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Diversity PSK signals in impulsive noise and generalized fading Chaiyakul, Thanes 2003

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D I V E R S I T Y P S K S I G N A L S I N I M P U L S I V E N O I S E A N D G E N E R A L I Z E D F A D I N G by T H A N E S C H A I Y A K U L B.Eng., Kasetsart University, Bangkok, Thailand, 1998 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A September 2003 © Thanes Chaiyakul, 2003 I n p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e it f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t is u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f b l e c t r i c a l E n g i n e e r i n g T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r , C a n a d a D a t e September 2 5 , 2QQ3 Abstract Diversity combining techniques are well-known and useful to improve the performance of digital communication systems that experience fading. However, most research topics deal with the performance evaluation of modulation schemes in fading and A W G N chan-nels, with diversity reception. Relatively few research topics cope with the performance evaluation of diversity techniques in non-Gaussian noise and fading. Therefore, to broaden research in the area of diversity combining techniques in a more realistic noise model, this thesis deals in particular with the performance evaluation of diversity combining techniques in the presence of impulsive noise and fading. Several contributions are made to the system model of modulation schemes in impulsive noise and fading, with diversity reception. In the first part, the PDFs of the sum of impulsive noises are derived. Then, a system model without fading is considered and theoretical expressions of the error rate performance of modulation schemes in impulsive noise, with or without diversity reception, are derived. It is demonstrated that the diversity technique chosen affects the P D F of the sum of impulsive noises and makes the signal performance with diversity reception different from the signal performance without diversity reception. This is in contrast to the signal performance in the A W G N channel. In the second part, for the digital communication system in impulsive noise and fading, analytical expressions of the signal performance in impulsive noise and fading are derived and validated through simulation. The final part contains a system in impulsive noise and fading with diversity reception. Performance evaluations of the error rate performance of modulation schemes in impulsive i i Abstract i i i noise and fading, with diversity reception, are derived and compared with the previously derived results. A l l analytical expressions are validated through numerous simulation results. Contents Abstract ii Contents iv List of Figures vii List of Abbreviations xiii Acknowledgements xv 1. Introduction 1 1.1 P S K Signals . . . 1 1.2 Non-Gaussian Noise Models 1 1.3 Fading Models 3 1.4 Diversity Combining Techniques 4 1.5 Research Objective and Thesis Organization 5 2. System Impulsive Noise and Simulation Modelling 7 2.1 Introduction 7 2.2 System Model Description 7 2.3 Impulsive Noise Modelling 9 2.4 Computer Simulation Methodology 19 2.5 Conclusions 20 3. Performance in Impulsive Noise with Diversity Reception 21 3.1 Introduction 21 3.2 B P S K without Diversity 22 3.3 B P S K with Dual-Branch E G C Diversity 24 iv Contents v 3.4 M P S K without Diversity 28 3.5 M P S K with Dual-Branch E G C Diversity 32 3.6 Conclusions 37 4. Performance in Fading and Impulsive Noise 38 4.1 Introduction 38 4.2 Rayleigh Fading 38 4.3 Rician Fading 41 4.4 Nakagami Fading 45 4.5 Conclusions 48 5. Performance of Diversity Combining Techniques 49 5.1 Introduction 49 5.2 Selection Combining (SC) 50 5.2.1 P D F with Rayleigh Fading 50 5.2.2 Performance in Rayleigh Fading 51 5.2.3 P D F with Rician Fading 55 5.2.4 Performance in Rician Fading 57 5.2.5 P D F with Nakagami Fading 66 5.2.6 Performance in Nakagami Fading 67 5.3 Equal-Gain Combining (EGC) 72 5.3.1 C H F with Rayleigh Fading 74 5.3.2 Performance in Rayleigh Fading . 76 5.3.3 C H F with Rician Fading 79 5.3.4 Performance in Rician Fading 81 Contents vi 5.3.5 C H F with Nakagami Fading 86 5.3.6 Performance in Nakagami Fading 88 5.4 SC and E G C Performance Comparisons 93 5.5 Conclusions 99 6. Conclusions and Suggestions for Future Research 101 6.1 Conclusions 101 6.2 Suggestions for Future Research 101 6.2.1 M P S K or M Q A M with Diversity Combining Techniques 102 6.2.2 M P S K or M Q A M in Correlated Fading with Diversity Combining Techniques 102 6.2.3 Performance of Diversity Combining Techniques with Coding . . . 102 References 104 A. Theoretical Derivations 110 A . l Derivation of Equation 3.10 110 A.2 Derivation of Equation 4.6 I l l List of Figures Figure 1.1 The signal space of (a) B P S K , (b) QPSK and (c) 8PSK 1 Figure 2.1 System model of P S K systems in fading and impulsive noise with dual-branch diversity reception 8 Figure 2.2 P D F of the impulsive noise envelope rji for T' = 10~4 and different values of A 16 Figure 2.3 P D F of the impulsive noise envelope rji for A = 0.01 and different values of T' 16 Figure 2.4 P D F of the in-phase amplitude component for V = 10~3 and differ-ent values of A 17 Figure 2.5 P D F of the in-phase amplitude component for A = 0.01 and different values of T' 17 Figure 2.6 Comparison between the P D F of the envelope of impulsive noise Prfiivi) and the P D F of the normalized envelope of the sum of two impulsive noises Pijiv) for T' = 10~4 and different values of A. . . . 18 Figure 2.7 Comparison between the P D F of the in-phase amplitude component Prjn {Vu) and the P D F of the normalized in-phase amplitude compo-nent of the sum of two impulsive noises Pri,(f}i) for V = 10~3 and different values of A 18 Figure 3.1 Performance of B P S K in impulsive noise for T' = 10~4 and different values of A without diversity reception 24 vii List of Figures v i i i F i g u r e 3.2 Performance of B P S K with dual-branch E G C diversity in impulsive noise for T' = 1CT4 and different values of A (A = Ax = A2) 26 F i g u r e 3.3 Performance of B P S K with dual-branch E G C diversity in impulsive noise for V = 1 0 - 4 and different values of A in each channel (Ai = 10, A2 = 0.01) . 27 F i g u r e 3.4 Comparison of the performance of B P S K with dual-branch E G C and without diversity reception in impulsive noise for T' = 10~4 and different values of A (A = Ax = A2) 27 F i g u r e 3.5 A geometrical representation of one of M P S K signals at point S with the envelope \fE~s and phase n added with impulsive noise at point Z with the envelope rjmini and phase 6^ to form the received signal with the envelope R and phase 7r — \I>. The line is the decision boundary of the signal with the phase TT 28 F i g u r e 3.6 Performance of M P S K without diversity in impulsive noise for T' = IO" 4 and A = 10 30 F i g u r e 3.7 Same caption as in Figure 3.6 but with A = 1. 31 F i g u r e 3.8 Same caption as in Figure 3.6 but with A = 0.35 31 F i g u r e 3.9 Same caption as in Figure 3.6 but with A — 0.01 32 s List of Figures ix F igure 3.10 A geometrical representation of the sum of two M P S K signals at point S with the total envelope 2\fEs and phase TT added with the sum of two impulsive noises at point Z with the total envelope r\min2 and phase 9V to form the received signal with the envelope R and phase ?r — \&. The line is the decision boundary of the signal with the phase ir 33 F igure 3.11 Performance of M P S K with dual-branch E G C diversity in impulsive noise for T' = 10~4 and A = 10 35 Figure 3.12 Same caption as in Figure 3.11 but with A — 1 35 Figure 3.13 Same caption as in Figure 3.11 but with A = 0.35 36 Figure 3.14 Same caption as in Figure 3.11 but with A = 0.01 36 Figure 4.1 Performance of B P S K in Rayleigh fading and impulsive noise for T' = 10~4 and different values of A 40 Figure 4.2 Performance of B P S K in Rician fading for different values of K and impulsive noise for T' = 10~4 and A = 10 43 Figure 4.3 Similar caption to Figure 4.2 but with A = 1 44 Figure 4.4 Similar caption to Figure 4.2 but with A = 0.35 44 Figure 4.5 Similar caption to Figure 4.2 but with A = 0.01 45 Figure 4.6 Performance of B P S K in Nakagami fading for m = 1, 2 and impulsive noise for T' = 1 0 - 4 and different values of A 47 Figure 5.1 Performance of B P S K in Rayleigh fading and impulsive noise for T' = 1 0 - 4 and A — 10 with SC diversity reception for different values of L 53 List of Figures X Figure 5.2 Same caption as in Figure 5.1 but with A = 1 53 Figure 5.3 Same caption as in Figure 5.1 but with A = 0.35 54 F igure 5.4 Same caption as in Figure 5.1 but with A = 0.01 54 Figure 5.5 Performance of B P S K in Rician fading for different values of K and impulsive noise for V = 10~4 and A = 10 with SC diversity reception for L = 2 60 Figure 5.6 Similar caption to Figure 5.5 but with L = 3 60 Figure 5.7 Similar caption to Figure 5.5 but with L = 4 61 Figure 5.8 Performance of B P S K in Rician fading for different values of K and impulsive noise for T' = 10~4 and A = 1 with SC diversity reception for L = 2 61 Figure 5.9 Same caption as in Figure 5.8 but with L = 3 62 Figure 5.10 Same caption as in Figure 5.8 but with L — 4 62 Figure 5.11 Performance of B P S K in Rician fading for different values of K and impulsive noise for V = I O - 4 and A = 0.35 with SC diversity recep-tion for L = 2 63 Figure 5.12 Similar caption to Figure 5.11 but with L = 3 63 Figure 5.13 Similar caption to Figure 5.11 but with L = 4 64 Figure 5.14 Performance of B P S K in Rician fading for different values of K and impulsive noise for V — 10~4 and A = 0.01 with SC diversity recep-tion for L = 2 64 Figure 5.15 Same caption as in Figure 5.14 but with L = 3 65 Figure 5.16 Same caption as in Figure 5.14 but with L = 4 65 List of Figures x i Figure 5.17 Performance of B P S K in Nakagami fading for rn = 2 and impulsive noise for V = 10-4 and A = 10 with SC diversity reception for different values of L 70 Figure 5.18 Same caption as in Figure 5.17 but with A = 1 70 Figure 5.19 Same caption as in Figure 5.17 but with A = 0.35 71 Figure 5.20 Same caption as in Figure 5.17 but with A — 0.01 71 Figure 5.21 Performance of B P S K in Rayleigh fading and impulsive noise for r' = 10-4 and A = 10 with dual-branch E G C diversity reception. . 77 Figure 5.22 Same caption as in Figure 5.21 but with A = 1 77 Figure 5.23 Same caption as in Figure 5.21 but with A — 0.35 78 Figure 5.24 Same caption as in Figure 5.21 but with A = 0.01 78 Figure 5.25 Performance of B P S K in Rician fading for different values of K and impulsive noise for V — 10-4 and A = 10 with dual-branch E G C diversity reception 84 Figure 5.26 Similar caption to Figure 5.25 but with A = 1 84 Figure 5.27 Similar caption to Figure 5.25 but with A = 0.35 85 Figure 5.28 Similar caption to Figure 5.25 but with A = 0.01 85 Figure 5.29 Performance of B P S K in Nakagami fading for different values of m and impulsive noise for V = 10~4 and A = 10 with dual-branch E G C diversity reception 91 Figure 5.30 Same caption as in Figure 5.29 but with 4^ = 1 91 Figure 5.31 Same caption as in Figure 5.29 but with A = 0.35 92 Figure 5.32 Same caption as in Figure 5.29 but with A = 0.01 92 List of Figures x i i F i g u r e 5.33 Comparison between the performance of B P S K with dual-branch SC and E G C diversity reception in Rayleigh fading and impulsive noise for T' = 10~4 and A = 10 93 F i g u r e 5.34 Similar caption to Figure 5.33 but with . 4=1 94 F i g u r e 5.35 Similar caption to Figure 5.33 but with A = 0.35 94 F i g u r e 5.36 Similar caption to Figure 5.33 but with A = 0.01 95 F i g u r e 5.37 Comparison between the performance of B P S K with dual-branch SC and E G C diversity reception in Rician fading for different values of K and impulsive noise for F ' = 1 0 - 4 and A = 10 95 F i g u r e 5.38 Same caption as in Figure 5.37 but with A = 1 96 F i g u r e 5.39 Same caption as in Figure 5.37 but with A = 0.35. 96 F i g u r e 5.40 Same caption as in Figure 5.37 but with A = 0.01 97 F i g u r e 5.41 Comparison between the performance of B P S K with dual-branch SC and E G C diversity reception in Nakagami fading for m = 2 and impulsive noise for V = 10~4 and A = 10 97 F i g u r e 5.42 Similar caption to Figure 5.41 but with A = 1 98 F i g u r e 5.43 Similar caption to Figure 5.41 but with A = 0.35 98 F i g u r e 5.44 Similar caption to Figure 5.41 but with A = 0.01 99 List of Abbreviations 8PSK 8-ary Phase Shift Keying A W G N Additive White Gaussian Noise B E R Bit Error Rate B P S K Binary Phase Shift Keying C B F S K Coherent Binary Frequency Shift Keying C D F Cumulative Distribution Function C H F Characteristic Function D P S K Differential Phase Shift Keying E G C Equal-Gain Combining E M Electromagnetic fad fading \) I In-Phase ImpA Class A Impulsive Noise ISM Industrial, Scientific and Medical M E D S Method of Exact Doppler Spread M P S K M-ary Phase Shift Keying M Q A M M-ary Quadrature Amplitude Modulation Nak Nakagami N C B F S K Non-Coherent Binary Frequency Shift Keying P C Personal Computer P D F Probability Density Function xiii List of Abbreviations xiv PSD Power Spectral Density P S K Phase Shift Keying Q Quadrature Q A M Quadrature Amplitude Modulation Q P S K Quadrature Phase Shift Keying Ray Rayleigh SC Selection Combining Sim Simulation SNR Signal-to-Noise Ratio Acknowledgements I would like to take this opportunity to express my greatest gratitude towards the following people. My father, Phairuch, my mother, Anong, and my brother, Mark, for their continuous moral support and constant encouragement. M y supervisor, Dr. P. Takis Mathiopoulos, for providing continuous encouragement, technical knowledge, very useful comments and suggestions for improvement of the thesis. My supervisor's graduate stu-dent, Mr. Cyril-Daniel Iskander, for his numerous technical supports in both theoretical derivations and computer simulations. Also, another of my supervisor's graduate students, Mr. X u Zhang, for his technical supports in computer simulations. Without their support, the completion of this thesis would not have been possible. xv C H A P T E R 1 I n t r o d u c t i o n 1.1 P S K Signals A commonly used technique for digital signal transmission is the Phase Shift Key-ing (PSK) technique [44, p. 349]. When using the PSK technique, signal amplitudes are the same, but signal phases are different, with equal spacing between phases. The general form of the PSK signals with M different phases is called M-ary Phase Shift Key-ing (MPSK) . When M = 2,4,8, the P S K signals are called Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK) and 8-ary Phase Shift Keying (8PSK), respectively. Figure 1.1 shows the geometrical representation of these P S K signals. Q w 0 W (a) Figure Q (b)  1.1: The signal space of (a) BPSK, (b) QPSK and (c) 8 P S K . < • Q • • 0 • • • < > (c) The signal information lies within these phases. Thus, the more phases the P S K signals have, the more binary information the P S K signals can represent. However, this causes a lower resistance of the PSK signals to noise. 