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Asymptotic performance analysis for EGC and SC over arbitrarily correlated Nakagami-m fading channels Li, Xianchang 2011-12-31

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ASYMPTOTIC PERFORMANCE ANALYSISFOR EGC AND SC OVER ARBITRARILYCORRELATED NAKAGAMI-m FADINGCHANNELSbyXIANCHANG LIB.Eng., Northwestern Polytechnical University, China, 2004A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(OKANAGAN)April 2011c Xianchang Li, 2011AbstractThe multi-branch diversity combining technique has been widely used in wireless com-munications systems to overcome the adverse effect of multipath fading. The three mostpopular diversity combining schemes are selection combining (SC), equal gain combining(EGC), and maximal ratio combining (MRC). In the performance analysis of multi-branchdiversity combining, the asymptotic technique is an attractive approach to obtain compactand 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 probabil-ity density function (PDF) of the instantaneous output SNR, or in the frequency domain byfinding the moment generating function (MGF) of the square root of the instantaneous outputSNR. In this thesis, the PDF of the instantaneous SNR at the output of selection combinerand the MGF of the square root of the instantaneous SNR at the output of equal gain com-biner over arbitrarily correlated Nakagami-m fading channels are derived and used to obtainasymptotic error rate and outage probability expressions of SC and EGC, respectively. Theseexpressions provide accurate and rapid estimation of error rates and outage probabilities. Theaccuracy of our analytical results is verified by computer simulation. More importantly, ouranalytical results provide physical insights into the transmission behavior of EGC and SCover arbitrarily correlated Nakagami-m fading channels. For instance, it is shown that theasymptotic error rates over correlated branches can be obtained by scaling the asymptoticerror rates over independent branches with a factor, detm(M), where det(M) is the deter-minant of matrix M whose elements are the square root of corresponding elements in thebranch power covariance correlation matrix R. A similar relationship can also be found forthe outage probabilities.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Multipath Fading and Diversity Combining Techniques . . . . . . . . . . . . 82.1 Multipath Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Rician Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Nakagami-m Distribution . . . . . . . . . . . . . . . . . . . . . . 102.2 Diversity Combining Techniques . . . . . . . . . . . . . . . . . . . . . . 11iiiTable of Contents2.2.1 Maximal Ratio Combining . . . . . . . . . . . . . . . . . . . . . 112.2.2 Equal Gain Combining . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Selection Combining . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Asymptotic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Construction of Multiple Correlated Nakagami-m RVs . . . . . . . . . . . 162.4.1 Construction of Multiple Correlated Nakagami-m RVs . . . . . . . 172.4.2 Joint PDF of Multiple Correlated Nakagami-m RVs . . . . . . . . 183 Asymptotic Performance Analysis of SC over Arbitrarily Correlated Nakagami-m Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 A Series Representation of Qm(a;b) . . . . . . . . . . . . . . . . . . . . 223.3 PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels . . 233.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 PDF of SC Output SNR . . . . . . . . . . . . . . . . . . . . . . . 253.4 Asymptotic Performance Analysis of SC . . . . . . . . . . . . . . . . . . 273.4.1 Asymptotic Error Rate . . . . . . . . . . . . . . . . . . . . . . . . 283.4.2 Asymptotic Outage Probabilities . . . . . . . . . . . . . . . . . . 283.5 Discussions and Numerical Results . . . . . . . . . . . . . . . . . . . . . 293.5.1 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Asymptotic Performance Analysis of EGC over Arbitrarily Correlated Nakagami-m Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 MGF of Square Root of Instantaneous SNR at the Output of EGC . . . . . 364.3 Asymptotic Performance Analysis of EGC . . . . . . . . . . . . . . . . . 374.3.1 Asymptotic Error Rate . . . . . . . . . . . . . . . . . . . . . . . . 38ivTable of Contents4.3.2 Asymptotic Outage Probabilities . . . . . . . . . . . . . . . . . . 384.4 Discussions and Numerical Results . . . . . . . . . . . . . . . . . . . . . 394.4.1 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48AppendicesA Derivation of Error Rate for Noncoherent Modulation . . . . . . . . . . . . 55B Derivation of (2.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58C Derivation of (2.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62vList of Tables2.1 Parameters p and q for different coherent and noncoherent modulation schemes 163.1 The exact and approximate values of Qm(a;b) for different values of a and b 24viList of Figures1.1 Q‡pb ¯g·as functions of b for ¯g = 0, 5, and 15 dB. . . . . . . . . . . . . . 62.1 Multi-branch diversity combining receiver. . . . . . . . . . . . . . . . . . . 123.1 The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 whenm = 0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branch correlated Nakagami-m channels with matrices M1 and M2 whenm = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 The asymptotic and simulated outage probabilities of SC over 3-branch cor-related Nakagami-m channel with matrices M1 and M2 when m = 0:5. . . . 333.4 The asymptotic and simulated outage probabilities of SC over 3-branch cor-related Nakagami-m channel with matrices M1 and M2 when m = 2. . . . . 344.1 The asymptotic and simulated BERs of BPSK for EGC over 3-branch corre-lated Nakagami-m channels with matrices M1 and M2 when m = 0:5. . . . 414.2 The asymptotic and simulated BERs of BPSK for EGC over 3-branch corre-lated Nakagami-m channels with matrices M1 and M2 when m = 2. . . . . 424.3 The asymptotic and simulated outage probabilities of EGC over 3-branchcorrelated Nakagami-m channel with matrices M1 and M2 when m = 0:5. . 434.4 The asymptotic and simulated outage probabilities of EGC over 3-branchcorrelated Nakagami-m channel with matrices M1 and M2 when m = 2. . . 44viiList of AcronymsAcronyms DefinitionsAWGN Additive White Gaussian NoiseBER Bit Error RateBDPSK Binary Differential Phase Shift KeyingBNCFSK Binary Noncoherent Frequency Shift KeyingBPSK Binary Phase Shift KeyingCDF Cumulative Distribution FunctionEGC Equal Gain CombiningLOS Line-of-SightMGF Moment Generating FunctionM-PAM M-ary Pulse Amplitude ModulationM-PSK M-ary Pulse Shift KeyingM-QAM M-ary Quadrature Amplitude ModulationMRC Maximal Ratio CombiningPDF Probability Density FunctionPSD Power Spectral DensityRV Random VariableSC Selection CombiningSER Symbol Error RateSNR Signal-to-Noise RatioviiiList of SymbolsSymbols Definitions[¢]T Matrix transpose operatord Transmitted data symboldet(M) Determinant of matrix Mexp(¢) Exponential functionE[¢] The expectation of a RVES Average symbol energy of the transmitted data symbolf(b) PDF of RV bf(g) PDF of instantaneous SNRf(x) PDF of RV XfgSC (g) PDF of gSCf (r1;r2;¢¢¢;rL) Joint PDF of R1;R2;¢¢¢;RLF(a;b;c;z) Hypergeometric function defined asF(a;b;c;z)= G(c)G(a)G(b)  ¥n=0 G(a+n)G(b+n)G(c+n) znn!F(rjz) Marginal CDF of R depending on ZF (r1;r2;¢¢¢;rLjz) Conditional CDF of R1;R2;¢¢¢;RL depending on ZF (r1;r2;¢¢¢;rL) Joint CDF of R1;R2;¢¢¢;RLFgSC (g) CDF of gSCGc Coding gainGd Diversity gainGil Zero-mean complex Gaussian RVixList of Symbolshi Channel fading amplitude in the i-th branchHi Nakagami-m RVIv(¢) vth-order modified Bessel function of the first kind defined asIv(x)=  ¥k=0 (x=2)v+2kk!G(v+k+1)Mh(s) MGF of RV hmaxfgigLi=1 Maximum of g1;g2;¢¢¢;gLni Complex AWGN in the i-th branchN0 Power spectral density of complex AWGNo() Small o notationp Parameter in the conditional SERpe(b) Conditional SER in terms of bpe(g) Conditional SER in terms of gPe Average SER for coherent or noncoherent modulationPEe Asymptotic SER of EGC reception with coherent modulationPS;Ce Asymptotic SER of SC reception with coherent modulationPS;Ne Asymptotic SER of SC reception with noncoherent modulation(PEGCe )asym Asymptotic SER of EGC over arbitrarily correlatedNakagami-m fading channels(PEGCe;i )asym Asymptotic SER of EGC over independent Nakagami-m