where γ (·, ·) is the incomplete gamma function defined as γ (m, z) =  (3.2)  z m−1 −t e dt 0t  [26, (8.350.1)].  Proof: By [33, eq. (5)-(8)], for integer-valued m the generalized Marcum Q-function can be expressed as  Qm (α , β ) =  ∞  β  x α  x  = 1−  β 0  x  m−1  e−(x  x α  m−1  2 +α 2  e−(x  )/2 I  m−1 (α x) dx  2 +α 2  )/2 I  m−1 (α x) dx.  (3.3)  For m-th order modified Bessell function of the first kind, its series expansion can be represented as [26, (8.445)] ∞  Im (z) =  1  ∑ Γ (m + k + 1) k!  k=0  z 2  m+2k  .  (3.4)  Substituting (3.4) into (3.3), we have β 0  x  =α  x α  m−1  2  1−m − α2  e  e−(x  2 +α 2  ∞  )/2 I  m−1 (α x) dx  1  ∑ Γ (m + k) k!  k=0  = e−  α2 2  ∞  α 2k  ∑ Γ (m + k) k! · 2k · γ  k=0  α 2  β  m+2k−1  m + k,  0  β2 2  x2  x2m+2k−1 e− 2 dx (3.5)  and (3.2) follows immediately. n u+n  (−1) y Using the series expression of the incomplete gamma function γ (u, y) = ∑∞ n=0 n!(u+n)  22  3.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels [26, (8.354.1)], we can rewrite Qm (α , β ) as  Qm (α , β ) = 1 − e  2 − α2  n  β2  m+k+n  ∞ (−1) 2 α 2k ∑ Γ (m + k) k! · 2k ∑ (m + k + n) · n! . n=0 k=0 ∞  (3.6)  An immediate consequence of (3.6) is that when β → 0+ , the m-th order Marcum Q-function Qm (α , β ) can be approximated as Qm (α , β ) = 1 −  β 2m α2 exp − 2m · m! 2  + o β 2m .  (3.7)  To demonstrate the accuracy of the above approximation, we calculate the exact values of Qm (α , β ) using the function NCX2CDF provided by Matlab and compare them with the approximate values obtained from (3.7) for different values of α and β . The numerical results are compared in Table 3.1. From Table 3.1, we can observe that the relative error becomes small when β → 0+ , and therefore we conclude that the small argument approximation of the Marcum Q-function is highly accurate.  3.3 PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels 3.3.1 System Model Assume that there are L available diversity branches experiencing frequency-nonselective and slow Nakagami-m fading with fading parameter m taking the same positive integer value. Let Hi be the instantaneous channel fading amplitude on the i-th branch, ES be the average symbol energy of the transmitted data symbol, and let N0 be the PSD of complex AWGN in the i-th branch. The instantaneous SNR of the i-th diversity branch, γi , is defined as  γi = |Hi |2  ES N0  (3.8) 23  3.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels  Table 3.1: The exact and approximate values of Qm (α , β ) for different values of α and β  α  0.1  1  5  β  Exact value  Approximate value  Relative error  0.1  0.999996887986722  0.999987562344010  9.325 × 10−6  0.05  0.999999805256228  0.999999222646501  5.826 × 10−7  0.01  0.999999999688286  0.999999998756234  9.32 × 10−10  0.005  0.999999999980518  0.999999999922265  5.825 × 10−11  0.001  0.999999999999969  0.999999999999876  9.314 × 10−14  0.1  0.999997569793885  0.999992418366754  5.151 × 10−6  0.05  0.999999847945917  0.999999526147922  3.217 × 10−7  0.01  0.999999999756628  0.999999999241837  5.147 × 10−10  0.005  0.999999999984789  0.999999999952615  3.217 × 10−11  0.001  0.999999999999976  0.999999999999924  5.151 × 10−14  0.1  0.999999993945953  0.999999999953417  6.007 × 10−9  0.05  0.999999999622624  0.999999999997089  3.744 × 10−10  0.01  0.999999999999397  0.999999999999995  5.986 × 10−13  0.005  0.999999999999962  1.000000000000000  3.741 × 10−14  0.001  0.9999999999999999  1  1.11 × 10−16  24  3.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels and the average SNR of the i-th diversity branch, Γi , is given by Γi = E |Hi |2  ES . N0  (3.9)  Recall from Chapter 2 that for SC, the instantaneous SNR at the output of the selection combiner is given by  γSC = max γi . i=1,··· ,L  (3.10)  3.3.2 PDF of SC Output SNR We now use the method described in Chapter 2 to construct correlated Nakagami-m RVs H1 , H2 , . . . , HL . Then the instantaneous SNR of the i-th diversity branch is γi = Ri NES0 . Let R denote the branch power covariance coefficient matrix, whose (ik)-th element (R)ik is defined as (R)ik =  E [γi γk ] − E [γi ] E [γk ] . Var [γi ] Var [γk ]  (3.11)  Using (2.18), we can show that    ρR ,R = ρi ρk , i = k i k (R)ik = .   