Performance Analysis of Wireless FadingChannels: A Unified Approach

by

Omar Alhussein

B.Sc., Khalifa University, 2013

Thesis Submitted in Partial Fulfillmentof the Requirements for the Degree of

Master of Applied Sciences

in theSchool of Engineering ScienceFaculty of Applied Sciences

© Omar Alhussein 2015SIMON FRASER UNIVERSITY

Summer 2015

All rights reserved.However, in accordance with the Copyright Act of Canada, this work may be

reproduced without authorization under the conditions for “Fair Dealing.”Therefore, limited reproduction of this work for the purposes of private study,

research, criticism, review and news reporting is likely to be in accordance withthe law, particularly if cited appropriately.

Approval

Name: Omar Alhussein

Degree: Master of Applied Sciences (Engineering Science)

Title: Performance Analysis of Wireless Fading Channels: AUnified Approach

Examining Committee: Rodney G. Vaughan (chair)Professor

Jie LiangSenior SupervisorAssociate Professor

Sami MuhaidatCo-SupervisorAssociate ProfessorKhalifa UniversityUnited Arab Emirates

Paul K. M. HoInternal ExaminerProfessor

Date Defended: 26 August 2015

ii

Abstract

This thesis presents two major contributions. First, we consider a unified approach tomodel and simplify wireless fading channels or potentially fading scenarios by mixture dis-tributions, namely using the mixture of Gaussian (MoG) and the mixture Gamma (MG)distributions. The approximation methodologies rely on maximum a posteriori and likeli-hood estimation techniques, such as the expectation-maximization and variational Bayes.Through the use of the mean-square error and the Kullback-Leibler divergence measures,we show that our models provide similar accuracy yet simpler representation than otherexisting models. In addition, we provide closed-form expressions or approximations for sev-eral performance metrics used in wireless communication systems, including the momentgenerating function, the raw moments, the amount of fading, the outage probability, andthe average channel capacity.

Second, through the use of the MoG and MG distributions, we provide a unifying andverstaile performance analysis over intricate generalized and composite fading channels inseveral contemporary wireless research topics, such as cognitive radio networks, cooperative-and diversity-based communications, and impulsive noise environments. The new approachand proposed distributions resolves intractable problems in many other fields, such as cog-nitive radio networks, cooperative networks, cascaded wireless applications and others.

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Dedication

To my parents, brothers and Ghina

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Acknowledgements

First and foremost, I would like to sincerely thank my two remarkable and startling advisors,Prof. Jie Liang and Prof. Sami Muhaidat, for their everlasting all-inclusive care and support.It indeed is a solace to be surrounded by advisors who have tremendous sheer of experience,motivation, special skills and insight; I was honoured to experience that and will remain todo so.

Second, I would like to thank Prof. George Karagiannidis for his continuous passionatesupport and technical guidance. I would also like to thank Prof. Imtiaz Ahmed for hiseverlasting support and advice in various technical and non-technical issues. I would alsolike to thank Prof. Paschalis Sofotasios for the nice and nurturing collaborative experiencewith him. I would like to thank my colleagues, Bassant Selim and Ahmed Al-Hammadi; itwas a benefecial collaborative experience.

My sincere thanks also goes to Prof. Paul Ho and and Prof. Rodney G. Vaughan fortheir keen comments and encouragement, which inspired me to widen my research fromvarious perspectives. Through out this degree, I have been enrolled in several informa-tive courses that were taught by Prof. Greg Mori, Prof. Faisal Beg, Prof. Paul Ho, andProf. Daniel C. Lee; I sincerely appreciate such an exemplary dedication and captivatingand thought provoking sessions. I also would like to thank Prof. Ivan Bajic for supportingand believing in me.

I would like to thank my colleagues, namely Koos van Nieuwkoop and Xiao Luo for theshort yet enjoyable collaborative experience. I would also like to thank my colleagues, labmates, and friends for the good times spent together. I am very grateful for the faculty andsta� of the school of engineering science and graduate student society for their support andhelp.

Lastly, I would like to say that no matter how many times I thank my beloved parentsand siblings, it would not be su�cient. Their patience, compassion, and advices–which aremost often not followed–is what urges me to better myself.

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ContentsApproval

Abstract

Dedication

Acknowledgements

Table of Contents

List of Figures

List of Symbols

List of Acronyms

1 Introduction 11.1 Introductory Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Mixture Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Diversity Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Modeling of Wireless Fading Channels using MoG Distribution . . . 51.2.2 Modeling of Wireless Fading Channels using MG Distribution . . . . 51.2.3 Energy Detection over Generalized and Composite Fading Channels 61.2.4 Analysis of Wireless Systems over Generalized and Composite Fading

Channels with Impulsive Noise . . . . . . . . . . . . . . . . . . . . . 61.3 Scholarly Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Journal Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Conference Publications . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Modeling of Wireless Fading Channels using MoG Distribution 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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2.2.1 The Nakagami/Lognormal Channel . . . . . . . . . . . . . . . . . . . 92.2.2 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 The Ÿ ≠ µ and ÷ ≠ µ Fading Models . . . . . . . . . . . . . . . . . . 112.2.4 The Ÿ ≠ µ Shadowed Fading model . . . . . . . . . . . . . . . . . . . 12

2.3 The MoG Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Parameter Estimation via Expectation-Maximization . . . . . . . . . 132.3.2 The pdf of the Instantaneous SNR of the MoG Model . . . . . . . . 162.3.3 Determining the optimal number of mixture components . . . . . . . 162.3.4 MoG Model Analysis and Comparisons . . . . . . . . . . . . . . . . 182.3.5 Parameter Estimation via Variational Bayes . . . . . . . . . . . . . . 202.3.6 Comparison between EM and VB . . . . . . . . . . . . . . . . . . . . 22

2.4 Performance Analysis of Wireless Channels . . . . . . . . . . . . . . . . . . 242.4.1 Moment Generating Function . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Raw Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 Amount of Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.4 Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.5 Average Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . 272.4.6 Symbol Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Modeling of Wireless Fading Channels using MG Distribution 343.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Proposed Model and Approximation Methodology . . . . . . . . . . . . . . 35

3.2.1 Proposed Model Analysis and Comparisons . . . . . . . . . . . . . . 373.3 Applications of Mixture Gamma . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.0.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Raw Moments of the end-to-end SNR . . . . . . . . . . . . . . . . . 40

3.3.1.1 Amount of Fading (AoF) of the end-to-end SNR . . . . . . 413.3.1.2 Average Channel Capacity of the end-to-end SNR . . . . . 41

3.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Energy Detection over Generalized and Composite Fading Channels 444.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Probability of Detection over Composite Fading Channels . . . . . . . . . . 47

4.3.1 Single-antenna Scenario . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Diversity Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.2.1 Square-Law Combining . . . . . . . . . . . . . . . . . . . . 48

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4.3.2.2 Square-Law Selection . . . . . . . . . . . . . . . . . . . . . 524.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 524.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Analysis of Wireless Systems over Generalized and Composite FadingChannels with Impulsive Noise 565.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Middleton’s Class-A and ‘-Mixture Impulsive Noise Models . . . . . . . . . 575.3 Approximating Fading Models with the MG Distribution . . . . . . . . . . 585.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.4.2 Pairwise Error Probability . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.2.1 Pairwise Error Probability for MRC . . . . . . . . . . . . . 615.4.2.2 Pairwise Error Probability for Selection Combining . . . . 655.4.2.3 Symbol Error Probability and Bit Error Probability . . . . 66

5.4.3 Average Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . 675.5 Simulations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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6 Conclusions and Future Work

Bibliography

Appendix A Mixture of Gaussian Parameters

Appendix B Sample Code for Section 2.3.3

Appendix C Mixture Gamma Parameters

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List of Figures

Figure 1.1 Pathloss, shadowing and multipath versus distance [1]. . . . . . . . 2