fadingchannels(PSCe )asym Asymptotic SER of SC over arbitrarily correlated Nakagami-mfading channels(PSCe;i )asym Asymptotic SER of SC over independent Nakagami-m fadingchannelsPEout (gth) Asymptotic outage probability of EGC receptionPSout (gth) Asymptotic outage probability of SC receptionq Parameter in the conditional SERxList of SymbolsQ(¢) Gaussian Q-function defined as Q(x)=R¥x 1p2p exp¡¡y2=2¢dyQv(a;b) Generalized Marcum Q-function defined asQv(a;b)=R¥b x¡xa¢v¡1 e¡x2+a22 Iv¡1(ax)dxri Received signal in the i-th branchrEGC Output signal of equal gain combinerrMRC Output signal of maximal ratio combinerS¡1 Inverse of matrix SVar[¢] Variance of a RVwi Weighting factor in the i-th branchb A RV depending on the channel statisticsg Instantaneous SNR at the output of the diversity combinergEGC Instantaneous SNR at the output of equal gain combinergMRC Instantaneous SNR at the output of maximal ratio combinergSC Instantaneous SNR at the output of selection combinergi Instantaneous SNR of the i-th branch¯g The average SNR at the combiner outputGi Average SNR of the i-th branchG(¢) Gamma function defined as G(x)=R¥0 tx¡1e¡tdtg (m;z) Incomplete Gamma function defined asg (m;z)=Rz0 tm¡1e¡tdtdu;v Kronecker delta functionri Correlation factorji Channel fading phase in the i-th branchN ¡m;s2¢ Gaussian distributed with mean m and variance s2cn¡0;s2i ¢ Central chi-square distribution with n degrees of freedom andcommon Gaussian variance s2ixiList of Symbolscn¡s;s2i ¢ Noncentral chi-square distribution with n degrees of freedom,noncentrality parameter s2 and common Gaussian variance s2ixiiAcknowledgmentsI 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 schol-arship, clarity in thinking, and professional integrity.I would like to express my thanks to Dr. Heinz Bauschke for his willingness to serve asmy external examiner. I would also like to thank Dr. Richard Klukas and Dr. Jonathan Holz-man for serving on the committee. I really appreciate their valuable time and constructivecomments 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 constructiveviewpoints generously with me during our discussions. Special thanks are given to XueguiSong for his long-lasting help in research and daily life since 2003. I would also like tothank 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, under-standing and support over all these years. I also want to thank my girlfriend and her parentsfor their encouragement and support. All my achievements would not have been possiblewithout their constant encouragement and support.xiiiChapter 1Introduction1.1 Background and MotivationIn the 1860s, James Clerk Maxwell proposed Maxwell’s equations, which form the foun-dation of electromagnetics and make wireless communications possible. However, it wasnot until 1895 that Guglielmo Marconi opened the way for modern wireless communica-tions by transmitting the Morse code for the letter ‘S’ over a distance of approximately oneand a half kilometers utilizing electromagnetic waves [1]. Since then, wireless communica-tions have become a significant element of modern society. From satellite broadcasting towireless Internet to the now ubiquitous cellular telephones, wireless communications haverevolutionized the way societies function. On the one hand, wireless communications havegained wide popularity around the world. For instance, it is reported that the number ofmobile phone subscribers is on the order of 4 billion currently and will rise to 5:6 billion in2013 worldwide. On the other hand, wireless communications are faced with a number ofchallenges, among which multipath fading is the main cause leading to system degredation.Multipath fading is mainly caused by the scattering, reflection, refraction, and diffrac-tion of radiowaves when wireless signals go through complex physical mediums. Multipathfading leads to a loss of signal power without reducing noise power, and hence causing poorperformance in wireless communication systems. The diversity concept was introduced toovercome the adverse effects of multipath fading on the performance of wireless communi-cation 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 deep11.1. Background and Motivationfades in all the replicas. In the spatial domain, diversity can be realized through multiple-receiver antennas, i.e., antenna diversity. Multi-branch diversity combining is an effectiveantenna diversity technique to combat the detrimental effects of multipath fading in wirelesscommunications [3]. The three most popular diversity combining schemes are maximal ra-tio combining (MRC), selection combining (SC), and equal gain combining (EGC). Amongthem, MRC is the optimal diversity combiner but with high implementation complexity; SCand EGC provide comparable performance to MRC with relatively lower complexity.In the study of diversity combining over correlated fading channels, analytical methodshave been used to obtain accurate expressions for the average error rate and outage probabil-ity. While exact, the results are often complicated and difficult to apply since they generallydemand one or two fold numerical integration and the exactness for the estimation of theaverage error rate and outage probability may be constrained by a selected numerical inte-gration routine. In addition, these analytical results usually cannot provide useful physicalinsights into the transmission characteristics of correlated fading channels [4]. However,the asymptotic technique is a powerful analytical tool that usually provides compact and ac-curate estimation for error rate and outage probability in large signal-to-noise ratio (SNR)regions.Besides the lower computational complexity, another significant advantage of asymptoticanalysis is that the resulting solutions can often reveal some important insights that can notbe easily obtained otherwise. For instance, Liu et al. observed that the asymptotic errorrates of MRC, EGC, and SC over arbitrarily correlated Rayleigh channels can be obtainedby scaling the asymptotic error rates over independent branches with the determinant of thenormalized channel correlation matrix [5].21.2. Literature Review1.2 Literature ReviewIn the study of multi-branch diversity combining, performance analyses over independentfading channels have been widely published (see references in [2]). However, correlated fad-ing among several diversity branches exists for many real systems [6], [7] due to the insuf-ficient separation between the antennas. Therefore, quantitative analysis of the degradationof diversity systems due to correlated fading is important from both theoretical and prac-tical standpoints. While a comprehensive theoretical performance analysis for MRC overvarious 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 func-tion (CDF) or probability density function (PDF) at the output of SC and EGC combinersoperating over correlated branches is generally unknown.A number of researchers have studied the performance of SC and EGC over correlatedfading channels. For SC, most of the existing performance analysis over correlated fadingchannels has primarily focused on two or three branches [8–12]. In [13], Zhang and Luproposed a general approach for studying L-branch SC in arbitrarily correlated fading. How-ever, their method requires an L-dimensional integration, and the computation complexityincreases exponentially with L [14]. In [15], Karagiannidis et al. obtained an expressionfor the joint CDF of multivariate Nakagami-m random variables (RVs) with exponential cor-relation, and this result was subsequently applied to the performance analysis of SC overcorrelated Nakagami-m fading channels. In another related work, Karagiannidis et al. pro-posed an efficient approach for evaluating the PDF and CDF of multiple Nakagami-m RVswith 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 convergencecan 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 correlatedNakagami-m fading by noting that a set of equally correlated complex Gaussian RVs can beobtained by linearly combining a set of independent Gaussian RVs [17]. Their technique31.2. Literature Reviewwas further applied by Zhang and Beaulieu for the performance analysis of generalized SCin 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 doubleintegration involving the m-th order generalized Marcum Q-function.The exact performance analysis of EGC over correlated fading channels with arbitrarydiversity