1, i=k  (3.12)  This model simplifies to the equally correlated case when ρi = ρk . By the symmetry property of R, ρi and ρk can be uniquely determined by  ρi =  (R)il (R)in (R)ln  (3.13)  ρk =  (R)kl (R)kn (R)ln  (3.14)  and  when n, l = i, n, l = k, l = n.  25  3.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels If we define M as the matrix whose (ik)-th entry is given by  (M)ik =  (R)ik .  (3.15)  Express M as a rank-one updated matrix M = S + uuT , where S = diag(1 − ρ1 , · · · , 1 − ρL ) √ √ √ and u = [ ρ1 , ρ2 , · · · , ρL ]. Using [56, eq. (6.2.3)], we can obtain the determinant of matrix M as L  ρi i=1 1 − ρi  det(M) = det (S) 1 + uT S−1 u = 1 + ∑  L  ∏ (1 − ρi) .  (3.16)  i=1  Following the construction of H1 , H2 , . . . , HL described in Chapter 2, we can obtain the mean square value of Hi as E |Hi |2 = mσi2 and the average SNR of the i-th diversity branch as Γi = mσi2 NES0 . Since the instantaneous SNR of the i-th diversity branch is γi = Ri NES0 , we can express γi in terms of as Γi , i.e., γi =  Ri Γ. mσi2 i  By (3.10), the CDF of the SC output SNR  γSC can be written as FγSC (γ ) = Pr (γsc ≤ γ ) = Pr (γ1 ≤ γ , · · · , γL ≤ γ ) = Pr  R1 RL Γ1 ≤ γ , · · · , ΓL ≤ γ 2 mσ 1 mσL2  = Pr R1 ≤  mσ12 mσL2 γ . γ , · · · , RL ≤ Γ1 ΓL  (3.17)  By the joint CDF of R1 , R2 , · · · , RL given in (2.23), we can obtain  FγSC (γ ) =  1 Γ (m)  ∞ L  ∏  0 i=1  1 − Qm  2ρi z , 1 − ρi  2mγ Γi (1 − ρi )  zm−1 e−z dz.  (3.18)  Eqn. (3.18) essentially generalizes the results in [17] and [14] to arbitrarily correlated SC over branches with different average SNR values. It is seen from (3.18) that the second parameter of the generalized Marcum Q-function 26  3.4. Asymptotic Performance Analysis of SC approaches zero when γ → 0+ , or when the CDF is near its origin. Using (3.7), we can obtain the asymptotic CDF of the SC output SNR as  FγSC (γ ) =  mmL  γ mL γ¯mL γ mL + o . mL mL ¯ γ ¯ γ [Γ (m + 1)]L detm (M) ∏Li=1 Γm i  (3.19)  Differentiating (3.19) with respect to γ , we obtain the PDF of instantaneous SNR at the output of the SC as  fγSC (γ ) =  mL · mmL  γ mL−1 γ¯mL γ mL−1 . + o mL γ¯mL [Γ (m + 1)]L detm (M) ∏Li=1 Γm i γ¯  (3.20)  The analytical expression obtained in (3.20) for the PDF of the output SNR near its origin will be useful in studying the asymptotic error rate performance of SC.  3.4 Asymptotic Performance Analysis of SC Comparing (3.20) with (2.13) in Chapter 2, we observe that  t = mL − 1  (3.21)  and c=  mL · mmL  γ¯mL . [Γ (m + 1)]L detm (M) ∏Li=1 Γm i  (3.22)  From (3.21), it is obvious that the diversity order Gd is mL.  27  3.4. Asymptotic Performance Analysis of SC  3.4.1 Asymptotic Error Rate The asymptotic SER of SC reception with coherent modulation can thus be expressed from (2.11) as PeS,C =  Γ (2mL + 1) mmL p  1  1  +o m  2mL+1 qmL Γ (mL + 1) [Γ (m + 1)]L detm (M) ∏Li=1 Γi  ∏Li=1 Γm i  .  (3.23)  In the special case when m = 1, it can be shown that (3.23) agrees with the asymptotic error rate of SC reception over arbitrarily correlated Rayleigh channels [5, eq. (7)]2 . Similarly, the asymptotic SER of SC reception with noncoherent modulation can be obtained from (2.12) as PeS,N =  mmL Γ (mL + 1) p L  kmL [Γ (m + 1)] det  m  1  (M) ∏Li=1 Γm i  +o  1 ∏Li=1 Γm i  .  (3.24)  3.4.2 Asymptotic Outage Probabilities The outage probability is defined as  Pout (γth ) = Pr (γ < γth ) =  γth 0  f (γ )d γ  (3.25)  where γth is a predefined outage threshold and f (γ ) is the PDF of the instantaneous SNR at the output of diversity combiner. Substituting (2.13) into (3.25), one obtains the asymptotic outage probabilities as c Pout (γth ) = t +1 2 When  γth γ¯  t+1  t+1 γth + o t+1 γ¯  .  (3.26)  m = 1, M is same as the matrix M defined in [5].  