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Improving and bounding asymptotic approximations for diversity combiners in correlated generalized rician… Schlenker, Joshua 2013

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Improving and Bounding Asymptotic Approximations for Diversity Combiners in Correlated Generalized Rician Fading by Joshua Schlenker B.A.Sc., The University of British Columbia, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical & Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2013 c© Joshua Schlenker 2013 Abstract Although relatively simple exact error rate expression are available for se- lection combining (SC) and equal gain combining (EGC) with independent fading channels, results for correlated channels are highly complex, requiring multiple levels of integration when more than two branches are involved. Not only does the complexity make numeric computation resource intensive, it obscures how channel statistics and correlation affect system performance. Asymptotic analysis has been used to derive simple error expressions valid in high signal-to-noise ratio (SNR) regimes. However, it is not clear at what SNR value the asymptotic results are an accurate approximation of the exact solution. In this thesis, we derive asymptotic results for SC, EGC, and max- imal ratio combining (MRC) in correlated generalized Rician fading chan- nels. By assuming generalized Rician fading, our results incorporate Rician, Rayleigh, and Nakagami-m fading scenarios as special cases. Furthermore, the asymptotic results for SC are expanded into an exact infinite series. Al- though this series grows quickly in complexity as more terms are included, truncation to even two or three terms has much greater accuracy than the first (asymptotic) term alone. Finally, we derive asymptotically tight lower ii and upper bounds on the error rate for EGC. Using these bounds, we are able to show at what SNR values the asymptotic results are valid. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis Outline and Contributions . . . . . . . . . . . . . . . . 6 2 Fading and Diversity Combining . . . . . . . . . . . . . . . . 8 2.1 Multipath Fading . . . . . . . . . . . . . . . . . . . . . . . . 8 iv 2.1.1 Rayleigh Distribution . . . . . . . . . . . . . . . . . . 11 2.1.2 Rician Distribution . . . . . . . . . . . . . . . . . . . 13 2.1.3 Nakagami-m Distribution . . . . . . . . . . . . . . . . 15 2.1.4 Generalized Rician Distribution . . . . . . . . . . . . . 16 2.2 Diversity Combining . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Generalized Correlation Model . . . . . . . . . . . . . . . . . 20 3 Asymptotic Performance Analysis of Combining Methods in Generalized Rician Fading . . . . . . . . . . . . . . . . . . . . 26 3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 First Order Joint Distributions . . . . . . . . . . . . . . . . . 30 3.4 Combiner Asymptotics . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 EGC . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Discussion and Numerical Results . . . . . . . . . . . . . . . 34 3.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . 37 4 Exact Series Form of the BER with SC in Generalized Rician Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 SNR Distribution . . . . . . . . . . . . . . . . . . . . . . . . 42 v 4.3 BER of Binary Modulations . . . . . . . . . . . . . . . . . . . 46 4.3.1 BER for Binary Coherent Modulations . . . . . . . . . 46 4.3.2 BER for Binary Noncoherent Modulations . . . . . . . 47 4.4 Convergence and Truncation Error . . . . . . . . . . . . . . . 47 4.4.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . 47 4.4.2 Truncation Error . . . . . . . . . . . . . . . . . . . . . 51 4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Asymptotically Tight Error Bounds for EGC with General- ized Rician Fading . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Bounds on Joint PDF . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . 60 5.2.2 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 68 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 vi Appendices A Derivation of the Correlation Coefficient ρGkGi . . . . . . . . 85 B Derivation of the Integral Identity (4.7) . . . . . . . . . . . . 87 vii List of Tables 3.1 Parameters p and q for various coherent modulations . . . . . 29 viii List of Figures 2.1 The probability density function of a Rayleigh RV with differ- ent values of Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The probability density function of a Rician RV with Ω = 1. . 14 2.3 The probability density function of a Nakagami-m RV with Ω = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Performance of EGC and SC relative to MRC. . . . . . . . . . 36 3.2 Asymptotic and simulated BERs of coherent BPSK of triple branch SC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Asymptotic and simulated BERs of coherent BPSK of triple branch EGC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Asymptotic and simulated BERs of coherent BPSK of triple branch MRC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. . . . . . . . . . . . . . . . . . . . . . . 40 ix 4.1 Approximate (N = 30) and simulated BERs of coherent BPSK of triple branch SC with Rician and Nakagami-m fading. Av- erage SNR is identical for each branch. For Rician fading, S = 1.5 and for Nakagami-m, m = 2. Power correlation is ac- cording to P1 and P2 for Rician and Nakagami-m, respectively. 55 4.2 Approximate and simulated BERs of coherent BPSK of triple branch SC with Rician fading (S = 1.5) and equal average branch SNR. Branches are correlated with power correlation matrix P1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Relative truncation error for coherent BPSK with triple branch SC in Nakagami-2 fading and equal average branch SNR. Branches are correlated with power correlation matrix P2. . . . . . . . . 57 5.1 Upper and lower BER bounds of coherent BPSK over Rayleigh fading channels with triple branch EGC and equal average branch SNR. λ = [0.9, 0.8, 0.9] and power correlation matrix P1. 71 5.2 Upper and lower BER bounds of coherent BPSK over Rayleigh fading channels with triple branch EGC and equal average branch SNR. λ = [0.5, 0.3, 0.8] and power correlation matrix P2. 72 5.3 Upper and lower BER bounds of coherent BPSK over Nakagami- 1.5 fading channels with triple branch EGC and equal average branch SNR. λ = [0.2, 0.3, 0.5] and power correlation matrix P3. 73 x 5.4 Upper and lower BER bounds of coherent BPSK over Rician fading channels (S = 2) with triple branch EGC and equal average branch SNR. λ = [0.6,−0.4, 0.5] and power correlation matrix P4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 xi List of Notation | · | Absolute value of a complex number Iv(·) Modified Bessel function of the first kind with order v, Iv(x) ,∑∞ i=0 (x/2)v+2i i!Γ(v+i+1) Jv(·) Bessel function of the first kind with order v, Jv(x) ,∑∞ i=0 (−1)i(x/2)v+2i i!Γ(v+i+1)( n k ) Binomial coefficient, ( n k ) , n! k!(n−k)! ΦX(·) Characteristic function of a random variable X, ΦX(ω) , E [ ejωX ] (·)∗ Conjugate of a complex number F (·) Dawson’s integral, F (x) , e−x2 ∫ x 0 et 2 dt det(·) Determinant of a matrix E[·] Expectation of a random variable N ! Factorial of a non-negative integer N , N ! , 1 ·2 · · · (N−1) ·N and 0! = 1 Γ(·) Euler’s Gamma function, Γ(x) , ∫∞ 0 tx−1e−tdt N (µ, σ2) Gaussian distribution with mean µ and variance σ2 xii 1F1(·; ·; ·) Kummer confluent hypergeometric function, 1F1(a; b;x) ,∑∞ i=0 (a)i (b)ii! xi =(·) Imaginary part of a complex number L (α) i (·) Generalized Laguerre polynomial with degree i and order α, L (α) i (x) , ∑i j=0 (j+α+1)i−j (i−j)!j! (−x)j, α > −1 Qv(·, ·) Generalized Marcum Q-function with order v, Qv(a, b) ,∫∞ b xv av−1 e −(x2+a2)/2Iv−1(ax)dx MX(·) Moment generating function of a random variable X, MX(s) , E [ e−sX ] N Set of non-negative integers 0, 1, 2, . . . (x)n Pochhammer’s symbol, (x)n , Γ(x+n)Γ(x) Q(·) Gaussian Q-function, Q(x) , 1√ 2pi ∫∞ x exp (−t2/2)dt R Set of real numbers <(·) Real part of a complex number o(·) Order of a function, a function g(x) is o(x) if lim x→0 g(x) x = 0 (·)T Transpose of a vector or matrix xiii List of Abbreviations AWGN Additive White Gaussian Noise BCFSK Binary Coherent Frequency Shift Keying BDPSK Binary Differential Phase Shift Keying BER Bit Error Rate BNCFSK Binary Noncoherent Frequency Shift Keying BPSK Binary Phase Shift Keying CDF Cumulative Distribution Function CHF Characteristic Function EGC Equal Gain Combining H-S/MRC Hybrid Selection/Maximal Ratio Combining i.i.d. Independent, Identically Distributed LOS Line-Of-Sight MGF Moment Generating Function M -PAM M -ary Pulse Amplitude Modulation M -PSK M -ary Phase Shift Keying M -QAM M -ary Quadrature Amplitude Modulation MRC Maximal Ratio Combining xiv OFDM Orthogonal Frequency Division Multiplexing PDF Probability Density Function RF Radio Frequency RV Random Variable SC Selection Combining SER Symbol Error Rate SNR Signal-to-Noise Ratio w.r.t. With Respect To xv Acknowledgements I would like to express deep gratitude to my superiors Julian Cheng and Robert Schober for their hard work and dedication to seeing me through to the end despite my doubts along the way. Without their insight and expertise I would scarcely have scratched the surface of what this project became. I also owe debts of thanks to many colleagues I have encountered in my time at UBC in Vancouver and the Okanagan, some of whom have become life long friends. Finally, I would like to thank my parents for their enduring support and my closest confidant, my girlfriend Kayla, for having the patience to stick with me through to the end. Funding for this research was provided by the Natural Sciences and En- gineering Research Council of Canada (NSERC). xvi Chapter 1 Introduction 1.1 Background and Motivation In recent decades the use of wireless devices has exploded, driven mainly by the increase in consumer electronics with wireless capabilities, such as cellular phones, laptops, and GPS units. The mobile nature of these devices and the multipath propagation of the wireless signal introduces a time variation in the quality of the channel between transmitter and receiver known as fading. Without adequate mitigation of this effect, the experience to the end user would be extremely detrimental, causing dropped phone calls, inability to load