1.2 Non-Gaussian Noise Models The simplest and most frequently used additive noise model in digital transmission is 1 1.2 Non-Gaussian Noise Models 2 Gaussian noise [23, Chapter 5]. Other types of noise are usually referred to as non-Gaussian noise. There are many types of non-Gaussian noise models [17] such as the alpha-stable process [37],[42] and mixture process [21]. [34] shows that the electrical ignition circuits of vehicles have produced non-Gaussian noise that interferes with communication systems. This study also shows that the noise intensity is higher in urban areas that have more traffic, than in rural areas. Aside from vehicular interference, [35] shows that microwave ovens also generate non-Gaussian noise that interferes with signals at high frequency band, such as the 2.45 GHz used in the Industrial, Scientific and Medical (ISM) band. More studies about other non-Gaussian noise environments can be found in [8],[50]. Thus, non-Gaussian noise represents man-made atmospheric noise, for example microwave ovens and vehicular electrical ignition circuit interferences, and natural noise, such as thunder and ice-breaking interferences. Some of these non-Gaussian types of noise have burst characteristics, and are thus usually called impulsive noise [12]. One of the most acceptable non-Gaussian noise models thus chosen to be studied in this thesis is the so-called class A impulsive noise model, introduced by Middleton [12] —[16]. The class A impulsive noise model is studied in many papers [29],[31],[46] —[49]. The analytical expressions of the performance of signals in class A impulsive noise models are derived in [10],[11],[27],[28],[30],[41],[43],[45]. In a recent paper by Middleton discussing his impulsive noise model [16], the class A impulsive noise model is compared with the alpha-stable noise model. The class A noise model is superior to the alpha-stable model, because it includes an additive Gaussian background component. Table I of [16] gives some examples of interferences that can be modelled as the class A impulsive noise model. Considering the acceptable and precise noise model for both man-made and natural inter-1.3 Fading Models 3 ference, it is worth studying how the digital communication systems perform in class A impulsive noise environments. It is well-known that when a signal propagates through a wireless channel, it is reflected and scattered, and arrives at the receiver with a slight time differential [23, p. 800]. The received signal is a combined version of these reflected and scattered signals. The received signal's envelope and phase fluctuate over time. This phenomenon is called fading [51],[54]. In this thesis, fading is modelled by using various types of statistical envelope probabilities, while the fluctuation of the phase of the received signal is assumed to be compensated for by a perfectly coherent detection system. The fading models used in this thesis are the Rayleigh, Rician and Nakagami fading. The Rayleigh fading is used to model multipath fading that doesn't have a line-of-sight path. The probability density function (PDF) of the Rayleigh fading envelope r is well-known to be given by [23, p. 44] where r > 0 and r2 = 2a 2 where (.) denotes an average of (.). The Rician fading is used to model the multipath fading that has a line-of-sight path and the P D F of the Rician fading 1.3 Fading Models (1.1) envelope r can be expressed as [23, p. 46] (1.2) 1.4 Diversity Combining Techniques 4 where r > 0, Ar is the specular amplitude, a2 is the Gaussian's variance and IQ(-) is the zero-order modified Bessel function of the first kind. For the special case of the Rician P D F with Ar = 0, Equation 1.2 simplifies to Equation 1.1, which is the Rayleigh P D F . The Nakagami-m fading, also used to model the multipath fading channel that has no line-of-sight path, is more general than the Rayleigh fading model, because it has the parameter m to control the severity of the fading. The smaller the value of m is, the more severe the Nakagami fading becomes. For the P D F of the Nakagami fading envelope r, it is known where r > 0, 7 = r 2 and 0.5 < m < 0 0 . When the parameter m = 1, Equation 1.3 reduces to Equation 1.1 that is the Rayleigh P D F . These fading channels degrade the signal performance substantially [23, Chapter 14] depending on the type of fading. For example, the Rician fading channel with Ar > 0 does not degrade the signal performance as much as the Rayleigh fading channel, because of the line-of-sight path. 1.4 Diversity Combining Techniques One of the most widely used techniques to combat fading is the diversity technique [22, Chapter 9]. By employing a combination of antennas (i.e? two or more), the received diversity signal usually gives a higher signal-to-noise ratio (SNR) than from one received signal alone. The diversity combining techniques used in this thesis are the Selection Combining that [23, p. 47] (1.3) 1.5 Research Objective and Thesis Organization 5 (SC) [4],[7],[18],[28],[32],[41], and Equal-Gain Combining (EGC) [1]-[3],[5],[33],[38],[39] techniques. These two techniques give very good performance improvement and are easily implemented in the receiver hardware [51, Chapter 6]. In the SC technique, the receiver chooses the instantaneous signal from an antenna that has the highest fading envelope and discards the rest of the received signals from the other antennas. In the E G C technique, the receiver combines all received signals and makes a decision from this combined signal. In a case where there are two receiver antennas in the E G C diversity, the system is called the dual-branch (L = 2) E G C diversity. 1.5 Research Objective and Thesis Organization In the past, there have been relatively few publications dealing with the subject of the performance evaluation of digital communication systems in the presence of class A impulsive noise (see Section 3.1) and fading (see Section 4.1) with diversity reception (see Section 5.1). Furthermore, to the best of our knowledge, the subject of diversity reception over impulsive channels has not been investigated in the open technical literature. Motivated by these observations, in this thesis we make the following contributions: 1) theoretical expressions of the error rate performance of P S K signals in class A im-pulsive noise, with or without diversity reception; 2) analytical expressions of the performance of B P S K in class A impulsive noise and different classes of fading; 3) performance evaluations of B P S K in class A impulsive noise and different kinds of fading, with the SC or E G C diversity reception. 1.5 Research Objective and Thesis Organization 6 Including this chapter, this thesis is composed of six chapters and an appendix. After this introductory chapter, the organization of this thesis is as follows. Chapter 2 introduces the system model description and computer simulation method-ology of the P S K signals in class A impulsive noise and fading, with diversity reception. Furthermore, the PDFs of class A impulsive noise in the SC and E G C diversity reception are investigated and derived. Chapter 3 is based on the previously derived PDFs, and consists of derivations of analytical expressions of the error rate performance of B P S K and M P S K in impulsive noise with or without diversity reception. Chapter 4 considers the effect of the fading channel, and comprises derivations of the performance of B P S K in the class A impulsive noise and the Rayleigh, Rician or Nakagami fading. Chapter 5 is composed of derivations for the expressions of the performance evaluation of B P S K in the class A impulsive noise and different classes of fading channels with the SC or E G C diversity reception. Chapter 6 concludes this thesis, and suggests future work. Finally, in Appendix A, some useful theoretical derivations are given. C H A P T E R 2 S y s t e m I m p u l s i v e N o i s e a n d S i m u l a t i o n M o d e l l i n g 2.1 Introduction This chapter presents the system design and simulation model of a PSK system assum-ing diversity combining reception with fading and impulsive noise. Furthermore, impulsive noise modelling in the receiver without (L = 1) or with dual-branch (L = 2) E G C diversity reception is investigated and novel expressions of the P D F of the sum of two impulsive noises are derived. The organization of this chapter is as follows. Section 2.2 presents the P S K system model in fading and impulsive noise with the dual-branch diversity re-ception. In Section 2.3, the impulsive noise in the receiver without and with dual-branch E G C diversity reception is modelled and the results are evaluated. Section 2.4 presents the employed computer simulation methodology. Conclusions can be found in Section 2.5. 2.2 System Model Description The block diagram of the system model of PSK systems in dual-branch diversity combining techniques with fading and impulsive noise is illustrated in Figure 2.1. It consists of an M P S K encoder (a binary to phase converter), multiplicative fading and additive impulsive noise channels, a diversity combining device, a phase detector, a decision device and an M P S K decoder (a phase to binary converter). The M P S K encoder converts the 7 2 . 2 System Mode l Description 8 binary sequences ak into transmitted M P S K signal Sk that can be expressed as Sk = ^Es exp(j9s) (2.1) where Eg = Energy/symbol, 9S = 2(ms — l)n/M and ms = 1, 2, • • •, M. The signal Sk is passed through two independent multiplicative fading and additive impulsive noise channels. Each channel multiplies the signal Sk with the fading envelope (i = 1, 2)1, and adds the signal Sk with impulsive noise r\i exp(j^?7j). Then, each received signal R{ exp(j9fu) enters the diversity combining device. In the SC system, the diversity combining device selects one of the received signals with the largest fading envelope to be the output. In the E G C system, the diversity combining device is merely a combiner, with its output being the sum of the two received signals. This output signal Rexp(j9R) is passed through the phase detector. The phase 9R is passed through the decision device to determine and regenerate the signal Sk- The regenerated signal Sk is passed through the M P S K decoder to convert the phase to the binary sequences a^. Transmitter I Channel 1 Receiver ~1 f n r ? ! ^ ! " ! M P S K Encoder Sk I I Channel 2 I I Diversity Combin-ing Device R 0R Sk Phase Decision M P S K 1 o,kt Detector Device Decoder J Figure 2.1: System model of PSK systems in fading and impulsive noise with dual-branch diversity reception. The noise in the SC diversity receiver is composed of one impulsive noise, no matter 1From now on i = 1, 2, unless otherwise is stated. 2.3 Impulsive Noise Modelling 9 how many branches of diversity reception are used. Therefore, the performance evaluation of SC diversity can be studied up to L branches of diversity. The results are investigated only up to L = 4, because of the difficulty in obtaining the theoretical and simulation results for L > 4. On the other hand, the noise in the E G C diversity receiver comprises the sum of impulsive noises from all channels. Thus, the P D F of impulsive noise in the E G C diversity reception must be numerically calculated for each total number of diversity. In this thesis, only the performance evaluation of dual-branch E G C diversity is studied, because the highest diversity gain is obtained when the diversity branch increases from L=l to 2, and only the P D F of the sum of two impulsive noises is derived. The P D F of one impulsive noise and the P D F of the sum of two impulsive noises are investigated in the next section. 2.3 Impulsive Noise Modelling Middleton's class A impulsive noise represents electromagnetic (EM) interference that has bandwidth comparable to or narrower than the receiver. The P D F of the impulsive noise envelope rji given by Equation 84 of [16] is expressed as where rji > 0, o2m. = (rrij/Aj-rT-)/(l + T-), Ai is impulsive index and is the Gaussian fac-tor. Ai is the product of the received average number of impulses per unit time, multiplied by the mean duration of the impulses. The value of Ai measures how non-Gaussian the noise is. On the other hand, when the value of Ai is small (i.e. Ai <C 10), the noise events (2.2) 2.3 Impulsive Noise Modell ing 10 and durations are less, and the noise becomes more impulsive or highly non-Gaussian. On the other hand, when the value of A is large (e.g. A% > 10), the noise tends to be more Gaussian. For instance, Ai = 10 means that non-Gaussian noise acts similarly to Gaussian noise. In contrast, Ai <C 10, such as 1, 0.35 and 0.01, means that the non-Gaussian noise is more impulsive. T- = O-Q/Q2A is the mean power ratio of the Gaussian noise compo-nent o2G to the non-Gaussian noise component Q 2 A - The. lower the value T- is, the more non-Gaussian the noise becomes. The P D F of the impulsive noise phase #m- is uniformly distributed in (0,2?r] and given by PeJ0Vr) = ^ - (2-3) Using Equations 2.2 and 2.3 leads to the joint P D F between the impulsive noise envelope rji and the impulsive noise phase 6n\ Pm.ejrh, M = e x p ( - ^ ) £ exp ( - J ? U . (2.4) This joint P D F of the impulsive noise is used in Chapter 3 to derive the performance of M P S K in impulsive noise without E G C diversity reception. The P D F of the in-phase (J) component of impulsive noise rju = Re{r/i exp(j^i)}, given by Equation 7 of [10], can be expressed as ~ A™1 i / \ M = e x p ( - ^ ) £ - A . - — ^ exp ( - ^ j 0 0 Ami IT7' = e x p ( - ^ ) exp (-Grfu) (2.5) m i = 0 2.3 Impulsive Noise Modell ing 11 where G\ = 1/(2(7^) and —co < rju < co. This P D F of the /-component of impulsive noise is used in Chapter 3 to derive an analytical expression of the error rate performance of B P S K in impulsive noise without E G C diversity reception. It is interesting to note