28  3.5. Discussions and Numerical Results With the values of t and c obtained in (3.21) and (3.22), we obtain S Pout (γth ) =  mL γth 1 + o . L L L m (m!) detm (M) ∏i=1 Γi ∏i=1 Γm i  mmL  (3.27)  When m = 1, it can be shown that (3.27) agrees with the asymptotic outage probability of SC reception over arbitrarily correlated Rayleigh channels [5, eq. (14)]. Although we have derived the asymptotic error rate and outage probability expressions (3.23), (3.24) and (3.27) assuming integer fading parameter values, our numerical results suggest these analytical expressions are also valid at least for m = 0.5.  3.5 Discussions and Numerical Results 3.5.1 Discussions Since det(M) = 1 for independent fading channels, we observe from (3.23) and (3.24) that the asymptotic SER of SC over arbitrarily correlated Nakagami-m fading channels can be expressed in terms of the asymptotic SER over independent Nakagami-m fading channels scaled by a factor detm (M), i.e., (PeSC )asym =  SC ) (Pe,i asym  detm (M)  (3.28)  SC ) where (Pe,i asym denotes the asymptotic SER of SC over independent Nakagami-m fading  branches. In (3.28), the factor detm (M), which is less than unity, can be considered as the loss factor due to channel correlation. It should be noted that the simple relationship in (3.28) also holds for MRC over arbitrarily correlated Nakagami-m fading channels [25]. Finally a relationship similar to (3.28) can also be seen for the outage probability from (3.27).  29  3.5. Discussions and Numerical Results  3.5.2 Numerical Results In this subsection, we present some numerical results of coherent BPSK and noncoherent BDPSK with three-branch SC over correlated Nakagami-m fading channels. We use the method proposed in [57] to generate correlated Nakagami-m RVs. For all the numerical results obtained here, we have assumed three-branch diversity reception with Γ1 = Γ2 = Γ3 and γth = 3 dB. We use ρ˜ = [ρ1 , ρ2 , ρ3 ] to denote a vector whose elements comprise the power covariance coefficient matrix. For comparison, we set ρ˜ 1 = [0.4571, 0.2296, 0.4571] and ρ˜ 2 = [0.3968, 0.5265, 0.7861]. By (3.12) and (3.15), we obtain     1 0.3240 0.4571      M1 =  0.3240 1 0.3240     0.4571 0.3240 1 and      1 0.4571 0.5585     M2 =  1 0.6433   0.4571 .   0.5585 0.6433 1 It follows that det(M1 )= 0.6771 and det(M2 )= 0.3937. From (3.28), the determinant inequality det(M1 ) > det(M2 ) predicts that more highly correlated fading channels will lead to worse SER performance, as one expects. This is confirmed by the bit error rate (BER) curves shown in Fig. 3.1 and Fig. 3.2 for m = 0.5 and m = 2 respectively. Fig. 3.3 and Fig. 3.4 plot the asymptotic and simulated outage probabilities of threebranch SC with matrices M1 and M2 for m = 0.5 and m = 2 respectively. Figs. 3.1- 3.4 indicate that the analytical asymptotic results have excellent agreement with the simulated result in large SNR regions. This implies that the analytical asymptotic results can be used to predict small values of error rate and outage probability accurately for large SNR regions, where Monte Carlo simulation becomes time-consuming. Note that Fig. 3.1 and Fig. 3.3 also suggest that (3.23), (3.24) and (3.27) are also valid at least for m = 0.5. 30  3.5. Discussions and Numerical Results  0  10  BPSK−Asymptotic BPSK−Simulation BDPSK−Asymptotic BDPSK−Simulation  −1  10  M2  −2  BER  10  −3  10  −4  10  M1  −5  10  −6  10  0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 SNR (dB)  Figure 3.1: The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 0.5.  31  3.5. Discussions and Numerical Results  0  10  BPSK−Asymptotic BPSK−Simulation BDPSK−Asymptotic BDPSK−Simulation  −1  10  −2  10  −3  10  M2  −4  BER  10  −5  10  −6  M1  10  −7  10  −8  10  −9  10  −10  10  −11  10  0  2.5  5  7.5  10 12.5 SNR (dB)  15  17.5  20  22.5  Figure 3.2: The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 2.  32  3.5. Discussions and Numerical Results  0  10  Asymptotic Simulation −1  Outage Probability  10  M2  −2  10  −3  M1  10  −4  10  −5  10  0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 SNR (dB)  Figure 3.3: The asymptotic and simulated outage probabilities of SC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 0.5.  