webpages, etc. when a deep fade occurs. One powerful method of com- bating fading is known as diversity combining. Diversity combining employs multiple antennas at the receiver, with each branch experiencing different in- stantaneous signal-to-noise ratios (SNRs) provided the antennas are spaced sufficiently far apart. The probability that each antenna simultaneously ex- periences poor channel conditions decreases significantly as more antennas are added. Although many diversity combining techniques have been studied in the literature, the three most popular are selection combining (SC), equal 1 gain combining (EGC), and maximal ratio combining (MRC). Although exact performance analysis of SC, EGC, and MRC has been widely studied, the results are generally complex, even for independent fad- ing. For correlated fading with more than two branches, the results are given in terms of multiple nested infinite series or integrations even for specialized correlation matrices. This complexity obscures how performance is affected by fading parameters and the correlation among branches. In response, a number of research papers have studied the performance when SNR ap- proaches infinity. This technique is known as asymptotic analysis. The assumption of high SNR results in a closed-form equation for performance metrics such as symbol error rate (SER) and outage probability. Although only strictly valid at infinite SNR, the exact performance can be approx- imated by the asymptotic results at moderate SNR levels. However, it is difficult to determine at what SNR value the approximation is valid without resorting to computationally expensive exact analytical analysis or Monte Carlo simulations, both of which are susceptible to inaccuracies at low SERs typical in the high SNR regime. 1.2 Literature Review The performance analysis of SC, EGC, and MRC for independently faded channels has been widely studied, and SER expressions are available for a wide variety of linear modulations which only require a single level of integra- 2 tion regardless of the number of branches. (See [1] and the references therein.) However, correlation among branches arises when the diversity antennas are in close proximity, which occurs for devices with space constrained form fac- tors such as mobile phones. For the general case of L diversity branches and arbitrary channel correlation, single integral forms and single infinite series exist for the error rate with MRC at the receiver [2–4]. However, error rate expressions for EGC and SC are much more complex, typically involving multi-level integration or infinite series. Expressions with a single integral or infinite series are only available when the number of branches is limited to two [5–10]. For L > 2, single integral or infinite series forms for the error rate are unavailable even for specialized correlation models. Regarding the SC literature, Karagiannidis et al. obtained a dual infi- nite series representation of the error rate for a triple branch SC undergoing Nakagami-m fading with exponential branch correlation [11]. The general result for arbitrary correlation can be obtained using the joint cumulative distribution function (CDF) of three Nakagami-m random variables (RVs) obtained in [12]; however this requires an additional two infinite series. The joint CDF of L exponentially correlated Nakagami-m RVs was obtained in [13], consisting of a L − 1 fold infinite series. The results of [13] were extended to arbitrarily correlated Nakagami-m RVs using a Green’s matrix approximation of the correlation matrix [14]. Exact results for the most gen- eral case of arbitrary L and correlation matrix are available for SC but the complexity of the error rate expressions increases exponentially in L. In [15] 3 the authors took a characteristic function (CHF) approach independent of the fading distribution; however, the Fourier inversion requires L infinite inte- grals. A multivariate joint probability density function (PDF) of the general α− µ distribution consisting of a single infinite sum of generalized Laguerre polynomials was derived in [16]; however, application to SC performance analysis requires an L-fold integral of the joint PDF. The literature for EGC with L > 2 and correlation among the branches is limited to special correlation models and approximations. In [17], the authors found the moments at the output of an EGC combiner with Nakagami-m fad- ing in terms of an L−1 fold infinite summation. The results are approximate for arbitrarily correlated branches and exact for exponential correlation. The moment generating function (MGF) can be found from the central moments via a Taylor series expansion or Padé approximation, from which the error rate follows with a single integration. A different approach was taken in [18], where the approximate PDF of a sum of correlated Nakagami-m RVs was found using moment matching. Using the joint PDF of exponentially corre- lated Nakagami-m RVs in [13], Sahu and Chaturvedi found the error rates for coherent BPSK [19] and noncoherent modulations [20]. However, the ex- pressions are highly complex and involve L−1 fold infinite summation along with several levels of finite summation. A novel approach was taken in [21] to find single integral representations for the CDF of the SNR at the output of a SC combiner with equally cor- related Nakagami-m, Rayleigh, and Rician fading channels. The correlated 4 RVs are modelled using a linear combination of independent Gaussian RVs. This allowed the authors to find the error rate using two nested integrals. Using the same model as [21], Chen and Tellambura found the moments at the output of an EGC combiner for equally correlated Rayleigh, Rician, and Nakagami-m fading channels [22]. The authors then presented an approach for finding the error rate using a single infinite series of the moments. Al- though the joint PDF was found in single integral form, the moment expres- sions are L-fold summations of a Lth order Lauricella function. In [23], the model was generalized for Nakagami-m fading channels and was termed the ‘generalized correlation model’, which was then utilized to derive error rate and outage probability expressions for hybrid selection/maximal ratio com- bining (H-S/MRC). Using the generalized correlation model, Beaulieu and Hemachandra [24] derived single integral forms of the joint PDF and CDF of Rayleigh, Rician, generalized Rician, Nakagami-m, and Weibull RVs. The results cited above all require multiple nested levels of integration to evaluate the exact SER for SC or EGC with correlated fading when L > 2. Asymptotic analysis allows this complexity to be circumvented by finding simple SER expressions which are valid only at asymptotically high SNR. Results for SC, EGC, and MRC are available in the literature. In [25], Li and Cheng conducted an asymptotic performance analysis for SC with Nakagami-m fading channels using the generalized correlation model. For arbitrary correlated Rician channels, the asymptotic results for SC and EGC can be found in [26]. For arbitrarily correlated Nakagami-m and Rician 5 channels the asymptotic technique was employed to find the SER of MRC in [27] and [28], respectively. Although it provides insight into how correlation and fading distribution parameters affect SER, the major shortcoming of the asymptotic technique is that it can not predict at what SNR value the asymptotic solutions can adequately approximate the exact solution. Assessment of the accuracy of asymptotic solutions for a particular SNR using Monte Carlo simulations or exact analytical analysis is problematic. Monte Carlo simulation is time consuming due to the low probability of a symbol error at high SNR, and the computation of exact analytical solutions requires complex multilevel nu- meric integration which is also susceptible to error due to the small numerical values or oscillatory nature of the integrand over the integration region. 1.3 Thesis Outline and Contributions The remainder of this thesis consists of five chapters. In Chapter 2, we intro- duce necessary background information relevant to the chapters that follow, including an overview of fading and diversity combining. Furthermore, the joint CDF and PDF of generalized Rician RVs correlated according to the generalized correlation model are derived. In Chapter 3, we find the asymp- totic error rate expressions for SC, EGC, MRC. In Chapter 4, an infinite series expansion of the bit error rate (BER) of SC is derived along with the truncation error when the series is terminated at a finite number of terms. 6 The termination of the series to the first term is the asymptotic error rate. In Chapter 5, we develop asymptotically tight error bounds for EGC. The bounds are in the form of a single integration, comparable to the complexity of computing the exact error rate if the branches were independent. Chap- ters 3-5 assume generalized Rician fading with generalized correlation among branches. The results of Chapter 4 and Chapter 5 can be utilized to show at what SNR the asymptotic error rate is guaranteed to be within a specified tolerance of the exact error rate. In Chapter 6, we conclude this thesis and suggest possibilities for related future work. 7 Chapter 2 Fading and Diversity Combining In this chapter, we present an overview of the challenges that fading places on reliable wireless communications and several statistical distributions com- monly used to model the fading channel. We then introduce several diver- sity combining techniques which are used to reduce the deleterious effects of multipath fading when the transmitted signal is sent over multiple fad- ing channels. In general, the multiple fading channels are correlated, and we introduce a method to construct generalized Rician RVs for a special correlation model known as the generalized correlation model. 