that this P D F of the /-component of impulsive noise is intentionally modelled on a one-dimensional scheme impulsive noise only. This P D F cannot be used to find the P D F of an envelope, which is a two-dimensional scheme noise, of impulsive noise, because the in-phase (/) and quadrature (Q) components of impulsive noise are dependent. In a dual-branch E G C diversity system, the impulsive noise in the receiver is the sum of two impulsive noises and can be expressed as where rj is the envelope of the sum of two impulsive noises and 9V is the phase of the sum of two impulsive noises. As shown in [6], the phase dn of the sum of random vectors is still uniformly distributed in (0,2TT] so that its P D F is given by Using Equations 9 and 10 of [6] leads to the P D F of the envelope of the sum of two impulsive noises 77 e x p ( j ^ ) = 771 e x p ( j ' ^ i ) + r)2 e x p ( j ^ 2 ) (2.6) (2.7) (2.8) where 2 A ( p ) = E, Y[ MviP) =  E m IMVIP)]  Ev2 IMmp)] (2.9) 2.3 Impulsive Noise Modell ing 12 and 771, r]2 are independent. The term EVi [Jo(r}ip)] can be rewritten as /oo MViPlPruiVi) drh •00 0 0 Ami f°° = e x p ( - A ) \ ViMVip) exp ( - G ^ ? ) d/fc (2.10) n TYli- Jo m,=0 where Gj — l / c ^ , . Applying Equation 6.631.1 of [19, p. 698] leads to the following equation K [MrhP)] = e x p ( - A ) E ^ T e x P ( ITT • ( 2 - U ) ^ Ami /_n2 mi=0 Substituting the above equation into Equation 2.9 we can obtain A(p) = « p [ - ( * + A,)] E E ^ e x p (2,2) m i = 0 7712=0 where G/ = G1G2/(G1 + G 2 ) = l / ^ c r ^ + cr^2)] and thus, Equation 2.8 can be rewritten as ^ ) = exp[-(A 1 + 4 2)] £ E ^ j - ^ / p J o M exp ( ^ - J dp. (2.13) m i = 0 m 2 = 0 • • 0 By using Equation 6.631.1 of [19, p. 698], the above equation yields the P D F of the envelope of the sum of two impulsive noises oo oo Ami A1712 P v ( V ) = e x p [ - ( A + A2)\ £ £ ^ M ^ 2 r ] C l e x ^ C ^ ^ mi=0 m2=0 Using Equations 2.7 and 2.14 leads to the joint P D F between the envelope rj and phase 9V 2.3 Impulsive Noise Modelling 13 of the sum of two impulsive noises 0 0 0 0 Ami Am.2 ^(-t P^(rj,6v) = e x p [ - ( ^ + A,)] £ E e x p ^ C , ) . (2.15) mi!m 2 ! m i = 0 m 2 = 0 This joint P D F is used in Chapter 3 to derive an analytical expression of the performance of M P S K in impulsive noise with dual-branch E G C diversity reception. Furthermore, the P D F of the /-component of the sum of two impulsive noises 77/ = Re{77exp(j0,,)} = Rejryx e x p ^ i ) } + Re{?72 exp(j^ 2 )} = Vu + V21 can be expressed as /o o PmimiiVi ~ x,x)dx •00 0 0 o ° Am\ Am2 si = e x P [ - ( ^ + A 2)] £ E 1 /V e x p C - G ^ ? ) ^ — ' m i ! m 2 ! TT 7711=0 TO2=0 Z /o o exp [-((?! + G2)x2 + 2/7/xGj dx •00 0 0 0 0 / l m i / J m 2 /7s*-= e x p [ - ( ^ + A,)) E E ^ Z , ^ « P (-Cfg) • (2.16) 7771=0 7772=0 i - 2 ' V The above P D F is used in Chapter 3 to derive an analytical expression of the error rate performance of B P S K in impulsive noise with dual-branch E G C diversity reception. Figures 2.2 and 2.3 illustrate the P D F of the impulsive noise envelope (see Equation 2.2) for different values of V and A.2 It can be seen that, when values of V or A are high, i.e. F' > 100 or A > 10, the P D F of the impulsive noise envelope is closed to the P D F of Rayleigh. On the other hand, when values of F' or A are low, i.e. F' <C 100 or A <C 10, the impulsive noise becomes more impulsive and has a long P D F tail. The values of F' = 10~4 — 1 0 - 3 are frequently used to represent the highly non-Gaussian noise 2 From now on V = T[ — T'2 and A = A\ = A2, unless r't and Ai are assigned specifically. 2.3 Impulsive Noise Modell ing 14 characteristic (e.g. [10]). Simulation results, which are obtained by means of computer simulation (see the next section), are seen to very well match the theoretical curves. Figures 2.4 and 2.5 show the P D F of the /-component of impulsive noise (see Equation r 2.5) for different values of T' and A. As it can be seen from the plots, similar remarks to Figures 2.2 and 2.3 can be made so that when the values of V or A are high, the P D F of the /-component of impulsive noise is very similar to the P D F of Gaussian. On the other hand, when the values of T' or A are low, the impulsive noise becomes more impulsive and has a long P D F tail. Simulation results obtained by computer simulation can be seen to match the theoretical curves very well. In order to see how the dual-branch E G C reception influences the impulsive noise, the P D F of the envelope of the sum of two impulsive noises (see Equation 2.14) is compared with the P D F of one impulsive noise envelope (see Equation 2.2). However, the P D F of the sum of two impulsive noises cannot be directly compared with the P D F of one impulsive noise, because the power of the sum of two impulsive noises is twice as much as the power of one impulsive noise. Thus, the envelope of the sum of two impulsive noises 77 is needed to be normalized and the P D F of the normalized envelope of the sum of two impulsive noises 77 = 77/\/2 is obtained by 00 00 . m i . m 2 ^(77) = V2pv (V2fj) = e x p H A + A2)} £ E - W 477 C7 exp(-277 2C 7). (2.17) m i = 0 m 2 = 0 1 z Figure 2.6 shows the P D F of the impulsive noise envelope (see Equation 2.2) versus the P D F of the normalized envelope of the sum of two impulsive noises (see Equation 2.17) for r ' = 10~4 and different values of A. It can be seen that, for the value of A = 10,7^(77) 2.3 Impulsive Noise Modell ing 15 is slightly lower than pVl{r]i) at high values of the envelope. On the other hand, Pfj(fj) is identical to pm(T}i) a t low values of the envelope. For low values of A (e.g. A <§C 10), Prjipi) is largely lower than pm(r]i) at high values of the envelope. On the other hand, Pfj(fj) is higher than pm(' r]i) at low values of the envelope. Thus, it can be concluded that the summation of impulsive noises reduces the P D F of their sum at high values of the envelope and increases the P D F of their sum at low values of the envelope. Similarly, in order to compare the P D F of the /-component of impulsive noise (see Equation 2.5) and the P D F of the /-component of the sum of two impulsive noises (see Equation 2.16), i]j needs to be normalized. The P D F of the normalized /-component of the sum of two impulsive noises 77} = rjj/y/2 can be expressed as 0 0 0 0 Ami Am2 IcpFT ^(m) = A / 2 ft,, ( V ^ , ) = exp[-(A1+A2)} £ Yl I 2 1 r exp(-277/2 C / ) . (2.18) — 777.1 !77lo! v 7T m i =0m 2 =0 1 ^ v Figure 2.7 compares the P D F of the /-component of impulsive noise pVl](vu) and the P D F of the normalized /-component of the sum of two impulsive noises Pfjj{fji) f ° r T' = 10~3 and different values of A. The plot illustrates that for the value of A = 10, Pfuivi) is slightly lower than PVu(Vu) at high values of the amplitude: On the other hand, p^ivi) is identical to pm,{Tjii) at low values of the amplitude. For low values of A (i.e. A <C 10), PfjiiVi) is largely lower than pVlI{vu) at high values of the amplitude. On the other hand, PrjiiVi) is higher than p,ni(r)u) at low values of the amplitude. Similar comments to those in Figure 2.6 can also be made to illustrate that the summation of impulsive noise reduces the P D F of the in-phase component of the sum of two impulsive noises at high values of the amplitude and increases the P D F of the in-phase component of the sum of two impulsive 2.3 Impulsive Noise Modell ing 16 0 5 10 „ 15 20 25 Figure 2.2: P D F of the impulsive noise envelope rji for V = 10~ 4 and different values of A. PmiVi) Figure 2.3: P D F of the impulsive noise envelope 771 for A = 0.01 and different values of V. 2.3 Impulsive Noise Modell ing 17 PvuiVu) Figure 2.4: P D F of the in-phase amplitude component for T' = 10 3 and different values of A. Figure 2.5: P D F of the in-phase amplitude component for A = 0.01 and different values of V 2.3 Impulsive Noise Modell ing 18 Figure 2.6: Comparison between the P D F of the envelope of impulsive noise pm(r)i) a n d t n e P D F o f t h e normalized envelope of the sum of two impulsive noises p^{n) for V = I O - 4 and different values of A. 10 10u P D F Gaussian a 2 •• Pvuimi) a—> A=10 b —> A=1 c —> A=0.35 d —> A=0.01 6 rjU or 77/ Figure 2.7: Comparison between the P D F of the in-phase amplitude component pVlI (771/) and the P D F of the normalized in-phase amplitude component of the sum of two impulsive noises Pfi,{f)i) for V = 10~3 and different values of-A. 2 . 4 Computer Simulation Methodology 19 noises at low values of the amplitude. It should be noted that in the case of Gaussian noise, the P D F of the normalized sum of two (or more) Gaussian noises is identical to the P D F of one Gaussian noise. 2.4 Computer Simulation Methodology This section presents the computer simulation methodology used to evaluate the per-formance of the P S K systems [24, Section 7.3.1] in impulsive noise and fading with diversity reception. The program codes for the simulation are written in C++ language, which can be run on both P C (i.e. Borland C++) and Unix. The simulation results are checked against the theoretical results on the same figure, drawn by using Matlab program. The basic idea behind the simulation of the performance evaluation is to generate the random variables, that have the required P D F , of impulsive noise and fading. In order to generate the correct random variable, the percentile transformation method [9, p. 226],[24, Section 2.2] is used, so any random variable can be generated with its cumulative distribution function (CDF). To generate the impulsive noise, the C D F of the class A impulsive noise envelope given by Equation 85 of [16] is used. However, this method requires the inverse of the CDF, which is difficult to derive. Thus, the trial-and-error method is used to evaluate the random variable with the percentile transformation method. For the fading simulator, the Rayleigh fading random variable is generated by us-ing the method of an exact doppler spread (MEDS) of [36, Section 5.1.6]. The random variables generated using this method have the Rayleigh P D F and Jake's power spectral density (PSD) [54, p. 19]. Then, the Rician fading random variable is generated from the 2.5 Conclusions 20 Rayleigh fading random variable added by a constant. The Nakagami fading random vari-able is generated by using the method in [52]. Furthermore, a simpler method of generating the fading random variable is to use the same method as the impulsive noise generation. This necessitates the writing of fewer program codes and less simulation run time. 2.5 Conclusions In this chapter, the PSK system model in fading and impulsive noise with diversity combining reception is presented in Section 2.2. In Section 2.3, the PDFs of impulsive noise with non-diversity reception are evaluated and verified with computer simulation results. Moreover, novel expressions of the P D F of the envelope and in-phase component of the sum of two impulsive noises are derived and evaluated. The results show that the sum of impulsive noises gives lower PDFs at high values of the envelope and in-phase amplitude component of impulsive noise, but it gives higher PDFs at low values of the envelope and in-phase amplitude component of impulsive noise. Finally, Section 2.4 presents the simulation methodology. C H A P T E R 3 Performance in Impulsive Noise with Diversity Reception 3.1 Introduction For over two decades, since Middleton introduced the non-Gaussian noise model [12], relatively few researchers have published work on performance evaluations of various mod-ulation schemes in impulsive noise. For instance, Spaulding et al. present the performance evaluations of B P S K , C B F S K [10] and N C B F S K [11] in impulsive noise. Then, Seo et al. [27] derive the analytical expression of the performance of M Q A M in class A impulsive noise by considering the independence of the I and Q components. On the other hand, Miyamoto et al. [45] derive the theoretical expression of the performance of Q A M in class A impulsive noise by considering the dependence of the / and Q components. In [43], Kos-mopoulos et al. present the performance evaluation of M Q A M in the presence of combined Gaussian and non-Gaussian noise. Prasad et al. [41] give the analytical expression of the performance of D P S K in class A impulsive noise. However, the performance evaluation of modulation scheme in impulsive noise with diversity reception has not been studied in any previous paper. This chapter presents an analytical expression of the performance of B P S K in im-pulsive noise without diversity reception. Furthermore, it presents a novel theoretical expression of the performance of B P S K in impulsive noise with dual-branch E G C diversity reception, and the new integral representations of the performance of M P S K in impul-sive noise without or with dual-branch E G C diversity reception. The organization of this 21 3.2 B P S K without Diversity 22 chapter is as follows. Section 3.2 presents the analytical expression of the performance of B P S K without diversity reception in impulsive noise. In Section 3.3, the derivation of the analytical expression of the performance of B P S K with dual-branch E G C diversity reception in impulsive noise is presented. Section 3.4 presents the integral representation of the performance of M P S K without diversity reception in impulsive noise. In Section 3.5, the integral representation of the performance of M P S K with dual-branch E G C diversity reception in impulsive noise is derived. Conclusions are presented in Section 3.6. 3.2 B P S K without Diversity The performance of B P S K in impulsive noise without diversity reception is easily derived with the knowledge of the P D F of the in-phase component of impulsive noise (see Equation 2.5) and can be expressed as r-vm ^ B P S K = Pr{Ru < 0) = Pr Uj <-VEb)= / pmi{x) dx ^ ' J-oo = e x P ( - A 1 ) , i - ^ - / exp - — dx (3.1) where Ru = Re{Ri ejdm}, assuming that signal Sk = y/Eb where Eb = Energy/bit. Chang-ing variable z = — xj ^2ami, the above equation becomes ^ B P S K = ^M-Ai) £ / e x p H 2 ) dx mi=0 1 - v J \ \ l ± -3.2 B P S K without Diversity 23 The SNR/bi t 7 b and Eb are related as follows: where Ni is the average power of the in-phase component of impulsive noise 7/f7 and can be calculated by fOO /C  x2pVlI(x) dx •oo • ~ A™ 2 f°° 2 / x2 \ , = exp[—Ai) > —, / x exp — n _ ax oo i r n , 2 = exp( -A a ) £ (3.4) Using Equations 3.2 and 3.3, the performance of B P S K in impulsive noise without diversity reception can be expressed mathematically as Figure 3.1 illustrates the bit error rate (BER) of B P S K in class A impulsive noise for T' = 1 0 - 4 and different values of A without diversity reception. As can be observed, on one hand, for A = 10 the performance of B P S K in impulsive noise is very close to the performance of B P S K in Gaussian noise. On the other, for low values of A (e.g. A -C 10), the performance of B P S K in impulsive noise is better than in Gaussian at low values of SNR, but poorer at high values of SNR. The BERs obtained by simulation are also included and are seen to match the theoretical curves very well. This plot is identical to 3.3 B P S K with Dual-Branch E G C Diversity 24 the plot in Figure 10 using Equation 30 of [10], although the two analytical expressions of the performance of B P S K in impulsive noise are derived using different approaches. 