order is equally challenging. Chen and Tellambura studied the performance ofEGC in equally correlated Nakagami-m fading channels in [18], where the equally correlatedNakagami-m fading channels are transformed into a set of conditionally independent chi-square RVs and the moments of the EGC output SNR are expressed in terms of the Appellhypergeometric function. This solution, while exact, is complex and can be difficult to applybecause it requires multiple numerical integrations. Karagiannidis analyzed the performanceof EGC by approximating the moment generating function (MGF) of the output SNR, wherethe moments are determined exactly only for exponentially correlated Nakagami-m channelsin terms of a multi-fold infinite series [19]. More recently, various simple and accurateapproximations to the PDF of the sum of an arbitrary number of correlated Nakagami-m RVsare proposed in [20–23], which then are used for analytical EGC performance evaluation. Allof the approaches proposed in [19–23] are complex because they require approximations forthe 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 obtainaccurate error rate and outage probability in large SNR regions. Let g = b ¯g be the instan-taneous SNR at the output of the diversity combiner, where b is a RV depending on thechannel statistics and ¯g is the average SNR at the combiner output. For coherent modu-lation, the conditional symbol error rate (SER) in terms of g is pe(g) = pQ¡pqg¢, whereQ(¢) is one-dimensional Gaussian Q-function defined as Q(x) =R¥x 1p2p exp¡¡y2=2¢dy [2,eq. (4.1)], and p and q are constants related to the modulation formats. In [25], Wang andGiannakis observed that the conditional SER curve for coherent modulation behaves likean impulse function when instantaneous SNR is close to zero and the conditional SER is41.3. Thesis Outlinenearly zero for large instantaneous SNR (see Fig. 1.1). For correlated fading channels, theexpression of f(g) can be difficult to obtain. Since the average SER is defined asPe =Z ¥0pe(g)f(g)dg (1.1)where f(g) is the PDF of instantaneous SNR, based on the observation of Wang and Gian-nakis, one can approximate the unknown PDF of the instantaneous SNR near its origin tocompute the average SER for large values of average SNR. The PDF near zero can be de-rived 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 overcorrelated fading channels. Another approach, which can also yield the same asymptoticsolution, is to calculate the MGF of the instantaneous SNR [25]. The asymptotic techniqueis suitable for providing highly accurate estimation for error rate and outage probability inlarge SNR regions, and for revealing physical insights into the transmission characteristicsof multi-branch diversity combining over correlated fading channels.1.3 Thesis OutlineThis thesis consists of five chapters. Chapter 1 presents background knowledge of wire-less communication developments and technologies. In modern wireless communication,mitigating the detrimental effects of multipath fading is important for any mature wirelesscommunication system. Multi-branch space diversity is commonly used to improve the errorrate performance of wireless communication systems. However, the analytical expressionsfor the error rate of diversity combining systems are often complex and unfeasible for corre-lated diversity branches. This motivates researchers to apply asymptotic techniques to studythe error rate and outage probability of diversity combining systems in large SNR regions.Chapter 2 provides detailed technical background for the entire thesis. Firstly, multipathfading is presented and the three basic and most widely used fading models, namely Raleigh,51.3. Thesis Outline0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.100.10.20.30.40.50.6β¯γ=0dBQparenleftbig√β¯γparenrightbig¯γ=15dB¯γ=5dBFigure 1.1: Q‡pb ¯g·as functions of b for ¯g = 0, 5, and 15 dB.61.3. Thesis OutlineRician and Nakagami-m fadings, are introduced. Secondly, the three most popular diversityschemes, namely MRC, EGC, and SC, are explained. Thirdly, the asymptotic technique isreviewed to show how the approximate MGFs or PDFs can be used to calculate the asymp-totic error rate. Finally, a single integral representation is presented for the joint PDF ofmultiple correlated Nakagami-m RVs with specified correlation matrix.In Chapter 3, we derive a new series representation of the generalized Marcum Q-functionand give a simple accurate approximation for Marcum Q-function Qm(a;b) when b ! 0+.Utilizing this approximation, we obtain the PDF of the instantaneous SNR at the output ofSC when SNR is near its origin and then derive closed form expressions for asymptotic SERand 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 gaincombiner, and then derive the asymptotic average SER and outage probability of EGC overarbitrarily correlated Nakagami-m channels.Chapter 5 summarizes the entire thesis and lists our contributions in this thesis. In addi-tion, future work related to our current research is suggested.7Chapter 2Multipath Fading and DiversityCombining TechniquesIn this chapter, we will present some background knowledge concerning multipath fad-ing, diversity combining techniques, the asymptotic technique, as well as construction ofmultiple Nakagami-m random variables with specified correlation matrix.2.1 Multipath FadingRadiowave propagation through wireless channels is a complicated process character-ized by various effects such as multipath fading and shadowing. Multipath fading is due tothe constructive and destructive combination of randomly delayed, reflected, scattered, anddiffracted signal components. This type of fading is relatively fast and therefore is responsi-ble for short-term signal variations, where both the signal envelope and signal phase fluctuateover time.Depending on the relative relation between the symbol period of the transmitted signaland the coherence time of fading channels, fading can be classified into slow fading and fastfading [2], [3]. Coherence time is defined as the time period over which we can consider thefading process to be correlated. Slow fading occurs when the symbol duration is less than thechannel coherence time, and fast fading is the opposite. Similarly, according to the relativerelation between the transmitted signal bandwidth and the channel coherence bandwidth,fading can also be classified into frequency-nonselective fading and frequency-selective fad-82.1. Multipath Fadinging. Coherence bandwidth is defined as the frequency range over which the fading process iscorrelated. If the transmitted signal bandwidth is much smaller than the channel coherencebandwidth, 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 statisticaldistributions to describe the random behavior of the received signal amplitude. Most widelyused statistical models include the Rayleigh, Rician, and Nakagami-m distributions.2.1.1 Rayleigh DistributionThe Rayleigh distribution is frequently used to model the time varying characteristicsof 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 ofthe sum of two independent and identically distributed (i.i.d.) Gaussian signals with zeromean and variance s2 obeys a Rayleigh distribution. The PDF of the Rayleigh distributionis given byf(x)= xs2 exp ¡ x22s2¶= 2xW exp ¡x2W¶; x ‚ 0 (2.1)where W = 2s2, is the mean square value of the received signal amplitude.2.1.2 Rician DistributionWhen a strong LOS path exists between the transmitter and receiver in addition to manyweaker random multipath signal components, the randomness of the received signal ampli-92.1. Multipath Fadingtude is modeled using the Rician distribution whose PDF is given byf(x)= 2x(K +1)W exp ¡K¡(K +1)x2W¶I0ˆ2xrK(K +1)W!; x ‚ 0 (2.2)where K = A2=(2s2) is the Rician factor defined as the ratio of the LOS power A2 to the scat-tered power 2s2, the average amplitude power is denoted by W = E[X2] = A2 +2s2 whereE[¢] denotes the expectation of a RV, and Iv(¢) is the vth-order modified Bessel function ofthe first kind defined as Iv(x)=  ¥k=0 (x=2)v+2kk!G(v+k+1) where G(¢) is the Gamma function defined asG(x) =R¥0 tx¡1e¡tdt [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 Kbecomes large, the fading effect tends to vanish.2.1.3 Nakagami-m DistributionIntroduced by Nakagami in the early 1940’s [27], the Nakagami-m distribution is a ver-satile distribution used to model multipath fading in wireless channels. Empirical