33  3.5. Discussions and Numerical Results  0  10  Asymptotic Simulation  −1  10  −2  10 Outage Probability  −3  10  M2  −4  10  −5  10  M1  −6  10  −7  10  −8  10  −9  10  −10  10  −11  10  0  2.5  5  7.5  10 12.5 SNR (dB)  15  17.5  20  22.5  Figure 3.4: The asymptotic and simulated outage probabilities of SC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 2.  34  Chapter 4 Asymptotic Performance Analysis of EGC over Arbitrarily Correlated Nakagami-m Fading Channels In this chapter, using an integral representation for the joint PDF of multiple correlated Nakagami-m RVs obtained in Chapter 2, we derive the MGF of the square root of instantaneous SNR at the output of equal gain combiner and obtain the asymptotic average SER and outage probability of EGC over arbitrarily correlated Nakagami-m channels.  4.1 System Model Assume that there are L available diversity branches experiencing frequency-nonselective and slow Nakagamim-m fading with fading parameter m taking the same positive integer value. Let Hi be the instantaneous channel fading amplitude on the i-th branch, ES be the average symbol energy of the transmitted data symbol, and let N0 be the PSD of complex AWGN in the i-th branch. The instantaneous SNR of the i-th diversity branch, γi , is defined as  γi = |Hi |2  ES N0  (4.1)  and the average SNR of the i-th diversity branch, Γi , is given by Γi = E |Hi |2  ES . N0  (4.2) 35  4.2. MGF of Square Root of Instantaneous SNR at the Output of EGC Recall in Chapter 2 that for EGC, the instantaneous SNR at the output of the equal gain combiner is given by  2  ∑Li=1 Hi ES . γEGC = LN0  (4.3)  4.2 MGF of Square Root of Instantaneous SNR at the Output of EGC We now use the method described in Chapter 2 to construct correlated Nakagami-m RVs H1 , H2 , . . . , HL . Then the joint PDF of H1 , H2 , . . . , HL can be expressed as (2.25). Let R denote the branch power covariance coefficient matrix whose (ik)-th element (R)ik is defined by (3.11). Define M as the matrix whose (ik)-th entry is given by (3.15), then the determinant of matrix M can be expressed as (3.16). Using a series expansion of first kind modified Bessel function in [26, eq. (8.445)], when x → 0, Im−1 (x) can be written as Im−1 (x) = Define HE =  1 x Γ (m) 2  m−1  + o xm−1 .  (4.4)  √ γEGC . By (4.3), when γEGC → 0, Hi → 0. Substituting (4.4) into (2.25), we  can write the joint PDF of H1 , H2 , · · · , HL as L h2i 1 2 1 2m−1 2m−1 f (h1 , h2 , · · · , hL ) = hi + o(hi ) exp − 2 . (4.5) 2m detm (M) ∏ σi (1 − ρi ) i=1 Γ(m) σi  36  4.3. Asymptotic Performance Analysis of EGC Using (4.5), by definition, the MGF of HE can be written as MHE (s) = EHE e−sHE  √  = EH1 ,H2 ,··· ,HL e =  −  s(H1 +H2 +···+HL ) √ LN0  1 2 m det (M) Γ(m)  ES  (4.6)  L L  ∏ gi (s) i=1  where gi (s) is given by  gi (s) =  ∞ 0  m NES0  m  h2m−1 + o(h2m−1 ) i i  Γi  exp −  h2i m NES0 Γi (1 − ρi )  ES dhi . LN0  exp −shi  (4.7) Using [26, eqs. (3.462) and (9.246)], we can show that when s → ∞, gi (s) can be written as mL gi (s) = Γ(2m) Γi  m  1 s2m  +o  1 s2m  .  (4.8)  Substituting (4.8) into (4.6), we obtain the MGF of HE as  MHE (s) =  (mL)mL 2Γ(2m) detm (M) Γ(m)  L  1  1  ∏Li=1 Γm i  s2mL  +o  1 s2mL  .  (4.9)  The MGF expression obtained in (4.9) will be used in studying the asymptotic error rate and outage probability of EGC.  4.3 Asymptotic Performance Analysis of EGC Comparing (4.9) with (2.15) in Chapter 2, we observe that  t = mL − 1  (4.10)  37  4.3. Asymptotic Performance Analysis of EGC and c=  2L−1 Γ(2m) Γ(2mL) Γ(m)  L  (mL)mL γ¯mL . detm (M) ∏Li=1 Γm i  (4.11)  From (4.10), it is obvious that the diversity order Gd is mL.  4.3.1 Asymptotic Error Rate The asymptotic SER of EGC reception with coherent modulation can be expressed from (2.11) as PeE  =  [Γ (2m)]L (mL)mL p  1  m 2mL+1−L Γ (mL + 1) [Γ (m)]L detm (M) ∏Li=1 (qΓi )  +o  1 ∏Li=1 Γm i  .  (4.12)  In the special case of m = 1, it can be shown that (4.12) agrees with the asymptotic symbol error rate of EGC reception over arbitrarily correlated Rayleigh channels [5, eq. (5)].  