10" DC LU m -20 A=0.35 A=0.01 Gaussian ImpA L=1 Sim of ImpA L=1 -10 0 40 10 SNR/bit (dB) Figure 3.1: Performance of BPSK in impulsive noise for T' = 1 0 - 4 and different values of A without diversity reception. 3.3 B P S K with Dual-Branch E G C Diversity As in the previous section, the performance of B P S K in impulsive noise with dual-branch E G C diversity reception can easily be derived with the knowledge of the P D F of the in-phase component of the sum of two impulsive noises (see Equation 2.16). It can be expressed as ^ B P S K C = Pr(Ri < 0) = Pr (vi < -2y/Eb) = -2yfWb Pri,{y)dy oo oo = exv[-{A1 + A2)] E E I71l=0 7712=0 A™1 A™2 \[C~i r~2y//E>> mi!m2!v /7r exp (-Ciy2) dy (3.6) 3 . 3 B P S K with Dual-Branch E G C Diversity 25 where /?/ = Re{RejdR}. Changing variable t = —\fC~iy, Equation 3.6 becomes ^ / B P S K = e x p [ - ( ^ + A2)\ V V ^ , V / exp (-t2) dt 1 0 0 Ami Am.2 . . = i e x p [ - ( A 1 + ^ ) ] £ ^ . ^ ( ^ f t f t ) . (3.7) m i ! m 2 ! 7771=0 7712=0 The £b and 7b are related as follows: 76 = 71 + 72 = ^ + 7 ^ Eb = (3.8) b iVi + iV 2 where 71. and 72 are the SNR/bit for channels 1 and 2, respectively, and Ni and _/V2 are the average powers of the /-component of impulsive noise for channels 1 and 2, respectively, (see Equation 3.4). Using Equations 3.7 and 3.8, the performance of B P S K in impulsive noise with dual-branch E G C diversity is obtained as W - ^ + M ± ± % & ' * ( / g f ) • (-) Figures 3.2 and 3.3 show the B E R of B P S K with dual-branch E G C reception in impulsive noise for T' = 1 0 - 4 and different values of A. In Figure 3.2, the impulsive noises in both channels are identical and have the same values of A (i.e. A = A\ = A2). In Figure 3.3, the impulsive noises are different for both channels and have different values of A (e.g. Ai = 10, A2 = 0.01). It can be seen from the plots that the simulations match the analytical curves very well. 3 . 3 B P S K with Dual-Branch E G C Diversity 26 Figure 3.4 compares the performance of B P S K without diversity reception with the performance of B P S K with dual-branch E G C reception in impulsive noise for T' = ICC 4 and different values of A. The plot shows that the dual-branch E G C diversity reception improves the performance of B P S K in impulsive noise at high values of SNR, but it degrades the performance at low values of SNR. This is because the dual-branch E G C diversity increases the P D F of the / component of impulsive noise at low values of amplitude, but it decreases the P D F of the / component of impulsive noise at high values of amplitude (see Figure 2.7). In the Gaussian noise case, it should be noted that the performance of B P S K with SNR/bit (dB) Figure 3.2: Performance of BPSK with dual-branch E G C diversity in impulsive noise for V = I O - 4 and different values of A (A = Ai = A2). 3 . 3 B P S K with Dual-Branch E G C Diversity 27 10 SNR/bil (dB) Figure 3.3: Performance of BPSK with dual-branch E G C diversity in impulsive noise for T' = 1 0 - 4 and different values of A in each channel (A\ = 10, A2 — 0.01). 10 -20 A=0.01 — Gaussian - - ImpA L=1 lmpAI_=2 A=0.35 -10 40 0 10 20 SNR/bit (dB) Figure 3.4: Comparison of the performance of BPSK with dual-branch E G C and without diversity reception in impulsive noise for V = 10~ 4 and different values of A (A = A\ = A2). 3 . 4 M P S K without Diversity 2 8 (L > 1) and without diversity reception in Gaussian noise are identical and can be ex-pressed as ^ s G K = ^ B P S K = ^ e r f c ( ^ ) - (3-10) G a u s s i a n G a u s s i a n ^ Equation 3.10 is derived in Appendix A . l . 3.4 M P S K without Diversity In [20], an integral representation of the performance of M P S K in Gaussian noise is derived with the knowledge of the joint P D F of the envelope and phase of Gaussian noise. Similarly, for the impulsive noise case, an integral representation of the performance of M P S K in impulsive noise can be derived with the knowledge of its joint P D F of the envelope and phase (see Equation 2.4). In Figure 3.5, a geometrical representation of the received signal with the envelope R and phase IT — ^ is composed of one of the M P S K signals with the envelope \fE~s and phase IT and impulsive noise with the envelope r\min\ and phase dn\. Using the law of sines, Vmini can be expressed as s in(^ i + ' (3.11) \ \ \ 7 T -s s \ \ Q Vmini / TT / / / S M / v o M Figure 3.5: A geometrical representation of one of MPSK signals at point S with the envelope i /E7 and phase TT added with impulsive noise at point Z with the envelope r / m i n i and phase 6vi to form the received signal with the envelope R and phase n — * . The line is the decision boundary of the signal with the phase 7r. 3 . 4 M P S K without Diversity 29 A n error occurs when the envelope of impulsive noise is 771 > r]mini and the phase of impulsive noise is — TT + * < 9vi < n — ^ . A n integral representation of the performance of M P S K in impulsive noise without diversity can be expressed as /•TT — V pOO •P^MPSK = 2 / / P(Vu M dm dOvi JO Jnm.M 0 J Vminl AT Vi ( vl =2L L^L^^-^)**"* = r l - p ( - , 1 ) | o f l e x p ( - ^ ) ^ , (3,2) Using Equation 3.11, the above equation becomes P e W = I ~ exp( -Ai ) E o ^ exp sin 2 \I> ^- — , d9vl. (3.13) ^ s i n 2 ^ - ^ ) / 7 , 1 The SNR/symbol js and signal energy/symbol Es are related as follow 7, - — (3-14) where the average power of impulsive noise Nvi is expressed as /oo ViPm (Vi)dm •00 °L 4 m i r°° n3 / n2 \ = e x p ( - A ) E ^ / i-dr)*> m i =o 1 • y ° m i ^ z % / <. m i ! . / n a l . " V za^, / 1 m i = 00 A 7711 = ^ ) E ^ r ( 3- 1 5 ) 7711=0 1 Substituting Equation 3.14 into Equation 3.13 leads to the following integral representation 3 . 4 M P S K without Diversity 30 of the performance of M P S K in impulsive noise without diversity reception P1 ^ e . M P S K - e x p ( - A 1 ) m i FIT—\P exp 7 , ^ 1 sin 2 2 < s in 2 (* + M d9vi. (3.16) In order to evaluate Equation 3.16, the Romberg integration of [40, p. 207] is used and the results are plotted in Figures 3.6-3.9. For the special case of M = 2 (BPSK), the results of Equation 3.16 are identical to the results of Equation 3.5. Similar remarks to those in Figure 3.1 can also be made in Figures 3.6-3.9. For example, as it can be seen from the plot, the performance of M P S K in impulsive noise for the high value of A (i.e. A = 10) is very similar to the performance of M P S K in Gaussian noise. However, for low values of A (e.g. A <C 10), the performance of M P S K is improved at low values of SNR, but it is degraded at high values of SNR. Simulation results are also included in the plots and are 10 SNR/bit (dB) Figure 3.6: Performance of M P S K without diversity in impulsive noise for r" = 1 0 - 4 and A = 10. 3 . 4 M P S K without Diversity 31 3 . 5 M P S K with Dual-Branch E G C Diversity 32 10" 10 10" 10 LU W 10 10 10 10" BPSK ImpA QPSKImpA 8PSK ImpA Sim of MPSK ImpA -20 -10 20 0 10 SNR/bit (dB) Figure 3.9: Same caption as in Figure 3.6 but with A = 0.01. 30 40 seen to very well match the theoretical curves. 3.5 M P S K with Dual-Branch E G C Diversity Using the same approach as in the previous section, an integral representation of the performance of M P S K in impulsive noise with dual-branch E G C diversity can be derived with the knowledge of the joint P D F of the envelope and phase of the sum of two impulsive noises (see Equation 2.15). Similarly to Figure 3.5, Figure 3.10 also shows a geometrical representation of an M P S K signal, impulsive noise and received signal. However, now the received signal with the envelope R and phase n — fr is composed of the sum of two M P S K signals with the total envelope 2\f~E~s and phase TX and the sum of two impulsive noises with the total envelope 3 . 5 M P S K with Dual-Branch E G C Diversity 33 Vmin2 and phase 9V. Comparable to the previous section by using the law of sines, r]min2 can be expressed as Vmin2 2^/Essm^ sm(6r, + *)' (3.17) s - * Q \ \ T)min2 . / / / S M Figure 3.10: A geometrical representation of the sum of two M P S K signals at point S with the total envelope 2sfEs and phase TT added with the sum of two impulsive noises at point Z with the total envelope r\mini and phase 0^ to form the received signal with the envelope R and phase TT — The line is the decision boundary of the signal with the phase TT. Similar remarks to those used in Figure 3.5 can be made in Figure 3.10 that an error occurs, when the envelope of the sum of impulsive noises is 77 > r]min2 and the phase of the sum of impulsive noises is — -rr + ^ < 9n < n — \&. The integral representation of the performance of M P S K in impulsive noise with dual-branch E G C diversity can be expressed as p / / - E G C • 'e .MPSK P7T— ^ POO 2 / / p(77, 0v)di1d9ri •J Vmin2 0 = 2 TT — <£< POO OO OO ATA™2 rjd Vmin2 mi=0TB2=0 0 0 0 0 Ami / l r »2 pTT—\P mi\m2\ TT exp (—?72C/) drjd9r] = ^ e x P [ - ( A 1 + A 2)] E E mi=0 m2=0 AT A2 mi\m2\ JQ exp (-rimwCi) d9v (3-18) Using Equation 3.17, the above equation becomes exp ( -—2777-^777 ) ddv (3-19) p£$az = \ exp[-(^i+A2)] E E ATA* ' ' ^S mi=0 m2=0 mi\m2\ JQ sm2{9v + tf) 3 . 5 M P S K with Dual-Branch E G C Diversity 34 The 7 S and Es are related as follow 7. = ^ (3.20) where the average power of impulsive noise Nv can be calculated by /oo r}2pv{rj)dr) -oo = e x p h ^ + A2)] Y £ ^ T 2 C / / ^ 6 X P d V 77ii=0m 2 =0 X ' 2 ' * ' 0 7711=0 7712=0 By using Equations 3.19 and 3.20 leads to the integral representation of the performance of M P S K with dual-branch E G C diversity in impulsive noise (3.22) Similarly to Equation 3.16, the above equation is also evaluated using the Romberg inte-gration and the results are plotted in Figures 3.11-3.14. For the special case of M = 2 (BPSK with diversity), the results of Equation 3.22 are identical to the results of Equation 3.9. Similar comments to Figure 3.4 can also be made to Figures 3.11-3.14 in that the dual-branch E G C diversity improves the performance of M P S K in impulsive noise at high values of SNR. However, the diversity reception degrades the performance of M P S K in impulsive noise at low values of SNR. This can be explained by noting the P D F of the envelope of the sum of two impulsive noises (see Figure 2.6). The E G C diversity reception increases the P D F of the envelope of impulsive noise at low values of amplitude, but 3 . 5 M P S K with Dual-Branch E G C Diversity 35 3 . 5 M P S K with Dual-Branch E G C Diversity 36 M P S K ImpA (L=1) 40 SNR/bit (dB) Figure 3.13: Same caption as in Figure 3.11 but with A = 0.35. 10 SNR/bit (dB) Figure 3.14: Same caption as in Figure 3.11 but with A = 0.01. 3 . 6 Conclusion 37 decreases the P D F of the envelope of impulsive noise at high values of amplitude. The SERs obtained by computer simulations are seen to match very well to the analytical curves. 3.6 Conclusions The goal of this chapter is to provide the performance evaluation of P S K signals in impulsive noise with or without diversity reception, especially the performance of mod-ulation schemes in impulsive noise with diversity reception that had not previously been studied and presented in any paper. The alternatively derived performance evaluation of B P S K in impulsive noise is presented. Then, the novel analytical expression of the error rate performance of B P S K in impulsive noise with dual-branch E G C diversity reception is derived. Furthermore, the new integral representations of the performance of M P S K in impulsive noise with or without dual-branch E G C diversity reception are presented. The results show that the E G C diversity reception improves the performance of modulation schemes at high values of SNR. However, the E G C diversity reception degrades the per-formance of modulation schemes at low values of SNR. In all cases, the theoretical results are thoroughly validated using computer simulations. C H A P T E R 4 Performance in Fading and Impulsive Noise 4.1 Introduction Although, in the past, the subject of performance evaluation of various modulation schemes in fading channels has been extensively investigated [23, p. 818],[25],[26],[53], relatively few researchers have studied their performance in a combination of fading and impulsive noise. To the best of our knowledge, in [30], an analytical expression is derived for the performance of high level Q A M with Nakagami fading and non-Gaussian noise by considering the independence of the / and Q components. This chapter presents novel analytical expressions for the performance of B P S K in Rayleigh, Rician or Nakagami fading with class A impulsive noise. The organization of the chapter is as follows. After this introduction, in Section 4.2, the analytical expression for the performance of B P S K with Rayleigh fading and impulsive noise is derived and evaluated. In Section 4.3, the derivation of the performance of B P S K with Rician fad-ing and impulsive noise is presented. Section 4.4 presents an analytical expression of the performance of B P S K with Nakagami fading and impulsive noise. The conclusions of this chapter can be found in Section 4.5. 