2.1 Multipath Fading When a radio frequency (RF) signal is propagated from a transmitter to receiver wirelessly it typically takes multiple paths due to scattering and reflections caused by objects in the surrounding environment include build- ings, roads and cars in an urban environment, and trees and hills in a more 8 rural area. The number of propagation paths can be large, especially in an urban setting. Each path experiences a different attenuation and phase delay. When the signals recombine at the receiver, their constructive and destructive addition results in a phenomenon known as small-scale fading. Small-scale fading, as opposed to large-scale fading which is the result of large objects such as buildings or hills located between transmitter and re- ceiver, is highly location specific. Even motion on the order of centimeters can cause large fluctuations in the signal strength at the receiver due to the high carrier frequencies in typical wireless communication systems. For ex- ample, if a mobile phone operates with a carries frequency of 2 GHz, with a corresponding wavelength of approximately 15 cm, even a small movement by the mobile user could cause the signal strength to transition from rel- atively strong to weak. In this thesis, we only consider small-scale fading, which we will simply refer to as fading. This is a common assumption in the literature. A deterministic treatment of the wireless channel is not practical due to large number of propagation paths and random movements of the users and surrounding objects. Thus, fading is typically modelled as a random process. The fading channel is characterized by its coherence time and band- width. The coherence time, roughly speaking, is the length of time the fading coefficient can be considered as approximately constant. Signals with a sym- bol duration smaller than the coherence time will experience slow fading, where the channel conditions are constant throughout the symbol duration. 9 A signal with a symbol duration longer than the channel coherence time will experience fast fading, where the fading coefficient changes during sym- bol transmission. Coherence bandwidth is a frequency-domain description of fading channels. A transmitted symbol with a bandwidth larger than the channel coherence bandwidth will experience frequency selective fading whereas when the opposite is true, each frequency in the transmitted symbol will experience approximately identical fading resulting in frequency nonse- lective or flat fading. In slow, frequency nonselective fading, the channel can be described by the complex gain hejθ for the duration of the transmitted symbol, where h and θ are the random fading envelope and phase respec- tively. We will assume slow, frequency nonselective fading in this thesis. In situations where the channel is frequency selective, it can be divided into multiple frequency nonselective channels using orthogonal frequency division multiplexing (OFDM) [30]. Though there exists a large number of statisti- cal models for h, each tailored to specific fading environments, three widely adopted fading models are Rayleigh, Rician, and Nakagami-m. The effect of fading on receiver performance is significant. In a constant channel gain, i.e. no fading, in the high SNR regime, a binary phase shift keying (BPSK) modulated signal has the BER Pe ≈ 1 2 √ piγ̄ e−γ̄ (2.1) where γ̄ is the average SNR at the receiver. The identical receiver in a 10 Rayleigh fading channel experiences a BER at large SNR of [31] Pe ≈ 1 4γ̄ . (2.2) In effect, the presence of fading has reduced the BER from being exponential in γ̄ to only linear. At typical SNRs, fading increases the error rate by several orders of magnitude. For example, at 10 dB a constant gain channel would experience a BER of approximately 5×10−5, while a Rayleigh faded channel would experience a BER of roughly 2×10−2. As this is also only the average error rate, the instantaneous error rate can even be worse. 2.1.1 Rayleigh Distribution When the complex channel gain is modelled as a zero mean Gaussian RV with independent, identically distributed (i.i.d.) real and imaginary parts, the fading envelop follows the Rayleigh distribution. This model best fits environments in which the transmitted signal is scattered and reflected mul- tiple times before reaching the receiver without a line-of-sight (LOS) path. By a central limit theorem, the sum of all signal paths at the receiver will have a zero-mean Gaussian distribution. The Rayleigh RV X has a PDF of fX(x) = 2x σ2 e− x2 σ2 (2.3) 11 where σ2/2 is the variance of the underlying Gaussian RVs. The average power of the Rayleigh distribution is Ω = E[X2] = σ2. In Fig. 2.1, the Rayleigh PDF is plotted for several values of Ω. When the fading envelope is Rayleigh distributed, it can be shown that the fading power follows an exponential distribution. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f X (x) x   Ω=1 Ω=3 Ω=8 Figure 2.1: The probability density function of a Rayleigh RV with different values of Ω. 12 2.1.2 Rician Distribution When a rich scattering environment similar to Rayleigh arises but with an additional LOS path between transmitter and receiver, a Rician distribution model accurately describes the fading envelope. A Rician RV can be con- structed from two Gaussian RVs, X1 ∼ N ( µ1, σ2 2 ) and X2 ∼ N ( µ2, σ2 2 ) , as X = √ X21 +X 2 2 . (2.4) The distribution of X is given by the Rician PDF fX(x) = 2x σ2 e− x2+s2 σ2 I0 ( 2sx σ2 ) (2.5) where s2 = µ21 +µ 2 2 is the power due to the LOS signal. The average power is Ω = E[X2] = s2 + σ2. The factor σ2 is known as the scattering power since it is the sum power contribution of the numerous scattering and reflection paths. The ratio of LOS power to scattering power is known as the Rice factor K = s2 σ2 . (2.6) The Rician distribution can be rewritten in terms of K and Ω as fX(x) = 2(K + 1) Ω xe− K+1 Ω x2−KI0 ( 2 √ K (K + 1) Ω x ) . (2.7) 13 In Fig. 2.2, we have plotted the Rician PDF for several values of K. When K = 0, the Rician distribution specializes to the Rayleigh distribution. On the other hand, as K is increased while Ω is held constant, the probability of deep fades decreases and the Rician distribution approaches a δ pulse, which can be seen in Fig. 2.2. 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 f X (x) x   K=0 K=2 K=8 Figure 2.2: The probability density function of a Rician RV with Ω = 1. 14 2.1.3 Nakagami-m Distribution Another distribution with more flexibility than Rayleigh is the Nakagami-m distribution. The PDF of a Nakagami-m RV X is given by [32] fX(x) = 2x2m−1 Γ(m) (m Ω )m e− m Ω x2 , m ≥ 1 2 (2.8) where Ω = E[X2]. The parameter m is known as the shape parameter and controls the severity of fading. When m=1, Nakagami-m reduces to the Rayleigh distribution, and 1 2 ≤ m < 1 and m > 1 corresponds to more and less severe fading, respectively, than Rayleigh as shown in Fig. 2.3. The square of a Nakagami-m RV is Gamma distributed with PDF fX2(x) = xm−1 Γ(m) (m Ω )m e− m Ω x. (2.9) The Nakagami-m distribution has been shown to fit empirical data in an urban setting better than Rayleigh or Rician distributions [33]. Also a theoretical basis for modelling the outdoor mobile radio channel with the Nakagami distribution was provided by [34]. 15 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 f X (x) x   m=0.5 m=1 m=2 m=3 Figure 2.3: The probability density function of a Nakagami-m RV with Ω = 1. 2.1.4 Generalized Rician Distribution The Rician model can be generalized by changing (2.4) from two to an arbi- trary n dimensions X = √√√√ n∑ l=1 X2l (2.10) 16 where Xl ∼ N (µl, σ2/2), l = 1, 2, . . . , n are independent. The generalized Rician PDF is given by fX(x) = 2x n 2 σ2s n 2 −1 e −x2+s2 σ2 In 2 −1 ( 2sx σ2 ) (2.11) where s2 = ∑n l=1 µ 2 l and Ω = E[X 2] = s2 + n 2 σ2. Although the generalized Rician model is not typically used to model fad- ing due to a lack of theoretical or empirical justification, it does incorporate the three previous models to some extent. Obviously with n = 2 it specializes to the Rician distribution, and with n = 2, s = 0 we obtain the Rayleigh dis- tribution. Although not as obvious, the generalized Rician distribution also specializes to the Nakagami-m distribution. Consider X2 with s = 0, which is the sum of n independent squared Gaussian RVs with identical variances and zero means, hence X2 follows the chi-square distribution and has a PDF of f(x) = 2xn−1 Γ ( n 2 ) ( n 2Ω )n 2 e− x σ2 (2.12) which matches (2.9) when m = n 2 . Thus, when m is a integer or half integer the Nakagami-m distribution is a special case of generalized Rician distribu- tion. In this thesis, we will assume generalized Rician fading since it includes the other three fading models as special cases. 17 2.2 Diversity Combining A common method for mitigating fading is to employ diversity at the receiver. Multiple copies of the transmitted signal are made available to the receiver, each experiencing different fading channels. The probability that all fading channels are in deep fade simultaneously decreases rapidly as diversity is increased. Receive diversity can be achieved by transmitting the same symbol over multiple frequencies or time slots. However, this requires additional bandwidth. To circumvent this problem, the receiver can employ multiple antennas, each receiving the transmitted signal over different fading channels. In general, the channels are correlated due to space constraints in small form factors, such as mobile phones, which requires antennas to be in close proximity. Branches in a system employing time or frequency diversity will be correlated if duplicate transmissions occur within the coherence time or coherence bandwidth respectively. There are several methods to combine the received signals, each with a different performance-complexity trade off. Suppose that the receiver has L branches of diversity available, and the transmitted symbol with energy ES is subject to a fading channel with complex channel gain hke jθk and additive white Gaussian noise (AWGN) with power spectral density N0 on the kth branch. The instantaneous SNR on branch k is then γk = h 2 k ES N0 . A linear combiner multiplies the signal on the kth branch by a complex weight wk, then sums the result. The instantaneous SNR at the output of a general 18 linear combiner is then γc = ∣∣∣∑Lk=1wkhkejθk∣∣∣2∑L k=1 |wk|2 ES N0 . (2.13) The simplest combining technique would be to only consider the branch with the largest SNR and perform signal detection on this branch exclu- sively. This technique is known as SC. For SC, wi = 1 and {wk = 0, k = 1, 2, . . . , L, k 6= i} where i = arg maxk=1,2,...,L γk. SC has an instantaneous SNR at the combiner output of γSC = max k=1,2,...,L γk. (2.14) Beyond its implementation simplicity, SC is also advantageous for noncoher- ent modulations since it does not require knowledge of the channel phases. Although all the other branches may have lower SNRs, they still contain valuable information about the transmitted signal. A more complex combin- ing technique that utilizes all available branches is known as EGC. EGC uses the weight wk = e −jθk , phase aligning each branch before summation. This technique requires phase knowledge for each branch and is suitable only for coherent modulations. EGC has an instantaneous SNR of γEGC = (∑L k=1 √ γk )2 L . (2.15) 19 Although performance of EGC is better in most cases than SC, this may not be true in severe fading [35] or exponentially decaying average branch SNRs [36]. With EGC, we treat each branch identically regardless of the individ- ual SNRs of each branch. A more sophisticated method is to weigh the contribution of each branch according to the severity of the fading by recog- nizing that branches with high SNR are more trustworthy than those with lower SNR. This combining method is known as MRC, which has the weight wk = hke −jθk . MRC has an instantaneous SNR at the output of the combiner of γMRC = L∑ k=1 γk. (2.16) Applying the Cauchy-Swartz inequality to (2.13), one can prove that MRC is the optimal linear combining scheme in terms of maximizing the combiner output SNR. Although MRC is optimal, it requires knowledge of the fading envelope and phase, thus analysis of the simpler EGC and SC is of practical interest. 