data showthat the Nakagami-m fading model often gives the best fit to land-mobile and indoor-mobilemultipath propagation [2]. The PDF of the Nakagami-m distribution is given byf(x)= 2G(m)‡mW·mx2m¡1 exp ¡mx2W¶; x ‚ 0;m ‚ 12 (2.3)where W is the mean square value of the amplitude. The fading severity parameter m isdefined as W2=E[(X2¡W)2]. Beyond its empirical justification, the Nakagami-m distributionis often used for the following reasons. First, the Nakagami-m distribution can be usedto model fading conditions more or less severe than Rayleigh fading. When m = 1, theNakagami-m distribution becomes the Rayleigh distribution. When m = 0:5, it becomesa one-sided Gaussian distribution. As the value of the parameter m increases, the fadingseverity decreases. Second, the Rician distribution can be approximated by the Nakagami102.2. Diversity Combining Techniquesdistribution with K =pm2¡m=(m¡pm2¡m) and m =(K +1)2=(2K +1) for m > 1 [3].The corresponding squared Nakagami-m fading amplitude has a Gamma PDF asf(x)= 1G(m)‡mW·mxm¡1 exp‡¡mxW·; x ‚ 0;m ‚ 12: (2.4)2.2 Diversity Combining TechniquesDiversity techniques were introduced to overcome the detrimental effects of multipathfading on wireless communication systems. The principle of diversity techniques is that, ifseveral copies of the same information bearing signal are available and they all experienceindependent fading, then the probability that all copies are in deep fading simultaneously issmall. If signal copies are appropriately combined at the receiver end, one can reduce theeffect of multipath fading and improve the performance of wireless communication systems.There are several known methods to obtain the independent copies of the signal: spacediversity, frequency diversity and time diversity. Among them, space diversity is widely usedbecause it is simple to implement and requires no additional bandwidth. In this thesis, wefocus only on diversity in the spatial domain with multi-branch reception. The structure ofmulti-branch diversity combining is shown in Fig. 2.1 where ri denotes the received signalin the i-th branch, hi is the channel fading amplitude, ji is the corresponding phase, d is thetransmitted data symbol, ni is the complex additive white Gaussian noise (AWGN) in the i-thbranch, and wi is the weighting factor.The three commonly used multi-branch reception schemes are MRC, SC, and EGC. Theyare briefly described in the following subsections.2.2.1 Maximal Ratio CombiningIn the maximal ratio combining mechanism, the weighting factors are the complex conju-gation of the channel gains, i.e., wi =hie jji. Therefore, the combiner eliminates the influence112.2. Diversity Combining TechniquesFigure 2.1: Multi-branch diversity combining receiver.of random phase, which is induced by multipath fading, on the output signal. The combineramplifies the strong signals and attenuates the weak ones to maximize the instantaneous out-put SNR at the combiner. Assuming the noise components of the input branches are mutuallyindependent, the output signal of maximal ratio combiner isrMRC =L i=1wiri =L i=1h2i d +hie jjini (2.5)and the instantaneous output SNR gMRC is given bygMRC = ( Li=1 h2i )2ES( Li=1 h2i )N0 =L i=1h2i ESN0 =L i=1gi (2.6)where ES is the average symbol energy of the transmitted data symbol, N0 is the powerspectral density (PSD) of complex AWGN in the i-th branch, and gi = h2i ESN0 denotes theinstantaneous SNR of the i-th branch.MRC is the optimal diversity combiner in the sense of maximizing the combiner outputSNR, in the absence of other interfering sources [28]. However, MRC is also known forhaving the highest complexity to implement due to the fact that phase-lock and amplitude122.2. Diversity Combining Techniquesweighting must be performed.2.2.2 Equal Gain CombiningIn equal gain combining, the weighting factors have constant amplitude value, but haveopposite phase to the channel gains, i.e., wi = e jji, which is known as the co-phasing op-eration. Therefore, the combiner only eliminates the influence of random phase with equalweights. Assuming equal noise powers in all branches, the instantaneous output signal ofEGC is given byrEGC =L i=1wiri =L i=1hid +e jjini (2.7)and the instantaneous output SNR gEGC is given bygEGC = ( Li=1 hi)2ESLN0 : (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 complexitythan MRC.2.2.3 Selection CombiningIn selection combining, the combiner only picks one best branch out of the L noisy re-ceived signals ri (i = 1;¢¢¢;L). The weighting factor for the selected branch is unity and theother weighting factors are zero. Suppose all branches have the same noise power spectraldensity N0. Then the output of SC can be expressed asrSC = rindex(maxfhigLi=1)(2.9)132.3. Asymptotic Techniquewhere index(yi) denotes the index i corresponding to yi. Then the instantaneous output SNRgSC is given bygSC = maxfgigLi=1 (2.10)where gi = h2i ESN0 denotes the instantaneous SNR of the i-th branch. Since SC processes only asingle branch, it has a much lower complexity compared to MRC and EGC. However, sinceSC ignores information provided by other diversity branches, its performance is poorer thanEGC and MRC. SC can be used with coherent modulations, noncoherent modulations, anddifferentially coherent modulations.2.3 Asymptotic TechniqueLet g = b ¯g be the instantaneous SNR at the output of the diversity combiner, where b isa RV depending on the channel statistics and ¯g is the average SNR at the combiner output.Suppose the PDF of b can be approximated by a single polynomial term for b ! 0+ asf (b)= cbt +o(bt).1For coherent modulation with conditional SER pe(b)= pQ‡pqb ¯g·, the average SERfor large SNR is given by [25]Pe = 2tcG¡t + 32¢ppp (t +1)(q ¯g)t+1 +o 1¯gt+1¶: (2.11)Many coherent modulation schemes, such as binary phase shift keying (BPSK), M-ary phaseshift keying (M-PSK), M-ary pulse amplitude modulation (M-PAM) and M-ary quadratureamplitude modulation (M-QAM), have conditional SERs of the form pe(b)= pQ‡pqb ¯g·,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 noncoherentmodulation in large SNR regions has not been derived. We present this result in the following1We write a function p(x) as o(x) if limx!0+ p(x)=x = 0.142.3. Asymptotic Techniqueproposition.Proposition 1: For noncoherent modulation with conditional symbol error rate of theform pe(b)= pexp(¡qb ¯g), the error rate at large SNR can be expressed asPe = cG(t +2) p(t +1)(q ¯g)t+1 +o 1¯gt+1¶(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 mod-ulations, like binary noncoherent frequency shift keying (BNCFSK) and binary differentialphase shift keying (BDPSK), are tabulated in Table 2.1.Therefore, to compute the asymptotic SER at large SNR, one needs to determine theparameters c and t from the PDF of b, or, from the MGF of b when s ! ¥. Anotherapproach is to consider the square root of g at the output of the diversity combiner. Leth = pg = pb ¯g. From the PDF of b, with a change of variable, it is straightforward toobtain the PDF of g asf(g)= c gt¯gt+1 +o gt¯gt+1¶: (2.13)The PDF of h isf(h)= 2ch2t+1¯gt+1 +o h2t+1¯gt+1¶(2.14)and the corresponding MGF of h can be expressed asMh(s)= 2cG(2t +2)¯gt+1s2t+2 +o 1¯gt+1s2t+2¶: (2.15)Therefore, one can simply obtain the asymptotic SER by extracting the parameters c and tfrom (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¢ ¯g)¡Gd +o 1¯gGd¶(2.16)152.4. Construction of Multiple Correlated Nakagami-m RVsTable 2.1: Parameters p and q for different coherent and noncoherent modulation schemesModulation Scheme Conditional SER p qBPSK Q(p2g) 1 2M-PSK M ‚ 4 … 2Q¡p2g sin pM¢ 2 2sin2 pMM-PAM 2(1¡ 1M)Q‡q 6gM2¡1·2(1¡ 1M) 6M2¡1M-QAM 4(1¡ 1pM)Q q3gM¡1¶4(1¡ 1pM) 3M¡1BNCFSK 12 exp¡¡12g¢ 12 12BDPSK 12 exp(¡g) 12 1where Gc is the coding gain, and Gd is the diversity order. The diversity order Gd determinesthe slope of the SER versus average SNR curve, at high SNR, in a log-log scale. Gc (in deci-bels) determines the shift of the curve in SNR relative to a benchmark SER curve of ¡¯g¡Gd¢.Comparing (2.16) with (2.11) and (2.12) respectively, we observe that Gd =t+1 for both co-herent and noncoherent modulation, Gc = q•2tcpG(t+32)pp(t+1)‚¡ 1t+1for coherent modulation, andGc = qhcpG(t+2)t+1i¡ 1t+1for noncoherent modulation.2.4 Construction of Multiple Correlated Nakagami-m RVsIn this section, we present a framework to obtain a single integral representation formultiple Nakagami-m distributions with a specified