Figure 2.1 Normalized BIC versus the number of components . . . . . . . . . . 17Figure 2.2 Optimal number of components versus the amount of fading . . . . 18Figure 2.3 MoG approximation for di�erent channel models. . . . . . . . . . . 19Figure 2.4 Bayesian networks representing EM (left) and VB (right). Both

models use three parameters to model the underlying mixture ofGaussians (mixing coe�cients as well parameters describing eachGaussian in the mixture). In the VB case these are treated as randomvariables, while in the EM case they are treated as deterministicparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 2.5 EM- (left) and VB-based (right) approximations of the NL distribu-tion with m = 2 shown in black. The blue, red, and green distribu-tions correspond to 100, 1000, and 10,000 data points respectively.In both, the number of classes C = 6. . . . . . . . . . . . . . . . . . 23

Figure 2.6 Low number of observations EM- (blue) and VB-based (red) approxi-mations of the NL distribution with m = 2 shown in black. The plotswere trained with C = 6 (left) and C = 2 (right). Both were trainedusing N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 2.7 Low number of observations EM- (blue) and VB-based (red) approxi-mations of the NL distribution with m = 2 shown in black. The plotswere trained with C = 6 (left) and C = 2 (right). Both were trainedusing N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 2.8 Analytical and simulated outage probability versus “th for two sce-narios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 2.9 Analytical and simulated outage probability versus “ for two scenarios. 30Figure 2.10 Analytical and simulated average channel capacity, B=1

2

. . . . . . . 31Figure 2.11 Analytical and simulation SER of 2-branch MRC diversity receiver

for BPSK signaling scheme for RL and NL fading channels. . . . . 31Figure 2.12 Analytical and simulation SER of L-branch MRC diversity receiver

for 16-QAM signaling scheme for for various Ÿ ≠ µ Shadowed fadingscenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Figure 3.1 Approximation of instantaneous SNR distribution of Lognormal, Weibull,and RL for M = 0 dB. . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 3.2 Approximation of instantaneous SNR distribution of several scenar-ios of NL distributions for M = 0 dB, ’ = 1

2

dB, with varying m. . 38Figure 3.3 Fixed-gain dual-hop cooperative communication system. . . . . . . 39Figure 3.4 Average channel capacity for the selected scenario with U = 0.5,

B = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 4.1 Complimentary ROC for various fading channels and no diversity,with “

0

= 10 dB, u = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.2 Complimentary ROC for composite NL fading scenarios and SLC

scheme, with L = 2, “0

= 10 dB, u = 2. . . . . . . . . . . . . . . . . 54Figure 4.3 Complimentary ROC for selected NL fading scenario and SLS scheme,

with varying L and “0

= 10 dB, u = 2. . . . . . . . . . . . . . . . . 54

Figure 5.1 The BIC versus number of components for various MG based fadingchannels using the BIC. The considered BIC is normalized to havezero mean and unit variance. . . . . . . . . . . . . . . . . . . . . . . 60

Figure 5.2 Analytical and simulated SEP of BPSK with 4-MRC and 2-SC schemesfor various MG based fading channels with MCA Noise of ⁄, A = 0.1,and C = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 5.3 Analytical and simulated SEP of BPSK with MRC scheme for NLfading contaminated with MCA Noise of ⁄ = 0.1, A = (0.1, 0.3, 0.9),and C = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Figure 5.4 Average channel capacity with and without MRC diversity for NLand ÷ ≠ µ fading channels contaminated with ‘-mixture noise with“ = 10 dB and › = 75. . . . . . . . . . . . . . . . . . . . . . . . . . 71

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List of Symbols

fX(.), pX(.) Probability density function of random variable X

FX(.) Cumulative density function of random variable X

Pr{.} Probability operatorE[.] Expected value{.}T Transposition{.} Average value{.} Estimated valueœ Belongs toR Real numbersN Natural numbersCN (µ, ‡2) Complex Gaussian random variable with mean µ and variance ‡2

W(x|W, ‹) Wishart distribution over X = x with scale matrix W and degreesof freedom ‹

N (x|µ, ‡2) Gaussian distribution over X = x with mean µ and variance ‡2

Dir(x|–) Dirchlet distribution over X = x with concentration parameter –

L(◊|y) Logarithm of the likelihood function given ◊ given some observationy

KL(pX (.) ||pY (.)) Kullback-Leibler divergence between probability distributions of X

and Y

V ar{X} Variance of X, that is E[(X ≠ E[X])2]o (.) One of the Landau symbols defined as f = o („) which implies that

f„ æ 0

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List of Acronyms

AIC Akaike Information CriterionAF Amplify-and-ForwardAoF Amount of FadingAWGN Additive White Gaussian NoiseBEP Bit Error ProbabilityBFSK Binary Frequency Shift KeyingBIC Bayesian Information CriterionBPSK Binary Phase Shift KeyingCDF Cumulative Distribution FunctionCR Cognitive RadioCSI Channel State InformationED Energy DetectionEGC Equal Gain CombiningEM Expectation-MaximizationGL Gamma-LognormalIG Inverse-Gaussiani.i.d. Independent and Identically DistributedIMI Instantaneous Mutual Informationi.n.i.d. Independent but not Identically DistributedKL Kullback-LeiblerLOS Line-of-SightLRT Likelihood Ratio TestMCA Middleton’s Class-AMGF Moment Generating FunctionMG Mixture GammaMoG Mixture of GaussianMIMO Multiple-Input-Multiple-OutputMLE Maximum Likelihood Estimation

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MPSK M-ary Phase Shift KeyingMRC Maximal Ratio CombiningMSE Mean-Square ErrorNL Nakagami-LognormalNLOS Non-Line-of-SightPDF Probability Density FunctionPEP Pairwise Error ProbabilityPU Primary UserQAM Quadrature Amplitude ModulationRV Random VariableRL Rayleigh-LognormalRIGD Rayleigh/Inverse-Gaussian DistributionROC Receiver Operating CharacteristicSC Selection CombiningSU Secondary UserSER Symbol Error RateSEP Symbol Error ProbabilitySISO Single-Input-Single-OutputSLC Square-Law CombiningSLS Square-Law SelectionVB Variational Bayes

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Chapter 1

Introduction

1.1 Introductory Background

1.1.1 Fading Channels

Modeling the terrestrial wireless propagation is of importance for the design and perfor-mance analysis of wireless communication systems. Unlike wired propagation scenarioswhich are stationary and predictable [2], in a typical mobile radio propagation scenario,the received signal presents both small-scale power fluctuations–also known as multipathfading–and large-scale power fluctuations–also known as shadowing. In the former, thetransmitted signal experiences reflection, di�raction, scattering, and delays, whereby thethe multipath phenomenon is caused by a relatively fast constructive and destructive ran-dom combinations of the received signal copies. Large-scale signal power fluctuations, onthe other hand, are caused by the presence of large obstacles between the transmitter andreceiver. Fig. 1.1 illustrates the received power versus the distance, and it illustrates thepath loss, small-scale fading and the large-scale fading phenomena [1].

The small-scale fading results in very rapid fluctuations around the mean signal level,while large-scale gives rise to relatively slow variations of the mean signal level [3]. The Log-normal distribution was shown adequate to model the large-scale phenomena [4, 5, 6]. Asfor the small-scale phenomenon, several conventional statistical models, such as Rayleigh,Nakagami-m, Weibull and Rice distributions have been widely used to characterize it de-pending on some environment conditions. For instance, the Rayleigh and the Nakagami-mdistributions are most often utilized in characterizing non-line-of-sight (NLOS) environ-ments. The Rice–also known as the Nakagami-n–distribution is used to model environ-ments containing one dominant component, which often is the line-of-sight (LOS) path.The Weibull distribution, which is also commonly used in reliability engineering and sur-vival analysis, was was shown to provide an excellent fit for indoor [7] and outdoor en-vironments [8][9]. Unlike the Rayleigh and Nakagami-m models, the Weibull distributionhas no phenomonological theory or theoretical statistical fading model that justifies it, but

1

Figure 1.1: Pathloss, shadowing and multipath versus distance [1].

rather it was shown adequate in fitting real experimental data among numerous candidatedistributions, largely because of the flexibility associated with this distribution [7]. Indeed,this was one of the observations that motivated us to carry out our line of work, as is goingto be explained later.