4.3.2 Asymptotic Outage Probabilities Substituting (4.10) and (4.11) into (3.26), we obtain the outage probability E Pout (γth ) =  mL γth 1 + o . L L L m Γ(2mL) [Γ(m)] detm (M) ∏i=1 Γi ∏i=1 Γm i  2L−1 [Γ(2m)]L (mL)mL−1  (4.13)  When m = 1, it can be shown that (4.13) agrees with the asymptotic outage probability of EGC reception over arbitrarily correlated Rayleigh channels [5, eq. (13)]. Equations (4.12) and (4.13) are important new results. Although we deduce these results for integer fading parameter m, our numerical results suggest they are also valid at least for m = 0.5.  38  4.4. Discussions and Numerical Results  4.4 Discussions and Numerical Results 4.4.1 Discussions Since det(M) = 1 for independent fading channels, we observe from (3.23) and (3.24) that the asymptotic SER of EGC over arbitrarily correlated Nakagami-m fading channels can be expressed in terms of the asymptotic SER over independent Nakagami-m fading channels scaled by a factor detm (M), i.e., (PeEGC )asym  =  EGC ) (Pe,i asym  detm (M)  (4.14)  EGC ) where (Pe,i asym denotes the asymptotic SER of EGC over independent Nakagami-m fad-  ing branches. In (4.14) the factor detm (M) can be considered as the loss factor due to channel correlation. It should be noted that the simple relationship in (4.14) also holds for MRC over arbitrarily correlated Nakagami-m fading channels [25]. Finally a relationship similar to (4.14) can also be seen for the outage probability from (3.27).  4.4.2 Numerical Results In this subsection, we present some numerical results of coherent BPSK with threebranch EGC over correlated Nakagami-m fading channels. We use the method proposed in [57] to generate correlated Nakagami RVs. For all the numerical results obtained here, we have assumed three-branch diversity reception with Γ1 = Γ2 = Γ3 and γth = 3 dB. We use ρ˜ = [ρ1 , ρ2 , ρ3 ] to denote a vector whose elements comprise the power covariance coefficient matrix. For comparison, we set ρ˜ 1 = [0.3968, 0.5265, 0.7861] and ρ˜ 2 = [0.5008, 0.6229, 0.8264].  39  4.4. Discussions and Numerical Results By (3.12) and (3.15), we obtain     1 0.4571 0.5585      M1 =  0.4571 1 0.6433     0.5585 0.6433 1 and      1 0.5585 0.6433     M2 =  1 0.7175  .  0.5585   0.6433 0.7175 1 It follows that det(M1 )= 0.3937 and det(M2 )= 0.2750. From (4.14), the determinant inequality det(M1 ) > det(M2 ) predicts that more highly correlated fading channels will lead to worse SER performance, as one expects. This is confirmed by the BER curves shown in Fig. 4.1 and Fig. 4.2 for m = 0.5 and m = 2 respectively. As seen from Fig. 4.1 and 4.2, the asymptotic error rates are accurate for SNR greater than 20 dB. Fig. 4.3 and Fig. 4.4 plot the asymptotic and simulated outage probabilities of threebranch EGC with matrices M1 and M2 for m = 0.5 and m = 2 respectively. Figs. 4.1- 4.4 indicate that the analytical asymptotic results have excellent agreement with the simulated result in large SNR regions. This implies that the analytical asymptotic results can be used to predict small values of error rate and outage probability accurately for large SNR regions, where Monte Carlo simulation becomes time-consuming. Note that Fig. 4.1 and Fig. 4.3 also suggest that (4.12) and (4.13) are also valid at least for m = 0.5.  40  4.4. Discussions and Numerical Results  0  10  BPSK−Asymptotic BPSK−Simulation  −1  10  M2  −2  BER  10  M  1  −3  10  −4  10  −5  10  0  2.5  5  7.5  10 12.5 15 17.5 20 22.5 25 SNR (dB)  Figure 4.1: The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 0.5.  41  4.4. Discussions and Numerical Results  0  10  −1  BPSK−Asymptotic BPSK−Simulation  10  −2  10  −3  10  M  −4  2  BER  10  −5  M1  10  −6  10  −7  10  −8  10  −9  10  −10  10  −11  10  0  2.5  5  7.5  10 12.5 SNR (dB)  15  17.5  20  Figure 4.2: The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 when m = 2.  42  4.4. Discussions and Numerical Results  0  10  BPSK−Asymptotic BPSK−Simulation −1  Outage Probability  10  M2 M1  −2  10  −3  10  −4  10  0  2.5  5  7.5  10 12.5 15 17.5 20 22.5 25 SNR (dB)  Figure 4.3: The asymptotic and simulated outage probabilities of EGC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 0.5.  