2.3 Generalized Correlation Model In general, the fading channels of a diversity combiner are correlated, thus we require the joint statistics of the channel to analyze combiner performance. In this section, we present the method of Beaulieu and Hemachandra for obtaining single integral representations of the multivariate joint PDF and 20 CDF of correlated generalized Rician RVs [24]. Consider the following1 Xkl = σk (√ 1− λ2kUkl + λkU0l ) , k = 1, 2, . . . , L, l = 1, 2, . . . , n (2.17) where −1 < λk < 1, {Ukl ∼ N (0, 12), k = 1, 2, . . . , L, l = 1, 2, . . . , n}, and {U0l ∼ N ( ml, 1 2 ) , l = 1, 2, . . . , n}, Ukl and Uij are independent for k 6= i or l 6= j. Each Xkl is the sum of independent Gaussian RVs and hence a Gaussian RV itself with Xkl ∼ N ( σkλkml, σ2k 2 ) . For a given l, U0l is common to Xkl for k = 1, 2, . . . , L, it then follows that the Xkl’s are correlated RVs. Define the correlation coefficient between two RVs X and Y as ρXY , E [XY ]− E [X] E [Y ]√ Var[X]Var[Y ] (2.18) where Var[X] is the variance of RV X. The correlation coefficient between Xkl and Xij is given by ρXklXij =  1 k = i, l = j λkλi k 6= i, l = j 0 l 6= j . (2.19) 1We have removed the imaginary part of [24, Eq. (11)] to match the conventional form of the generalized Rician model given in [31]. 21 Since for a given k, the Xkl are a set of independent Gaussian RVs with identical variance σk and non-zero means, Gk = ∑n l=1 X 2 kl and Rk = √ Gk, k = 1, 2, . . . L, are a noncentral χ2 and generalized Rician RV, respectively. It is shown in Appendix A that the correlation between Gi and Gk is ρGkGi = ρR2kR2i = λ2kλ 2 i ( n 2 + 2S2)√ n 2 + 2λ2kS 2 √ n 2 + 2λ2iS 2 , k 6= i (2.20) where S2 = ∑n l=1m 2 l . For Nakagami-m fading, (2.20) reduces to ρGkGi = ρR2kR2i = λ 2 kλ 2 i , k 6= i. (2.21) To find the joint PDF of R = [R1, R2, . . . , RL], we remove the dependence by conditioning Xkl on U0l, which is a Gaussian RV with E[Xkl|U0l] = σkλkU0l and Ωk = Var[Xkl|U0l] = σ 2 k(1−λ2k) 2 . It follows then that Gk conditioned on U0l, l = 1, 2, . . . , n is noncentral χ 2 distributed with PDF fGk|T (gk) = 1 2Ω2k ( gk σ2kλ 2 kT )n−2 4 e −x+λ 2 kσ 2 kT 2Ω2 k In 2 −1 (√ gkσ2kλ 2 kT Ω2k ) (2.22) where T = ∑n l=1 U 2 0l, and we have used the fact that conditioning on U0l, l = 1, 2, . . . , n is identical to conditioning on T . T is a noncentral χ2 RV with PDF fT (t) = ( t S2 )n−2 4 e−(t+S 2)In 2 −1 ( 2S √ t ) . (2.23) SinceGk’s conditioned on T are independent, the joint PDF of G = [G1, G2, . . . , GL] 22 conditioned on T is then fG|T (g1, g2, . . . , gL) = L∏ k=1 1 2Ω2k ( gk σ2kλ 2 kT )n−2 4 e − gk+λ 2 kσ 2 kT 2Ω2 k In 2 −1 (√ gkσ2kλ 2 kT Ω2k ) . (2.24) We obtain the joint PDF of G by taking the expectation of (2.24) with respect to (w.r.t.) the PDF of T fG(g1, g2, . . . , gL) = ∞∫ 0 ( t S2 )n−2 4 e−(t+S 2)In 2 −1 ( 2S √ t ) × L∏ k=1 1 2Ω2k ( gk σ2kλ 2 kt )n−2 4 exp ( −gk + λ 2 kσ 2 kt 2Ω2k ) In 2 −1 (√ gkσ2kλ 2 kt Ω2k ) dt. (2.25) By definition, the joint CDF of G is FG(g1, g2, . . . , gL) = g1∫ 0 g2∫ 0 · · · gL∫ 0 fG(x1, x2, . . . , xL)dx1dx2 · · · dxL = ∞∫ 0 ( t S2 )n−2 4 e−(t+S 2)In 2 −1 ( 2S √ t ) L∏ k=1 [ 1−Qn 2 (√ 2λ2kt 1− λ2k , √ gk Ωk )] dt (2.26) where we have used the property Qv(a, 0) = 1. We can now obtain the joint 23 CDF of R from (2.26) as follows FR(r1,r2, . . . , rL) =Pr [ r1 ≤ √ G1, r2 ≤ √ G2, rL ≤ √ GL ] =FG(r 2 1, r 2 2, . . . , r 2 L) = ∞∫ 0 ( t S2 )n−2 4 e−(t+S 2)In 2 −1 ( 2S √ t ) L∏ k=1 [ 1−Qn 2 (√ 2λ2kt 1− λ2k , rk Ωk )] dt. (2.27) Taking the partial differentiation of (2.27) w.r.t. r1, r2, . . . , rL, we obtain the joint PDF of R as fR(r1, r2, . . . , rL) = exp (−S2) S n 2 −1 ∞∫ 0 t n−2 4 exp (−t)In 2 −1(2S √ t) × L∏ k=1 r n 2 k (λ2kσ 2 kt) n−2 4 Ω2k exp { −r 2 k + λ 2 kσ 2 kt 2Ω2k } In 2 −1 ( rk √ σ2kλ 2 kt Ω2k ) dt. (2.28) Although [24] justifies the relevance of the generalized correlation model by pointing out the lack of simple PDF and CDF expressions for commonly used fading models with L > 2, it does not discuss the limitations of the model. Let us examine the marginal PDF of Rk, which can be found by setting ri =∞ for i = 1, 2, . . . , L, i 6= k in (2.27), then differentiating w.r.t. 24 to rk to obtain fRk(rk) = 2r n 2 k (λkσkS) n 2 −1σ2k exp ( −λ2kS2 − r2k σ2k ) In 2 −1 ( 2λkSrk σk ) (2.29) where we have also used the integral identity [37, Eq. (3.25.17.1)]. Compar- ing (2.29) to the generalized Rician PDF (2.11), we see that s2k = λ 2 kσ 2 kS 2 in the generalized correlation model. Since the parameter S is identical in the marginal PDF of each Rk, k = 1, 2, . . . , L, si cannot be chosen independently of sk for i 6= k. For Rician fading, this translates to a dependence among the Rice factors. Of course, this limitation does not apply to the Rayleigh and Nakagami-m special cases since sk = 0. The second limitation is that the power correlation coefficients of R (2.20) cannot be arbitrarily chosen for L > 3. There are (L2−L)/2 correlation coefficients, but only L number of λ coefficients. When L > 3, (L2 − L)/2 > L and there are not enough degrees of freedom available to arbitrarily choose each correlation coefficient. The generalized correlation model does however include the special case of equal correlation when λk = λ, k = 1, 2, . . . L. Despite the limitations we have outlined here, we will adopt the generalized correlation model for this thesis in the absence of a tractable analytical model for arbitrary correlation. 25 Chapter 3 Asymptotic Performance Analysis of Combining Methods in Generalized Rician Fading In this chapter, we derive the asymptotic error rates of SC, EGC, and MRC for coherent and noncoherent modulations in correlated generalized Rician fading channels. Although using our results, asymptotic error rates can be found for all combinations of signal modulations and combining schemes we consider, some of combinations are not useful in practice, e.g. asymptotic error rates of noncoherent modulations with EGC and MRC combiners. As mentioned in Chapter 1, some results derived here are not original, as asymptotic error rate expressions for Rician and Nakagami-m fading with generalized correlation are a special case of those in [26–28]. The case of Nakagami-m fading with SC is also identical to the results in [25]. How- ever, the asymptotic SC results are expanded into an exact infinite series in Chapter 4 with an upper bound on the truncation error. In addition, asymp- totically tight bounds on the error rate under EGC are derived in Chapter 5. 26 The results of Chapters 4 and 5 can guarantee the minimum SNR at which the asymptotic error rates given here are within a tolerance of the exact error rate. The results for MRC are derived here for completeness and for comparison to SC and EGC. 3.1 System Model Consider an L-branch diversity combiner where the transmitted signal is im- paired by slow, frequency nonselective fading and AWGN on each branch. The instantaneous SNR on the kth branch is γk = R 2 k, where Rk is a general- ized Rician RV. We assume correlation among branches fits the generalized correlation model discussed in Chapter 2, then the average SNR on the kth branch is γk = E[γk] = σ 2 k ( n 2 + λ2kS 2 ) , and power correlation between the kth and ith branch is ργkγi = λ2kλ 2 i ( n 2 + 2S2)√ n 2 + 2λ2kS 2 √ n 2 + 2λ2iS 2 , k 6= i. (3.1) 3.2 Asymptotic Analysis Let us assume a generic receiver where γ is the instantaneous SNR at the demodulator with average SNR γ = E [γ] and PDF fγ(γ). If the conditional SER for a constant (non fading) channel is Pe|γ, then the SER for a fading 27 channel is given by Pe = ∞∫ 0 Pe|γfγ(γ)dγ. (3.2) It turns out that when the average SNR approaches infinity, the error prob- ability is solely determined by the shape of the fγ(γ) at the origin [27]. This can be intuitively explained as follows. The conditional error rate decreases exponentially in SNR due to the nature of Gaussian noise, thus even for moderate γ̄, Pe is mostly determined by the probability that γ is small since Pe|γ will be several orders of magnitude greater at γ = 0 than at γ = γ̄. Let the first-order approximation of fγ(γ) at the origin be given by fγ(γ) = aγ t + o(γt). (3.3) Using (3.3) in conjunction with (3.2) we can produce an approximation of Pe. As γ̄ → ∞, Pe is increasingly dominated by the probability that γ is near zero, and the approximation approaches the exact solution. At high SNR, many coherent modulations have a conditional error proba- bility in the form of pQ( √ qγ) including BPSK, M -ary pulse amplitude mod- ulation (M -PAM), M -ary phase shift keying (M -PSK), and M -ary quadra- ture amplitude modulation (M -QAM) [31]. The parameters p and q for these modulations are listed in Table 3.1. The asymptotic SER with con- ditional error probability pQ( √ qγ) and the first-order approximation of the 28 SNR PDF (3.3) is given by [27] P∞e,c = 2taΓ(t+ 3 2 )p√ pi(t+ 1)qt+1 . (3.4) Table 3.1: Parameters p and q for various coherent modulations Modulation p q BPSK 1 2 M -PAM 2 ( 1− 1 M ) 6 M2−1 M -PSK (M ≥ 4) 2 2 sin2 pi M M -QAM 4 ( 1− 1√ M ) 3 M−1 On the other hand, the noncoherent modulations binary differential phase shift keying (BDPSK) and binary noncoherent frequency shift keying (BNCFSK) have a conditional error probability of the form 1 2 exp (−qγ) with q = 1 for BDPSK, and q = 1/2 for BNCFSK. When the conditional error rate is given by 1 2 exp (−qγ) and first-order approximation of the SNR PDF is given by (3.3), the asymptotic error rate becomes [25] P∞e,nc = aΓ (t+ 1) 2qt+1 . (3.5) Since we are considering diversity combiners, γ is the output SNR of the diversity combiner, which we denote γXX where XX = SC, EGC, or MRC depending on the combiner under consideration. It is straight forward to determine the asymptotic SER of many coherent and noncoherent modula- tions by obtaining the parameters a and t from (3.3) provided that the PDF 29 at the output of the combiner, fγXX (γ), is available. While this approach is feasible for SC, it is difficult to directly determine the PDF of the sum of RVs directly, as is the case with MRC and EGC. In these cases, a Laplace transform approach is often employed. For this approach, we require the MGF of γ, defined as Mγ(s) = E [e−sγ], to be Mγ(s) = aΓ(t+ 1) st+1 + o ( 1 st+1 ) (3.6) as s → ∞. Alternatively, we can also find a, and t from the MGF of the equivalent channel at the output of the combiner, h = √ γ. The MGF of h when s→∞ is Mh(s) = 2aΓ(2t+ 2) s2t+2 + o ( 1 s2t+2 ) . (3.7) 3.3 First Order Joint Distributions In order to determine the first-order approximation of γ at the origin we first find the joint PDF and the joint CDF of R = [R1, R2, . . . , RL] near the origin. From a Taylor series expansion of the exponential function, lim x→0 exp(x) = 1 + o(1) and from the series expansion of the modified Bessel function of the first kind [39, Eq. (9.6.10)], lim x→0 Iv(x) = 1 Γ(v+1) ( 1 2 x )v + o(xv+1). Substituting these asymptotic expression into the joint PDF of R (2.28) we find fR(r1, . . . , rL) = 2Le− Λ Λ+1 S2 ΓL ( n 2 ) det n 2 (M) (∏L k=1 σ 2 k )n 2 L∏ k=1 (rn−1k + o(r n−1 k )) (3.8) 30 where Λ = ∑L k=1 λ2k 1−λ2k . In obtaining (3.8), we have used an integral iden- tity [40, Eq. (6.643.2)] and performed algebraic simplifications using [40, Eq. (9.220.2)] and [39, Eq. (13.6.12)]. We have also introduced the correlation matrix M M =  1 λ1λ2 · · · λ1λL λ1λ2 1 · · · ... ... ... . . . λL−1λL λ1λL · · · λL−1λL 1  (3.9) where det(M) = [ 1 + ∑L k=1 λ2k 1−λ2k ]∏L k=1(1 − λ2k). This result can be found by expressing M as a rank 1 update matrix and using a matrix determinant lemma [25]. The off diagonal elements of M are the correlation coefficients of the underlying Gaussian RVs in R which were given in (2.19). The CDF of R can be found by integrating (3.8) over r1, r2, . . . , rL, resulting in FR(r1, . . . , rL) = e− Λ Λ+1 S2 ΓL ( n 2 + 1 ) det n 2 (M) (∏L k=1 σ 2 k )n 2 L∏ k=1 (rnk + o(r n k )). (3.10) 31 3.4 Combiner Asymptotics 3.4.1 SC From (2.14), the CDF of γSC can be related to FR(r1, . . . , rL) via FγSC (γ) = Pr[γ1 ≤ γ, γ2 ≤ γ, . . . , γL ≤ γ] = Pr[R1 ≤ √γ,R2 ≤ √γ, . . . , RL ≤ √γ] = FR ( √ γ, . . . , √ γ) . (3.11) Substituting (3.10) into (3.11) and differentiating w.r.t. γ we obtain the PDF of γSC as γ → 0 as fγSC (γ) = nLe− Λ Λ+1 S2γ nL 2 −1 2ΓL ( n 2 + 1 ) det n 2 (M) (∏L k=1 σ 2 k )n 2 + o(γ nL 2 −1). (3.12) Extracting the parameters a and t from (3.12), we have aSC = nLe− Λ Λ+1 S2 2ΓL ( n 2 + 1 ) det n 2 (M) (∏L k=1 σ 2 k )n 2 (3.13) and tSC = nL 2 − 1. (3.14) 32 3.4.2 EGC For EGC, we consider the equivalent channel gain at the output of the EGC hEGC , √ γEGC = 1√ L L∑ k=1 Rk. (3.15) By definition, the MGF of hEGC is MhEGC (s) = E [ e−shEGC ] = ∫ ∞ 0 · · · ∫ ∞ 0︸ ︷︷ ︸ L e − s√ L ∑L k=1 rkfR(r1, . . . , rL)dr1 . . . drL. (3.16) If we apply the first-order joint PDF (3.8) to (3.16) and convert the L-fold integral to a product of L integrals we obtain MhEGC (s) for s→∞ as MhEGC (s) = e− Λ Λ+1 S2 det n 2 (M) ( 2L n 2 Γ(n) Γ ( n 2 ) )L( L∏ k=1 1 σ2k )n 2 1 snL + o ( 1 snL ) . (3.17) Comparing (3.17) to (3.7) we find aEGC = e− Λ Λ+1 S2 2Γ(nL) det n 2 (M) ( 2L n 2 Γ(n) Γ ( n 2 ) )L( L∏ k=1 1 σ2k )n 2 (3.18) and tEGC = nL 2 − 1. (3.19) 33 3.4.3 MRC Since (2.16) is a linear sum, we can easily find the MGF of γMRC using (3.8) and convert the resulting the L-fold integral to a product of L easily evaluated integrals. This results in MγMRC (s) = e− Λ Λ+1 S2 det n 2 (M) ( L∏ k=1 1 σ2k )n 2 1 s nL 2 + o ( 1 s nL 2 ) (3.20) for s→∞. From (3.6) and (3.20) parameters a and t can be calculated as aMRC = e− Λ Λ+1 S2 Γ ( nL 2 ) det n 2 (M) ( L∏ k=1 1 σ2k )n 2 (3.21) and tMRC = nL 2 − 1. (3.22) 3.5 Discussion and Numerical Results 3.5.1 Discussion As the t parameter (t+1 is also known as the diversity gain and indicates the slope on a log-log plot of P∞e vs SNR) is identical for each diversity scheme we can directly evaluate the relative asymptotic error probability with identical modulation among the combining schemes solely by comparing the parameter a. It can be seen from (3.13), (3.18), and (3.21) that for each combining scheme, the parameters aSC , aEGC , and aMRC share several common terms. 34 If we isolate the commonality, we can redefine aXX as aXX = µXXc, where c = e − Λ Λ+1 S2 det n 2 (M) (∏L k=1 1 σ2k )n 2 is the common component among all combining schemes and µXX is the combining specific factor given by µSC = nL 2ΓL ( n 2 + 1 ) (3.23) µEGC = 1 2Γ(nL) [ 2L n 2 Γ(n) Γ ( n 2 ) ]L (3.24) µMRC = 1 Γ ( nL 2 ) . (3.25) Note that µXX only depends on the severity of the fading and number of branches, and does not depend on the channel correlation or average branch power. If we plot the µXX of SC and EGC relative to µMRC (the optimum linear combiner), i.e. µMRC/µXX , as in Fig. 3.1, we see that for dual diversity (L=2) and n = 1, SC and EGC achieve identical performance with µSC = µEGC = 4/pi. This result has also been shown to apply at all SNRs for independent Nagakami-m fading channels [41]. For L > 2 and n > 1 we see that EGC is able to outperform SC while achieving similar performance to MRC. The performance gap between both SC and EGC relative to MRC increases with the number of branches and as the fading severity decreases. Interestingly in Fig. 3.1, it appears as though as n → ∞, the ratio µMRC/µEGC approaches a constant. This is indeed the case and can be 35 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ M R C/ µ X X n   SC, L=2 EGC, L=2 SC, L=3 EGC, L=3 SC, L=4 EGC, L=4 Figure 3.1: Performance of EGC and SC relative to MRC. shown using Stirling’s formula, limx→∞ Γ(x) = √ 2pie−xxx−1/2, to be lim n→∞ µMRC µEGC = 2 1−L 2 . (3.26) Also as L → ∞ there is also a fixed relationship between the average SNR required for EGC and MRC to have equal performance. If γ̄EGC and γ̄MRC are the average SNR on each branch for EGC and MRC respectively, then γ̄EGC γ̄MRC = ( µEGC µMRC ) 2 nL . (3.27) 36 Again, using Sterling’s formula, we obtain lim L→∞ γ̄EGC γ̄MRC = e n21− 2 n [ Γ(n) Γ ( n 2 )] 2n (3.28) = 10 log10 ( e n21− 2 n ) + 20 n log10 ( Γ(n) Γ ( n 2 )) [dB]. (3.29) This represents the additional power required at the transmitter if EGC is implemented over MRC to reduce the combiner complexity. Similar results for SC do not exist, with limn→∞ µMRC/µSC = 0 and limL→∞ γ̄SC/γ̄MRC = ∞. 3.5.2 Numerical Results In order to validate our results we have plotted P∞e,c for BPSK and equal average branch SNR along with Monte Carlo simulations in Fig. 3.2, Fig. 3.3, and Fig. 3.4 for SC, EGC, and MRC respectively. In all cases we let λ = [0.9, 0.3, 0.8]. We can see from Figs. 3.2-3.4 that for each combining method the simulated results approach the asymptotic error rates as SNR increases. 37 0 5 10 15 20 25 30 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 B ER Average Branch SNR (dB)   n=1 n=2 n=3 Simulation Figure 3.2: Asymptotic and simulated BERs of coherent BPSK of triple branch SC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. 38 0 5 10 15 20 25 30 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 B ER Average Branch SNR (dB)   n=1 n=2 n=3 Simulation Figure 3.3: Asymptotic and simulated BERs of coherent BPSK of triple branch EGC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. 39 0 5 10 15 20 25 30 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 B ER Average Branch SNR (dB)   n=1 n=2 n=3 Simulation Figure 3.4: Asymptotic and simulated BERs of coherent BPSK of triple branch MRC with generalized Rician fading channels with S = 3 and λ = [0.9, 0.3, 0.8]. 40 Chapter 4 Exact Series Form of the BER with SC in Generalized Rician Fading In this chapter we develop an exact infinite series representation for the PDF and CDF of the SNR for SC. The statistical distributions are then used to obtain an exact infinite series expression for the BER of binary coherent and noncoherent modulations. We also find an upper bound on the truncation error introduced when the infinite series is truncated to a finite number of terms. 4.1 Channel Model Consider an L-branch diversity combiner where the transmitted signal is im- paired by slow, frequency nonselective fading and AWGN on each branch. The instantaneous SNR on the kth branch is γk = R 2 k, where Rk is a gen- eralized Rician RV. Assume correlation among branches fits the generalized 41 correlation model, then the average SNR on the kth branch is γk = E[γk] = σ2k ( n 2 + λ2kS 2 ) and the power correlation between the kth and ith branch is ργkγi = λ2kλ 2 i ( n 2 + 2S2)√ n 2 + 2λ2kS 2 √ n 2 + 2λ2iS 2 , k 6= i (4.1) as found in Chapter 2. Furthermore, let the average SNR of the kth branch be given by γ̄k = gkγ̄ where gk is the gain of the kth branch relative to some baseline SNR value γ̄. For instance, γ̄ could be the average SNR across all branches, in which case γ̄ = ∑L k=1 γ̄k. 4.2 SNR Distribution We start by deriving an infinite series expression for the CDF at the output of the SC combiner. Using the joint CDF of generalized Rician RVs with generalized correlation (2.27), the CDF of γSC is given in single integral form as FγSC (γ) =FR (R1 ≤ √ γ,R2 ≤ √γ . . . , RL ≤ √γ) (4.2) = exp (−S2) S n 2 −1 ∞∫ 0 t n−2 4 exp (−t)In 2 −1(2S √ t) × L∏ k=1 [ 1−Qn 2 (√ 2λ2kt 1− λ2k , √ 2dkγ γ̄ )] dt. (4.3) where dk = (n+2λ2kS 2) 2gk(1−λ2k) . 