Although the aformentioned conventional statistical models were shown to provide widerange of applicability, they are still not able to exactly fit all existing experimental data[6]. Additionally, the adequacy of the Nakagami-m and the Rayleigh distributions havebeen questioned [10], due to the fact that they assume homogeneous di�use scattering fieldswhich inevitably is not a realistic assumption especially in spatially correlated non-linearenvironments [6]. Due to the aformentioned facts and with the emergence of relativelynew wireless mediums, such as which occurs in underwater acoustic [11, 12] and bodycommunication [13, 14, 15, 16] fading channels, the wireless research community have hada reincarnated interest in finding more accurate and generalized fading models that providea better fit to new and realistic measurements. Consequently, new generalized small-scalefading models, such as the –≠µ [17], Ÿ≠µ, and ÷≠µ [18] distributions, were proposed. The–≠µ model was derived with the assumption of spatially correlated surfaces to characterizethe nonlinearity in propagation mediums, and it includes several conventional models, suchas the Gamma, Nakagami-m, exponential, Weibull, and one-sided Gaussian. The Ÿ≠µ andthe ÷ ≠ µ distributions are general fading distributions for LOS and NLOS applications,respectively. These distributions can represent the Rice (Nakagami-n), the Nakagami-m,the Rayleigh, the One-Sided Gaussian, and the Hoyt (Nakagami-q) distributions as specialcases.

Composite fading scenarios occur when both the small-scale the large-scale fading arepresent, where typically the Lognormal distribution is mixed with some small-scale fading

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distribution. Composite models can be derived in two ways, either by assuming that thetotal power is subject to random fluctuations or by assuming that only dominant compo-nents are subject to random fluctuations. The former class is called multiplicative shadowfading model whereas the latter class is called LOS shadow fading models [12].

A common example of composite fading channels is the Nakagami/Lognormal (NL).In this case, the density function is obtained by averaging the instantaneous Nakagami-mfading average power over the conditional probability density function (pdf) of the Lognor-mal shadowing, resulting in a complicated pdf that has no closed form expression. The K

[19] and generalized-K (KG) distributions [20] have been introduced as relatively simplermodels to characterize the NL fading model, in which the Lognormal distribution is re-placed by the Gamma distribution in the Rayleigh/Lognormal (RL) and NL distributions,respectively. The K and KG distributions contain the modified Bessel function of the firstor second kind, which complicates further analytical performance measures. In [5], theLognormal distribution was replaced by the Inverse-Gaussian (IG) distribution, resultingin the Rayleigh/IG (RIGD) model, followed by its generalized versions, the G-distribution[21], the Ÿ ≠ µ/IG [22] and the ÷ ≠ µ/IG [23]. The drawback of these distributions is theirincreased complexity due to the presence of the modified Bessel function of the second kind.

Recently, an interesting work has been proposed by Atapattu et al. [24], where sev-eral channel models were expressed as a mixture Gamma (MG) distribution via Gauss-Quadrature approximations. The MG distribution is more accurate than the aforemen-tioned alternatives, and it has the advantage of simplicity as well. The MG distribution isproposed to approximate several fading models, namely NL, K, KG, ÷ ≠ µ, Ÿ ≠ µ, Hoyt,Rician, and Nakagami-m. One caveat to keep in mind is that, similar to the Weibull dis-tribution, the MG distribution is proposed using a purely mathematical approach and itdoesn’t rely on any phenomenological theory.

In this thesis, similar to the aformentioned works, the main objective is to simplifythe representation of composite fading models and thereby provide verstaile and unifiedperformance analysis over all fading channels. Unlike all the existing works, this work isable to generalize to arbitrarily any fading channel or even function of fading channels. Thisis performed by generalizing this problem as a maximum likelihood estimation problem andsolving it by utilizing mixture distributions, such as the Mixture of Gaussian (MoG) andMG, in conjuction with advanced iterative clustering techinques, such as the expectation-maximization (EM) and the variational Bayes (VB). The objectives and contributions of thisresearch is resumed with greater detail in Section 1.2, while some more essential introductoryinformation is provided in Sections 1.1.2 and 1.1.3.

1.1.2 Mixture Distributions

Mixture distributions in general and Gaussian mixtures in particular provide a mathematical-based approach to statistical modeling of a wide variety of random phenomena [25]. They

3

are widely used in areas spanning adaptive signal processing, machine learning, patternrecognition, and statistical modeling. A distribution f(x) is a mixture distribution of C

components if it can be written as a convex linear combination as follows,

f(x) =Cÿ

i=1

Êi fi(x),

where f1

, f2

, ..., fC are not necessarily identical distributions, and Êi is the mixing weightwith the constraints

qCi=1

Êi = 1, and 0 < Êi Æ 1, ’ i = 1, ..., C. Commonly, the distribu-tions f

1

, f2

, ..., fC are chosen from the same parametric family, which in most applicationsare picked up from the exponential family of distributions. The choice of the compo-nent distributions is an application dependent problem. As a rough guideline, one wouldaim at choosing mixture components that yield: (1) feasible or appropriate approximationmethodology, (2) accurate modeling performance, and (3) relatively tractable algebraic rep-resentation if any further tools or statistics are needed to be derived afterwards.

In this thesis, the MoG and MG distributions are utilized to model generalized andcomposite fading channels. The approximation methodologies are based on the EM andthe VB algorithms.

1.1.3 Diversity Schemes

The composite e�ect of shadowing and multipath have determintal e�ects on the perfor-mance of communication systems. Diversity schemes is a viable approach to combat fad-ing by exploiting the availability of independent multiple copies of the transmitted signal.Diversity schemes rely on the fact that obtaining a deep fade, that is low instantaneoussignal-to-noise ratio (SNR), in all signal replicas is highly unlikely [4]. Diversity combiningcan be classified according to the method by which the signal replicas are obtained into:space diversity, cooperative diversity, time diversity, frequency diversity, polarization di-versity, etc. In this thesis, space and cooperative diversity schemes are considered. Spacediversity is perhaps the most intuitive scheme, where the signal is transmitted through dif-ferent physical propagation paths. This method works by employing multiple antennas atthe transmitter or the receiver, assuming that the seperation of the antennas is far enoughsuch that the received replicas have low correlation. Cooperative diversity can be thoughtof as space diversity where the multiple antennas belong to di�erent nodes. Diversity canbe also classified according to the combining technique used at the receiver. Several pureand hybrid combining techniques are proposed in the literature [4], of which maximal-ratiocombining (MRC), selection combining (SC), square-law combining (SLC), and square-lawselection (SLS) are considered in this work.

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1.2 Objectives and Contributions

In this thesis, the main objective is to provide a versatile and unifying performance analysisover generalized and composite fading channels, which mostly are intricate or cumbersometo deal with. To that end, we represent arbitrarily any fading channel–be it multipath,generalized, or composite–by the MG or MoG distributions. The approximation methodol-ogy relies on maximizing the a posteriori and likelihood estimates, where the EM and VBare utilized. The appropriateness and flexibility of the proposed frameworks and the newdistributions are asserted, while several analytical tools and statistics essential for the per-formance analysis of digital communication systems are derived. In addition, performanceanalysis that spans wide area of contemporary applications, such as diversity analysis, spec-trum sensing, and impulsive noise environments are carried out. In what follows is an outlineof our contributions presented in each chapter.

1.2.1 Modeling of Wireless Fading Channels using MoG Distribution

In Chapter 2, we consider a unified approach to model wireless channels by a MoG distribu-tion. The proposed approach provides an accurate approximation for the envelope and thesignal-to-noise (SNR) ratio distributions of wireless channels. First, we utilize the EM algo-rithm to estimate the parameters of the MoG distribution and further utilize the Bayesianinformation criterion (BIC) to determine the number of mixture components automatically.The Kullback-Leibler (KL) divergence and the mean-square error (MSE) criteria are used todemonstrate the accuracy of the proposed approximation. In addition, we explore potentialimprovements using the VB algorithm instead of EM. In contrast to EM, VB can providethe optimal number of mixture components without resorting to the BIC approach. Weshow that VB has other considerable advantages over EM, particularly when dealing withsmall number of observations.

Additionally, we provide closed-form expressions or approximations for several perfor-mance metrics used in wireless communication systems, including the moment generatingfunction (MGF), the raw moments, the amount of fading (AoF), the outage probability,and the average channel capacity. Numerical Analysis and Monte Carlo simulation resultsare presented to corroborate the analytical results.