4.2 Rayleigh Fading It is well-known that the performance of signals in the presence of fading can be derived by averaging the error rate probability Pe of signal over the P D F of fading envelope square 38 4.2 Rayleigh Fading 39 7 = r 2 as shown below P. /•oo = / PeP.il) JO d"f. (4.1) Using Equation 1.1, the P D F of 7 for the Rayleigh fading channel is given by p 7 ( 7 ) = P r i ^ = ~ exp ( - = -Zy/l lb V lb (4.2) where 7 b = 2o2,= SNR/bit . By averaging Equation 3.5 with Equation 4.2, the performance of B P S K in Rayleigh fading and impulsive noise can be expressed as p /-Ray •'e.BPSK PiBPSKP~f{l) dj = fleM-A)± £ eric (^g)i exp (- _1_ lb 1 v--T Ami 1 f°° 2 r°° I 7 - exp(-A) > — —= exp ( - t 2 ) dt exp ( - — ) dj (4.3) where B = Ni/a^. Changing variable t = y/^yz, Equation 4.3 can be rewritten as 1 0 0 ^BRp1K = ^ e x P ( - A ) £ Ami 1 2 7 3 / 2 *exp + — 7 7 b d7dz. (4.4) Using Equation 3.381.4 of [19, p. 342], the above equation becomes 1 JUL 4 m i 1 r Pewk = l exp(-A) Y ±- - \ dz lb JVB (z 2 + l / 7 6 ) 1 + 1/2' (4.5) 4.2 Rayleigh Fading 40 Appendix A.2 presents the derivation for the closed-form solution for the integral dx a (x2 + 6) C +V2 = 2 2c-1 26 1 -a2 + b c-i 1 ^ fe! fc=0 2\ V a2 + b 1 k r(fc + c)r(c) r(2c) (4.6) where a and 6 are real numbers, and c is a positive integer. Using Equation 4.6 into Equation 4.5 leads to the following compact expression of the performance of B P S K in Rayleigh fading and impulsive noise p/ -Ray -"e.BPSK 2 (4.7) Figure 4.1 shows the B E R (as a function of the SNR/bit) of B P S K in Rayleigh fading and 0 10 20 SNR/bit (dB) Figure 4.1: Performance of BPSK in Rayleigh fading and impulsive noise for T' = 1 0 - 4 and different values of A. 4.3 Rician Fading 4 1 impulsive noise for T' — 10 4 and different values of A. Simulation results are also included and very well validate the theoretical curves. 4.3 R i c i a n F a d i n g Similar to the previous section, the performance of B P S K in Rician fading and im-pulsive noise can be derived by using the Equation 4.1. Using Equation 1.2, the P D F of Rician envelope square 7 = r 2 is given by ftH=-vr=—exp {—%—r [y^h—1 (4'8) where yb = A2, + 2a2 and the specular-to-random ratio K = A2/(2a2) — power in steady component/power in random component. By averaging Equation 3.5 with Equation 4.8, the performance of B P S K in Rician fading and impulsive noise can be expressed as ^ B R P C S K = J ^BPSKP7(7 )<*7 = h - M ) | o £ - ( ^ ) ^ e X p ( - l i l ± f ± ^ ) = - exp [-(A + K)\ y — r ^ ^ - 7 -= / exp - t 2 dt 4.3 Rician Fading 42 where B = Ni/a^ . Changing variable t = ^fyz to the previous equation results in 1 0 0 P^fK = ^^p[-(A + K)]J2 Ami 1 + K 2 2 x exp mi = 0 mil 7t V^JVBJO OO /-OO 7 1/2 - Z' l + K 7 76 J (4.10) Using Equation 6.643.2 of [19, p. 701], the above equation can be rewritten as oo P^fK = ^exp[-(A + K)} m i mi=0 A_ l + K K lb / e x P (l + K)K ^ \2(z^h + l + K)\ 1 x z 2 7 6 + 1 + K , (l + K)K . , M _ l i 0 21_ , . ' 1 dz z^7 f c + 1 + K (4.11) where M A > A t (z) is the Whittaker function. Applying Equation 9.220.2 of [19, p. 1014] leads to the following equation 1 0 0 Ami f PlfviK = 5 e x P [ - ( A + K)] £ — (1 + K)fb/2 / x 1*1 ^;1 rri i=0 ; I + ^ ) A -V5 ( z 2 7 b + 1 + ^ ) 3 / 2 2' ' z ^ + l + i f dz = - e x p [ - ( ^ + AT)] £ £ ^ 7iJnrt mi=0 n=0 mi! X dz ^[z2 + (l + K)/jb]n+w/2 (4.12) where iFi(.) is a confluent hypergeometric function given by Equation 9.210.1 of [19, p. 1013]. By using Equation 4.6 into Equation 4.12, we can obtain the following closed-form 4 .3 Ric ian Fading 43 solution for the performance of B P S K in Rician fading and impulsive noise OO OO Tl = exp[-(A + K)) £ £ £ mi=0 n=0 g=0 Ami Kn {n + q)\ m i ! n! n\q\ 5 ' 1 ^76 fi7b + 1 + i f n + l 1 1 + B ^ (4.13) For the special case of K = 0 (i.e. Rayleigh fading), Equation 4.13 simplifies to Equation 4.7. Figures 4.2-4.5 illustrate the performance of B P S K in Rician fading and impulsive noise for V — 1 0 - 4 and different values of K and A. Simulation results are seen to very well match the theoretical curves. 10 DC LU ffl a —> K=0 b —> K=6dB c —> K=8dB d—> K=10dB e—> K=12dB f —> no fading -20 -10 Gaussian — Impulsive A • Simulation of ImpA Rice 0 10 SNR/bit (dB) Figure 4.2: Performance of BPSK in Rician fading for different values of K and impulsive noise for V = I O - 4 and A = 10. 4 . 3 Rician Fading 44 SNR/bit (dB) Figure 4.3: Similar caption to Figure 4.2 but with A = 1. 4.4 Nakagami Fading 45 4.4 Nakagami Fading Using Equation 1.3, the P D F of Nakagami envelope square 7 = r 2 is given by P i ( 7 ) = ? ^ * (^)V l e x p (4.14) 2^7 T(m) V7&/ V lb J By substituting Equations 3.5 and 4.14 into Equation 4.1, the performance of B P S K in 4.4 Nakagami Fading 46 Nakagami fading and impulsive noise can be obtained as follows roo ^ B P S K = J ^ B P S K ^ ( 7 ) d 7 2 ^ 0 mx! r(m) VT"J io V ^ U 1 ' x exp (~^p) ^7 (4-15) where B = Ni/a^ . Changing variable t = ^fyz, the above equation can be rewritten as p / -Nak _ exp(-4) ^ A™l_L_ fm\ , , m + x / 2 - i e ' B P S K " 2 ^ m x I T M U J v^-Wo 7 x exp[-(z 2 + 771 / 7^ )7 ] ofydz. (4-16) Using Equation 3.381.4 of [19, p. 342], the above equation becomes p/-N a k _ exp(-^ ) A^_J_ / m \ m _ 2 _ /°° r(m + 1/2) Considering the case where m 6 /+ substituting T(m+ 1/2) = T{2m)T(l/2)/[T(m)22m-1} and using Equation 4.6 into Equation 4.17, the performance of B P S K in Nakagami fading 4.4 Nakagami Fading 47 and impulsive noise can be expressed as oo m— 1 ^B N P as kK = exp(-A) J2 m i = 0 fc=0 Ami (m + k- 1)! mi ! (m — l)\k\ B B + m/-yb 2 + y B + m/jb (4.18) For the special case of m = 1 (i.e. Rayleigh fading), Equation 4.18 becomes Equation 4.7. Figure 4.6 shows the B E R of B P S K in Nakagami fading and impulsive noise for V = I O - 4 and different values of m and A. It can be seen that the B E R obtained from the simulator very well match the analytical curves. 10 10' 10" 10" ID CO 10" 10" 10" 10" a—> A=10 b --> A=1 c — > A=0.35 d — > A=0.01 -20 -10 \ • \ \ \ \ \ Gaussian no fading ImpA no fading ImpA Nakagami Sim of ImpA Nak m=1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ 1 \ \ 1 \ \ 'a \b t a,b,c,( m=2 ,d a\ 30 40 0 10 20 SNR/bit (dB) Figure 4.6: Performance of BPSK in Nakagami fading for m = 1, 2 and impulsive noise for V = 1 0 - 4 and different values of A. 4.5 Conclusions 48 4.5 Conclusions This chapter presents several new theoretical expressions for the performance of B P S K in Rayleigh, Rician or Nakagami fading and class A impulsive noise. Furthermore, in order to derive the performance in fading, the PDFs of Rayleigh, Rician or Nakagami envelope ' square 7 are also presented. It can be seen from the plots that the Rayleigh fading degrades the performance of B P S K in impulsive noise for all values of A. For Rician fading, the degree of degradation from the fading depends on the value of K. The lower the value of K is, the higher the degree of degradation from the Rician fading causes to the performance of B P S K with impulsive noise. In the most general cases of fading, the Nakagami fading channel for the low value of m impairs the B P S K performance more than the Nakagami fading for the high value of m. C H A P T E R 5 Performance of Diversity Combining Techniques 5.1 Introduction Numerous research topics presented in the past decade deal with the performance evaluations of various modulation schemes in Gaussian noise and fading channels with diversity combining reception [1] —[5],[7],[18],[32],[33],[38],[39]. However, to the best of our knowledge, very few research topics have tackled the performance evaluations of modula-tion schemes in impulsive noise and fading with diversity reception. More specifically in [41], the analytical expression of the performance of D P S K in Rician fading and impulsive noise with the SC system is derived, while in [28] the theoretical expressions of the perfor-mance of D P S K in Rician fading and impulsive noise with SC or E G C diversity reception are presented. This chapter presents the novel theoretical expressions for the performance evaluation of B P S K in impulsive noise and different types of fading (e.g. Rayleigh, Rician and Nak-agami) with SC or E G C diversity reception: The organization of this chapter is as follows. After this introduction, in Section 5.2 the analytical expressions for the performance of B P S K in impulsive noise and different classes of fading with L-branch SC diversity recep-tion are derived. In order to derive the performance in SC diversity reception, the PDFs of SC technique with different kinds of fading are also presented. In Section 5.3, the theoret-ical expressions for the performance evaluation of B P S K in impulsive noise and different kinds of fading with dual-branch E G C diversity reception are derived. In order to derive 49 5.2.1 P D F with Rayleigh Fading 50 the performance of E G C diversity reception, the characteristic function (CHF) method is used and the E G C CHFs with different sorts of fading are presented. In Section 5.4, a comparison for the performance of B P S K in impulsive noise and fading with dual-branch SC and E G C diversity reception is made. Conclusions can be found in Section 5.5. 5.2 Selection Combining (SC) To derive analytical expressions for the SC system, the same methodology as the pre-vious chapter (See Equation 4.1) is used. First, the P D F of SC with fading needs first to be derived. Then, the error probability of B P S K in fading and impulsive noise with SC diversity is derived by averaging the error rate probability of B P S K in impulsive noise with the previously derived SC P D F with fading. 5.2.1 P D F with Rayleigh Fading In a L-branch SC system, there is only one channel with the largest fading envelope r or largest 7 = r 2 and the remainder in L — 1 channels have lower 7. The SC P D F with Rayleigh fading can be expressed as Psc,Ray(7) = M 7 ) ^ ( 7 < 7 ) L - 1 (5-1) where p(j) is the Rayleigh P D F (see Equation 4.2) and P(j < 7) is the Rayleigh C D F at 7 that can be calculated by P ( , < 7 ) = ^ l e x p ( _ i ) r f , = 1 _ e x p ( _ 2 ) ( , 2 ) 5.2.2 Performance in Rayleigh Fading 51 where 7 C = SNR/bit/channel. Substituting Equation 5.2 into Equation 5.1 results in the SC P D F with Rayleigh fading P S C , R a y ( 7 ) = =~ exp ( 7 C V 7 C - L v ( L ~ 1 I c 1=0 1 — exp ( — — 7c/ j 7 n L-l I (-1)' exp (1 + 0 7 7 C (5.3) 5.2.2 Performance in Rayleigh Fading The performance of B P S K in impulsive noise and Rayleigh fading with SC diversity reception, derived by averaging Equation 3.5 with Equation 5.3, can be expressed as p L - S C , R a y _ - ' " e . B P S K ~ 0 ^ B P S K P S C . R a y f r ) ^7 Jo 2 m\=0 - m i / 7 c ; = 0 ( 1 + 0 7 \ , x exp I - _ dq 7 C L ^ ^ A m > a - v 7 r <—' ' m i ! V I l c mi=0 1=0 1 X /•CO cy pOO (-1) ' / ^ / exp ( - i 2 ) eft (1 + 0T\ , x exp ( - _ d7 7 C (5.4) where B = Ni/a^ . Changing variable t = \fyz, the above equation becomes -i-<£sS(V)<-tf*jGr p L - S C , R a y - ' e . B P S K 7 .3/2-1 x exp 2 (1 + 0' 7 C 7 ) <i7<iz. (5.5) Using Equation 3.381.4 of [19, p. 342], the previous equation can be written alternatively 5.2.2 Performance in Rayleigh Fading 52 as ,L-SC,Ray e.BPSK [z2 + (1 + l)/lc\ •oo dz 1+1/2 - (5.6) Applying Equation 4.6 into Equation 5.6 yields the performance of B P S K in impulsive noise and Rayleigh fading with L-branch SC diversity reception and impulsive noise Figures 5.1-5.4 show the B E R of B P S K in impulsive noise and Rayleigh fading with SC diversity reception for V — 10~4 and different values of A and L. It can be seen from these results that the L-branch SC reception improves the performance of B P S K in impulsive noise and Rayleigh fading at high values of SNR. The greatest performance improvement is obtained when the diversity branch increases from L — 1 to 2. On the other hand, the SC diversity degrades the performance at some values of SNR, as the value of L increases. It happens, because the power of a fading signal is not completely dominating the power of noise. When the SC diversity is employed to increase the chance of receiving the higher fading signal power, the chance of receiving the higher noise power increases as well. Thus, these degradations come from receiving the noise with higher power than the fading signal. The theoretical curves and computer simulation results are in very close agreement. P. ,L-SC,Ray e.BPSK B + (l + 0 / 7 c B (5.7) 5 . 2 . 2 Performance in Rayleigh Fading 53 SNR/bit (dB) Figure 5.1: Performance of BPSK in Rayleigh fading and impulsive noise for T' = I O - 4 and A = 10 with SC diversity reception for different values of L. SNR/bit (dB) Figure 5.2: Same caption as in Figure 5.1 but with A = 1. 5 . 2 . 2 Performance in Rayleigh Fading 54 10° SNR/bit (dB) Figure 5.3: Same caption as in Figure 5.1 but with A = 0.35. -10 0 10 20 30 40 SNR/bit (dB) Figure 5.4: Same caption as in Figure 5.1 but with A = 0.01. 5.2.3 P D F with Rician Fading 55 5.2.3 P D F w i t h R i c i a n Fading Similarly to Equation 5.1, the SC P D F with Rician fading is given by Psc ,Rice(7) = Lpfr) P ( 7 < I T ' 1 (5.8) where 39 (7 ) is the Rician P D F (see Equation 4.8) and P ( 7 < 7 ) is the Rician C D F at 7 that can be calculated by P ( 7 < 7 ) = jT ^ exp + y + ^)/ 0 ^ l + _ K ^ j ^ = Pcexp{-K) J\M~M) h {2y/K0^"j d^y (5.9) where (3C= (K + 1)/T"c. Changing variable 7 = z 2/(2/3 c), the above equation becomes P ( 7 < 7 ) = J z exp I _ i £ _ ± A | ^ M _ J j / o ( ^ 2 / ^ ^  ( 5 . 1 0 ) Using the well-known Marcum's Q function Equations 2.1-122 and 2.1-123 of [23, p. 43-44], the above equation can be written alternatively as P ( 7 < 7 ) = l - Q i ( v ^ ) V / 2 ^ c 7 ) n/2 / K \ n / = 1 - exp[-{K + fa)]^ — Uy/AKKn). (5.11) „ = o \ P c 7 / 5.2.3 P D F with Rician Fading 56 Applying Equation 5.11, the term [P(7 < ^)]L in Equation 5.8 can be expressed as [ P ( 7 < 7) ] L - X = °° / K \ n / 2 i _ e x p [ - ( / r + M)\ E b r UyfiKfa n=0 L - l L - l i=0 E ( L ^ 1 l b 1 ) ' exp [ - (^ + /?c7)/] ?i=0 (5.12) Expanding the nth-order modified Bessel function of the first kind by using Equation 2.1-120 of [23, p. 43], the following equation is obtained L - l [Pin < i)\L~l = E ( L , 1 ) ( - 1 ) ' « p [ - ( t f + Mi\ 1=0 oo / J( \ n / 2 0 0 X (VKp7r)n+2q qW(n + q + 1) L - l = E ( L / 1 ) ( - l ) ' e x p [ - ( i , + / ? c 7 ) / ] x ( t f & 7 ) 9 00 / oo E E q=0 \n=0 Kr' q\T(n + q + 1) (5.13) By using Equation 0.314 of [19, p. 17], the above equation reduces to L - l OO / r _ - | \ [Pin < 7 ) ] L _ 1 = E E ( i ) ( - 1 ) ' c * e x P [ - ( ^ + Ml] (Kpd)h (5.14) /=0 h=0 ^ ' w h e r e Co = exp(Kl) a n d /l oo Ch E E [s(f + 1) - h]Kn hexp(K)^^n s\(n + s)\ K ' s=\ n=0 v ' Ch- (5.15) 5.2.4 Performance in Rician Fading 57 with h = 1, 2, 3, • • •. The SC P D F with Rician fading can now be expressed as PSC,RM = L P c Y Y [ i ) ( - l ) ^ ( ^ c 7 ) f t e x P [ - ( ^ + /?c7)a + l ) ] / o ( 2 v ^ c 7 ) -1=0 h=0 ^ ' (5.16) 5.2.4 Performance i n R i c i a n Fading Similarly to Section 5.2.2, the performance of B P S K in Rician fading and impulsive noise with SC diversity reception derived by averaging Equation 3.5 with Equation 5.16 can be expressed as ^ B P C S K ^ = J ^ B P S K P S C . R i c e W ^ -h—ss-WiKgtv)'-* x (Kpcl)hexp[-(K + Pcl)(l + l)]I0(2y/KfcY) ^ o o L-l o o . m i / r _ - , \ = ^ L f t e x p M ) Y EE^TT I K-Vl<*Wc)H™P[-K(l mi=0 1=0 h=0 1 - ^ ' r\ /"OO / * 0 O + l ) ] -7=/ / exp(- t 2 )d t 7 ' l exp[- /? c 7( / + l ) ] / o (2v / A 7 ^c7 )d7 (5.17) V* Jo J^m where B = Nx/a^ . Changing variable t = the above equation becomes mi=0 ;=o /i=0 1 - ^ ' o / ' C O / • o o + 1)1-7= / / 7 " + 1 " 1 / 2 e x P ( - [ z 2 + /3 c ( / + l ) ] 7 ) ^ 2 V / A 7 ^ c 7 ) d 7 ^ . (5.18) 7 J O 5.2.4 Performance in Rician Fading 58 Using Equation 6.643.2 of [19, p. 701], the above equation can be written alternatively as m i = 0 1=0 h=0 1 - ^ ' + DI A r r ( / l + 3 / 2 ) e x p f ^ ^ \8(i+i)+zrih+i) x M _ ( , + 1 ) , 0 ( m ™ + 3 f l ) dz. (5.19) Applying Equation 9.220.2 of [19, p. 1014] to the above equation yields = \L(3ceM-A) E E E ^ T I , ) ( - 1 ) ^ * 7 ^ e x p h ^ Z m i = 0 / = 0 / i = 0 X ' ^ ' + l)]-^=T(h + 3/2) f ° [z2 + & ( i + l ) ] - ^ 3 / 2 ) x F i (fc +1; 1; V7T J^B V 2 z 2 + &(( + i) oo L—l oo = 2 L ^ e x P ( - ^ ) E E E ^ - 7 j )(-l)'c f c(/C)9 t) f cexp[-A:(Z mi=0 /=0 /i=0 1 - ^ ' + l)]±T(h + 3/2) T [,2 + &(J + l ) ] - ( f t + 3 / 2 ) E Substituting ( l ) t = £!, (h + 3/2)t = (2h + 2t + l)\hl/[22t(2h + l)l{h + a n d T(h +3/2) = T(2h + 2)r(l/2)/[r(/i + l ) 2 2 / l + 1 ] , the following expression is obtained OO L-l OO . m i / r _ , \ ^ B P C S K C E = ^ C e x P ( - A ) ^ ^ X ] ^ T / )(-l)lch(Kpc)heM-K(l + l)} m i = 0 ( = 0 / i = 0 1 " ^ ' i oo X 1 ^{2h + 2t + l)\(Kf3cy r dz 2 fc+i 22'(fc + r.)