43  4.4. Discussions and Numerical Results  0  10  BPSK−Asymptotic BPSK−Simulation  −1  10  −2  10 Outage Probability  −3  10  M2  −4  10  −5  M1  10  −6  10  −7  10  −8  10  −9  10  −10  10  −11  10  0  2.5  5  7.5  10 12.5 SNR (dB)  15  17.5  20  Figure 4.4: The asymptotic and simulated outage probabilities of EGC over 3-branch correlated Nakagami-m channel with matrices M1 and M2 when m = 2.  44  Chapter 5 Conclusions In this chapter, we first summarize the contributions of this work, and then suggest some future work.  5.1 Summary of Contributions This thesis makes the following contributions: 1. An asymptotic error rate expression for noncoherent modulation in large SNR regions has been derived. This expression can be used to estimate the high SNR SER or BER of various noncoherent modulation schemes, such as BNCFSK and BDPSK. 2. A new series representation of the generalized Marcum Q-function and a simple approximation for the Marcum Q-function Qm (α , β ) when β → 0+ have been derived. Numerical results show that the relative error between the exact value obtained by using the function NCX2CDF and the approximate value becomes small when β → 0+ , and therefore the small argument approximation of the Marcum Q-function is highly accurate. 3. Closed-form error rate and outage probability expressions are derived for multi-branch EGC and SC over arbitrarily correlated Nakagami-m fading channels. They can be used to provide accurate and rapid estimation of error rates and outage probabilities in large SNR regions, where Monte Carlo simulation becomes time-consuming. These simple expressions will allow wireless system engineers to estimate the required fading  45  5.2. Future Work margin in link budget analysis without resorting to time-consuming computer simulations. 4. Simple relationships have been established for the asymptotic error rates and outage probabilities of EGC and SC over arbitrarily correlated and independent Nakagami-m fading channels. The same relationship also holds for MRC. It has been shown that the asymptotic error rate and outage probability over correlated branches can be obtained by scaling the asymptotic error rate and outage probability over independent branches with a factor, detm (M), where det(M) is the determinant of matrix M whose elements are the square root of corresponding elements in the branch power covariance correlation matrix R. The factor detm (M) can be considered as the loss factor due to channel correlation. These significant relationships explain theoretically the observation that more highly correlated fading channels lead to worse error rate and outage probability performance.  5.2 Future Work In this work, it is assumed that all the correlated Nakagami-m fading channels have the same integer-valued fading parameter m. However, the correlated Nakagami-m fading channels with different fading parameters can exist in some practical transmission scenarios. Because of the validity and feasibility of asymptotic techniques for large SNRs, the asymptotic technique can be extended to the study of the performance of multi-branch diversity combinings over correlated Nakagami-m fading channels with arbitrarily-valued fading parameters. In addition, so far there exists no method that can accurately generate multiple correlated Nakagami-m RVs with arbitrary fading parameters and mean square values. In [57], Zhang proposed a decomposition technique to generate multiple correlated Nakagami-m RVs with same fading parameters. Zhang’s algorithm is inaccurate when 2m is not an integer. In our future research, we will develop an accurate method to generate multiple correlated 46  5.2. Future Work Nakagami-m RVs with arbitrary fading parameters and mean square values.  47  Bibliography [1] L. Anderson, Nikola Tesla on His Work With Alternating Currents and Their Application to Wireless Telegraphy, Telephony, and Transmission of Power: An Extended Interview. Breckenridge, Colo: Twenty-First Century Books, 2002. [2] M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, 2nd ed. New York: Wiley-IEEE Press, 2004. [3] G. L. St¨uber, Principles of Mobile Communications. Norwell, MA: Kluwer, 1996. [4] L. 