1.2.2 Modeling of Wireless Fading Channels using MG Distribution

In Chapter 3, the MoG distribution is utilized to model wireless fading channels via theuse of EM and VB. We propose a new approach to represent di�erent fading distributionsby MG distribution. The new approach relies on the EM algorithm in conjunction withthe Newton-Raphson maximization algorithm. We show that our model provides similarperformance to other existing state-of-the-art models in both accuracy and simplicity, whereaccuracy is analyzed by means of MSE. In addition, we demonstrate that this algorithm may

5

potentially approximate any fading channel, and thus we utilize it to model both generalizedand composite fading models. A novel closed-form expression of the raw moments of a dual-hop fixed-gain cooperative network is derived, which paves the way to study the e�ectiveend-to-end average channel capacity in such networks. Numerical simulation results areprovided to corroborate the analytical findings.

1.2.3 Energy Detection over Generalized and Composite Fading Chan-nels

In Chapter 4, we evaluate energy detection based spectrum sensing over di�erent multipathfading and shadowing conditions. This is realized by means of a unified and versatile ap-proach that is based on the MG distribution. To this end, novel analytic expressions arefirstly derived for the probability of detection over MG fading channels for the conventionalsingle-channel communication scenario. These expressions are subsequently employed inderiving closed-form expression for the case of SLC and SLS diversity methods. The valid-ity of the o�ered expressions is verified through comparisons with results from respectivecomputer simulations. Furthermore, they are employed in analyzing the performance ofenergy detection over multipath fading, shadowing and composite fading conditions, whichprovides useful insights on the performance and design of future cognitive radio based com-munication systems.

1.2.4 Analysis of Wireless Systems over Generalized and Composite Fad-ing Channels with Impulsive Noise

In Chapter 5, we consider a single-input-multiple-output (SIMO) system over generalizedand composite MG-based fading channels in the presence of impulsive noise, which is mod-eled by Middleton’s Class-A (MCA) and ‘-mixture noise models. First, we develop a simpleand e�ective information theoretic approach to determine the optimal number of compo-nents for the MG distribution based on the Bayesian information criterion. We then derivenovel analytic pairwise error probability (PEP) expressions for the considered system byemploying MRC and SC schemes at the receiver. The o�ered PEP expressions involve finitesingle-fold integrals, which are further simplified to rather more tractable expressions ap-plicable for the special case of integer values of the involved scale parameter —k. Likewise,we provide analytical tractable expressions for the average channel capacity under the im-pulsive noise assumption for the considered system. Numerical analyses and Monte Carlosimulations are presented to validate the analytical results.

6

1.3 Scholarly Publications

The contributions in this thesis resulted in numerous publications; some of which are alreadypublished, while the rest are still under the review process. The publications are listedbelow:

1.3.1 Journal Publications

1. O. Alhussein, I. Ahmed, B. Selim, S. Muhaidat, J. Liang, G. K. Karagiannidis,“SIMO Diversity Receivers over Composite Fading Channels and Impulsive Noise: AUnified Approach,” IEEE Trans. Vehicul. Technol, submitted.

2. B. Selim, O. Alhussein, S. Muhaidat, G. K. Karagiannidis, and J. Liang, “ModelingAnd Analysis of Wireless Channels via the Mixture of Gaussians Distribution,” IEEETrans. Vehicul. Technol., accepted with minor revision.

3. A. A. Hammadi, O. Alhussein, S. Muhaidat, M. Al-Qutayri, S. Al-Araji, and G. K.Karagiannidis, “Unified Analysis of Cooperative Spectrum Sensing over Compositeand Generalized Fading Channels,” IEEE Trans. Vehicul. Technol., Mar. 2015,accepted with minor revision.

1.3.2 Conference Publications

1. O. Alhussein, A. Al Hammadi, P. C. Sofotasios, S. Muhaidat, J. Liang, M. Al-Qutayri, and G. K. Karagiannidis, “Performance Analysis of Energy Detection overMixture Gamma based Fading Channels with Diversity Reception,” in IEEE Int.Conf. Wireless and Mobile Comput., Netw. and Commun., Abu Dhabi, UAE, Oct.2015, accepted for presentation.

2. O. Alhussein, B. Selim, T. Assaf, S. Muhaidat, J. Liang, and G. K. Karagiannidis,“A Generalized Mixture of Gaussians for Fading Channels,” in IEEE Conf. Vehicul.Tech., Stocholm, May 2015, pp. 1–6.

3. O. Alhussein, S. Muhaidat, J. Liang, and P. D. Yoo, “A unified approach for repre-senting wireless channels using EM-based finite mixture of gamma distributions,” inIEEE Global Telecommun. Conf., Texas, Dec. 2014, pp. 1008–1013.

7

Chapter 2

Modeling of Wireless FadingChannels using MoG Distribution

2.1 Introduction

As mentioned in Section 1.1.1, existing alternative distributions and models are cumbersomeeither because they can only be represented as infinite integrals or due to the presence ofspecial functions, such as the modified Bessel function of the second kind. In this chapter,an alternative model, that represents both generalized and composite fading channels by theMoG distribution, is presented. The approximation methods are based on the EM and VBframeworks. The EM algorithm, which was coined by Dempster et al. in their seminal paper[26], is essentially a set of algorithms exceptionally useful for finding the maximum likelihoodestimator (MLE) of any distribution in the exponential family [27], and widely used for themissing data problem, i.e. modeling mixture distributions. VB can be thought of as the fullBayesian variant of EM, and therefore, it inherits several advantages as compared to theEM1. In contrast to EM, VB shall provide one with the ability to determine the numberof components without resorting to information theoretic approaches, which can be verytime-consuming and impractical for wireless channel modeling based applications, such asreal-time channel estimation and noise estimation applications. In addition, the over-fittingphenomenon or singularities that occur with the MLE based approach, namely in EM, areavoided.

The main contributions of this chapter can be summarized as follows:

• We propose the MoG distribution to model both the envelope of wireless channels us-ing either the EM or VB approaches. The proposed algorithms are shown to accuratelymodel both generalized and composite channels through a very simple expression.

1

In this thesis, although VB and EM frameworks essentially comprise set of algorithms, we will refer to

them as ’algorithms’.

8

• For the EM algorithm, we determine the appropriate number of components usingthe BIC.

• Qualitative and quantitative experiments and comparisons between the EM and VBalgorithms are carried out.

• We demonstrate the importance and tractability of our model by deriving severaltools for the performance analysis of single-user communications, such as the outageprobability and raw moments. Moreover, we derive the MGF, of which the symbolerror rate (SER) of L-branch MRC diversity system is presented for various signalingschemes.

• Numerical analysis and Monte Carlo simulation results are presented to corroboratethe derived analytical results.

The rest of this chapter is organized as follows: Section 2.2 gives a description of severalwireless fading channel models of interest. In Section 2.3, the MoG distribution is introducedtogether with a description of the EM and VB algorithms. In Section 2.4, performancemetrics, such as the MGF, the raw moments, the amount of fading, the outage probability,and the average channel capacity are derived using the MoG distribution. Simulation resultsand numerical analysis are presented in Section 2.5, while Section 2.6 concludes this work.It is noted this work in the present chapter is already published in [28, 29].

2.2 Fading Channels

Radio-wave propagation through wireless channels undergoes detrimental e�ects character-ized by multipath fading and shadowing. Modeling of such fading channels is typically acomplex process and often leads to intractable solutions. Considerable e�orts have focusedon the statistical modeling which resulted in a wide range of statistical models for fadingchannels. In this section, we give a brief description of some fading channels that are consid-ered in our approximations. It is noted that the approach presented within can be appliedto more fading models and scenarios.

2.2.1 The Nakagami/Lognormal Channel

The NL fading model is a mixture of Nakagami-m distribution and Lognormal distributionobtained by averaging the instantaneous Nakagami-m fading average power over the pdf ofthe Lognormal shadowing as follows

f–(–) =ˆ Œ

0

f–(–|‡)f‡(‡) d‡, (2.1)

where f–(–|‡) is the Nakagami-m distribution given by

9

f–(–|‡) = 2mm

‡m�(m)–2m≠1e≠m –

2

‡ . (2.2)

where �(.) is the gamma function [30] and m = E2{“}Var{“} is the fading parameter, which is

inversely proportional to multipath fading severity i.e., as m æ Œ, multipath severity di-minishes. The average power ‡ follows a Lognormal distribution, contributing to shadowingat longer routes, expressed as

f‡(‡) = ⁄Ô2fi‡’

e≠(10 log(‡)≠M)

2

2’

2 , (2.3)

where ⁄ = ln 10

10

, M and ’2, measured in dB, are the mean and variance of the GaussianRV V = 10 log

10

(‡), respectively. In order to compare (2.3) with that of the Gaussian RVX = ln(‡), the following relations apply [31]

X = ⁄V,

MX = ⁄M,

’X = ⁄’.