42 A novel infinite series form of the generalized Marcum Q-function in terms of generalized Laguerre polynomials was recently proposed by András et al. [42] as Qv(a, b) = 1− ∞∑ i=0 (−1)ie−a 2 2 L (v−1) i ( a2 2 ) Γ(v + i+ 1) ( b2 2 )v+i . (4.4) The availability of well known inequalities for the generalized Laguerre poly- nomial makes this series suitable for a truncation error analysis. Substituting (4.4) into (4.3) and interchanging the order of summation and integration we can express FγSC (γ) as an infinite series FγSC (γ) = [ L∏ k=1 dk ]n 2 ∞∑ i=0 Ci ( γ γ̄ )i+nL 2 (4.5) where the coefficient Ci is given by Ci = (−1)i e−S2 S n 2 −1 ∞∫ 0 t n−2 4 exp (− (Λ + 1) t) In 2 −1 ( 2S √ t ) × ∑ j1+···+jL=i L∏ k=1 L (n2−1) jk ( λ2kt 1−λ2k ) Γ(n 2 + jk + 1) djkk dt. (4.6) To evaluate the integral in Ci we consider the integral identity ∞∫ 0 ti+ µ−1 2 e−αtIµ−1 ( 2β √ t ) dt = i! ( β αi )µ exp ( β2 α ) L (µ−1) i ( −β 2 α ) (4.7) 43 where i ∈ N and µ > 0. The derivation of (4.7) is located in Appendix B. The integral in Ci can then be evaluated using a finite series form of the generalized Laguerre polynomial [39, Eq. (22.3.9)] L (α) i (x) = i∑ j=0 (j + α + 1)i−j (i− j)!j! (−x) j, α > −1 (4.8) and the integral identity (4.7) along with some algebraic simplifications to obtain Ci = (−1)i e− ΛΛ+1S2 (Λ + 1) n 2 ΓL ( n 2 ) ∑ j1+···+jL=i L∏ k=1 djkk jk! ( n 2 + jk ) × j1,...,jL∑ l1=0,...,lL=0 (∑L k=1 lk ) ! (Λ + 1) ∑L k=1 lk L (n2−1)∑L k=1 lk ( − S 2 1 + Λ ) × L∏ k=1 ( jk lk ) 1( n 2 ) lk ( −λ2k 1− λ2k )lk . (4.9) The PDF of γSC can be found by differentiating (4.5) w.r.t. γ, upon which we obtain fγSC (γ) = [ L∏ k=1 dk ]n 2 ∞∑ i=0 ( nL 2 + i ) Ciγ̄ ( γ γ̄ )i+nL 2 −1 . (4.10) It is easy to verify that the first term in (4.10) is the asymptotic PDF at the output of the SC combiner (3.12) derived in Chapter 3. As a sanity check, we examine the special case when L = 1. For L = 1, 44 Ci is simplifies to Ci = (−d)i(1− λ2)n2 +ie−λ2S2 Γ ( n 2 ) i! ( n 2 + i ) i∑ l=0 ( i l )( 1 + λ2 1− λ2 )i−l( −λ2 1− λ2 )l × L (n2−1) l (−(1− λ2)S2) L (n2−1) l (0) = (−d)i(1− λ2)n2 +ie−λ2S2 Γ ( n 2 ) i! ( n 2 + i ) L(n2−1)i (λ2S2) L (n2−1) i (0) (4.11) where we have used the fact that L (α) n (0) = (α+1)n n! and a Laguerre sum formula [43, Eq. (18.18.12)]. Substituting (4.11) into (4.10) we obtain the following fγSC (γ) = e−λ 2S2 Γ ( n 2 ) σ2 ( γ σ2 )n 2 −1 ∞∑ i=0 ( − γ σ2 )i L(n2−1)i (λ2S2) i!L (n2−1) i (0) (4.12) = γ n−2 4 e−λ 2S2− γ σ2 σ2 (λSσ) n 2 −1 In2−1 ( 2λS σ √ γ ) (4.13) where we have used the formula [42] ∞∑ i=0 L (α) i (x) L (α) i (0) (−z)i i! = Γ(α + 1)e−z(xz)− α 2 Iα(2 √ xz). (4.14) This matches the result found by differentiating (4.2) w.r.t. γ, substituting (2.29) and letting L = 1. 45 4.3 BER of Binary Modulations 4.3.1 BER for Binary Coherent Modulations For binary coherent modulations, the conditional error rate is given by Pe|γ = Q ( √ qγ) . (4.15) where q = 2 for BPSK and q = 1 for binary coherent frequency shift keying (BCFSK). Substituting (4.15) into (3.2) and changing the order of integration we obtain Pe,c = 1√ 2pi ∞∫ 0 exp ( −x 2 2 ) Fγ ( x2 q ) dx. (4.16) An infinite series form for Pe,c can then be found by substituting (4.5) into (4.16) and changing the order of summation and integration to arrive at Pe,c = 1 2 √ pi [ L∏ k=1 dk ]n 2 ∞∑ i=0 2 nL 2 +iCiΓ ( i+ nL+1 2 ) (γ̄q) nL 2 +i . (4.17) As γ̄ →∞, Eq. (4.17) is dominated by the first term in the summation: Pe,c ≈ 1 2 √ pi L∏ k=1 ( n+ 2λ2kS 2 2gk )n 2 (−1)i e− ΛΛ+1S2Γ (nL+1 2 ) (Λ + 1) n 2 ΓL ( n 2 + 1 ) 1 γ̄ nL 2 (4.18) which matches the asymptotic BER (3.4) when t and a are given by (3.14) and (3.13) respectively. Although we have only derived the BER of coherent binary modulations, 46 the results can be generalized to many M -ary linear digital modulations following an MGF approach [1]. 4.3.2 BER for Binary Noncoherent Modulations As mentioned in Chapter 3, the conditional error rate for binary noncoherent modulations is given by Pe|γ = 12 exp (−qγ), where q = 1/2 for BNCFSK and q = 1 for BDPSK. We then obtain the average BER as Pe,nc = 1 2 ∞∫ 0 exp (−qγ) fγ (γ) dγ. (4.19) Substituting (4.10) into (4.19) and changing the order of integration we ob- tain an infinite series solution for the BER Pe,nc = 1 2 [ L∏ k=1 dk ]n 2 ∞∑ i=0 CiΓ ( i+ nL 2 + 1 ) (γ̄q) nL 2 +i . (4.20) 4.4 Convergence and Truncation Error 4.4.1 Convergence Thus far, we have proceeded with interchanging the order of summation and integration without considering whether the resulting series converges or not. While we will show the infinite series forms of FγSC (γ) and fγSC (γ) are convergent for all 0 ≤ γ < ∞, the infinite series for the BER for both 47 coherent and noncoherent modulations are only convergent for a sufficiently high average SNR. We start by finding an upper bound on Ci. To achieve this, we return to the form of Ci prior to carrying out the integration (4.6), and apply a well known inequality for the generalized Laguerre polynomial [39, Eq. (22.14.13)] ∣∣∣L(α)i (x)∣∣∣ ≤ (α + 1)ii! exp(x2) , α ≥ 0, x ≥ 0 (4.21) along with an integral identity [37, Eq. (3.15.2.8)] to obtain |Ci| ≤ exp ( − ΛS2 (Λ+2) ) ΓL ( n 2 ) ( Λ 2 + 1 )n 2 ∑ j1+···+jL=i L∏ k=1 djkk( n 2 + jk ) jk! ≤ exp ( − ΛS2 (Λ+2) ) ΓL ( n 2 + 1 ) ( Λ 2 + 1 )n 2 ∑ j1+···+jL=i L∏ k=1 djkk jk! = exp ( − ΛS2 (Λ+2) ) ΓL ( n 2 + 1 ) ( Λ 2 + 1 )n 2 ηi i! (4.22) where η = ∑L k=1 dk and we have used the multinomial identity to get (4.22). Note that due to the use of (4.21), which is valid only for α ≥ 0, Eq. (4.22) applies for n > 1 only. For n = 1 we use the inequality ∣∣∣L(α)i (x)∣∣∣ ≤ 2 exp(x2) − 1 < α ≤ 0, x ≥ 0 (4.23) which is a relaxed form of [39, Eq. (22.14.14)]. Using a similar methodology 48 to how (4.25) was obtained, we find for n = 1 |Ci| ≤ 2L exp ( − ΛS2 (Λ+2) ) √ Λ 2 + 1 ∑ j1+···+jL=i L∏ k=1 djkk Γ(jk + 3 2 ) ≤ 2L exp ( − ΛS2 (Λ+2) ) √ Λ 2 + 1 ∑ j1+···+jL=i L∏ k=1 djkk jk! = 2L exp ( − ΛS2 (Λ+2) ) √ Λ 2 + 1 ηi i! . (4.24) Since (4.22) and (4.24) vary only by a constant coefficient, we define a new term valid for all n = 1, 2, . . . |Ci| ≤ Aη i i! (4.25) A =  2 exp ( − ΛS2 (Λ+2) ) pi L 2 (Λ2 +1) n 2 n = 1 exp ( − ΛS2 (Λ+2) ) ΓL(n2 +1)( Λ 2 +1) n 2 n = 2, 3, . . . . (4.26) Using (4.25), we can now bound the series form of FγSC (γ) (4.5) as follows |FγSC (γ)| ≤ [ L∏ k=1 dk ]n 2 ∞∑ i=0 |Ci| ( γ γ̄ )i+nL 2 ≤ A ( γ γ̄ )nL 2 [ L∏ k=1 dk ]n 2 ∞∑ i=0 1 i! ( ηγ γ̄ )i = A ( γ γ̄ )nL 2 [ L∏ k=1 dk ]n 2 exp ( ηγ γ̄ ) . (4.27) 49 Eq. (4.27) converges for 0 ≤ γ <∞, thus convergence of (4.5) is guaranteed for the same interval. A similar exercise can be performed to show that the PDF in (4.10) also converges for 0 ≤ γ <∞. Due to the presence of the Γ(·) term in the numerator of the summation in (4.17), convergence of the series is conditional. Finding the entire region of convergence of (4.17) is extremely complex due to the nested summations. However, we can find a subset of the region of convergence using the upper bound on |Ci|. Using (4.25), we find |Pe,c| ≤ 2 nL 2 −1A√ pi(qγ̄) nL 2 [ L∏ k=1 dk ]n 2 ∞∑ i=0 Γ ( i+ (nL+1) 2 ) i! ( 2η qγ̄ )i (4.28) which converges by the ratio test for 2η qγ̄ < 1. Similarly, it can be shown (4.20) converges when η qγ̄ < 1. The requirement 2η qγ̄ < 1 or η qγ̄ < 1 is satisfied provided that average branch power is sufficiently high. For instance, for branches experiencing equally correlated Nakagami-m fading, equal average branch SNR, and BPSK modulation we require η γ̄ < 1, which is achieved when γ̄ > mL (1−√ρ) where ρ = λ4 is the power correlation coefficient between any two branches. From this we can infer that the minimum γ̄ required to guarantee the infinite series expressions for Pe converge increases with the number of branches and branch correlation, but decreases with the fading severity. 50 4.4.2 Truncation Error In this section we derive an upper bound on the truncation errors when the infinite series for SNR CDF (4.5) and BER for binary coherent mod- ulations (4.17) are terminated at a finite number of terms. We omit the results for binary noncoherent modulations, but the truncation error can be bounded in a similar manner. The truncation of (4.5) and (4.17) and to the first N + 1 terms is given by F̂ (N)γSC (γ) = [ L∏ k=1 dk ]n 2 N∑ i=0 Ci ( γ γ̄ )i+nL 2 (4.29) P̂ (N)e,c = 1 2 √ pi [ L∏ k=1 dk ]n 2 N∑ i=0 2 nL 2 +iCiΓ ( i+ nL+1 2 ) (qγ̄) nL 2 +i . (4.30) We then define the associated truncation errors as  (N) FγSC (γ) , ∣∣∣FγSC (γ)− F̂ (N)γSC (γ)∣∣∣ and  (N) Pe,c (γ) , ∣∣∣Pe,c(γ)− P̂ (N)e,c (γ)∣∣∣ respectively. (N)FγSC (γ) can be bounded us- ing (4.25) to obtain  (N) FγSC (γ) ≤A ( γ γ̄ )nL 2 [ L∏ k=1 dk ]n 2 ∞∑ i=N+1 1 i! ( ηγ γ̄ )i (4.31) =A ( γ γ̄ )nL 2 [ L∏ k=1 dk ]n 2 { exp ( ηγ γ̄ )i − N∑ i=0 1 i! ( ηγ γ̄ )i} . (4.32) 51 Likewise we bound  (N) Pe,c (γ) as  (N) Pe,c ≤ 2 nL 2 −1A√ pi(qγ̄) nL 2 [ L∏ k=1 dk ]n 2 ∞∑ i=N+1 Γ ( i+ nL+1 2 ) i! ( 2η qγ̄ )i (4.33) = 2 nL 2 −1Γ ( nL+1 2 ) A √ pi(qγ̄) nL 2 [ L∏ k=1 dk ]n 2  1(1− 2η qγ̄ )nL+1 2 − N∑ i=0 ( nL+1 2 ) i i! ( 2η qγ̄ )i (4.34) where we have used the binomial series 1 (1−x)α = ∑∞ i=0 (α)i i! xi, |x| < 1 to obtain (4.34). It is easy to see from (4.33) that the upper bound on the truncation error is o ( γ̄−( nL 2 +N+2) ) , the same order as the first term ignored when the series is truncated to (4.30). Thus, we can conclude our truncation error bound decreases relative to the truncated error rate as γ̄ →∞. 