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Alouini, “Some new results for integrals involving the generalized Marcum Q-function and their application to performance evaluation over fading channels,” IEEE Transactions on Wireless Communications, vol. 2, pp. 611–615, July 2003. [41] G. Weinberg, “Poisson representation and monte carlo estimation of generalized Marcum Q-function,” IEEE Transactions on Aerospace and Electronic Systems, vol. 42, pp. 1520–1531, Oct. 2006.  52  Chapter 5. Bibliography [42] A. Annamalai and C. Tellambura, “A simple exponential integral representation of the generalized Marcum Q-function QM (a, b) for real-order Q with applications,” in Proc. IEEE Military Communications Conference MILCOM 2008, San Diego, CA, USA, Nov. 17–19 2008, pp. 1–7. [43] A. Annamalai, C. Tellambura, and J. Matyjas, “A new twist on the generalized Marcum Q-function QM (a, b) with fractional-order Q and its applications,” in Proc. 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International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, Saint Julians, Malta, Mar. 12–14 2008, pp. 1351–1356. [53] A. Baricz and Y. Sun, “New bounds for the generalized Marcum Q-function,” IEEE Transactions on Information Theory, vol. 55, pp. 3091–3100, July 2009. [54] A. Baricz, “Tight bounds for the generalized Marcum Q-function,” Journal of Mathematical Analysis and Applications, vol. 360, pp. 265–277, Dec. 2009. [55] Y. Sun, A. Baricz, and S. Zhou, “On the monotonicity, log-concavity and tight bounds of the generalized Marcum and Nuttall Q-functions,” IEEE Transactions on Information Theory, vol. 56, pp. 1166–1186, Mar. 2010. [56] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000. [57] Q. Zhang, “A decomposition technique for efficient generation of correlated Nakagami fading channels,” IEEE Journal on Selected Areas in Communications, vol. 18, pp. 2385–2392, Nov. 2000.  54  Appendix A Derivation of Error Rate for Noncoherent Modulation Let A be a small fixed positive number such that when β < A, the PDF of β can be written as f (β ) = cβ t + o (β t ). Then the average SER for noncoherent modulation with conditional SER pe (β ) = p exp(−qβ γ¯) can be caculated as  Pe = =  ∞ 0  ∞  0  =p =p −p +p  pe (β ) f (β )d β p exp(−qβ γ¯) f (β )d β A  0 0  exp(−qβ γ¯) f (β )d β + p  ∞  ∞  A  exp(−x) cβ t + o β t  qβ γ¯ ∞ ∞  A qβ γ¯ ∞ A  ∞  exp(−x) cβ t + o β t  exp(−qβ γ¯) f (β )d β  exp(−qβ γ¯) f (β )d β dxd β dxd β (A.1)  55  Appendix A. Derivation of Error Rate for Noncoherent Modulation ∞ −y x e dy.  where in obtaining the last equality we have used a fact e−x =  The first integral in  (A.1) can be written as ∞  ∞  p = cp  qβ γ¯ 0 ∞ ∞  qβ γ¯  0  0  0  ∞ 0  exp (−qβ γ¯) o β t d β  β t d β exp (−x) dx + o γ¯−(t+1) ∞  cp  =  dxd β  exp(−x)β t dxd β + p  x qγ¯  ∞  = cp  exp(−x) cβ t + o β t  t+1  xt+1 exp (−x) dx + o γ¯−(t+1)  0 (t + 1) (qγ¯) cΓ (t + 2) p + o γ¯−(t+1) . = t+1 (t + 1) (qγ¯)  (A.2)  Ignoring the term o (β t ), the second integral in (A.1) can be expressed as  cp = cp = cp  ∞  ∞  exp(−x)β t dxd β  A qβ γ¯ x ∞ qγ¯ Aqγ¯ ∞ Aqγ¯  A  β t d β exp (−x) dx  β t+1 t +1  cp  =  t+1  (t + 1) (qγ¯)  x qγ¯  exp (−x) dx A  ∞  Aqγ¯  exp (−x) xt+1 − (Aqγ¯)t+1 dx.  (A.3)  The integral in (A.3) becomes zero as γ¯ → ∞. Therefore, the second integral can be written as o γ¯−(t+1) . For the third integral in (A.1) we have the following relation  p  ∞ A  exp(−qβ γ¯) f (β )d β ≤ p  ∞ A  exp (−qAγ¯) f (β ) d β  = p · exp (−qAγ¯) ≤ p · exp (−qAγ¯)  ∞ A  f (β ) d β (A.4)  56  Appendix A. Derivation of Error Rate for Noncoherent Modulation and exp (−qAγ¯) γ¯t+1 = 0. = lim γ¯→∞ γ¯−(t+1) γ¯→∞ exp (qAγ¯) lim  (A.5)  Therefore, the third integral can be written as o γ¯−(t+1) . Finally, it follows that the average SER for noncoherent modulation at large SNR can be expressed as (2.12).  