(2.4)

An important remark regarding the Lognormal distribution is that while ’ essentially definesdi�erent Lognormal distributions, M is e�ectively a scaling factor [31]. Denote Mn = 10M/10,then it is straightforward to show that

f–(–Mn) = 1Mn

f–(–|M = 0). (2.5)

Therefore, it is only su�cient to perform an approximation for M = 0 dB, and general-ize the results for other scaling factors. Let Es denote the energy per symbol, and N

0

be the single sided power spectral density of the complex additive white Gaussian noise(AWGN). Assuming E[|–2|] = 1, where E[.] denotes the expectation operator. By applyingthe following transformation to (2.1)

“ = –2“, (2.6)

where “ = E[“] = Es

N0

is the average signal-to-noise ratio (SNR), we obtain the Gamma/Log-normal (GL) distribution expressed as

f“(x) = 2⁄mm

�(m)Ô

2fi’

ˆ Œ

0

xm≠1

“m‡m+1

e≠ mx

“‡ e≠(10 log ‡)

2

2’

2 d‡. (2.7)

One should note that the Suzuki distribution [32] is also a composite RL fading model, butwith having

Ô‡ modeled as a Lognormal distribution. The SNR density function is not

expressed in a closed form, making the performance analysis of wireless communications

10

under this particular channel very complicated or intractable. Note that the RL distributionis a special case of NL distribution with m = 1.

2.2.2 The Weibull Distribution

The Weibull distribution provides an excellent fit for indoor [7] and outdoor environments[8][9]. For instance, it is was shown adequate in characterizing a multipath environmentassociated with mobile radio systems operating in the 800/900 MHz [4], and it is expressedas

f–(–) = m[�(1 + 2

m)� ]–m≠1e≠[

–

2

�

�(1+

2

m

)]

m

2 , (2.8)

where following (2.6), the corresponding instantaneous SNR distribution is written as

f“(“) = m

2 (�(1 + 2/m)“

0

)—/2“m/2≠1e

≠[

“

“

0

�(1+

2

m

)]

m/2

, (2.9)

where m is the Weibull fading parameter.

2.2.3 The Ÿ ≠ µ and ÷ ≠ µ Fading Models

The Ÿ≠µ fading model is mostly used to represent the multipath fading with LOS conditionand includes the following fading models as a special case: the Rice, the Nakagami-m,the Rayleigh, and the One-sided Gaussian. The instantaneous Ÿ ≠ µ SNR distribution isexpressed as [18]

f“(x) =µ(1+Ÿ

“ )µ+1

2

Ÿµ≠1

2 exp(µŸ)x

µ≠1

2 (2.10)

◊ exp(≠µ(1 + Ÿ)“

x) Iµ≠1

(2µ

ÛŸ(1 + Ÿ)x

“),

where Ÿ > 0 is the ratio between the total power of the dominant components and the totalpower of the scattered waves, µ > 0 is given by µ = E2{“}

Var{“}(1+2Ÿ)

(1+Ÿ)

2

, and Iµ(.) is the modifiedBessel function of the first kind and order µ [30, eq. (8.445)]. It is worth mentioning that asŸ tends to zero, the Ÿ ≠ µ distribution degenerates to the exact Nakagami-m distribution,with µ = m = E2{“}

Var{“} . In a similar manner, by setting µ = 1, the Ÿ ≠ µ distributiondegenerates to the exact Nakagami-n distribution, with Ÿ = n. Complementing the Ÿ ≠ µ

model, the ÷ ≠ µ model was proposed to represent NLOS multipath environments, where itincludes the Nakagami-q, the Nakagami-m, and the One-Sided Gaussian as a special case.

11

The instantaneous ÷ ≠ µ SNR distribution is expressed as [18]

f“(x) = 2Ô

fiµµ+

1

2 hµ

�(µ)Hµ≠ 1

2 “µ+

1

2

(2.11)

◊ xµ≠ 1

2 exp(≠2µhx

“) Iµ≠ 1

2

(2µH

“x),

where µ > 0 is given by µ = 1

2

E2{“}Var{“} + 1

2

E2{“}Var{“}(H

h )2, and parameters h and H can havetwo di�erent formats corresponding to two di�erent physical phenomena as follows: In For-mat1, h = 2+÷≠1

+÷4

and H = ÷≠1≠÷4

, where 0 < ÷ < Œ is interpreted as the power ratiobetween the independent in-phase and quadrature components. In Format2, the in-phaseand quadrature components are correlated and have a power ratio of unity. The two corre-sponding parameters are defined by h = 1

1≠÷2

and H = ÷1≠÷2

, where ≠1 < ÷ < 1 representsthe correlation between the in-phase and quadrature components. It is noteworthy that thetwo formats can be obtained from each other using the relation ÷

Format1

= 1≠÷Format2

1+÷Format2

. It isworth mentioning that the ÷ ≠ µ distribution degenerates to the Nakagami-q distributionby setting µ = 0, with q = Ô

÷ in Format1 and q =Ò

1≠÷1+÷ in Format2.

2.2.4 The Ÿ ≠ µ Shadowed Fading model

The Ÿ ≠ µ Shadowed fading model, was firstly proposed as an LOS shadow fading model[33], where unlike the NL formulation above, here it is assumed that only the dominantcomponents of the multipath clusters are subject to random fluctuations. The unconditionalinstantaneous SNR distribution of the Ÿ ≠ µ Shadowed model is obtained by averaging theconditional Ÿ ≠ µ distribution over the Nakagami-m distribution as follows,

f“(“) =ˆ Œ

0

f“|›(“; ›) f›(›) d›

= µ(1 + Ÿ)µ+1

2

“Ÿµ≠1

2

(“

“)

µ≠1

2 exp(≠µ(1 + Ÿ)““

) mm

�(m) �(“), (2.12)

where

�(“) ,ˆ Œ

0

2 exp(≠›2(µŸ + m))›2m≠µIµ≠1

(2µ›

ÛŸ(1 + Ÿ)“

“) d›,

which results in the following closed-form expression,

f“(“) = µµmm(1 + Ÿ)µ

�(µ)“(µŸ + m)m(“

“)µ≠1 (2.13)

◊ exp(≠µ(1 + Ÿ)““

)1

F1

(m, µ; µ2Ÿ(1 + Ÿ)µŸ + m

“

“),

where1

F1

(.) is the confluent hypergeometric function [30, eq. (9.210.1)]. Interestingly in avery recent work [34], it has been shown that under new formulation the Ÿ ≠ µ Shadowed

12

fading model can represent the ÷ ≠ µ distribution as a special case with µ = 2µ, Ÿ = 1≠÷/2÷,and m = m, where the underlined symbols belong to the Ÿ ≠ µ Shadowed model for thesake of clarity.

2.3 The MoG Distribution

We consider the problem of estimating the wireless channels’ density functions. Gaussianmixtures [35, 36, 37, 38, 39] are often used due to the fact that their individual densitiesare e�ciently characterized by the first two moments [40, 41]. The MoG distribution is at-tributed to have the “Universal-approximation” property, as it has been proven by Weiner’sapproximation theorem [35], which states that the MoG distribution can approximate anyarbitrarily shaped non-Gaussian density. The objective of this section is to provide a unifiedMoG distribution that can accurately represent di�erent fading channels.