correlation matrix. The joint PDF ofcorrelated Nakagami-m RVs is expressed explicitly in terms of single integral solutions.A remarkable feature of these expressions is that the computational complexity reduces toa single integral computation for an arbitrary number of dimensions. The basic idea forthe construction of multiple correlated Nakagami-m RVs is that a set of equally correlatedcomplex Gaussian RVs can be obtained by linearly combining a set of independent GaussianRVs [30]. The original construction approach was proposed in [17] to obtain the CDF of L-branch SC output SNR in equally correlated Nakagami-m fading. In [14], this approach wasused to evaluate the performance of diversity combiners with positively correlated branches.162.4. Construction of Multiple Correlated Nakagami-m RVs2.4.1 Construction of Multiple Correlated Nakagami-m RVsSimilar to [17, eq. (5)], [30] for integer m we express the L correlated Nakagami-m RVsthrough Lm zero-mean complex Gaussian RVs byGil = si‡p1¡riXil +priX0l·+ jsi‡p1¡riYil +priY0l·(2.17)for i = 1;¢¢¢;L and l = 1;¢¢¢;m, where j2 = ¡1, 0 • ri • 1, and X0l, Xil, Y0l and Yil areindependent Gaussian RVs with distribution N (0;1=2). That is, for any u;v 2f0;1;¢¢¢;Lg,and l;n 2 f1;¢¢¢;mg, E[XulYvn] = 0, E[XulXvn] = E[YulYvn] = du;vdl;n=2, where du;v is theKronecker delta function.Let Ri denote the summation of the absolute square of Gil, i.e., Ri =  ml=1jGilj2. Then,it can be shown Ri is the sum of squares of m independent Rayleigh envelopes with centralchi-square distribution c2m(0;s2i =2) [14]. The cross-correlation between Ri and Rk can beshown to berRi;Rk = E[RiRk]¡E[Ri]E[Rk]pVar[Ri]Var[Rk]= rirk; i 6= k (2.18)where Var[¢] denotes the variance of a RV. The proof of (2.18) is given in Appendix B.Let Hi = pRi, then it can be shown H1;H2;¢¢¢;HL are L correlated Nakagami-m RVswith identical fading parameter m and mean-square value ms2i . The relationship betweenthe correlation of Hi and Hk and the correlation of Ri and Rk, denoted by rHi;Hk and rRi;Rkrespectively, is [27]rHi;Hk = F¡¡12;¡12;m;rRi;Rk¢¡1y (m)¡1 (2.19)where y (m)= G(m)G(m+1)=G2(m+1=2) and F(¢) is the hypergeometric function definedas F(a;b;c;z)= G(c)G(a)G(b)  ¥n=0 G(a+n)G(b+n)G(c+n) znn! [31, eq. (15.1.1)].172.4. Construction of Multiple Correlated Nakagami-m RVs2.4.2 Joint PDF of Multiple Correlated Nakagami-m RVsLet Z =  ml=1¡X20l +Y 20l¢. When X0l = x0l and Y0l = y0l (l = 1;¢¢¢;m) are fixed, the realand imaginary parts of Gil have equal variance of s2i (1¡ri)=2 and means siprix0l andsipriy0l, respectively. Therefore, R1;R2;¢¢¢;RL are independent noncentral chi-square RVswith distribution c2m(qri  ml=1¡x20l +y20l¢;s2i (1¡ri)=2) and whose marginal CDF is givenby [29, eq. (2-1-124)]F (rijz)= Pr(Ri • rijz)= 1¡Qmˆs2zri1¡ri;s2ris2i (1¡ri)!(2.20)where Qm(¢;¢) denotes the m-th order Marcum Q-function defined asQm(a;b)=R¥b x¡xa¢m¡1 e¡x2+a22 Im¡1(ax)dx [32]. The joint marginal CDF of R1;¢¢¢;RL isgiven byF (r1;r2;¢¢¢;rLjz)=L i=1"1¡Qmˆs2zri1¡ri;s2ris2i (1¡ri)!#: (2.21)Z =  ml=1¡X20l +Y 20l¢ follows a central chi-square distribution c2m(0;1=2) and its PDF isgiven byf (z)= zm¡1e¡zG(m) ; z ‚ 0: (2.22)Averaging the joint marginal CDF in (2.21) with respect to the PDF of Z, the joint CDF ofR1;R2;¢¢¢;RL becomesF (r1;r2;¢¢¢;rL)= 1G(m)Z ¥0L i=1"1¡Qmˆs2zri1¡ri;s2ris2i (1¡ri)!#zm¡1e¡zdz: (2.23)Taking partial derivatives of (2.23) with respect to r1;r2;¢¢¢;rL, one can obtain the joint PDF182.4. Construction of Multiple Correlated Nakagami-m RVsof R1;R2;¢¢¢;RL as followsf (r1;r2;¢¢¢;rL)= 1G(m)Z ¥0L i=11s2i (1¡ri) rizris2i¶m¡12exp•¡ ris2i (1¡ri)‚£Im¡1ˆ21¡riszriris2i!zm¡1e¡‡1+ Li=1 ri1¡ri·zdz:(2.24)Eqns. (2.23) and (2.24) essentially are the joint CDF and PDF of L correlated Gamma RVsrespectively, and will be useful in studying the asymptotic performance of SC.Using variable transformation, we can obtain the joint PDF of H1;H2;¢¢¢;HL asf (h1;h2;¢¢¢;hL)= 1G(m)Z ¥0L i=12his2i (1¡ri) h2izris2i¶m¡12exp•¡ h2is2i (1¡ri)‚£Im¡1ˆ21¡riszrih2is2i!zm¡1e¡‡1+ Li=1 ri1¡ri·zdz:(2.25)The proof of (2.25) is given in Appendix C. Eqn. (2.25) is the joint the PDF of L correlatedNakagami-m RVs and will be useful in studying the asymptotic performance of EGC.19Chapter 3Asymptotic Performance Analysis of SCover Arbitrarily Correlated Nakagami-mFading Channels3.1 IntroductionThe 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 ob-servations [32]. This special function is also frequently used in the performance analysisinvolving noncoherent and differential detection of multichannel narrowband signals overfading channels [2], [29]. The generalized Marcum Q-function is given by [32]Qv(a;b)=Z ¥bx‡ xa·v¡1e¡(x2+a2)=2Iv¡1(ax)dx (3.1)where a and b are non-negative real numbers, and v is a positive real number. Although v cantake any positive real number, when v is a positive integer m, Qm(a;b) is the complementarycumulative 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 ofconsiderable research over the past several decades. The precise computation of Qv(a;b)can be difficult especially for large a and v values, because the integral in (3.1) involves the203.1. Introductionmodified Bessel function of the first kind [34]. In order to calculate the generalized MarcumQ-function, power series expansion methods were used in [35–37] and the Neuman series ex-pansion method was proposed in [38]. Since direct computation of (3.1) is difficult, variousalternative representations of Qv(a;b) were proposed. For example, several integral formsof Qv(a;b) 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) orby 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 kindIm¡1(x) we present a new series representation of Qm(a;b) in terms of the incompletegamma function g(m;x) and obtain a simple accurate approximation of Qm(a;b) whenb ! 0+. Utilizing this approximation of Qm(a;b), we study the asymptotic performanceof 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 errorrate and outage probability estimation. Of theoretical interest, we reveal some physical in-sights into the transmission characteristics of SC over correlated Nakagami-m channels, andin particular, how the branch power covariance coefficient matrix can influence the averageerror rate and outage probability in large SNR regions.The rest of this chapter is organized as follows. In Section 3.2, an alternative series repre-sentation and a new approximation of Qm(a;b) when b ! 0+ are presented. In Section 3.3,the PDF of L-branch SC output SNR near its origin over arbitrarily correlated Nakagami-mfading channels is derived. Section 3.4 derives the asymptotic error rate and outage prob-ability of SC. Section 3.5 discusses some important insights and provides some numericalresults.213.2. A Series Representation of Qm(a;b)3.2 A Series Representation of Qm(a;b)We first present a new series representation of Qm(a;b) as follows.Proposition 2: The generalized Marcum Q-function can be written asQm(a;b)= 1¡e¡a22¥ k=0a2kG(m+k)k!¢2k g m+k; b22¶(3.2)where g(¢;¢) is the incomplete gamma function defined as g (m;z)=Rz0 tm¡1e¡tdt [26, (8.350.1)].Proof: By [33, eq. (5)-(8)], for integer-valued m the generalized Marcum Q-function canbe expressed asQm(a;b)=Z ¥bx‡ xa·m¡1e¡(x2+a2)=2Im¡1(ax)dx= 1¡Z b0x‡ xa·m¡1e¡(x2+a2)=2Im¡1(ax)dx: (3.3)For m-th order modified Bessell function of the first kind, its series expansion can be repre-sented as [26, (8.445)]Im(z)=¥ k=01G(m+k+1)k!‡z2·m+2k: (3.4)Substituting (3.4) into (3.3), we haveZ b0x‡ xa·m¡1e¡(x2+a2)=2Im¡1(ax)dx= a1¡me¡a22¥ k=01G(m+k)k!‡a2·m+2k¡1Z b0x2m+2k¡1e¡x22 dx= e¡a22¥ k=0a2kG(m+k)k!¢2k ¢g m+k; b22¶(3.5)and (3.2) follows immediately.Using the series expression of the incomplete gamma function g (u;y) =  ¥n=0 (¡1)nyu+nn!(u+n)223.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channels[26, (8.354.1)], we can rewrite Qm(a;b) asQm(a;b)= 1¡e¡a22¥ k=0a2kG(m+k)k!¢2k¥ n=0(¡1)n‡b 22·m+k+n(m+k+n)¢n! : (3.6)An immediate consequence of (3.6) is that when b !0+, the m-th order Marcum Q-functionQm(a;b) can be approximated asQm(a;b)= 1¡ b2m2m¢m! exp ¡a22¶+o¡b 2m¢: (3.7)To demonstrate the accuracy of the above approximation, we calculate the exact values ofQm(a;b) using the function NCX2CDF provided by Matlab and compare them with the ap-proximate values obtained from (3.7) for different values of a and b. The numerical resultsare compared in Table 3.1. From Table 3.1, we can observe that the relative error becomessmall when b ! 