!(r<)2 7^ [ 2 2 + ^ + 1 } ] 5.2.4 Performance in Rician Fading 59 Using Equation 4.6, the performance of B P S K in Rician fading and impulsive noise with L-branch SC diversity reception is obtained as pL-SC,Rice ^e .BPSK r ( * ^ ^ ^ ^ V * A m i (L - A (-VlchKh+t r „ „ Lexp(-A) ^ ^ L L L ^ - ( i ) UM-, ex P [-K(Z + l)]T(h + t + g + l) ma=0 1=0 h=0 t = 0 p=0 / 1 (i!)^! 2(/ + l) 1 -B + Pc{l + l) h+t+l X 2 + y B+Pc(I + I) (5.22) For the special case of K = 0 (i.e. Rayleigh fading), Equation 5.22 becomes Equation 5.7. The above equation has been evaluated for V — 1 0 - 4 and different values of K, L and A and the results are summarized in Figures 5.5-5.16. It can be observed that, the SC technique improves the performance for all values of K only at the high values of SNR. However, at some values of SNR, SC diversity degrades the performance for the high value of K more than the performance for the low value of K. Similar remarks to Figures 5.1-5.4 can be made in that using the SC diversity at the low values of SNR will degrade the performance, because the chance of receiving the higher power of noise increases, as the value of L increases. Thus, it can be concluded that the SC diversity technique is not an efficient technique to combat fading that has a line-of-sight path such as Rician fading. The simulated BERs obtained from computer simulations are seen to match the analytical curves very well. 5 . 2 . 4 Performance in Rician Fading 60 5.2.4 Performance in Rician Fading 61 10 SNR/bit (dB) Figure 5.7: Similar caption to Figure 5.5 but with L = 4. 10 10" a —> K=0 b — > K=6dB c —> K=8dB d—> K=10dB e — > K=12dB -20 -10 — Impulsive A (no fading) ImpA w/ Rice ImpA in SC w/ Rice • Sim ot ImpA in SC w/ Rice 20 30 40 0 10 SNR/bit (dB) Figure 5.8: Performance of BPSK in Rician fading for different values of K and impulsive noise for V = 10~ 4 and A = 1 with SC diversity reception for L = 2. 5.2.4 Performance in Rician Fading 62 Impulsive A (no fading) SNR/bit (dB) Figure 5.9: Same caption as in Figure 5.8 but with L = 3. SNR/bit (dB) Figure 5.10: Same caption as in Figure 5.8 but with L = 4. 5 . 2 . 4 Performance in Rician Fading 63 -20 -10 0 10 20 30 40 SNR/bit (dB) Figure 5.12: Similar caption to Figure 5.11 but with L = 3. 5 . 2 . 4 Performance in Rician Fading 64 10 10 10 10" rx m 10" 10 10" 10 a —> K=0 b — > K=6dB c — > K=8dB d—> K=10dB e—> K=12dB Impulsive A (no fading) ImpA w/ Rice ImpA in SC w/ Rice Sim of ImpA in SC w/ Rice -20 -10 20 30 40 0 10 SNR/bit (dB) Figure 5.14: Performance of BPSK in Rician fading for different values of K and impulsive noise for V = 1 0 - 4 and A = 0.01 with SC diversity reception for L — 2. 5.2.4 Performance in Rician Fading 65 -20 -10 0 10 20 30 40 SNR/bit (dB) Figure 5.16: Same caption as in Figure 5.14 but with L = 4. 5.2.5 P D F with Nakagami Fading 66 5.2.5 P D F with Nakagami Fading Similarly to Equations 5.1 and 5.8, the SC P D F with Nakagami fading is obtained by Psc,Nak(7) = Lp(7) P ( 7 < l)L~l (5.23) where p( 7 ) is the Nakagami P D F (see Equation 4.14) and P ( 7 < 7 ) is the Nakagami C D F at 7 that can be calculated by m\m 1 r ~m-i ____ f m j \ \ 7 c / r M Jo V 7 C / Using Equation 3.381.1 of [19, p. 342], the above equation becomes 1 (m) where 7 / n is the incomplete gamma function given by Equation 8.350.1 of [19, p. 890]. Equation 5.23 can be written alternatively as Psc,Nak(7) = ( j ? ) r ^ T ^ T l fr/"(m' m 7 / 7 c ) ] L " 1 exp ( ~ ^ ) • ( 5 - 2 6 ) Considering m is a positive integer and using Equation 8.352.1 of [19, p. 890] yields the 5.2.6 Performance in Nakagami Fading 67 following equation P s c M T ) = ( ? f £ S exp ( - ^ ) [(ra - (1 - exp ( - 2 1 X 7 C 7 [T(ro)]* ' m - l 1 / \ fc' V — I m 7 i 7727 A x exp I - - ^ 3 — 7 C / m—1 E l / ?rt7N fc fc=0 v / c (5.27) Using Equation 0.314 of [19, p. 17] and Equation 16 of [32] leads to the SC P D F with Nakagami fading PSC,Nak(7) L ^ ^ V m \ m + 9 a-i T(m E E ; ;=o o=o v / c / I M l )(-!)'exp •(I + 1)7717' 7 C 7 m+q—l (5.28) where the coefficient cq can be computed recursively as c 0 = 1, C\ = I, Q(m_i) = [l/(m — and m i n ( m - l , o ) f , . , 1 ^ fc I + 1) - g „ , V = 7 n C1~h (5-29) fc! Cq-h with 2 < g < l(m - 1) - 1. 5.2.6 Performance in Nakagami Fading Similarly to Sections 5.2.2 and 5.2.4, the performance of B P S K in Nakagami fading and impulsive noise with SC diversity reception derived by averaging Equation 3.5 with 5.2.6 Performance in Nakagami Fading 68 Equation 5.28 can be expressed as 3 L - S C , N a k e.BPSK poo = / P e ^ B P S K PSC,Nak(7) Jo x I L { J ( - l ) ' e x p (/ + 1)777.7 7 C 00 L - l i ( m - l ) 7 m + 9 - l ^ mi=0 ;=0 9=0 m+9 9=0 L - l X noo exp (—t2) dt exp (1+ 1)7777' 7 C (5.30) where 5 = Ni/a^ . Changing variable t = ^ z , the above equation becomes 0 L - S C , N a k e.BPSK T ( m ) ^ ^ 777x1 V 7 J C ' v ' mi=0 /=0 . g=0 1 v / c / 00 poo VB JO yn+9+1 /2 -1 e x p ( _ z2 + (I + 1) 777 7 C 7 ) d'ydz. - 1 (5.31) Using Equation 3.381.4 of [19, p. 342], the above equation can be written alternatively as 1 j 00 L - l l(m-l) . . m + g / . v ; mi=0 Z=0 9=0 1 \ ' c / \ / V x F> r(m + 9 + l /2) A / B (z 2 + (Z + l ) m / 7 c ) m + 9 + 1 / 2 ' 1 ' j Substituting T(m + g +1 /2) = r(2[m + §]) r(l/2)/[r(m + q) 2 2 ( m +«)- 1 ] and using Equation 4.6 leads to the performance of B P S K in Nakagami fading and impulsive noise with L -5.2.6 Performance in Nakagami Fading 69 branch SC diversity reception >I,-SC,Nak e.BPSK 2 I V B + (I + l)m/% (5.33) For the special case of m = 1 (i.e. Rayleigh fading), Equation 5.33 simplifies to Equation 5.7. The above equation has been numerically evaluated for V = 10 - 4 , m = 2 and different values of L and A and the obtained results are presented in Figures 5.17-5.20. As it can be seen from the plots, SC diversity improves the performance in Nakagami fading at high values of SNR. The greatest diversity gain is received, when the diversity branch increases from L = 1 to 2. However, at some values of SNR, the SC diversity technique degrades the performance in Nakagami fading. Similar comments to Figures 5.1-5.4 can be made here that SC diversity increases chances of receiving the noise with higher power than the fading signal. Simulation results are also included and are seen to very well validate the theoretical curves. 5.2.6 Performance in Nakagami Fading 70 5.2.6 Performance in Nakagami Fading 71 5.3 Equal-Gain Combining (EGC) 72 5.3 Equal-Gain Combining (EGC) To find analytical expressions of the performance of B P S K in E G C , the performance of B P S K in impulsive noise with diversity is averaged with the E G C P D F . However, as compared to the SC case, the derivation of the E G C P D F is much more difficult. Instead, a characteristic function (CHF) approach, introduced in [3],[39], is used to find the performance of E G C diversity. The E G C C H F is more easily derived than the E G C P D F . The performance of a modulation scheme in impulsive noise and fading with E G C diversity, is derived by averaging (see Equation 4.1) the performance of a modulation scheme in impulsive noise and diversity reception with the E G C PDF , needs to be rewritten into a function of the E G C C H F instead. Then, the earlier derived E G C C H F can be substituted and the performance of E G C diversity is obtained. The E G C C H F is given by /oo PEGc(r) exp(jur) dr. (5.34) -oo In order to use the E G C C H F approach in the averaged integral of B P S K , the performance of B P S K in impulsive noise with dual-branch E G C reception (see Equation 3.9) is required to be alternatively written as 1 °° ° ° Ami / r a 2 ^ P E ? K C = i o p R * +M £ £ ^ e r f c ( y i ^ ) m i = 0 m 2 = 0 1 °° °° Aml Am2 r , N -, = i + £ £ - ^ j l - e r f ( r v ^ ) (5.35) rai=0 m 2 = 0 where r = and B2 = SN^d/i^ + N2). Using Equation 3.952.6 of [19, p. 497], the 5 . 3 Equal-Gain Combining (EGC) 73 above equation becomes ^ oo oo Ami Am2 r 9 r°° mi=0m 2=0 ^ 1 / 0 x sin ^2rtx/B^) di 7 (5.36) Changing variable y = t2, the performance of B P S K in impulsive noise with dual-branch E G C diversity reception can be expressed as oo oo m i =0 rri2=0 ^BP E S G K C = ^ X P [ - ( A 1 + A 2 ) ] £ £ x sin (2r\jB2y^ dy A™1 A™2 mi!m2! I r°° _ 1 - - / y _ 1 exp( - j / ) (5.37) Applying Equation 4.1 leads to the performance of B P S K in impulsive noise and fading with dual-branch E G C diversity reception / •CO p / / - E G C , f a d _ / p J J - E G C / - . \ „ / _ \ J_ e,BPSK - / K,BPSK (r) PEGc[r) ar Jo 2 ^ O r ^ O m i ! m 2 ' L W o ./o x sin (2ry/Bty^pEGc(r) dydr (5.38) Substituting the term JQ sin(2r y/B2y) PEGC(^) dr, which is the imaginary part of the E G C C H F (see Equation 5.34), in the above equation leads to the performance of B P S K 5.3.1 C H F with Rayleigh Fading 74 in impulsive noise and fading with dual-branch E G C diversity reception using the C H F as R / / - E G C . f a d e.BPSK . c o oo Ami Am2 r -i poo Iexp[-(^ 1 + ^)] £ E " I I " y^M-y) x I m a g j ^ E G C ( 2 v / S ^ / ) | dy (5.39) The previous equation will be used to derive analytical expressions of the error rate per-formance of B P S K in impulsive noise and Rayleigh, Rician and Nakagami fading with dual-branch E G C diversity reception in Sections 5.3.2, 5.3.4 and 5.3.6, respectively. 5.3.1 C H F with Rayleigh Fading By using Equation 5.34, the E G C C H F with Rayleigh fading can be calculated by ^EGCRayM = E[exp(jur)] = E L poo =n/0 (TF) exp dr, (5.40) where r is a normalized combined envelope, rt is a Rayleigh envelope in the Ith channel and the Rayleigh P D F of rt (see Equation 1.1) is given by 5.3.1 CHF with Rayleigh Fading 75 Equation 5.40 can be written alternatively as ? > E G C , R a y M = TJ / Pnfa) 1=1 J ° L wrj \ . . I tori cos I —^= I + j sin = JI [ ^ E G C , R e , R a y ( U ; ) + ^ E G C , I m , R a y ( a ; ) dri (5.42) where 0 EGc , R e , R a y ( a ; ) a n d ^ E G C i m . R a y C ^ ) a r e t n e r e a l a n c l imaginary parts of the E G C -Rayleigh C H F of the Ith channel, respectively. Using Equations 5.41 and 5.42, the < / ) E G C R e R a y ( t 0 ' ) c a n D e expressed mathematically as ' E G C P W - ) = ^ " f t , W o o . gl) dn = [ I exp ( - | ) oos g | ) (5.43) Using Equation 3.952.8 of [19, p. 497] reduces the above equation to Ith f \ _ IT I 1 U 1c ( P E G C , R e , R a y l a ' ) ~ 1-^ 1 \ L'' y (5.44) Similarly, the 0 E G C , i m , R a y ( u ; ) c a n ^ e expressed as h l t h 9 EGC,Im,Ray sm | —— I dn. (5.45) Applying Equation 3.952.7 of [19, p. 497], the above equation becomes ,,th , , co /7T7C / a ; 2 7 c $ 3 G C , I m . R a y M = 77 V — e X P 2 V L 4L (5.46) By substituting Equations 5.44 and 5.46 into Equation 5.42, the following equation is 5.3.2 Performance in Rayleigh Fading 76 obtained 0EGC,Ray(w) = JJ 1=1 I T? i i 1 w 2 7 c A , - W / ^ T T ( u27c (5.47) In the above equation, considering dual-branch diversity (L = 2) results in the imaginary part Of 0 E G C , R a y M Imag{0EGC,Ray(w)} = UJ\J — exp I — 1 i F i I I ; - ; — (5.48) 5.3.2 Performance in Rayleigh Fading Substituting Equation 5.48 into Equation 5.39 yields the performance of B P S K in Rayleigh fading and impulsive noise with dual-branch E G C reception 1 oo oo l m . Am? r / o R ^7 r°° o / / - E G C , R a y 1 „ , r / , A \ I ^ 1 ^ 2 •, *£>2lc / i i ^ . B P S K - j 6 X P h ( A l + M ) ] ^ ^ n ^ « ^ " V ~ J0 ' mi=0 m2= 0 x exp (5.49) Using Equation C l of [3], the following analytical expression of the performance of B P S K in Rayleigh fading and impulsive noise with dual-branch E G C diversity reception can be obtained 1 oo oo , m i , m 2 D / / - E G C , R a y „ „ r / A i A M A \ A 2 ^ e . B P S K = n exp[ - (^ l + A 2 )] ^ 2 ^ mi=0 m2= 0 m\\ m 2 ! y/!327c(2 + P 2 7 c ) ( l + £ 2 7 c ) (5.50) Figures 5.21-5.24 illustrate the theoretical B E R of B P S K in Rayleigh fading and impulsive noise with dual-branch E G C diversity versus the SNR/bit . It can be seen from the plots 5.3.2 Performance in Rayleigh Fading 77 -20 ImpA EGC no fading (L=2) ImpA Rayleigh (L=1) ImpA EGC Rayleigh (L=2) Sim of ImpA EGC Ray (L=2) -10 0 10 SNR/bit (dB) Figure 5.21: Performance of BPSK in Rayleigh fading and impulsive noise for V = 1 0 - 4 and A = 10 with dual-branch E G C diversity reception. -20 -10 0 10 20 30 40 SNR/bit (dB) Figure 5.22: Same caption as in Figure 5.21 but with A = 1. 5.3.2 Performance in Rayleigh Fading 78 0 10 20 SNR/bit (dB) Figure 5.24: Same caption as in Figure 5.21 but with A = 0.01. 5.3.3 C H F with Rician Fading 79 that E G C diversity improves the performance at high values of SNR. However, when the value of A decreases (i.e. A <C 10, see Figures 5.22-5.24), the E G C diversity degrades the performance at low values of SNR as compared to the performance in fading without diversity reception. This is because the E G C diversity increases the P D F of the sum of two impulsive noises at the low values of amplitude (see Figure 2.7). Simulated BERs are also included and are seen to match the theoretical curves very well. 5.3.3 C H F with Rician Fading Using a similar methodology as the one presented in Section 5.3.1, Equation 5.42 can be rewritten for the Rician fading as E G C cos — 7 = + j sin —= y/lj \y/L ^ poo Rice(w) = ]T / Pn(ri) 1=1 J ° L = E[ [ ^ E G C , R e , R i c e M + ^ E G C . I m . R i c e M dri (5.51) where 0 E G C , R e , R i c e ( a ; ) a n d ^ E G C j m . R i c e ^ ) a r e the real and imaginary parts of EGC-Rician C H F of the Ith channel, respectively, rt is a Rician envelope in the Ith channel. The Rician P D F of ri (see Equation 1.2) is given by Vri(n) = 2(3cn exp (-K - 0cr?) /„ fay/Wc (5.52) with PC = (1 + A ' ) /7 C . Using Equations 5.51 and 5.52, </>EGC Re RiceC^) c a n D e expressed 5 . 