2.3.1 Parameter Estimation via Expectation-Maximization

Let the ith entry of a random data vector y = {y1

, .., yN }, which represents the channelfading amplitude2 of the composite models, be regarded as incomplete data and modeledas a finite mixture of Gaussians as follows

pY (yi|◊) =Cÿ

j=1

Êj„(yi, ◊j), yi Ø 0, (2.14)

where i = 1, ..., N and C represents the number of components. Each jth component isexpressed as

„ (yi,◊j) = 1Ô2fi÷j

expA

≠(yi ≠ µj)2

2÷2

j

B

, (2.15)

where the weight of the jth component is Êj > 0, withqC

j Êj = 1. The parametersµj and ÷2

j correspond to the mean and variance of the jth component, respectively. Let◊ = ({Êj , µj , ÷2

j }Cj=1

), and the complete data X be the joint probability between Y and Z,where Z œ {1, .., C} is a hidden (latent) discrete RV that defines which Gaussian componentthe data vector Y comes from, namely

Pr{Z = j} = Êj , j = 1, .., C. (2.16)2

In this thesis, we do not indulge in modeling the phase of fading channels assuming flat fading channels.

13

Ideally, one would like to maximize the log-likelihood function as follows

◊MLE = arg max◊œ�

L(MoG)

(◊|y, C) = arg max◊œ�

ln p(y|◊, C),

= arg max◊œ�

Nÿ

i=1

ln# Cÿ

j=1

ÊjÔ2fi÷2

j

exp!≠(yi ≠ µj)2

2÷2

j

"$. (2.17)

One caveat to keep in mind when maximizing the log-likelihood is that it may have sin-gularities. This occurs when the kth component of the model has some mean µk equal toone of the data points yn. In this case, the likelihood function will degenerate to 1Ô

2fi÷k

.Thus, the maximization occurs when ÷k æ 0, which implies that each Gaussian componentis merely a Dirac delta function pointing at the particular yn; or in other words, it results inover-fitting. This issue is dealt with in Sections 2.3.3 and 2.3.5. Maximizing L

(MoG)

(◊|y, C)analytically is not tractable and is di�cult to optimize due to having a summation afterthe natural logarithm. Instead, the EM algorithm solves the MLE problem by maximizingthe complete-data log-likelihood function or the so-called Q-function3 as follows [42]

◊(t+1) = arg max◊œ�

Q1◊|◊(t)

2= arg max

◊œ�

EX|y,◊(m)

[log pX (X|◊)], (2.18)

where t is the iteration index. The reason beyond choosing to maximize Q1◊|◊(t)

2instead

of L(MoG)

(◊|y, C) is that it facilitates an iterative approach. Here the Q1◊|◊(t)

2can be

written as

Q(◊|◊(t)) =Nÿ

i=1

EZi

|yi

,◊(t)

[ln[pX(Zi|◊) pX(yi|Zi, ◊)]

=Nÿ

i=1

EZi

|yi

,◊(t)

[ln[CŸ

j=1

fiZi

j N Zi(yi|µj , ÷2

j )]

=Nÿ

i=1

Cÿ

j=1

EZi

|yi

,◊(t)

ËZi

!ln fij ≠ ln N (yi|µj , ÷2

j )"È

=Nÿ

i=1

Cÿ

j=1

EZi

|yi

,◊(t)

ËZi

È!ln fij ≠ ln N (yi|µj , ÷2

j )". (2.19)

Using existing result in [43, eq. (9.39)], the above equation simplifies to

Q(◊|◊(t)) =Nÿ

i=1

Cÿ

j=1

fl(t)ij

!ln fij ≠ ln N (yi|µj , ‡j)

", (2.20)

3

It is noted that the Q-function here is di�erent than the Gaussian Q-function, which appears in (2.45)

14

where fl(t)ij are the tth posterior probabilities (membership probabilities) given by

fl(t)ij = Pr{Zi = j |Yi = yi, ◊(t)} =

Ê(t)j „

1yi|µ(t)

j , ÷(t)j

2

qCl=1

Ê(t)l „

1yi|µ(t)

l , ÷(t)l

2 . (2.21)

The EM algorithm is performed by two iterative steps, namely the expectation step (E-step), and the maximization step (M-step). We set initial guesses of the MoG coe�cients,i.e. Ê(0), µ(0), ÷(0), whereby in the E-step, we compute the posterior probabilities using(2.21).

In the M-step, the coe�cients are updated by di�erentiating the Q-function with respectto Ê, µ, and ÷, resulting in the following analytical (t + 1)th estimates:

Ê(t+1)

j = 1N

Nÿ

i=1

fl(t)ij , j = 1, ..., C, (2.22)

µ(t+1)

j = 1NÊ

(t+1)

j

Nÿ

i=1

fl(t)ij yi, j = 1, ..., C, (2.23)

÷(t+1)

j = 1NÊ

(t+1)

j

Nÿ

i=1

fl(t)ij

1yi ≠ µ

(t)j

22

, j = 1, .., C. (2.24)

This iterative procedure is terminated upon convergence, that is when

L(t+1)

(MoG)

(◊|y, C) ≠ L(t)(MoG)

(◊|y, C) < ”, (2.25)

where ” is a preset threshold.The EM algorithm is guaranteed not to get worse as it iterates, i.e. L(t+1)

(MoG)

(◊|y, C) ÆL(t)

(MoG)

(◊|y, C) [26]. Hence, the lower ” is set, the more accurate the approximation wouldbe. In addition, one can always increase the accuracy by increasing the number of compo-nents. Though, this technique might be stuck in a local maxima, since the likelihood is amarginal distribution. However, one could mitigate this problem by heuristics and multipleinitial guesses. In this regard, Do et al. [44] suggest to initialize parameters in a way thatbreaks symmetry in mixture models. Finally, it is noteworthy to point out that the EMalgorithm has an advantage of being a completely unsupervised learning algorithm, whichmakes it very convenient for our density estimation application. For more details, one canrefer to [27, 42] and references therein.

15

2.3.2 The pdf of the Instantaneous SNR of the MoG Model

With the aid of the EM algorithm, all fading channels’ amplitudes can be represented as

f– (–) =Cÿ

j=1

ÊjÔ2fi÷j

expA

≠(– ≠ µj)2

2÷2

j

B

, – Ø 0. (2.26)

By taking the change of variables “ = “ x2, the pdf of the instantaneous SNR of the MoGdistribution can be written as

f“ (“) =Cÿ

j=1

ÊjÔ8fi“÷j

1Ô“

exp

Q

ca≠

1Ò““ ≠ µj

22

2÷2

j

R

db , “ Ø 0. (2.27)

2.3.3 Determining the optimal number of mixture components

Generally, when fitting a finite mixture distribution, the determination of an appropriatenumber of mixture components is inevitably a necessity. Choosing a small number of com-ponents would yield an inaccurate representation, while a very large number of componentswould unnecessarily increase the complexity of the distribution and may cause over-fitting.In addition, Chen [45] has shown that knowledge of the the number of components yields afaster optimal convergence rate for the estimates of a finite mixture than it normally wouldwhen the number of components is unknown.

In this subsection, in order to derive an appropriate number of mixture components, weadopt a simple yet e�ective unsupervised information theoretic criterion, called the BIC,which was introduced by Gideon Schwarz in [46].

Recall that y = {y1

, ..., yi, ..yN } correspond to N independent and identically distributed(i.i.d.) samples, drawn from any of the envelope distributions of the actual aforementionedfading models, then the log-likelihood function of the MoG distribution can be expressedas

L(MoG)

1◊|y, C

2= ln pY (y|◊, C) =

Nÿ

i=1

ln

Y]

[

Cÿ

j=1

ÊjÔ2fi÷2

j

expA

≠(yi ≠ µj)2

2÷2

j

BZ^

\ , (2.28)

where ◊ = (Ê, Ê2

, ..., ÊC , µ1

, µ2

, ..., µC , ÷1

, ÷2

, ..., ÷C) are the estimated parameters and C isthe corresponding number of components. The corresponding BIC score can be computedas

BICC = ≠2L(MoG)

1◊|y, C

2+ C ln N. (2.29)

It can be seen that the BIC penalizes the model complexity by adding the regularizationcoe�cient C ln(N). It is worth noting that although the EM algorithm maximizes the log-

16

0 2 4 6 8 10 12 14 16 18 2010

0

101

102

103

104

105

Number of Components

log 1

0(B

IC−BICoptimal+

1)

m=1,ζ=1

m=2,ζ=1

κ=1,µ=0.5

κ=1,µ=1

κ=7,µ=3

κ=1,µ=1,m=1

κ=1,µ=1,m=3

η=5,µ=10

η=0.5,µ=0.2

Figure 2.1: Normalized BIC versus the number of components

likelihood distribution, the BIC is an asymptotic approximation to the transformation of theBayesian a posteriori probability p(◊|y, C). As such, in a large-sample setting, the numberof components determined by the BIC is asymptotically optimal from the perspective of theBayesian posterior probability. Here we select the candidate model satisfying the minimumBIC score, satisfying asymptotically the maximum Bayesian posterior probability as

Copt = arg minCœN

BICC . (2.30)

Fig. 2.1 depicts the normalized BIC versus the number of components for some fadingscenarios selected from Section 2.2. The corresponding optimal number of components, Copt,indicated in the legend, will be adopted in the simulations and numerical results hereafterand will be denoted by C.