0+, and therefore we conclude that the small argument approximation ofthe Marcum Q-function is highly accurate.3.3 PDF of SC Output SNR Over Correlated Nakagami-mFading Channels3.3.1 System ModelAssume that there are L available diversity branches experiencing frequency-nonselectiveand 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 averagesymbol energy of the transmitted data symbol, and let N0 be the PSD of complex AWGN inthe i-th branch. The instantaneous SNR of the i-th diversity branch, gi, is defined asgi =jHij2 ESN0(3.8)233.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading ChannelsTable 3.1: The exact and approximate values of Qm(a;b) for different values of a and ba b Exact value Approximate value Relative error0:10:1 0:999996887986722 0:999987562344010 9:325£10¡60:05 0:999999805256228 0:999999222646501 5:826£10¡70:01 0:999999999688286 0:999999998756234 9:32£10¡100:005 0:999999999980518 0:999999999922265 5:825£10¡110:001 0:999999999999969 0:999999999999876 9:314£10¡1410:1 0:999997569793885 0:999992418366754 5:151£10¡60:05 0:999999847945917 0:999999526147922 3:217£10¡70:01 0:999999999756628 0:999999999241837 5:147£10¡100:005 0:999999999984789 0:999999999952615 3:217£10¡110:001 0:999999999999976 0:999999999999924 5:151£10¡1450:1 0:999999993945953 0:999999999953417 6:007£10¡90:05 0:999999999622624 0:999999999997089 3:744£10¡100:01 0:999999999999397 0:999999999999995 5:986£10¡130:005 0:999999999999962 1:000000000000000 3:741£10¡140:001 0:9999999999999999 1 1:11£10¡16243.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading Channelsand the average SNR of the i-th diversity branch, Gi, is given byGi = E£jHij2⁄ ESN0: (3.9)Recall from Chapter 2 that for SC, the instantaneous SNR at the output of the selectioncombiner is given bygSC = maxi=1;¢¢¢;Lgi: (3.10)3.3.2 PDF of SC Output SNRWe now use the method described in Chapter 2 to construct correlated Nakagami-m RVsH1;H2;:::;HL. Then the instantaneous SNR of the i-th diversity branch is gi = Ri ESN0 . LetR denote the branch power covariance coefficient matrix, whose (ik)-th element (R)ik isdefined as(R)ik = E[gigk]¡E[gi]E[gk]pVar[gi]Var[gk]: (3.11)Using (2.18), we can show that(R)ik =8><>:rRi;Rk = rirk; i 6= k1; i = k: (3.12)This model simplifies to the equally correlated case when ri = rk. By the symmetry propertyof R, ri and rk can be uniquely determined byri =s(R)il (R)in(R)ln (3.13)andrk =s(R)kl (R)kn(R)ln (3.14)when n;l 6= i;n;l 6= k;l 6= n.253.3. PDF of SC Output SNR Over Correlated Nakagami-m Fading ChannelsIf we define M as the matrix whose (ik)-th entry is given by(M)ik =q(R)ik: (3.15)Express M as a rank-one updated matrix M = S+uuT , where S = diag(1¡r1;¢¢¢;1¡rL)and u = [pr1;pr2;¢¢¢;prL]. Using [56, eq. (6.2.3)], we can obtain the determinant ofmatrix M asdet(M)= det(S)¡1+uT S¡1u¢="1+L i=1ri1¡ri# L i=1(1¡ri): (3.16)Following the construction of H1;H2;:::;HL described in Chapter 2, we can obtain themean square value of Hi as E£jHij2⁄= ms2i and the average SNR of the i-th diversity branchas Gi = ms2i ESN0 . Since the instantaneous SNR of the i-th diversity branch is gi = Ri ESN0 , wecan express gi in terms of as Gi, i.e., gi = Rims2iGi. By (3.10), the CDF of the SC output SNRgSC can be written asFgSC (g)= Pr(gsc • g)= Pr(g1 • g;¢¢¢;gL • g)= Pr R1ms21 G1 • g;¢¢¢;RLms2L GL • g¶= Pr R1 • ms21G1 g;¢¢¢;RL •ms2LGL g¶: (3.17)By the joint CDF of R1;R2;¢¢¢;RL given in (2.23), we can obtainFgSC (g)= 1G(m)Z ¥0L i=1"1¡Qmˆs2riz1¡ri;s2mgGi(1¡ri)!#zm¡1e¡zdz: (3.18)Eqn. (3.18) essentially generalizes the results in [17] and [14] to arbitrarily correlated SCover branches with different average SNR values.It is seen from (3.18) that the second parameter of the generalized Marcum Q-function263.4. Asymptotic Performance Analysis of SCapproaches zero when g ! 0+, or when the CDF is near its origin. Using (3.7), we canobtain the asymptotic CDF of the SC output SNR asFgSC (g)= mmL[G(m+1)]L detm(M)gmL ¯gmL Li=1 Gmi ¯gmL +o gmL¯gmL¶: (3.19)Differentiating (3.19) with respect to g, we obtain the PDF of instantaneous SNR at theoutput of the SC asfgSC (g)= mL¢mmL[G(m+1)]L detm(M)gmL¡1 ¯gmL Li=1 Gmi ¯gmL +o gmL¡1¯gmL¶: (3.20)The analytical expression obtained in (3.20) for the PDF of the output SNR near its originwill be useful in studying the asymptotic error rate performance of SC.3.4 Asymptotic Performance Analysis of SCComparing (3.20) with (2.13) in Chapter 2, we observe thatt = mL¡1 (3.21)andc = mL¢mmL[G(m+1)]L detm(M)¯gmL Li=1 Gmi : (3.22)From (3.21), it is obvious that the diversity order Gd is mL.273.4. Asymptotic Performance Analysis of SC3.4.1 Asymptotic Error RateThe asymptotic SER of SC reception with coherent modulation can thus be expressedfrom (2.11) asPS;Ce = G(2mL+1)mmL p2mL+1qmLG(mL+1)[G(m+1)]L detm(M)1 Li=1 Gmi +o 1 Li=1 Gmi¶: (3.23)In the special case when m = 1, it can be shown that (3.23) agrees with the asymptotic errorrate of SC reception over arbitrarily correlated Rayleigh channels [5, eq. (7)]2. Similarly, theasymptotic SER of SC reception with noncoherent modulation can be obtained from (2.12)asPS;Ne = mmLG(mL+1) pkmL[G(m+1)]L detm(M)1 Li=1 Gmi +o 1 Li=1 Gmi¶: (3.24)3.4.2 Asymptotic Outage ProbabilitiesThe outage probability is defined asPout (gth)= Pr(g < gth)=Z gth0f(g)dg (3.25)where gth is a predefined outage threshold and f(g) is the PDF of the instantaneous SNR atthe output of diversity combiner. Substituting (2.13) into (3.25), one obtains the asymptoticoutage probabilities asPout (gth)= ct +1 gth¯g¶t+1+oˆgt+1th¯gt+1!: (3.26)2When m = 1, M is same as the matrix M defined in [5].283.5. Discussions and Numerical ResultsWith the values of t and c obtained in (3.21) and (3.22), we obtainPSout (gth)= mmL(m!)L detm(M)gmLth Li=1 Gmi +o 1 Li=1 Gmi¶: (3.27)When m = 1, it can be shown that (3.27) agrees with the asymptotic outage probability ofSC 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 resultssuggest these analytical expressions are also valid at least for m = 0:5.3.5 Discussions and Numerical Results3.5.1 DiscussionsSince 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 canbe expressed in terms of the asymptotic SER over independent Nakagami-m fading channelsscaled by a factor detm(M), i.e.,(PSCe )asym = (PSCe;i )asymdetm(M) (3.28)where (PSCe;i )asym denotes the asymptotic SER of SC over independent Nakagami-m fadingbranches. In (3.28), the factor detm(M), which is less than unity, can be considered as theloss 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 arelationship similar to (3.28) can also be seen for the outage probability from (3.27).293.5. Discussions and Numerical Results3.5.2 Numerical ResultsIn this subsection, we present some numerical results of coherent BPSK and noncoherentBDPSK with three-branch SC over correlated Nakagami-m fading channels. We use themethod proposed in [57] to generate correlated Nakagami-m RVs. For all the numericalresults obtained here, we have assumed three-branch diversity reception with G1 = G2 = G3and gth = 3 dB. We use ˜r = [r1;r2;r3] to denote a vector whose elements comprise thepower covariance coefficient matrix. For comparison, we set ˜r1 = [0:4571;0:2296;0:4571]and ˜r2 =[0:3968;0:5265;0:7861]. By (3.12) and (3.15), we obtainM1 =0BBBB@1 0:3240 0:45710:3240 1 0:32400:4571 0:3240 11CCCCAandM2 =0BBBB@1 0:4571 0:55850:4571 1 0:64330:5585 0:6433 11CCCCA:It follows that det(M1)= 0:6771 and det(M2)= 0:3937. From (3.28), the determinant in-equality det(M1) > det(M2) predicts that more highly correlated fading channels will leadto 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 three-branch SC with matrices M1 and M2 for m = 0:5 and m = 2 respectively. Figs. 3.1- 3.4indicate that the analytical asymptotic results have excellent agreement with the simulatedresult in large SNR regions. This implies that the analytical asymptotic results can be usedto 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 alsosuggest that (3.23), (3.24) and (3.27) are also valid at least for m = 0:5.303.5. Discussions and Numerical Results0 2.5 5 7.51012.51517.52022.52527.53010−610−510−410−310−210−1100SNR (dB)BER  BPSK−AsymptoticBPSK−SimulationBDPSK−AsymptoticBDPSK−SimulationM1M2Figure 3.1: The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branchcorrelated Nakagami-m channels with matrices M1 and M2 when m = 0:5.313.5. Discussions and Numerical Results0 2.5 5 7.5 10 12.5 15 17.5 20 22.510−1110−1010−910−810−710−610−510−410−310−210−1100SNR (dB)BER  BPSK−AsymptoticBPSK−SimulationBDPSK−AsymptoticBDPSK−SimulationM1M2Figure 