3 . 3 C H F with Rician Fading 80 mathematically as 0 E G C , R e , R i c e M = JQ Pr, fa) COS (^=j drl = j f 2Pcn exp (-K - (3cr2) I0 fa^/Wc) cos dn. (5.53) Expanding IQ(.) in the above equation by using Equation 8.447.1 of [19, p. 909] leads to the following equation ^ . l U t f U c e M = 2 / ^ ^ dr, (5.54) Using Equation 3.952.8 of [19, p. 497] reduces to the following closed-form solution for the real part of EGC-Rician C H F of the Ith channel 0 E G C , R e , R i c e ( ^ ) = exp (-AT) ^ — XFX \nt + 1; - ; -—J . (5.55) Similarly, 0EGC im RiceC^) c a n D e expressed as C ;c , im,RiceM = J Pnfa) sin (^j= ) dn = 2Pcexp(-K)J\lexp (-(3crf) sin (^j J„ ( 2 r / v ^ ) dr,. (5.56) Expanding I0(.) by using Equation 8.447.1 of [19, p. 909], the above equation becomes ^ G C , i r a , R i c e ( ^ ) = 2pcexp(-K) f ; < ^ £ i J \ ^ + L exp (-f tr?) sin dr,. (5.57) 5 . 3 . 4 Performance in Rician Fading 81 Applying Equation 3.952.7 of [19, p. 497] to the above equation, the imaginary part of EGC-Rician C H F of the Ith channel simplifies to < / , EGC,Im,Rice ( u ; ) ojexp(-K)^ K n i f 3 \ _ / 3 3 u 2 £ ^ ^ r ( n ; + - ) lF1 (n z + - ; - ; 2' 2' 4/3cL (5.58) Substituting Equations 5.55 and 5.58 into Equation 5.51 yields the following expression 'EGC.Rice H = II exp(-^)X; — x F x U + l ; -J=l L ni=0 l ' ^ u;exp(-AT) ^ A""' 4 & L • / 3 \ ^ / 3 3 oj2 (5.59) In the previous equation, considering dual-branch diversity (L = 2), the imaginary part of 0EGC,Rice(k-O can be expressed as I m a g { 0 E G C , R i c e ( ^ ) } = ni=0 ri2=0 (n x!) 2n 2! 3 3 u 2 \ „ ( 1 UJ2 X l F l ( n i + 2 ; 2 ; - 8 ^ J lFir + 1 ; 2 ; - 8 ^ (5.60) 5.3.4 Performance in Rician Fading Similarly to Section 5.3.2, by substituting Equation 5.60 into Equation 5.39, the per-formance of B P S K in Rician fading and impulsive noise with dual-branch E G C diversity 5 . 3 . 4 Performance in Rician Fading 82 reception can be expressed as .. oo oo Ami ATT \mi A m 2 m\\ m2\ mi=0 m2=0 x E E / - . r ( n i + o ) / 2/2 e x p ( - y ) V Pc ni=0 n2=0 (n x!) 2n 2! oo 0 _ . 3 3 B2y\ ( 1 B2y X l F l ( n i + 2 ; 2 ; - 2 ^ J l F l l n 2 + 1 ; 2 ; - 2 A M y (5.61) Using Equation C l of [3], the above equation becomes oo oo . m i D / / - E G C , R i c e 1 N r / A , A W ^1 ^2 ^ e . B P S K = ~0 e X P [ - ( A X + A 2 )] ^ ^ — — -mi=0 m2=0 i™1 A™2 m\ \ m2\ 2' 2' * E E (SSr + ^  r(i/2) « ( i : ^ + + 1 ; 11 ni=0 n2=0 _ Eh _B1 2 /V 2/5c (5.62) where F2(.) is a hypergeometric function of two variables given by Equation 9.180.2 of [19, 5 . 3 . 4 Performance in Rician Fading 83 p. 1008]. Applying Equation 9.183.2 of [19, p. 1010] to the above equation yields T7l 2 2 oo co *m\ A ' D / / - E G C , R i c e 1 „ r / A , A 1 mi=0 m2=0 x E E ^ H h ^ Q m = 0 n 2 = 0 V i / ^ \ / \ 1 3 1 Bo Bo 2(pc + B2),2(pc + B2) oo oo ami Am2 - ^ H * + * > i E E £ f £ r mi=0 rri2=0 oo oo n i X 7r y pc + B2 ~ ~ m ^ 1 + r a 2 , 3 \ ( l / ^ - m ) , X 5, 2(& + £ 2 ) . _ / i 1 1 2*1 ( 75 + 9 , - ^ 2 - 7 5 ^ ; S o 2'2 '2(/3 c + £ 2 ) (5.63) where 2Fi (.) is a hyper geometric function given by Equation 9.100 of [19, p. 995]. Substi-tuting (1/2), = (2q)\/(22«q\), (3/2), = (2q + l)!/(2>!) and ( - n a ) , = ( - l ^ n J / K - q)\ to the above equation leads to the performance of B P S K in Rician fading and impulsive noise with dual-branch E G C diversity reception 1 °° 0 0 Ami A.m2 mj=0 m2=0 1 - - exp(-2A r ) 7T oo oo ni X ^ ^ 1 + n 2 r ( W l +1) r( |)(-i)g 1 1, ff2 2 ; 2 ! 2( /? c + 5 2 ) Bo 2(Pc + B2)\ x 2-F1 ( Q + 7j> _ n 2 (5.64) For the special case of K = 0 (e.g. Rayleigh fading), Equation 5.64 simplifies to Equation 5.50. The above equation has been numerically evaluated for T' = 10" 4 and different values of K and A. The obtained results are illustrated in Figures 5.25-5.28. As can be seen from 5 . 3 . 4 Performance in Rician Fading 84 5 . 3 . 4 Performance in Rician Fading 85 5 . 3 . 5 C H F with Nakagami Fading 86 the plots, the E G C diversity very well improves the performance in Rician fading and impulsive noise, especially at high values of SNR. Similar remarks to Figures 5.21-5.24 can also be made in that the E G C diversity degrades the performance in Rician fading and impulsive noise, particularly at low values of SNR and A (i.e. A <C 10), because the E G C diversity increases the P D F of the sum of two impulsive noises at low values of amplitude. Simulation results included in the plots are seen to be very close to the theoretical ones. 5.3.5 C H F with Nakagami Fading Using a similar methodology as the one presented in Section 5.3.1, Equation 5.42 can be rewritten for the Nakagami fading as ^ E G C . N a k ^ , w r , \ . . / uri cos + j sin —= y poo ) = n/ 1=1 J ° L = II tec,He,Nak(w) + ^ E G C , I m , N a k ( W ) 1=1 (5.65) where <£EGC Re N a k ( ^ ) a n d ^ E G C . i m . N a k ^ ) a r e ^ e r e a ^ a n c * imaginary parts of EGC-Nakagami C H F of the Ith channel, respectively, r; is a Nakagami envelope in the Ith channel. The Nakagami P D F of rt (see Equation 1.3) is given by 5 . 3 . 5 CHF with Nakagami Fading 87 Using Equations 5.65 and 5.66, ^ E G C R e N a k ^ ) c a n D e expressed mathematically as </>EGC,Re,NakH = J Pr , fa) COS (^=j drt I 2 f m\ 2r„_, ( mrf\__fujri Applying Equation 3.952.8 of [19, p. 497] to the above equation results in the real part of EGC-Nakagami C H F of the Ith channel 0 E G C , R e , N a k M = 1^1 \™>\ Tj! " T ^ g ) • ( 5 - 6 8 ) Similarly, the 0 E G C , i m , N a k ( t i ; ) c a n D e expressed as CjC.Im.NakM = J Pr.fa) sin l^j= Using Equation 3.952.7 of [19, p. 497], the imaginary part of EGC-Nakagami C H F of the Ith channel is obtained as rf^M = ^ g ) " r (m + i) (m + i; | . (5.70) 5 . 3 . 6 Performance in Nakagami Fading 88 Substituting Equations 5.68 and 5.70 into Equation 5.65 yields the following equation </>EGC,Nak(^) = Y]_ 1=1 2' 4mLj ' J T{m) \n^L) T'[m + 2 / 1 ^ 7 C \ , . _ . 1 3 o ; 2 7 c \ X l F l ( m + 2 ; 2 ; - 4 r 7 I (5.71) By considering dual-branch diversity (L = 2) in the above equation, the imaginary part of 0EGC ,Nak(w) can be expressed as 2U / — \ 1/2 / Imag {0EGC ,Nak(o;)} = r [ m + - ) iFx ( m; ^ 1 o;27( 8m (5.72) 5.3.6 Performance in Nakagami Fading Similarly to Sections 5.3.2 and 5.3.4, by substituting Equation 5.72 into Equation 5.39, the performance of B P S K in Nakagami fading and impulsive noise with dual-branch E G C diversity reception can be expressed as oo oo p 7 / - E G C , N a k _ 1 , , ) } f > f > A? A? \ 4 T (m + | ) J W 2 ^ . B P S K - 2 e x P t - ^ + ^ 2 . T r ^ l 1 " * r(m) V 2m mi=0m2=0 L v ' x / y^ 1 e x p ( - y ) i F 1 ( m ; - ; i F i ( m + - ; - ; ^ — ) dy 2m 2 ' 2 ' 2m (5.73) 5 . 3 . 6 Performance in Nakagami Fading 89 Using Equation C l of [3], the above equation becomes 1 oo oo Ami Am2 ' m i = 0 m 2 = 0 i - z ' . / I ^ 1 1 3 %B2 %B2 4 r ( m + J) / ^ B 0 r T(m) V 2m (5.74) Applying Equation 9.183.2 of [19, p. 1010] to the above equation results in the performance of B P S K in Nakagami fading and impulsive noise with dual-branch E G C diversity reception - oo oo Ami Am2 D / / - E G C , N a k 1 „ r / A , A \ i NT^ A l ^2 ^ e . B P S K = e x P [ - ( A ! + A 2)] ^ — — mi=0 777,2=0 TTli! m2\ \ 4 r ( m + j) r(m) x 7 C 5 2 / l l 1 3 %B2 7c#2 2 ( 7 C £ 2 + m) V2' 2 2' 2' 2 ( 7 C £ 2 + m) ' 2 ( 7 C £ 2 + rn) (5.75) Considering m is a positive integer, the above equation can be expressed alternatively as 1 00 0 ° Ami A m 2 mi=0 7712=0 mi! m 2 ! 4 r ( m + i ) - 1 0F T(m) ^ ?=o x (1 -m)q (2q + l) q\ %B2 2{%B2 + m) 1 P / 1 1 1 %B2 2F1 \q+ - , - ~m--; 2 '2 ' 2' 2 ( 7 C £ 2 + m) (5.76) For the special case of m = 1 (i.e. Rayleigh fading), the above equation simplifies to Equation 5.50 which is indeed the equivalent results for the Rayleigh fading channel. For 5.3.6 Performance in Nakagami Fading 90 m = 1/2, 3/2, 5/2, • • •, Equation 5.75 can be rewritten as 2 ^—'„ mi! m 2 ! mi=0 1)12=0 ( l - m ) 0 r 7 C 5 2 4 r ( m + i ) ^ j q=0 X 2(%B2 + m) 9+1 2*1 9 + o . l ni 2' 2(7 c 5 2 + m) (5.77) From the above equation for the special case of Equation 5.77 at m = 1/2, it simplifies to °° °° A1711 A"12 mj =0 7Ti2=0 mi! m 2 ! . 4 . _! / %B2 1 S i l l [ A 7 TT" (5.78) which can be compared with the Gaussian expression presented in Equation 34 of [3]. Figures 5.29-5.32 illustrate the performance of B P S K in Nakagami fading and impulsive noise with dual-branch E G C diversity reception for T' = 1 0 - 4 and considering different values of m and A. As expected and similar to the Rayleigh and Rician fading channels, E G C diversity significantly improves the performance in Nakagami fading and impulsive noise, especially at high values of SNR. However, as the value of A decreases (i.e. A <C 10, see Figures 5.30-5.32), E G C diversity degrades the performance in Nakagami fading and impulsive noise at low values of SNR, because E G C diversity increases the P D F of the sum of two impulsive noises at low values of amplitude. As shown in the figures, the validity of the theoretical results have been verified by means of computer simulation. 5.3.6 Performance in Nakagami Fading 91 SNR/bit (dB) Figure 5.29: Performance of BPSK in Nakagami fading for different values of m and impulsive noise for f = 10~ 4 and A = 10 with dual-branch E G C diversity reception. 5 . 3 . 6 Performance in Nakagami Fading 92 10° -20 -10 0 10 20 30 40 SNR/bit (dB) Figure 5.31: Same caption as in Figure 5.29 but with A = 0.35. SNR/bit (dB) Figure 5.32: Same caption as in Figure 5.29 but with A = 0.01. 5 . 4 SC and E G C Performance Comparisons 93 5.4 SC and E G C Performance Comparisons The purpose of this section is to compare the B E R performance of SC and E G C diversity methods for the various fading channels, i.e. Rayleigh (see Figures 5.33-5.36), Rician (see Figures 5.37-5.40) and Nakagami (see Figures 5.41-5.44). It can be seen from the plots that E G C diversity has better performance than the SC diversity for all cases of fading at high values of SNR, because the E G C diversity raises the mean of the received signal and also reduces the P D F of the received impulsive noise at high values of amplitude (see Figure 2.7). However, as the values of A decreases (e.g. A <^  10), the E G C diversity degrades the B E R at low values of SNR, because it increases the P D F of the received impulsive noise at low values of amplitude, while the SC diversity does nothing to the P D F of the received impulsive noise. SNR/bit (dB) Figure 5.33: Comparison between the performance of BPSK with dual-branch SC and E G C diversity reception in Rayleigh fading and impulsive noise for T' = 10 - 4 and A = 10. 5 . 4 SC and E G C Performance Comparisons 94 SNR/bit (dB) Figure 5.34: Similar caption to Figure 5.33 but with A = 1. ImpA SC Rayleigh ImpA EGC Rayleigh -20 -10 0 10 20 SNR/bit (dB) Figure 5.35: Similar caption to Figure 5.33 but with A = 0.35. 5 . 4 SC and E G C Performance Comparisons 95 10u 10" 10" io"3^ tr LU m 10 10" 10" 10" ImpA SC Rayleigh ImpA EGC Rayleigh -20 -10 0 20 10 SNR/bit (dB) Figure 5.36: Similar caption to Figure 5.33 but with A = 0.01 30 40 10" 10 10" 10" DC LU CD 10" 10" 10 10" a —> K=0 b —> K=6dB c —> K=8dB ImpA in SC w/ Rice ImpA in EGC w/ Rice -20 -10 20 30 40 0 10 SNR/bit (dB) Figure 5.37: Comparison between the performance of BPSK with dual-branch SC and E G C diversity reception in Rician fading for different values of K and impulsive noise for r' = I O - 4 and A = 10. 5 . 4 SC and E G C Performance Comparisons 96 5 . 4 SC and E G C Performance Comparisons 97 10 10" 10"' 10 ' DC LU m 10 10" 10" 10" a —> K=0 b —> K=6dB c —> K=8dB d —> K=10dB ImpA in SC w/ Rice ImpA in EGC w/ Rice -20 -10 0 10 20 SNR/bit (dB) Figure 5.40: Same caption as in Figure 5.37 but with A = 0.01. 30 40 10 SNR/bit (dB) Figure 5.41: Comparison between the performance of BPSK with dual-branch SC and E G C diversity reception in Nakagami fading for m = 2 and impulsive noise for F' = 1 0 - 4 and A = 10. 5.4 SC and E G C Performance Comparisons 98 5.5 Conclusions 99 -20 - ImpA SC Nakagami — ImpA EGC Nakagami -10 0 10 20 SNR/bit (dB) Figure 5.44: Similar caption to Figure 5.41 but with A = 0.01. 5.5 Conclusions This chapter presents several new expressions for the performance of diversity combin-ing techniques. After an introduction, Section 5.2 analytical expressions are derived for the performance evaluation of B P S K in impulsive noise and Rayleigh, Rician and Nakagami fading with L-branch SC diversity reception and the results are evaluated and compared with simulation results. Additionally, in this section the SC PDFs are presented with different kinds of fading. In Section 5.3, several theoretical expressions of the error rate performance of B P S K in impulsive noise and different types of fading with dual-branch E G C diversity reception are presented. Furthermore, the E G C CHFs with different classes of fading are introduced. The numerically evaluated theoretical and computer simulated results are presented together in the plots. In Section 5.4, the theoretical results from 5.5 Conclusions 100 Sections 5.2 and 5.3 are compared in the same figures by considering identical fading and impulsive noise channels. C H A P T E R 6 Conclusions and Suggestions for Future Research 6.1 Conclusions The major contribution of this thesis is the evaluation of the performance of mod-ulation schemes in the appearance of class A impulsive noise and fading with diversity reception through analysis and simulation. The contributions made throughout this thesis are summarized as follows. • The PDFs of the envelope and the /-component of the sum of two impulsive noises are obtained. Then, performance evaluations of B P S K and M P S K in impulsive noise with dual-branch E G C or without diversity reception are presented. The theoretical results are thoroughly validated by computer simulation. • Theoretical expressions of the error rate performance of B P S K in impulsive noise and different types of fading (i.e. Rayleigh, Rician and Nakagami) are proposed. The theoretical BERs very well match the simulated BERs. • Analytical expressions of the performance of B P S K in impulsive noise and different classes of fading with L-branch SC or dual-branch E G C diversity reception are de-rived and the theoretical results are verified by means of simulation. 