4.5 Numerical Results In order to verify that (4.30) converges to the exact error rate, we have plotted it for BPSK with N = 30 along with Monte Carlo simulations for Nakagami-m (m = 2) and Rician (S = 1.5) fading with equal average branch SNR in Fig. 4.1. In both cases we have used λ = [0.6, 0.4, 0.9], resulting in 52 power correlation matrices, where the (k, i)th element is given by (4.1), of P1 =  1 0.149 0.460 0.149 1 0.252 0.460 0.252 1  (4.35) and P2 =  1 0.058 0.292 0.058 1 0.130 0.292 0.130 1  (4.36) for Rician and Nagakami-m respectively. We see fem Fig. 4.1 that in both cases, the approximation is highly accurate where the series converges. The convergence criterion for BPSK, η γ̄ < 1, is satisfied when γ̄ > 12.85 dB for Rician fading and γ̄ > 12.05 dB for Nakagami-m fading. This is very close to the true value for the Nakagami-m fading; however, the convergence region of Rician fading is underestimated by roughly 2 dB. Although we have chosen N = 30 in the previous example, the asymptotic error rate can be greatly improved by including only a couple extra terms. This is shown in Fig. 4.2, where we have replotted the asymptotic (N = 0) error rate along with N = 1 and N = 2 for Rician fading. The parameters are the same as the for the Rician case in Fig. 4.1. The asymptotic error rate is not within 1% of the exact error rate until 28.9 dB. By adding a single additional term this occurs at 18.5 dB, and two additional terms improves it further to 14.8 dB. Thus, we can extend the SNR region where the asymptotic 53 expression is valid with little additional complexity by including the first several terms of the infinite series. Fig. 4.3 shows the relative truncation error for Nakagami-2 fading, where the relative truncation error is the ratio of (4.34) to (4.30). As we would expect, the number of terms required to satisfy a truncation error target di- minishes as SNR increases. For instance, to guarantee a relative truncation error less than 1%, we require N = 40 at 14.3 dB, but this drops to N = 10 at 17.4 dB, and further to N = 5 at 19.7 dB. Although it is tempting to increase N to achieve an arbitrary accuracy, computation of CN can be very intensive for large N and numerical integration of (4.16) may be more effi- cient. However, it should be noted that provided the relative SNRs between branches remains constant, Ci, i = 1, 2, . . . , N need only be computed once, whereas numerical integration would have to be performed at each SNR. 54 11 12 13 14 15 16 17 18 19 20 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 B ER Average Branch SNR (dB)   Approx Rician Approx Nakagami Simulation Figure 4.1: Approximate (N = 30) and simulated BERs of coherent BPSK of triple branch SC with Rician and Nakagami-m fading. Average SNR is identical for each branch. For Rician fading, S = 1.5 and for Nakagami- m, m = 2. Power correlation is according to P1 and P2 for Rician and Nakagami-m, respectively. 55 9 10 11 12 13 14 15 16 17 18 19 20 10−6 10−5 10−4 10−3 10−2 B ER Average Branch SNR (dB)   N=0 N=1 N=2 Simulation Figure 4.2: Approximate and simulated BERs of coherent BPSK of triple branch SC with Rician fading (S = 1.5) and equal average branch SNR. Branches are correlated with power correlation matrix P1. 56 10 15 20 25 30 35 40 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 U pp er  B ou nd  o n th e Re la tiv e Tr un ca tio n Er ro r Average Branch SNR (dB)   N=0 N=2 N=5 N=10 N=20 N=40 Figure 4.3: Relative truncation error for coherent BPSK with triple branch SC in Nakagami-2 fading and equal average branch SNR. Branches are cor- related with power correlation matrix P2. 57 Chapter 5 Asymptotically Tight Error Bounds for EGC with Generalized Rician Fading In this chapter, we derive single integral upper and lower bounds on the average error rate of a receiver employing EGC in correlated generalized Rician fading. These bounds are asymptotically exact, i.e. as average SNR approaches infinity. Furthermore, the bounds reduce to the exact error rate when the branches are independent. 5.1 Channel Model Consider an L-branch diversity combiner where the transmitted signal is im- paired by slow, frequency nonselective fading and AWGN on each branch. The instantaneous SNR on the kth branch is γk = R 2 k, where Rk is a gen- eralized Rician RV. Assume correlation among branches fits the generalized correlation model, then the average SNR on the kth branch is γk = E[γk] = 58 σ2k ( n 2 + λ2kS 2 ) and power correlation between the kth and ith branch is ργkγi = λ2kλ 2 i ( n 2 + 2S2)√ n 2 + 2λ2kS 2 √ n 2 + 2λ2iS 2 , k 6= i (5.1) as found in Chapter 2. 5.2 Bounds on Joint PDF We first find bounds on the joint PDF of the generalized Rician distributed fading amplitudes R = [R1, R2, . . . , RL]. As derived in Chapter 2, the joint PDF is given by fR(r1, r2, . . . , rL) = exp (−S2) S n 2 −1 ∞∫ 0 t n−2 4 exp (−t)In 2 −1(2S √ t) × L∏ k=1 r n 2 k (λ2kσ 2 kt) n−2 4 Ω2k exp { −r 2 k + λ 2 kσ 2 kt 2Ω2k } In 2 −1 ( rk √ σ2kλ 2 kt Ω2k ) dt (5.2) where Ω2k = 1 2 σ2k(1 − λ2k). In order for these bounds to be asymptotically exact, we must preserve the shape of (5.2) at the origin, that is as r2k γ̄k → 0 or equivalently as r2k σ2k → 0. However, the bounds must also satisfy two ad- ditional objectives in order to provide a meaningful reduction of compu- tational burden over the exact analysis. The first, and the most obvious one, is to simplify (5.2) by removing the integral over t. The impediment to finding a closed form solution of this integral is the product of Bessel 59 functions. Thus, a key component of our derivations is that both upper and lower bounds of the Bessel function approach equality at the origin in order to preserve the asymptotic behaviour. The second objective is that the joint PDF bounds must appear as those of independent RVs, i.e. as fR(r1, r2, . . . , rL) Q A ∏L k=1W (rk), where A is some constant term. This allows for bounds on the CHF, which we use to find the average error rate via well known methods, to be found as A ∏L k=1 ΦW , where ΦW is the CHF of W . 5.2.1 Lower Bound A simple lower bound on Iv(x) can be found by observing that each term in its series expansion, Iv(x) = ( 1 2 x )v ∞∑ i=0 ( 1 2 x )2i i!Γ(v + i+ 1) (5.3) is positive when v ≥ −1 2 , x ≥ 0. Thus, we can lower bound the modified Bessel function by truncating the series expansion to the first term, i.e. Iv(x) ≥ 1 Γ(v + 1) ( 1 2 x )v , v ≥ −1 2 , x ≥ 0. (5.4) As it is the first term in the series expansion, Eq. (5.4) approaches equality as x → 0. Substituting (5.4) into (5.2) and using the integral identity [37, 60 Eq. (3.15.2.8)] we obtain fR(r1, . . . , rL) ≥ e − Λ Λ+1 S2 det n 2 (M) L∏ k=1 2rn−1k Γ ( n 2 ) σnk e − r 2 k 2Ω2 k (5.5) where Λ = ∑L k=1 λ2k 1−λ2k and the matrix M is defined in (3.9). Eq. (5.5) satisfies both our objectives, and thus it is a suitable lower bound. For the special case of Nakagami-m fading, we note that (5.5) is a scaled product of Nakagami-m PDFs fR(r1, . . . , rL) ≥ 1 (1 + Λ)m L∏ k=1 21−mr2m−1k Γ (m) Ω2mk e − r 2 k 2Ω2 k (5.6) and thus the equality is also achieved for independent branches since Λ = 0. 5.2.2 Upper Bound While derivation of a suitable lower bound of the joint PDF is straightfor- ward, this is not the case for the upper bound. We start by considering an upper bound on the Bessel function of the first kind [39, Eq. (9.1.62)] |Jv(x)| ≤ exp |=(x)| Γ(v + 1) ∣∣∣∣12x ∣∣∣∣ , v ≥ −12 . (5.7) Using the relation Iv(x) = j −vJv(jx), where j2 = −1, we obtain for x ∈ R Iv(x) ≤ exp(x) Γ(v + 1) ( 1 2 x )v , v ≥ −1 2 , x ≥ 0 (5.8) 61 which approaches equality as x→ 0. Substituting (5.8) into (5.2) we obtain an upper bound on fR(r1, . . . , rL) as fR(r1, . . . , rL) ≤ G1 L∏ k=1 2rn−1k Γ(n 2 )2 n 2 Ωnk exp ( − r 2 k 2Ω2k ) (5.9) where G1 = 2 exp (−S2) S n−2 2 ∫ ∞ 0 t n 2 exp ( − (1 + Λ) t2 + L∑ k=1 rk √ σ2kλ 2 k Ω2k t ) In 2 −1(2St)dt (5.10) and we have also made the integral substitution t2 for t. If we define u =∑L k=1 rk √ σ2kλ 2 k 2 √ 1+ΛΩ2k , and make the integral substitution z = √ 1 + Λt−u, we arrive at an expression for G1 which can be split into two integrals G2 and G3 as G1 = exp (u2 − S2) S n 2 −1(1 + Λ) n 4 + 1 2 [ 2 0∫ −u (z + u) n 2 exp (−z2) In 2 −1 ( 2S (z + u)√ 1 + Λ ) dz ︸ ︷︷ ︸ G2 + 2 ∞∫ 0 (z + u) n 2 exp (−z2) In 2 −1 ( 2S (z + u)√ 1 + Λ ) dz ︸ ︷︷ ︸ G3 ] . (5.11) The term G2 can be upper bounded by observing that the integrand is max- imized when z = 0, and we get G2 ≤ 2un2 +1In 2 −1 ( 2Su√ 1 + Λ ) . (5.12) 62 Applying the Bessel function inequality in (5.8) to (5.12), we arrive at a suitable upper bound for G2 as G2 ≤ 2u n Γ ( n 2 ) ( S√ 1 + Λ )n 2 −1 exp ( 2Su√ 1 + Λ ) . (5.13) Let us now consider G3. The integral in G3 cannot be expressed in closed form due to the linear shift of the integrand by u in the power term and Bessel function. Using the fact that x−vIv(x)e−x is a monotonic decreasing function over 0 ≤ x <∞, v ≥ −1 2 [44], we write In 2 −1 ( 2S (z + u)√ 1 + Λ ) ≤ ( z + u z )n 2 −1 In 2 −1 ( 2Sz√ 1 + Λ ) exp ( 2Su√ 1 + Λ ) . (5.14) Applying (5.14) to G3, we obtain an upper bound for G3 as G3 ≤ 2 exp ( 2Su√ 1 + Λ ) ∞∫ 0 (z + u)n−1 z n 2 −1 e −z2In 2 −1 ( 2Sz√ 1 + Λ ) dz. (5.15) We note that since n ∈ N, we can expand (z + u)n−1 using the binomial theorem, obtaining an integral solvable in closed form with the integral iden- tity [40, Eq. (6.643.2)] and [40, Eq. (9.220.2)], resulting in G3 ≤ ( S√ 1 + Λ )n 2 −1 exp ( 2Su√ 1 + Λ + S2 1 + Λ ) × n−1∑ i=0 ( n− 1 i ) ui1F1 ( i 2 ; n 2 ;− S 2 1 + Λ ) . (5.16) 63 The results in (5.13) and (5.16) allow us to find an upper bound on G1 as G1 ≤ 1 (1 + Λ) n 2 exp ( u2 + 2Su√ 1 + Λ − ΛS 2 1 + Λ )[ n∑ i=0 aiu i ] (5.17) ai =  ( n−1 i ) 1F1 ( i 2 ; n 2 ;− S2 1+Λ ) i = 0, . . . , n− 1 2 Γ(n2 ) exp ( −S2 1+Λ ) i = n . (5.18) Although we have found an upper bound for G1 which is in the desired closed form, we cannot partition (5.17) into a product of L independent functions. Since ui for i ∈ N, u ≥ 0 is a convex function we can obtain, via Jensen’s inequality, the following ( L∑ k=1 rk √ σ2kλ 2 k 2 √ 1 + ΛΩ2k )i ≤ L∑ k=1 λ2k Λ (1− λ2k) ( Λrk√ (1 + Λ) σ2kλ 2 k )i . (5.19) Eq. (5.19), when combined with the inequality 1 + ∑N k=1 bk ≤ ∏N k=1(1 + bk), bk ≥ 0, allows us to find a suitable upper bound for G1 as G1 ≤ exp (− Λ Λ+1 S2 ) (1 + Λ) n 2 L∏ k=1 exp ( Λ 1 + Λ r2k 2Ω2k + S 1 + Λ rk √ λ2kσ 2 k Ω2k ) × 1 + λ2k Λ (1− λ2k) n∑ i=1 ai ( Λrk√ 1 + Λ √ σ2kλ 2 k )i . (5.20) Finally, by substituting (5.20) into (5.9) we obtain the upper bound of joint 64 PDF of the generalized Rician distribution as fR(r1, . . . , rL) ≤ exp (− Λ Λ+1 S2 ) (1 + Λ) n 2 ΓL ( n 2 ) L∏ k=1 1 + λ2k Λ (1− λ2k) n−1∑ i=1 ai ( Λrk √ 1− λ2k√ 1 + Λ √ 2Ω2kλ 2 k )i × 2r n−1 k 2 n 2 Ωnk exp ( − 1 1 + Λ r2k 2Ω2k + S 1 + Λ rk √ λ2kσ 2 k Ω2k ) . (5.21) As in the case with the lower bound of the joint PDF, by letting λk = 0, k = 1, . . . , L, we can show the equality in (5.21) is obtained for the Nakagami-m fading with independent branches. 