3.2: The asymptotic and simulated BERs of BPSK and BDPSK for SC over 3-branchcorrelated Nakagami-m channels with matrices M1 and M2 when m = 2.323.5. Discussions and Numerical Results0 2.5 5 7.5 1012.51517.52022.52527.53010−510−410−310−210−1100SNR (dB)Outage Probability  AsymptoticSimulationM1M2Figure 3.3: The asymptotic and simulated outage probabilities of SC over 3-branch corre-lated Nakagami-m channel with matrices M1 and M2 when m = 0:5.333.5. Discussions and Numerical Results0 2.5 5 7.5 10 12.5 15 17.5 20 22.510−1110−1010−910−810−710−610−510−410−310−210−1100SNR (dB)Outage Probability  AsymptoticSimulationM1M2Figure 3.4: The asymptotic and simulated outage probabilities of SC over 3-branch corre-lated Nakagami-m channel with matrices M1 and M2 when m = 2.34Chapter 4Asymptotic Performance Analysis ofEGC over Arbitrarily CorrelatedNakagami-m Fading ChannelsIn this chapter, using an integral representation for the joint PDF of multiple correlatedNakagami-m RVs obtained in Chapter 2, we derive the MGF of the square root of instanta-neous SNR at the output of equal gain combiner and obtain the asymptotic average SER andoutage probability of EGC over arbitrarily correlated Nakagami-m channels.4.1 System ModelAssume that there are L available diversity branches experiencing frequency-nonselectiveand slow Nakagamim-m fading with fading parameter m taking the same positive integervalue. Let Hi be the instantaneous channel fading amplitude on the i-th branch, ES be theaverage symbol energy of the transmitted data symbol, and let N0 be the PSD of complexAWGN in the i-th branch. The instantaneous SNR of the i-th diversity branch, gi, is definedasgi =jHij2 ESN0(4.1)and the average SNR of the i-th diversity branch, Gi, is given byGi = E£jHij2⁄ ESN0: (4.2)354.2. MGF of Square Root of Instantaneous SNR at the Output of EGCRecall in Chapter 2 that for EGC, the instantaneous SNR at the output of the equal gaincombiner is given bygEGC =¡ Li=1 Hi¢2 ESLN0 : (4.3)4.2 MGF of Square Root of Instantaneous SNR at theOutput of EGCWe now use the method described in Chapter 2 to construct correlated Nakagami-m RVsH1;H2;:::;HL. Then the joint PDF of H1;H2;:::;HL can be expressed as (2.25). Let Rdenote the branch power covariance coefficient matrix whose (ik)-th element (R)ik is definedby (3.11). Define M as the matrix whose (ik)-th entry is given by (3.15), then the determinantof matrix M can be expressed as (3.16).Using a series expansion of first kind modified Bessel function in [26, eq. (8.445)], whenx ! 0, Im¡1(x) can be written asIm¡1(x)= 1G(m)‡x2·m¡1+o¡xm¡1¢: (4.4)Define HE =pgEGC. By (4.3), when gEGC ! 0, Hi ! 0. Substituting (4.4) into (2.25), wecan write the joint PDF of H1;H2;¢¢¢;HL asf (h1;h2;¢¢¢;hL)= 1detm(M)L i=12G(m)1s2mi£h2m¡1i +o(h2m¡1i )⁄exp•¡ h2is2i (1¡ri)‚: (4.5)364.3. Asymptotic Performance Analysis of EGCUsing (4.5), by definition, the MGF of HE can be written asMHE(s)= EHE £e¡sHE⁄= EH1;H2;¢¢¢;HL"e¡s(H1+H2+¢¢¢+HL)pESpLN0#= 1detm(M)• 2G(m)‚L L i=1gi(s)(4.6)where gi(s) is given bygi(s)=Z ¥0ˆm ESN0Gi!m£h2m¡1i +o(h2m¡1i )⁄exp"¡ h2i m ESN0Gi(1¡ri)#exp ¡shir ESLN0¶dhi:(4.7)Using [26, eqs. (3.462) and (9.246)], we can show that when s ! ¥, gi(s) can be written asgi(s)= G(2m) mLGi¶m• 1s2m +o 1s2m¶‚: (4.8)Substituting (4.8) into (4.6), we obtain the MGF of HE asMHE(s)= (mL)mLdetm(M)•2G(2m)G(m)‚L 1 Li=1 Gmi• 1s2mL +o 1s2mL¶‚: (4.9)The MGF expression obtained in (4.9) will be used in studying the asymptotic error rate andoutage probability of EGC.4.3 Asymptotic Performance Analysis of EGCComparing (4.9) with (2.15) in Chapter 2, we observe thatt = mL¡1 (4.10)374.3. Asymptotic Performance Analysis of EGCandc = 2L¡1G(2mL)•G(2m)G(m)‚L (mL)mLdetm(M)¯gmL Li=1 Gmi : (4.11)From (4.10), it is obvious that the diversity order Gd is mL.4.3.1 Asymptotic Error RateThe asymptotic SER of EGC reception with coherent modulation can be expressed from(2.11) asPEe = [G(2m)]L(mL)mL p2mL+1¡LG(mL+1)[G(m)]L detm(M)1 Li=1(qGi)m +o 1 Li=1 Gmi¶: (4.12)In the special case of m = 1, it can be shown that (4.12) agrees with the asymptotic symbolerror rate of EGC reception over arbitrarily correlated Rayleigh channels [5, eq. (5)].4.3.2 Asymptotic Outage ProbabilitiesSubstituting (4.10) and (4.11) into (3.26), we obtain the outage probabilityPEout (gth)= 2L¡1[G(2m)]L(mL)mL¡1G(2mL)[G(m)]L detm(M)gmLth Li=1 Gmi +o 1 Li=1 Gmi¶: (4.13)When m = 1, it can be shown that (4.13) agrees with the asymptotic outage probability ofEGC reception over arbitrarily correlated Rayleigh channels [5, eq. (13)].Equations (4.12) and (4.13) are important new results. Although we deduce these resultsfor integer fading parameter m, our numerical results suggest they are also valid at least form = 0:5.384.4. Discussions and Numerical Results4.4 Discussions and Numerical Results4.4.1 DiscussionsSince 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 canbe expressed in terms of the asymptotic SER over independent Nakagami-m fading channelsscaled by a factor detm(M), i.e.,(PEGCe )asym = (PEGCe;i )asymdetm(M) (4.14)where (PEGCe;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 channelcorrelation. It should be noted that the simple relationship in (4.14) also holds for MRC overarbitrarily 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 ResultsIn this subsection, we present some numerical results of coherent BPSK with three-branch EGC over correlated Nakagami-m fading channels. We use the method proposedin [57] to generate correlated Nakagami RVs. For all the numerical results obtained here, wehave assumed three-branch diversity reception with G1 =G2 =G3 and gth =3 dB. We use ˜r =[r1;r2;r3] to denote a vector whose elements comprise the power covariance coefficient ma-trix. For comparison, we set ˜r1 =[0:3968;0:5265;0:7861] and ˜r2 =[0:5008;0:6229;0:8264].394.4. Discussions and Numerical ResultsBy (3.12) and (3.15), we obtainM1 =0BBBB@1 0:4571 0:55850:4571 1 0:64330:5585 0:6433 11CCCCAandM2 =0BBBB@1 0:5585 0:64330:5585 1 0:71750:6433 0:7175 11CCCCA:It follows that det(M1)= 0:3937 and det(M2)= 0:2750. From (4.14), the determinant in-equality det(M1) > det(M2) predicts that more highly correlated fading channels will leadto worse SER performance, as one expects. This is confirmed by the BER curves shown inFig. 4.1 and Fig. 4.2 for m = 0:5 and m = 2 respectively. As seen from Fig. 4.1 and 4.2, theasymptotic 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 three-branch EGC with matrices M1 and M2 for m = 0:5 and m = 2 respectively. Figs. 4.1- 4.4indicate that the analytical asymptotic results have excellent agreement with the simulatedresult in large SNR regions. This implies that the analytical asymptotic results can be usedto 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 alsosuggest that (4.12) and (4.13) are also valid at least for m = 0:5.404.4. Discussions and Numerical Results0 2.5 5 7.5 10 12.515 17.520 22.52510−510−410−310−210−1100SNR (dB)BER  BPSK−AsymptoticBPSK−SimulationM1M2Figure 4.1: The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlatedNakagami-m channels with matrices M1 and M2 when m = 0:5.414.4. Discussions and Numerical Results0 2.5 5 7.5 10 12.5 15 17.5 2010−1110−1010−910−810−710−610−510−410−310−210−1100SNR (dB)BER  BPSK−AsymptoticBPSK−SimulationM1M2Figure 4.2: The asymptotic and simulated BERs of BPSK for EGC over 3-branch correlatedNakagami-m channels with matrices M1 and M2 when m = 2.424.4. Discussions and Numerical Results0 2.5 5 7.5 10 12.515 17.520 22.52510−410−310−210−1100SNR (dB)Outage Probability  BPSK−AsymptoticBPSK−SimulationM1M2Figure 4.3: The asymptotic and simulated outage probabilities of EGC over 3-branch corre-lated Nakagami-m channel with matrices M1 and M2 when m = 0:5.434.4. Discussions and Numerical Results0 2.5 5 7.5 10 12.5 15 17.5 2010−1110−1010−910−810−710−610−510−410−310−210−1100SNR (dB)Outage Probability  BPSK−AsymptoticBPSK−SimulationM1M2Figure 4.4: The asymptotic and simulated outage probabilities of EGC over 3-branch corre-lated Nakagami-m channel with matrices M1 and M2 when m = 2.44Chapter 5ConclusionsIn this chapter, we first summarize the contributions of this work, and then suggest somefuture work.5.1 Summary of ContributionsThis thesis makes the following contributions:1. An asymptotic error rate expression for noncoherent modulation in large SNR regionshas been derived. This expression can be used to estimate the high SNR SER or BERof various noncoherent modulation schemes, such as BNCFSK and BDPSK.2. A new series representation of the generalized Marcum Q-function and a simple ap-proximation for the Marcum Q-function Qm(a;b) when b ! 