6.2 Suggestions for Future Research 101 6.2 Suggestions for Future Research 102 The work done in this thesis can be complemented by several extensions suggested in the following sections. 6.2.1 M P S K or M Q A M with Diversity Combining Techniques The theoretical expression of the error rate performance of M Q A M in class A impul-sive noise with the dependence of the I and Q parts, with or without dual-branch E G C diversity reception, have yet to be derived. Then, the analytical expressions of the per-formance of M P S K or M Q A M in class A impulsive noise and fading in the L-branch SC or dual-branch E G C diversity can be derived. Furthermore, deriving performance evalu-ations for M P S K or M Q A M in impulsive noise and fading with the L = 3 E G C diversity reception is an interesting and challenging open research problem. 6.2.2 M P S K or M Q A M in Correlated Fading with Diversity Combining Tech-niques Assuming close spacing between the antennas and using diversity techniques, the re-ceived diversity signals are more likely subject to dependent fading than to independent fading. Thus, it would be interesting and useful to investigate the performance of M P S K and M Q A M in SC or E G C techniques with correlated fading and impulsive noise and compare the difference between the performance in non-correlated and correlated fading. 6.2.3 Performance of Diversity Combining Techniques with Coding It is well-known that coding techniques improve the B E R and channel bandwidth efficiency. It will be interesting to find out how coding techniques improve the performance 6 . 2 Suggestions for Future Research 103 of modulation schemes (i.e. M P S K or M Q A M ) in impulsive noise and fading (or correlated fading) with diversity reception. References [1] A. Annamalai, C. Tellambura and V . K. Bhargava, "Unified analysis of equal-gain diversity on Rician and Nakagami fading channels, " in Proc. IEEE WCNC'99, pp. 21-24, Sep. 1999. [2] A . Annamalai, C. Tellambura and V . K . Bhargava, "Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary Q A M on Nakagami fading channels," IEEE Trans. Commun., Vol. 47, No. 9, pp. 1335-1344, Sep. 1999. [3] A . Annamalai, C. Tellambura and V . K . Bhargava, "Equal-gain diversity receiver performance in wireless channels, " IEEE Trans. Commun., Vol. 48, No. 10, pp. 1732-1745, Oct. 2000. [4] A . Annamalai and C. Tellambura, "Error rates for Nakagami-m fading multichannel reception of binary and M-ary signals," IEEE Trans. Commun., Vol. 49, No. 1, pp. 58-68, Jan. 2001. [5] A . Annamalai, V . Ramanathan and C. Tellambura, "Analysis of equal-gain diversity receiver in correlated fading channels," in Proc. IEEE VTC Spring 2002, Vol. 4, pp. 2038-2041, May 2002. [6] A . Abdi and H. Hashemi, "On the P D F of the sum of random vectors," IEEE Transc. Commun., Vol. 48, No. 1, pp. 7-12, Jan 2000. [7] A . A . Abu-Dayya and N . C. Beaulieu, "Microdiversity on Rician fading channels, " IEEE Trans. Commun., Vol. 42, No. 6, pp. 2258-2267, Jun. 1994. [8] A . Chandra and S. Bhawan, "Measurements of radio impulsive noise from various sources in an indoor environment at 900 MHz and 1800 MHz, "Proc. 13th IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, pp. 639 -643, 2002. [9] A . Papoulis, Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 1991. 104 References 105 [10] A . D. Spaulding and D. Middleton,"Optimum reception in an impulsive interference environment-Part I: coherent detection," IEEE Trans. Commun., vol. COM-25, pp. 910-934, Sept. 1977. [11] A . D. Spaulding and D. Middleton, "Optimum reception in an impulsive interference environment-Part II: incoherent reception," IEEE Trans. Commun., Vol. COM-25, No. 9, pp. 924-934, Sep. 1977. [12] D. Middleton, "Statistical-physical models of electromagnetic interference," IEEE Trans, on Electromagn. Compat, Vol. EMC-19, No. 3, pp. 106-127, Aug. 1977. [13] D. Middleton, "Procedures for determining the parameters of the first-order canonical models of class A and class B electromagnetic interference," IEEE Trans. Electromagn. Compat, Vol. EMC-21, No. 3, pp. 190-208, Aug. 1979. [14] D. Middleton, "Canonical non-Gaussian noise models: their implications for measure-ment and for prediction of receiver performance," IEEE Trans. Electromagn. Compat., Vol. EMC-21, No. 3, pp. 209-220, Aug. 1979. [15] D. Middleton, "Canonical and quasi-canonical probability models of class A inter-ference," IEEE Trans. Electromagn. Compat, Vol. EMC-25, No. 2, pp. 76-106, May. 1983. [16] D. Middleton, "Non-Gaussian noise models in signal processing for telecommunica-tions: new methods and results for class A and class B noise models," IEEE Trans. Inform. Theory, vol. 45, No. 4, pp. 1129-1149, May. 1999. [17] E. J. Wegman, S. C. Schwartz and J. B. Thomas, Topics in Non-Gaussian Signal Processing , New York: Springer-Verlag, 1989. [18] G. Fedele, "Error probability for diversity detection of binary signals over Nakagami fading channels," in Proc. IEEE PIMRC'91 pp. 607-611, Sep. 1994. [19] I. S. Gradshteyn and I. M . Ryzhik, Table of Integrals, Series and Products, Academic Press, 6th edition, 2000. References 106 [20] J. W. Craig, "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations," in Proc. IEEE MILCOM'91, Conf., pp. 571-575, Nov. 1991. [21] J. H . Miller and J . B. Thomas, "The detecton of signals in impulsive noise modeled as a mixture process," IEEE Trans. Commun., vol. COM-24, pp. 559-563, May. 1976. [22] J. S. Lee, L. E. Miller, CDMA Systems Engineering Handbook, Boston, Mass. : Artech House, 1998. [23] J. G. Proakis, Digital Communications, McGraw-Hill Book Company, fourth ed., 2000. [24] J. G. Proakis and M . Salehi, Contemporary Communication Systems using Matlab, Brooks/Cole, 2000. [25] J. A . Roberts and J. M . Bargallo, "DPSK performance for indoor wireless Rician fading channels," IEEE Trans. Commun., Vol. 42, No. 2/3/4, pp. 592-596, Feb.-Apr. 1994. [26] J. Sun and I. S. Reed, "Performance of M D P S K , M P S K and non-coherent M F S K in wireless Rician fading channels," IEEE Trans. Commun., Vol. 47, No. 6, pp. 813-816, Jun. 1999. [27] J. S. Seo, S. J. Cho and K . Feher, "Impact of non-Gaussian impulsive noise on the performance of high-level Q A M , " IEEE Trans, on Electromagn. Compat., Vol. 31, No. 2, pp. 177-180, May. 1989. [28] J. F. Weng and S. H. Leung, "On the performance of D P S K in Rician fading channels with class A noise," IEEE Trans. Veh. Technoi, Vol. 49, No. 5, pp. 1934-1949, Sep. 2000. [29] K . S. Vastola, "Threshold detection in narrow-band non-Gaussian noise," IEEE Trans. Commun., Vol. COM-32, No. 2, pp. 134-139, Feb. 1984. References 107 [30] K . Yoo-Shin, "Performance of high level Q A M in the presence of impulsive noise and co-channel interference in multipath fading environment," IEEE Trans. Broadcast., Vol. 36, No. 2, pp. 170-174, Jun. 1990. [31] L. A . Berry, "Understanding Middleton's canonical formula for class A noise," IEEE Trans. Electromagn. Compat, Vol. EMC-23, No. 4, Nov. 1981. [32] M . Alouini and M . K . Simon, "Performance of coherent receivers with hybrid S C / M R C over Nakagami-m, fading channels," IEEE Trans. Veh. Technoi, Vol. 48, No. 4, pp. 1155-1164, Jul. 1999. [33] M . Alouini and M . K . Simon, "Performance analysis of coherent equal gain combining over Nakagami-m fading channels," IEEE Trans. Veh. Technoi, Vol. 50, No. 6, pp. 1449-1463, Nov. 2001. [34] M . D. Button and J. G. Gardiner, "Measurement of the impulsive noise environment for satellite-mobile radio systems at 1.5 GHz," IEEE Trans. Veh. Technoi, vol. 51, No. 3, pp. 551-560, May. 2002. [35] M . Nie and L. Lu, "Simulation of microwave oven interference on digital radio commu-nication systems," Proc. IEEE 3rd Int. Symp. on Electromagn. Compat, pp. 513-516, 2002. [36] M . Patzold, Mobile Fading Channels, J. Wiley, 2002. [37] M . Shao and C. L. Nikias, "Signal processing with fractional lower order moments: Stable processes and their applications," Proc. IEEE, Vol. 81, No. 7, pp. 986-1010, Jul. 1993. [38] N . C. Beaulieu and A. A . Abu-Dayya, "Analysis of equal gain diversity on Nakagami fading channels," IEEE Trans. Commun., vol. 39, No. 2, pp. 225-234, Feb. 1991. [39] Q. T. Zhang, "Probability of error for equal-gain combiners over rayleigh channels: some closed-form solutions," IEEE Trans. Commun., Vol. 45, No. 3, pp.270-273, Mar. 1997. References 108 [40] R. L. Burden and J. D. Faires, Numerical Analysis, CA:Brooks/Cole Pub. Co., 7th edition, 2001. [41] R. Prasad, A . Kegel and A. de Vos, "Performance of microcellular mobile radio in a cochannel interference, natural, and man-made noise environment," IEEE Trans. Veh. Technoi, Vol. 42, No. 1, pp. 33-40, Feb. 1993. [42] S. Ambike, J. How and D. Hatzinakos, "Detection for binary transmission in a mixture of Gaussian noise and impulsive noise modeled as an alpha-stable process," IEEE Sign. Proc. Letters, Vol. 1, No. 3, Mar. 1994. [43] S. A . Kosmopoulos, P. T. Mathiopoulos and M . D. Gouta, "Fourier-bessel error per-formance analysis and evaluation of M-ary Q A M schemes in an impulsive noise envi-ronment," IEEE Trans, on Commun., Vol. 39, No. 3, pp. 398-404, Mar. 1991. [44] S. Haykin, Digital Communication, J. Wiley & Sons, 2000. [45] S. Miyamoto, M . Katayama and N . Morinaga, "Performance analysis of Q A M systems under class A impulsive noise environment," IEEE Trans, on Electromagn. Compat., Vol. 37, No. 2, pp. 260-267, May. 1995. [46] S. M . Zabin and H. V . Poor, "Parameter estimation for Middleton class A interference processes," IEEE Trans, on Commun., Vol. 37, No. 10, pp. 1042-1051, Oct. 1989. [47] S. M . Zabin and H. V . Poor, "Recursive algorithms for identification of impulsive noise channels," IEEE Trans. Info. Theor., Vol. 36, No. 3, pp. 559-578, May. 1990. [48] S. M . Zabin and H. V . Poor, "Efficient estimation of class A noise parameters via the E M algorithm," IEEE Trans, on Commun., Vol. 37, No. 1, pp. 60-72, Jan. 1991. [49] S. M . Zabin and H. V . Poor, "Nonparametric density estimation and detection in impulsive interference channels part I: estimators," IEEE Trans, on Commun., Vol. 42, No. 2-4, pp. 1684-1697, Feb-Apr. 1994. References 109 [50] T. K . Blankenship, D. M . Kriztman and T. S. Rappaport, "Measurements and sim-ulation of radio frequency impulsive noise in hospitals and clinics," Proc. IEEE 47th Veh. Technoi. Conf., pp. 1942-1946, May. 1997. [51] T. S. Rappaport, Wireless Communications, Englewood Cliffs, NJ : Prentice-Hall 1996. [52] U . Dersch and R. J. Riiegg, "Simulations of the time and frequency selective outdoor mobile radio channel," IEEE Trans. Veh. Technoi, Vol. 42, No. 3, pp. 338-344, Aug. 1993. [53] V . Aalo and S. Pattaramalai, "Average error rate for coherent M P S K signals in Nak-agami fading channels," IEEE Electro. Letters, Vol. 32, pp. 1538-1539, Aug. 1996. [54] W. C. Jakes, Jr., Microwave Mobile Communications. New York, N Y : J. Wiley & Sons, 1974. Reprint by IEEE Press, 1994. APPENDIX A Theoretical Derivations A . l Derivation of Equation 3.10 In L-branch E G C diversity with Gaussian noise, the sum y of Gaussian noises from all channels can be expressed as y = x1 + x2-\ YxL (A.l) where Xi is Gaussian noise in the ith channel and i = 1,2, • • • , L . The P D F of Xi is given by h W = i;ap(4)' (A-2) The characteristic function of Xi can be calculated by 1 f°° f x 2 ^ X I ^ = VZTUT J 6 X P \ 2 o * + iUXi ) d X i ' (A'3) Using Equation 3.323.2 of [19, p. 333] to the above equation leads to / 2 2 \ 0 x . H = e x p ( - ^ - ) . (A.4) Then, the characteristic function of y is obtained by </>» = [<t>xM]L = exp • (A.5) 110 A.2 Derivation of Equation 4.6 1 1 1 The P D F of y can be expressed as I [°° 1 / y2 \ P y i v ) = 2 7 ]_„ ^ M e x ^ " ^ ) duJ = T l ^ f e x P J • ( A - 6 ) The performance of B P S K in Gaussian noise with L-branch E G C diversity can be derived by ^ B P T K = Pr{Ri <-0) = Pr (y < -L^Eb) = f ^py{y)dy Gaussian J —OO A ^ . (A .7 ) The relation between signal energy per bit Eb and SNR/bit ~fb is lb = 7 i + 72 H h 7L ~ 2a 2 2a 2 2cr 2 L K 2a 2 (A .8 ) where 7 * is the signal-to-noise ratio per bit of the ith channel. Substituting Equation A . 8 into Equation A . 7 , Equation 3.10 follows. A . 2 D e r i v a t i o n o f E q u a t i o n 4 . 6 To find the closed-form solution for the integral A.2 Derivation of Equation 4.6 112 where a and b are real numbers, and c is a positive integer. The variable y = [1 — \Jx1 j{x2 + b) ]/2 is introduced. Then, Equation A.9 can be rewritten as [•p^-^f^-vr**, (A.io) where d = [1 — \Ja2/(a2 + b) ]/2. Using Equation 8.391 of [19, p. 900], the above equation becomes d x =22c-lb-c-2F1(c,l-c;c+l;d) a (x2 + 6)C+V2 c = 2 2 c - 1 6 - c - y ( ^ ) r a ( 1 , C ) , n dn. ( A . l l ) c £ ^ (c+l)nn! n=0 Substituting (c) n = (c + n - l)!/(c - 1)!, (1 - c)n = ( - l ) n ( c - l)!/(c - n - 1)! and (c + l ) n = in + c)!/c! into Equation A . l l gives f00 d x - 2 2 c~ V c < f V ( - i ) n r ( c ) r ( c + n) 7a { X 2 + h ) c + i / 2 - 2 b d ^ , r ( c _ n ) r ( c + n + 1 ) i 1 U rf)J • The last term [1 — (1 — d)]n can be expanded by using power series Equation 1.111 of [19, p. 25] and then the following expression is obtained r dx 0 2 0 - 1 . - ^ y ( - i ) * r ( c ) r ( n + c ) A I K Ja (x 2 +6)^/2 Z . r ( C - n ) r ( n + c + l ) l^k\{n-k)\K > x (1 - d)k t,o „ = t r ( c - n ) r ( n + c+l)*!(n-*)! A.2 Derivation of Equation 4.6 113 Changing variable n = I + k in Equation A. 13 results in f00 ^ = 2 ^ b - ^ T C T r (c)r ( ; + fc + c ) ( - i y g d)> i A U ) Substituting the term (-l)'/r(c - I - k) with r(/ + k - c + l)/[T(c - k)T(k - c + 1)] into the above equation leads to f°° dx = 22c-u-c,c (1 - d)T(c) c ^ k r(l + k + c)T(l + k-c+l) Ja (x 2 + 6)=+V2 k\ ^ T(l + k + c+l)T(c-k)l\ x r- ( A-!5) r(fc-c+l) . At / > c — the term r(Z + /c — c + 1) is zero, and the above equation can be expressed alternatively as f°° dx = ^ ( l - d ) * T ( c ) T(k + c) Ja (x 2 + 6 ) c +V2 k \ T(c- k)T(k + c+l) V r ( * + fc + c ) r ( ^ +  k -  c + !) r(fc + c + 1) v X Applying Equation 9.122.1 of [19, p. 998] to the last summation term, the above equation is reduced to a (a:2 + 6) c+ 1/2 ^ fc!T(2c) Rewriting Equation A. 17 by substituting d yields the closed-form solution for Equation ) A.2 Derivation of Equation 4.6 1 1 4 A . 9 J a dx (x2 + 6) C +V2 ->2c-l a2 + b c - i x k=0 X r(fc + c)r(c) r(2c) (A.18) 

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