5.3 Error Bounds In this section we use the lower and upper bounds on the joint PDF in (5.5) and (5.21) respectively, to bound the error rate of an EGC combiner. As in Chapter 3, we perform error analysis on the equivalent channel at the output of the combiner hEGC = √ γEGC = 1 L ∑L k=1 Rk. An alternate equation for the average error probability can be found by applying Parseval’s theorem to (3.2) [45]: Pe = 1 pi ∞∫ 0 <{G∗ (ω) ΦhEGC (ω)} dω (5.22) where ΦhEGC (ω) is the CHF of hEGC and G (ω) is the Fourier transform of 65 the conditional error probability. A full list of G(ω) for various coherent and noncoherent modulations is given in [Table 1][45]. For example, G(ω) for BPSK is given by G (ω) = 1 2ω { 2√ pi F (ω 2 ) + j [ 1− exp ( −ω 2 4 )]} . (5.23) It is apparent that using upper or lower bounds on ΦhEGC (ω) in the integration of (5.22) results in upper or lower bounds on Pe as Pe ≥ Pe,L = 1 pi ∞∫ 0 <{G∗ (ω) ΦhEGC ,L (ω)} dω (5.24) Pe ≤ Pe,H = 1 pi ∞∫ 0 <{G∗ (ω) ΦhEGC ,H (ω)} dω (5.25) where ΦhEGC ,L (ω) and ΦhEGC ,H (ω) are the upper and lower bounds respec- tively on ΦhEGC (ω) found using (5.5) and (5.21). From the definition of the CHF we have ΦhEGC (ω) = ∞∫ 0 · · · ∞∫ 0︸ ︷︷ ︸ L e j ω√ L ∑L k=1 rkfR(r1, r2, . . . , rL)dr1dr2 · · · rL. (5.26) In both cases, substituting (5.5) or (5.21) into (5.26) to find ΦhEGC ,L (ω) or 66 ΦhEGC ,H (ω), respectively requires integral identity ∞∫ 0 xn+i−1 αn+iΓ(n) exp ( − x 2 2α2 + βx ) dx = (n)i D̃−n−i (−αβ) (5.27) which can be found from [40, Eq. (3.462.1)]. Here in (5.27), D̃−n (z) is the normalized parabolic cylinder function2 and can be expressed in terms of the confluent hypergeometric function using [40, Eq. (9.240)] D̃−n (z) = 1 2 n 2 [ √ pi Γ ( n+1 2 )1F1(n 2 ; 1 2 ; z2 2 ) − √ 2piz Γ ( n 2 ) 1F1(n+ 1 2 ; 3 2 ; z2 2 )] . (5.28) From (5.5) we obtain the lower bound of the characteristic function ΦhEGC ,L(ω) = exp (− Λ Λ+1 S2 ) (1 + Λ) n 2 ( Γ ( n+1 2 ) 2 n 2√ pi )L L∏ k=1 D̃−n (−jωΩk√ L ) (5.29) and from (5.21), we obtain the upper bound of the characteristic function ΦhEGC ,H(ω) = exp (− Λ Λ+1 S2 ) (1 + Λ)(1−L) n 2 ( 2 n 2 Γ ( n+1 2 ) √ pi )L L∏ k=1 [ D̃−n (−βk) + λ2k Λ (1− λ2k) n∑ i=1 ( Λ √ 1− λ2k√ λ2k )i ai (n)i 2 i 2 D̃−n−i (−βk) ] (5.30) where βk = S√ 1+Λ √ 2λ2k 1−λ2k + jωΩk √ 1+Λ L . We have made use of the Gauss duplication formula [40, Eq. (8.335.1)] in the derivation of (5.29) and (5.30). 2D̃p (z) is related to the parabolic cylinder function, Dp (z), in [40] by D̃p (z) = e− z2 4 Dp (z). We have elected to use this alternate notation for simplicity. 67 Both error bounds, (5.24) and (5.25), consist of a single integral of closed form functions with implementations in popular mathematical software. There- fore, they are similar in computational complexity to the exact analysis for independent branches found in [45] and far simpler than the exact results in [22] and [46] which use the generalized correlation model with λk = λ, k = 1, 2 . . . , L, i.e. equally correlated branches. 5.4 Numerical Results The upper and lower bounds we have developed, in addition to being asymp- totically tight for correlated fading, are exact for independent fading. Thus we expect the bounds to be tighter for weaker correlation among branches. We can see this is indeed the case by comparing Fig. 5.1 and Fig. 5.2 which plot Pe,L and Pe,H along with the asymptotic results found in Chapter 3 for Rayleigh fading channels with coherent BPSK and equal average branch SNRs. The numerical results in Fig. 5.1 and Fig. 5.2 have power correlation matrices, where the (k, i)th element is given by (5.1), of P1 =  1 0.518 0.656 0.518 1 0.518 0.656 0.518 1  (5.31) 68 and P2 =  1 2.25× 10−2 2.03× 10−1 2.25× 10−2 1 7.29× 10−2 2.03× 10−1 7.29× 10−2 1  (5.32) respectively. We observe that while both bounds approach simulation results asymptotically, the lower bound is a much tighter fit especially at the higher correlation of P2 in Fig. 5.1. Additional scenarios with Nakagami-m and Rician fading channels are shown in Fig. 5.3 and Fig. 5.4 with power correlation matrices P3 =  1 3.5× 10−3 1× 10−2 3.5× 10−3 1 2.25× 10−2 1× 10−2 2.25× 10−2 1  (5.33) and P4 =  1 0.174 0.237 0.174 1 0.138 0.237 0.138 1  (5.34) respectively. We can see in Figs. 5.1-5.4, that while both bounds approach simulation results asymptotically, the lower bound is a much tighter fit especially at higher correlation. This is to be expected given the extra inequalities needed to produce a suitable upper bound. In situations were the SER-SNR curve is concave, as is the case in Figs. 5.1-5.4, the asymptotic error rate approx- 69 imation underestimates the actual error rate at low SNR. Thus, we only require Pe,L to determine the validity of the asymptotic approximation. Us- ing just Pe,L, the asymptotic approximation approaches the exact error rate at around 24 dB and 20 dB for the Rayleigh fading scenarios in Fig. 5.1 and Fig. 5.2 respectively, at 22 dB for both the Nakagami-1.5 fading in Fig. 5.3 and Rician fading in Fig. 5.4. However, the SER-SNR curve is not always concave, as is the case for Rician fading with larger Rice factors as is shown in Fig. 3.3 in Chapter 3. In such cases, we will require Pe,H to determine at what SNR the asymptotic error rate adequately approximates the exact error rate. 70 0 5 10 15 20 25 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 B ER Average Branch SNR (dB)   Upper Bound Lower Bound Asymptotic Simulation Figure 5.1: Upper and lower BER bounds of coherent BPSK over Rayleigh fading channels with triple branch EGC and equal average branch SNR. λ = [0.9, 0.8, 0.9] and power correlation matrix P1. 71 0 5 10 15 20 25 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 B ER Average Branch SNR (dB)   Upper Bound Lower Bound Asymptotic Simulation Figure 5.2: Upper and lower BER bounds of coherent BPSK over Rayleigh fading channels with triple branch EGC and equal average branch SNR. λ = [0.5, 0.3, 0.8] and power correlation matrix P2. 72 0 2 4 6 8 10 12 14 16 18 20 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 B ER Average Branch SNR (dB)   Upper Bound Lower Bound Asymptotic Simulation Figure 5.3: Upper and lower BER bounds of coherent BPSK over Nakagami- 1.5 fading channels with triple branch EGC and equal average branch SNR. λ = [0.2, 0.3, 0.5] and power correlation matrix P3. 73 0 5 10 15 20 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 B ER Average Branch SNR (dB)   Upper Bound Lower Bound Asymptotic Simulation Figure 5.4: Upper and lower BER bounds of coherent BPSK over Rician fading channels (S = 2) with triple branch EGC and equal average branch SNR. λ = [0.6,−0.4, 0.5] and power correlation matrix P4. 74 Chapter 6 Conclusion 6.1 Summary of Results In this thesis we derived methods for improving and/or bounding the asymp- totic error rates of SC and EGC over correlated generalized Rician fading channels. We first derived asymptotic error rates for SC, EGC, and MRC and compared their relative performance. The asymptotic results for SC were then extended into an exact infinite series for the SNR distributions and er- ror rate for both coherent and noncoherent modulations. We proved that the series expansions for the SNR distributions converge unconditionally, and the error rate series expansion converges provided that the average SNR is suf- ficiently high. For all series, we provided truncation error bounds when the series is terminated at a finite number of terms. As the first term of the series expansions are the asymptotic results, this also provides an asymptotically tight bounds on the asymptotic results. Furthermore, we developed asymp- totically tight bounds on the joint PDF of correlated generalized Rician RVs. Using this result, we then derived corresponding upper and lower bounds on the CHF of the root of the combined SNR and error rates for the EGC 75 combiner. The asymptotically tight bounds we have derived can be used to predict at what SNR value the asymptotic error rate can adequately approx- imate the exact error rate. This avoids time consuming multilevel numeric integration or Monte Carlo simulations that would otherwise be necessary. All our results have been verified by Monte Carlo simulations. 6.2 Future Work The following is a list of possible future research opportunities: • The Laguerre polynomial expansion of the Marcum Q-function could possibly be used to develop an infinite series expansion of the error rate of the EGC combiner. • It may be feasible to extend the results for SC in Chapter 4 to the more general H-S/MRC. Using the joint PDF of generalized Rician RVs, the SNR MGF of an MRC can be found in closed form. With results for SC and MRC available, it is possible that similar results for H-S/MRC could be derived. • Investigate other fading models. 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New York, NY: Cambridge University Press, 2010. 83 [44] A. Baricz and Y. Sun, “New bounds for the generalized Marcum Q- function,” IEEE Transactions on Information Theory, vol. 55, pp. 3091– 3100, Jul. 2009. [45] A. Annamalai, C. Tellambura, and V. Bhargava, “Equal-gain diversity receiver performance in wireless channels,” IEEE Transactions on Com- munications, vol. 48, pp. 1732–1745, Oct. 2000. [46] Y. Chen and C. Tellambura, “Performance of L-branch diversity com- biners in equally correlated Rician fading channels,” in Proc. IEEE GLOBECOM 2004, vol. 5, Dallas, Tx, Nov. 29-Dec. 3 2004, pp. 3379– 3383. 84 Appendix A Derivation of the Correlation Coefficient ρGkGi The correlation coefficient by definition between Gk and Gi is ρGkGi , E [GkGi]− E [Gk] E [Gi]√ Var[Gk]Var[Gi] . (A.1) Since Gk = ∑n l=1X 2 kl, the term in numerator can be written as E [GkGi]− E [Gk] E [Gi] = n∑ l=1 n∑ j=1 E [ X2klXij ]− E [X2kl]E [X2ij] (A.2) = n∑ l=1 E [ X2klX 2 il ]− E [X2kl]E [X2il] (A.3) where we have used the fact that X2kl and X 2 ij are independent for l 6= j. By examining (2.17) we note that Xkl and Xil consist of a component which is independent for k 6= i, i.e. Ukl, and a common component, U0l. After substitution of (2.17) into (A.3) the independent components will cancel out 85 and the cross terms are zero since E [Ukl] = 0 for k ≥ 1, we are left with E [GkGi]− E [Gk] E [Gi] = σ2kσ2i λ2kλ2i n∑ l=1 E [ U40,l ]− E [U20,l]2 (A.4) = σ2kσ 2 i λ 2 kλ 2 i (n 2 + 2S2 ) (A.5) where S2 = ∑n l=1m 2 l . Since Gk is noncentral χ 2 distributed its variance is Var[Gk] = σ 4 k (n 2 + 2λ2kS 2 ) . (A.6) Upon substituting (A.5) and (A.6) into (A.1) we obtain ρGkGi = λ2kλ 2 i ( n 2 + 2S2)√ n 2 + 2λ2kS 2 √ n 2 + 2λ2iS 2 , k 6= i. (A.7) 86 Appendix B Derivation of the Integral Identity (4.7) Define I = ∞∫ 0 ti+ µ−1 2 e−αtIµ−1 ( 2β √ t ) dt (B.1) where i ∈ N and µ > 0, then from the integral identity [40, Eq. (6.643.2)] and [39, Eq. (13.1.32)] we have I = (µ)i ( β αi )µ 1F1 ( µ+ i;µ; β2 α ) . (B.2) Using Kummer’s transform 1F1(a; b; z) = e z 1F1(b − a; b;−z) [39], (B.2) be- comes I = (µ)i ( β αi )µ exp ( β2 α ) 1F1 ( −i;µ;−β 2 α ) . (B.3) Since i is a nonnegative integer, 1F1 ( −i;µ;−β2 α ) reduces to a generalized Laguerre polynomial [39, Eq. (13.6.9)] and we have I = i! ( β αi )µ exp ( β2 α ) L (µ−1) i ( −β 2 α ) . (B.4) 87

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