0+ have been derived.Numerical results show that the relative error between the exact value obtained by us-ing the function NCX2CDF and the approximate value becomes small when b ! 0+,and therefore the small argument approximation of the Marcum Q-function is highlyaccurate.3. Closed-form error rate and outage probability expressions are derived for multi-branchEGC and SC over arbitrarily correlated Nakagami-m fading channels. They can beused to provide accurate and rapid estimation of error rates and outage probabilities inlarge SNR regions, where Monte Carlo simulation becomes time-consuming. Thesesimple expressions will allow wireless system engineers to estimate the required fading455.2. Future Workmargin in link budget analysis without resorting to time-consuming computer simula-tions.4. Simple relationships have been established for the asymptotic error rates and outageprobabilities of EGC and SC over arbitrarily correlated and independent Nakagami-mfading channels. The same relationship also holds for MRC. It has been shown that theasymptotic error rate and outage probability over correlated branches can be obtainedby scaling the asymptotic error rate and outage probability over independent brancheswith a factor, detm(M), where det(M) is the determinant of matrix M whose elementsare the square root of corresponding elements in the branch power covariance correla-tion matrix R. The factor detm(M) can be considered as the loss factor due to channelcorrelation. These significant relationships explain theoretically the observation thatmore highly correlated fading channels lead to worse error rate and outage probabilityperformance.5.2 Future WorkIn this work, it is assumed that all the correlated Nakagami-m fading channels have thesame integer-valued fading parameter m. However, the correlated Nakagami-m fading chan-nels with different fading parameters can exist in some practical transmission scenarios. Be-cause of the validity and feasibility of asymptotic techniques for large SNRs, the asymptotictechnique can be extended to the study of the performance of multi-branch diversity combin-ings over correlated Nakagami-m fading channels with arbitrarily-valued fading parameters.In addition, so far there exists no method that can accurately generate multiple correlatedNakagami-m RVs with arbitrary fading parameters and mean square values. In [57], Zhangproposed a decomposition technique to generate multiple correlated Nakagami-m RVs withsame fading parameters. Zhang’s algorithm is inaccurate when 2m is not an integer. Inour future research, we will develop an accurate method to generate multiple correlated465.2. Future WorkNakagami-m RVs with arbitrary fading parameters and mean square values.47Bibliography[1] L. 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Zhang, “A decomposition technique for efficient generation of correlated Nakagamifading channels,” IEEE Journal on Selected Areas in Communications, vol. 18, pp.2385–2392, Nov. 2000.54Appendix ADerivation of Error Rate forNoncoherent ModulationLet A be a small fixed positive number such that when b < A, the PDF of b can be writtenas f(b)= cbt +o(bt). Then the average SER for noncoherent modulation with conditionalSER pe(b)= pexp(¡qb ¯g) can be caculated asPe =Z ¥0pe(b)f(b)db=Z ¥0pexp(¡qb ¯g)f(b)db= pZ A0exp(¡qb ¯g)f(b)db + pZ ¥Aexp(¡qb ¯g)f(b)db= pZ ¥0Z ¥qb ¯gexp(¡x)£cbt +o¡bt¢⁄dxdb¡ pZ ¥AZ ¥qb ¯gexp(¡x)£cbt +o¡bt¢⁄dxdb+ pZ ¥Aexp(¡qb ¯g)f(b)db (A.1)55Appendix A. Derivation of Error Rate for Noncoherent Modulationwhere in obtaining the last equality we have used a fact e¡x =R¥x e¡ydy. The first integral in(A.1) can be written aspZ ¥0Z ¥qb ¯gexp(¡x)£cbt +o¡bt¢⁄dxdb= cpZ ¥0Z ¥qb ¯gexp(¡x)btdxdb + pZ ¥0exp(¡qb ¯g)o¡bt¢db= cpZ ¥0•Z xq ¯g0btdb‚exp(¡x)dx+o‡¯g¡(t+1)·= cp(t +1)(q ¯g)t+1Z ¥0xt+1 exp(¡x)dx+o‡¯g¡(t+1)·= cG(t +2) p(t +1)(q ¯g)t+1 +o‡¯g¡(t+1)·: (A.2)Ignoring the term o(bt), the second integral in (A.1) can be expressed ascpZ ¥AZ ¥qb ¯gexp(¡x)btdxdb= cpZ ¥Aq ¯g•Z xq ¯gAbtdb‚exp(¡x)dx= cpZ ¥Aq ¯g•bt+1t +1‚ xq ¯gAexp(¡x)dx= cp(t +1)(q ¯g)t+1Z ¥Aq ¯gexp(¡x)hxt+1¡(Aq ¯g)t+1idx: (A.3)The integral in (A.3) becomes zero as ¯g ! ¥. Therefore, the second integral can be writtenas o‡¯g¡(t+1)·.For the third integral in (A.1) we have the following relationpZ ¥Aexp(¡qb ¯g)f(b)db • pZ ¥Aexp(¡qA ¯g) f (b)db= p¢exp(¡qA ¯g)Z ¥Af (b)db• p¢exp(¡qA ¯g) (A.4)56Appendix A. Derivation of Error Rate for Noncoherent Modulationandlim¯g!¥ exp(¡qA ¯g)¯g¡(t+1) = lim¯g!¥ ¯gt+1exp(qA ¯g) = 0: (A.5)Therefore, the third integral can be written as o‡¯g¡(t+1)·.Finally, it follows that the average SER for noncoherent modulation at large SNR can beexpressed as (2.12).57Appendix BDerivation of (2.18)According to (2.17), for i2f1;¢¢¢;Lg and l; p2f1;¢¢¢;mg;l 6= p, we have the followingexpressionsjGilj2 = s2ih(1¡ri)X2il +riX20l +2pri(1¡ri)XilX0li+s2ih(1¡ri)Y 2il +riY 20l +2pri(1¡ri)YilY0li(B.1)andjGilj4s4i =(1¡ri)2X4il +r2i X40l +4ri(1¡ri)X2il X20l +(1¡ri)2Y 4il +r2i Y 40l+4ri(1¡ri)Y 2il Y 20l +2ri(1¡ri)X2il X20l +4(1¡ri)pri(1¡ri)X3il X0l+2(1¡ri)2X2ilY 2il +2ri(1¡ri)X2ilY 20l +4(1¡ri)pri(1¡ri)X2ilYilY0l+4ripri(1¡ri)X30lXil +2ri(1¡ri)X20lY 2il +2r2i X20lY 20l+4ripri(1¡ri)X20lYilY0l +4(1¡ri)pri(1¡ri)XilX0lY 2il+4ripri(1¡ri)XilX0lY 20l +8ri(1¡ri)XilX0lYilY0l +2ri(1¡ri)Y 2il Y 20l+4(1¡ri)pri(1¡ri)Y 3il Y0l +4ripri(1¡ri)YilY 30l (B.2)58Appendix B. Derivation of (2.18)andjGilj2jGipj2s4i =(1¡ri)2X2il X2ip +ri(1¡ri)X2il X20p +2(1¡ri)pri(1¡ri)X2il XipX0p+(1¡ri)2X2ilY 2ip +ri(1¡ri)X2ilY 20p +2(1¡ri)pri(1¡ri)X2ilYipY0p+ri(1¡ri)X20lX2ip +r2i X20lX20p +2ripri(1¡ri)X20lXipX0p+ri(1¡ri)X20lY 2ip +r2i X20lY 20p +2ripri(1¡ri)X20lYipY0p+2(1¡ri)pri(1¡ri)XilX0lX2ip +2ripri(1¡ri)XilX0lX20p+4ri(1¡ri)XilX0lXipX0p +2(1¡ri)pri(1¡ri)XilX0lY 2ip+2ripri(1¡ri)XilX0lY 20p +4ri(1¡ri)XilX0lYipY0p+(1¡ri)2Y 2il X2ip +ri(1¡ri)Y 2il X20p +2(1¡ri)pri(1¡ri)Y 2il XipX0p+(1¡ri)2Y 2il Y 2ip +ri(1¡ri)Y 2il Y 20p +2(1¡ri)pri(1¡ri)Y 2il YipY0p+ri(1¡ri)Y 20lX2ip +r2i Y 20lX20p +2ripri(1¡ri)Y 20lXipX0p+ri(1¡ri)Y 20lY 2ip +r2i Y 20lY 20p +2ripri(1¡ri)Y 20lYipY0p+2(1¡ri)pri(1¡ri)YilY0lX2ip +2ripri(1¡ri)YilY0lX20p+4ri(1¡ri)YilY0lXipX0p +2(1¡ri)pri(1¡ri)YilY0lY 2ip+2ripri(1¡ri)YilY0lY 20p +4ri(1¡ri)YilY0lYipY0p: (B.3)For any u 2 f0;1;¢¢¢;Lg, and l 2 f1;¢¢¢;mg, Xul and Yul are independent Gaussian RVswith distribution N (0;1=2), hence we have E[Xul] = E[Yul] = 0, E£X2ul⁄ = E£Y 2ul⁄ = 12,E£X3ul⁄= E£Y 3ul⁄= 0, and E£X4ul⁄= E£Y 4ul⁄= 34. Taking the expectation with respect to (B.2),(B.2), and (B.3), we can obtain E£jGilj2⁄= s2i , E£jGilj4⁄= 2s4i , and E£jGilj2jGipj2⁄= s4i .Since Ri =  ml=1jGilj2 and Rk =  ml=1jGklj2, it follows that E[Ri]= ms2i and59Appendix B. Derivation of (2.18)E[Rk]= ms2k . The mean square of Ri can be calculated asE£R2i ⁄= E24ˆ m l=1jGilj2!235= m l=1E£jGilj4⁄+m l=1m p6=lp=1E£jGilj2jGipj2⁄= m(m+1)s4i(B.4)and the variance of Ri is given byVar[Ri]= E£R2i ⁄¡E[Ri]2 = ms4i : (B.5)Similarly, we have Var[Rk]= ms4k .According to the definition of Ri and Rk, we haveRiRk =m l=1m p=1jGilj2jGkpj2: (B.6)Since Xil, X0l, Yil, and Y0l are independent zero mean Gaussian RVs, we can ignore themwhen calculating E[RiRk]. By symmetry, we can obtainE[RiRk]= 2s2i s2km l=1m p=1Eh(1¡ri)X2ilh(1¡rk)X2kp +rkX20p +(1¡rk)Y 2kp +rkY 20pii+2s2i s2km l=1m p6=lp=1EhriX20lh(1¡rk)X2kp +rkX20p +(1¡rk)Y 2kp +rkY 20pii+2s2i s2km l=1E£riX20l£(1¡rk)X2kl +rkX20l +(1¡rk)Y 2kl +rkY 20l⁄⁄= 2s2i s2k264 m l=1m p=11¡ri2 +2s2i s2km l=1m p6=lp=1ri2 +2s2i s2km l=1ri +rk2375= s2i s2k ¡m2 +mrirk¢: (B.7)Noting that E[RiRk]= s2i s2k ¡m2 +mrirk¢, E[Ri]= ms2i , E[Rk]= ms2k , Var[Ri]= ms4i , and60Appendix B. Derivation of (2.18)Var[Rk] = ms4k , by the definition of cross-correlation of Ri and Rk, we can finally obtain(2.18).61Appendix CDerivation of (2.25)Using variable transformation, we havef (h1;h2;¢¢¢;hL)= f (r1;r2;¢¢¢;rL)¢jJ(h1;h2;¢¢¢;hL)j (C.1)where j¢j is the absolute-value operator and J(h1;h2;¢¢¢;hL) is the Jacobian of the transfor-mation, which is defined asJ(h1;h2;¢¢¢;hL)=266664dr1dh1 ¢¢¢dr1dhL... ... ...drLdh1 ¢¢¢drLdhL377775: (C.2)In our case, ri = h2i , dridh j = 2hi when i = j; otherwise, dridh j = 0. Therefore, J(h1;h2;¢¢¢;hL)= Li=1 2hi. It is straightforward to obtain (2.25).62

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