57  Appendix B Derivation of (2.18) According to (2.17), for i ∈ {1, · · · , L} and l, p ∈ {1, · · · , m}, l = p, we have the following expressions |Gil |2 = σi2 (1 − ρi )Xil2 + ρi X0l2 + 2  ρi (1 − ρi )Xil X0l  + σi2 (1 − ρi )Yil2 + ρiY0l2 + 2 ρi (1 − ρi )Yil Y0l  (B.1)  and |Gil |4 = (1 − ρi )2 Xil4 + ρi2 X0l4 + 4ρi (1 − ρi )Xil2 X0l2 + (1 − ρi )2Yil4 + ρi2Y0l4 σi4 + 4ρi (1 − ρi )Yil2Y0l2 + 2ρi (1 − ρi )Xil2 X0l2 + 4(1 − ρi ) ρi (1 − ρi )Xil3 X0l + 2(1 − ρi )2 Xil2Yil2 + 2ρi (1 − ρi )Xil2Y0l2 + 4(1 − ρi ) ρi (1 − ρi )Xil2YilY0l + 4ρi  ρi (1 − ρi )X0l3 Xil + 2ρi (1 − ρi )X0l2 Yil2 + 2ρi2 X0l2 Y0l2  + 4ρi  ρi (1 − ρi )X0l2 YilY0l + 4(1 − ρi ) ρi (1 − ρi )Xil X0lYil2  + 4ρi  ρi (1 − ρi )Xil X0lY0l2 + 8ρi (1 − ρi )Xil X0lYilY0l + 2ρi (1 − ρi )Yil2Y0l2  + 4(1 − ρi ) ρi (1 − ρi )Yil3Y0l + 4ρi  ρi (1 − ρi )Yil Y0l3  (B.2)  58  Appendix B. Derivation of (2.18) and |Gil |2 |Gip |2 2 2 = (1 − ρi )2 Xil2 Xip + ρi (1 − ρi )Xil2 X0p + 2(1 − ρi ) ρi (1 − ρi )Xil2 Xip X0p σi4 2 + (1 − ρi )2 Xil2Yip2 + ρi (1 − ρi )Xil2Y0p + 2(1 − ρi ) ρi (1 − ρi )Xil2YipY0p 2 2 + ρi (1 − ρi )X0l2 Xip + ρi2 X0l2 X0p + 2ρi  ρi (1 − ρi )X0l2 Xip X0p  2 + ρi (1 − ρi )X0l2 Yip2 + ρi2 X0l2 Y0p + 2ρi  ρi (1 − ρi )X0l2 YipY0p  2 + 2ρi + 2(1 − ρi ) ρi (1 − ρi )Xil X0l Xip  2 ρi (1 − ρi )Xil X0l X0p  + 4ρi (1 − ρi )Xil X0l Xip X0p + 2(1 − ρi ) ρi (1 − ρi )Xil X0l Yip2 + 2ρi  2 ρi (1 − ρi )Xil X0lY0p + 4ρi (1 − ρi )Xil X0lYipY0p  2 2 + (1 − ρi )2Yil2 Xip + ρi (1 − ρi )Yil2 X0p + 2(1 − ρi ) ρi (1 − ρi )Yil2 Xip X0p 2 + (1 − ρi )2Yil2Yip2 + ρi (1 − ρi )Yil2Y0p + 2(1 − ρi ) ρi (1 − ρi )Yil2YipY0p 2 2 + ρi (1 − ρi )Y0l2 Xip + ρi2Y0l2 X0p + 2ρi 2 + ρi (1 − ρi )Y0l2 Yip2 + ρi2Y0l2 Y0p + 2ρi  ρi (1 − ρi )Y0l2 Xip X0p ρi (1 − ρi )Y0l2 YipY0p  2 + 2(1 − ρi ) ρi (1 − ρi )YilY0l Xip + 2ρi  2 ρi (1 − ρi )YilY0l X0p  + 4ρi (1 − ρi )YilY0l Xip X0p + 2(1 − ρi ) ρi (1 − ρi )YilY0l Yip2 + 2ρi  2 ρi (1 − ρi )YilY0lY0p + 4ρi (1 − ρi )Yil Y0l YipY0p .  (B.3)  For any u ∈ {0, 1, · · · , L}, and l ∈ {1, · · · , m}, Xul and Yul are independent Gaussian RVs with distribution N (0, 1/2), hence we have E [Xul ] = E [Yul ] = 0, E Xul2 = E Yul2 = 21 , E Xul3 = E Yul3 = 0, and E Xul4 = E Yul4 = 34 . Taking the expectation with respect to (B.2), (B.2), and (B.3), we can obtain E |Gil |2 = σi2 , E |Gil |4 = 2σi4 , and E |Gil |2 |Gip |2 = σi4 . m 2 2 2 Since Ri = ∑m l=1 |Gil | and Rk = ∑l=1 |Gkl | , it follows that E [Ri ] = mσi and  59  Appendix B. Derivation of (2.18) E [Rk ] = mσk2 . The mean square of Ri can be calculated as  E  R2i  = E  2  m   =  2  ∑ |Gil |  l=1  m  ∑E  l=1  4  m  |Gil | + ∑  m  ∑E  |Gil |2 |Gip |2 = m(m + 1)σi4  l=1 p=l p=1  (B.4) and the variance of Ri is given by Var [Ri ] = E R2i − E [Ri ]2 = mσi4 .  (B.5)  Similarly, we have Var [Rk ] = mσk4 . According to the definition of Ri and Rk , we have m  Ri Rk =  m  ∑ ∑ |Gil |2|Gkp|2.  (B.6)  l=1 p=1  Since Xil , X0l , Yil , and Y0l are independent zero mean Gaussian RVs, we can ignore them when calculating E [Ri Rk ]. By symmetry, we can obtain m  m  E [Ri Rk ] = 2σi2 σk2 ∑  ∑E  2 2 2 2 (1 − ρi )Xil2 (1 − ρk )Xkp + ρk X0p + (1 − ρk )Ykp + ρkY0p  + 2σi2 σk2 ∑  ∑E  2 2 2 2 ρi X0l2 (1 − ρk )Xkp + ρk X0p + (1 − ρk )Ykp + ρkY0p  l=1 p=1 m m  l=1 p=l p=1 m  + 2σi2 σk2 ∑ E ρi X0l2 (1 − ρk )Xkl2 + ρk X0l2 + (1 − ρk )Ykl2 + ρkY0l2 l=1   m    m m m 1 − ρi ρi ρi + ρk  2 2 2 2 σ σ + 2 σ + 2 σ  ∑ 2 i k ∑ ∑ i k ∑ 2 l=1 p=1 l=1 p=l 2 l=1   = 2σi2 σk2  ∑  m  p=1  = σi2 σk2 m2 + mρi ρk .  (B.7)  Noting that E [Ri Rk ] = σi2 σk2 m2 + mρi ρk , E [Ri ] = mσi2 , E [Rk ] = mσk2 , Var [Ri ] = mσi4 , and  60  Appendix B. Derivation of (2.18) Var [Rk ] = mσk4 , by the definition of cross-correlation of Ri and Rk , we can finally obtain (2.18).  61  Appendix C Derivation of (2.25) Using variable transformation, we have  f (h1 , h2 , · · · , hL ) = f (r1 , r2 , · · · , rL ) · |J(h1 , h2 , · · · , hL )|  (C.1)  where | · | is the absolute-value operator and J(h1 , h2 , · · · , hL ) is the Jacobian of the transformation, which is defined as    dr1 dh1    . . J (h1 , h2 , · · · , hL ) =   .   drL dh1  In our case, ri = h2i ,  dri dh j  = 2hi when i = j; otherwise,  ··· .. .  dr1 dhL  ···  drL dhL  dri dh j   ..  .  .   (C.2)  = 0. Therefore, J(h1 , h2 , · · · , hL ) =  ∏Li=1 2hi . It is straightforward to obtain (2.25).  62  

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