Fig. 2.2 shows the optimal number of components as a function of the AoF, which is ameasure of the severity of the respective fading channel. It is observed that as the fadingbecomes more severe, the mixture requires more components to accurately characterize thechannel.

In [47], Kass and Raftery has shown that as N tends to infinity, the quantity BICC1

≠BICC

2

can be roughly approximated by ≠2 ln BF12

, where BF12

= Pr(y|ˆ◊,C1

)

Pr(y|ˆ◊,C2

)

is called theBayes factor. Here the Bayes factor can be regarded as evidence provided by the datain favor of the model having C

1

components as opposed to the other model having C2

components. Furthermore, they showed that an evidence for one hypothesis in favor of theother is considered strong when ≠2 ln BF

12

> 6. Therefore, although here the minimum

17

0 2 4 6 8 10 12 14−0.5

0

0.5

1

1.5

2

2.5

3

Number of Components favored by BIC

Am

ou

nt

of

Fa

din

g

Figure 2.2: Optimal number of components versus the amount of fading

BIC score is selected, one can terminate the simulations if the change in two consecutiveBIC scores is not strong.

Note that Appendix B presents a useful sample code, written in MATLAB language,for the approximation of fading channels using the BIC assisted EM approach, as explainedin Sections 2.3.1 and 2.3.3.

2.3.4 MoG Model Analysis and Comparisons

In this section, several scenarios of the aforementioned fading channels are approximatedusing the MoG distribution, as in (2.27). The number of components was selected automat-ically using the BIC method explained in Section 2.3.3. We point out that higher accuracycan be achieved by increasing the number of components. In order to validate the accuracyof the approximations, we use two criteria of error, namely the MSE and the KL, definedas

MSE = EËf“ (x) ≠ f“ (x)

È(2.31)

and

KL1f“ (x) || f“ (x)

2=ˆ Œ

0

f“ (x) log f“ (x)f“ (x)

dx, (2.32)

18

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

f γ(x)

m=1, ζ=3dB, C=7, KL=2.2e−4, MSE=8.0e−6

m=4, ζ=1dB, C=4, KL=1.1e−4, MSE=8.3e−6

κ=1, µ=1, C=6, KL=1.6e−4, MSE=1.3e−5

κ=3, µ=1, C=5, KL=1.2e−4, MSE=1.0e−5

η=0.7, µ=0.4, C=8, KL=6.1e−4, MSE=6.9e−5

η=5, µ=10, C=3, KL=1.1e−4, MSE=2.3e−5

κ=1, µ=3, m=3, C=4, KL=1.2e−4, MSE=9.8e−6

Exact Distribution

Figure 2.3: MoG approximation for di�erent channel models.

respectively. Here f“ (x) is the exact pdf, and f“ (x) is the approximated pdf (MoG). TheKL divergence, also known as relative entropy, is an information theoretic measure thatquantifies the information lost when f“(x) is used to approximate f“ (x) [48]. Note that theMSE and KL measures are used in several related works, see e.g., [5, 21, 24].

Fig. 2.3 provides the approximation results for several scenarios of the NL, Ÿ ≠ µ,÷ ≠ µ, and Ÿ ≠ µ Shadowed fading models. The corresponding number of components areindicated in the legend. As shown from the MSE and KL measures, the approximationis very accurate when both increasing the shadowing and the multipath fading severity,whereas for large amount of fading, the number of components increases. For instance,for the ÷ ≠ µ distribution, when ÷ = 5, and µ = 10, the amount of fading and numberof components were 0.0721 and 3, respectively, whereas when ÷ = 0.7 and µ = 0.4, theamount of fading and number of components increased to 1.2999 and 8, respectively. Theparameters of the approximations are tabulated in Appendix A.

Further verification of the accuracy via numerical means is addressed in section 2.5.The purpose of this approach is not to increase the accuracy of the approximation, as ithas already been achieved in all aforementioned fading alternatives, but rather to provideanother unifying and simplifying distribution that has the potential of approximating allcontemporary fading, composite and non-composite, models via the EM algorithm.

19

2.3.5 Parameter Estimation via Variational Bayes

VB or variational inference is a bayesian approximation technique that eliminates someof the challenges that arise when working with the maximum likelihood approach. Forexample, the divergences we observed in the log-likelihood for EM are not an issues for theVB approach. In this subsection, we focus on the application of VB to the MoG distribution.Similar to Section 2.3, we have a random observations y = {y

1

, .., yN }, which representsthe channel fading amplitude of the composite models. Our ultimate goal is to fit the datausing the MoG distribution with C components. For each observation yn, let us introducea C-dimensional latent variable en = {en1

, en2

, ..., enC} which has one element enk = 1 andall other elements equal to 0. Therefore, it’s probability distribution is defined in termsof the mixing coe�cients such that Pr{enk = 1} = fik. Note that for any k, 0 Æ fik Æ 1and

qCk=1

fik = 1. A given en identifies exactly one Gaussian from the mixture. Definethe matrix E as an a N ◊ C matrix to represent the latent variables where the nth row ise

T

n . Therefore, one can write the conditional probability of the matrix E given the mixingcoe�cients as well as the conditional probability of the data vector y given hidden variablesand parameter values as:

p(E|fi) =NŸ

n=1

CŸ

k=1

fienk

k , p(y|E, µ, ‡

2) =NŸ

n=1

CŸ

k=1

N (yn|µk, ‡2

k)enk , (2.33)

where fi = {fi1

, ..., fiC}, µ = {µ1

, . . . , µC}, and ‡

2 = {‡2

1

, . . . , ‡2

C} are the weights, means,and variances respectively. We also choose priors for all the parameters, where the problemsimplifies significantly with conjugate priors. Consequently, Dirchelet distribution is usedfor the mixing coe�cients:

p(fi) = Dir(fi|–0

) = �(–0

)CŸ

k=1

fi–0

≠1

k , (2.34)

where �(–0

) is the normalization constant defined in terms of –0

which assigns a weight tothe prior relative to the data. The larger the value of –

0

, the more influence the prior hason the posterior distribution. For the mean and precision, Gaussian-Wishart prior is usedwhich has parameters m

0

, —0

, W0

, ‹0

. Like –0

, both —0

and ‹ assign weights to the priordistributions relative to the data, where

p(µ, ‡) = p(µ|‡)p(‡) =CŸ

k=1

N (µk)|m0

, (—0

‡k)≠1)W(‡k|W0

, ‹0

). (2.35)

Next, we define the joint distribution of the random variables defined above as follows

p(y, E, fi, µ, ‡) = p(y|E, µ, ‡)p(E|fi)p(fi)p(µ|‡)p(‡). (2.36)

20

In order to simplify, the latent variables and the parameters are assumed to be factoredas q(E, fi, µ, ‡) = q(E)q(fi, µ, ‡). The functional form of these factors is determined byan iterative procedure which update our parameters and latent variables. The derivationof this iterative procedure is rather involved and for the details, the reader is referredto [43, Section (10.2)]. Su�ce it to say that the functional form of q(E) and q(fi, µ, ‡)is similar to the priors for p(fi) and p(µ, ‡) respectively. During the derivation, severaladditional variables are defined which are ultimately used to update the parameters andlatent variables:

ln flnk = E[ln fik] + 12E[ln |‡k|] ≠ D

2 ln(2fi) ≠ 12Eµ

k

,‡k

[(yn ≠ µk)T‡k(yn ≠ µk)]. (2.37)

The above expression is then used to calculate the responsibilities using

rnk = flnkqKj=1

flnj

. (2.38)

Using this expression, one can define statistics reminiscent of those derived with EM whenmaximizing the likelihood:

Nk =Nÿ

n=1

rnk, yk = 1Nnk

Nÿ

n=1

rnkyn, Sk = 1Nk

Nÿ

n=1

rnk(yn ≠ yk)(yn ≠ yk)T (2.39)

These expressions appear in the following equations which are used to update the parametersand latent variables:

–k = –0

+ Nk, —k = —0

+ Nk, ‹k = ‹0

+ Nk (2.40)

mk = 1—k

(—0

m0

+ Nkyk), W ≠1

k = W ≠1

0

+ NkSk + —0

Nk

—0

+ Nk(yk ≠ m

0

)(yk ≠ m0

)T (2.41)

It is noted that as N æ Œ, the priors are ignored. VB iterates over these set of equationsin a similar fashion to EM as follows:

• Define the priors using the parameters: –0

, m0

, —0

, W0

, and ‹0

. Also set the initialconditions for the parameters fi, µ, ‡.

• Calculate the responsibilities using equation (2.38). This step is analogous to expec-tation stage in EM.

21

• Using the responsibilities calculated in step 2, update the values for –k, mk, —k,Wk, and ‹k using equations (2.40) and (2.41). Consequently, this updates q(E) andq(fi, µ, ‡). This step is analogous to the maximization stage in EM.

• Return to step 2 until convergence is achieved.

xn

µ

zn

fi

÷

N

xn µ

enfi

–0

÷

—0

m0

‹0

W0

N

Figure 2.4: Bayesian networks representing EM (left) and VB (right). Both models usethree parameters to model the underlying mixture of Gaussians (mixing coe�cients as wellparameters describing each Gaussian in the mixture). In the VB case these are treated asrandom variables, while in the EM case they are treated as deterministic parameters.

2.3.6 Comparison between EM and VB

Here we perform some interesting experiments to highlight the di�erences and similaritiesbetween the EM and VB approaches. For the VB framework, an existing MATLAB projectpresented by Chen [49] was used. As for the EM framework, Appendix B presents a usefulsample code for the approximation of fading channels using the BIC assisted EM approach,as explained in Sections 2.3.1 and 2.3.3.

For the first experiment, the goal is to explore the impact of the size of the observation seton the EM- and VB-based approaches. Fig. 2.5 shows the underlying NL distribution withm = 2 in black plotted with EM- and VB-based approximations for various dataset sizes.In both cases, the fit improves as N increases. However, for low number of observations,the VB approximations are significantly better than the EM approximations.

The aim of our second experiment was to explore the overfitting problem in EM whichis evident in Fig. 2.5. The plot on the left of Fig. 2.6 compares the EM and VB estimatefor low number of observations with C = 6. Note that EM is severely overtrained while theVB shows no signs of overtraining even though they both start out with C = 6. In thiscase, the VB algorithm indicates that the e�ective number of classes is C = 2. The ploton the right of Fig. 2.6 shows the EM and VB approximations under the assumption that

22

x0 1 2 3 4 5 6

f α(x

)

0

0.5

1

1.5EM-based Approximation for NL (Varying N)

Empirical DistributionN=100 MSE=0.11732N=1000 MSE=0.0040776N=10000 MSE=0.00027594

x0 1 2 3 4 5 6 7

f α(x

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7VB-based Approximation for NL (Varying N)

Empirical Distribution(VB) N=100 MSE=0.11255(VB) N=1000 MSE=0.0049067(VB) N=10000 MSE=0.00044465

Figure 2.5: EM- (left) and VB-based (right) approximations of the NL distribution withm = 2 shown in black. The blue, red, and green distributions correspond to 100, 1000, and10,000 data points respectively. In both, the number of classes C = 6.

x0 1 2 3 4 5 6 7

f α(x

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1EM- and VB-based Approximation (Small N)

Empirical Distribution(EM) N=100 MSE=0.098207(VB) N=100 MSE=0.10789

x0 1 2 3 4 5 6 7

f α(x

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8EM- and VB-based Approximation (Small N)

Empirical Distribution(EM) N=100 MSE=0.10746(VB) N=100 MSE=0.11206

Figure 2.6: Low number of observations EM- (blue) and VB-based (red) approximations ofthe NL distribution with m = 2 shown in black. The plots were trained with C = 6 (left)and C = 2 (right). Both were trained using N = 100.

C = 2. The overtraining in the EM model is significantly reduced due since the degrees offreedom in the model is smaller.

Thirdly, we also experimented with di�erent values for the prior parameter –0

. Thisparameter is a measure of how confident we are of our prior on the initial assumption ofthe mixing coe�cients. We created a random Gaussian mixture with N = 200 and C = 10.When –

0

was set to 0.001, the VB algorithm converged to 2 e�ective mixture components,whereas setting –

0

to 2000 resulted in 3 e�ective mixture components. This is because as–

0

decreases, the data become more powerful in rejecting one of the 3 centers.For the fourth experiment, we looked into Frequentist methods of determining the num-

ber of components so that we can validate the VB results. Fig. 2.7 shows the AIC and BICas a function of the number of components used in the model for both low and high number

23

Table 2.1: Impact Of –0

on VB

–0

E�ective C fi

2000 3 0.30, 0.55, 0.160.001 2 0.37, 0.63

Number of Components1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Info

rmatio

n C

rite

rion

950

960

970

980

990

1000

1010

1020

1030

1040

1050Frequentist Determination of Optimal Number of Components

AIC N=5e2BIC N=5e2

Number of Components1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Info

rmatio

n C

rite

rion

×105

1.9

1.91

1.92

1.93

1.94

1.95

1.96

1.97Frequentist Determination of Optimal Number of Components

AIC N=1e4

BIC N=1e4

Figure 2.7: Low number of observations EM- (blue) and VB-based (red) approximations ofthe NL distribution with m = 2 shown in black. The plots were trained with C = 6 (left)and C = 2 (right). Both were trained using N = 100.

of observations. For the former, the optimal number of classes is C = 2 which agrees withthe prediction from VB. Both AIC and BIC increase for C > 2 due to the penalty whichattempts to prevent over-fitting. Note that this rise is not as apparent in the case of highnumber of observations. From VB, we saw that the e�ective number of components forlarge statistics (i.e. N = 1, 000 and N = 10, 000) was C = 3. For large statistics, bothAIC and BIC do not decrease significantly for C > 3 which agrees with our results fromVB. In addition, also note that BIC applies a stronger penalty for more complex models asexpected.

2.4 Performance Analysis of Wireless Channels

The MoG distribution provides a simplifying and unifying analysis for wireless communica-tion systems over various composite and multipath fading channel models. In this section,we first derive several performance metrics, which can be used for the evaluation of wirelesscommunication systems in a generalized manner. In particular, we derive expressions forthe raw moments of the MoG model, the amount of fading (AoF), the outage probability,the channel capacity, and the MGF. We further derive expressions for the SER performanceof L-branch MRC diversity system and the average probability of detection for cognitiveradio networks.

24

2.4.1 Moment Generating Function

By definition, the MGF M“ (s) = E [e≠s“ ] is given by

M“ (s) =Cÿ

i=1

ÊiÔ8“fi÷i

ˆ Œ

0

1Ô“

exp

Q

ca≠1Ò

““ ≠ µi

22

2÷2

i

R

db e≠“s d“. (2.42)

Applying the change of variables x =Ò

““ , and after expanding the exponentials and con-

siderable mathematical simplifications, we get

M“ (s) =Cÿ

i=1

ÊiÔ2fi÷i

ˆ Œ

0

exp

Q

cca≠(2 ≠ —i)

3x2 ≠ 2µ

i

— x + µ2

i

—

4

2÷2

i

R

ddb dx, (2.43)

where —i = 1 + 2÷2

i “s. Then, after some mathematical manipulations, we obtain

M“ (s) =Cÿ

i=1

Êi exp3

µ2

i

!1

—

≠1

"

2÷2

i

4

Ô—ifi

ˆ Œ≠µ

i

÷

i

Ô2—

exp1≠z2

2dz, (2.44)

which leads to the following expression

M“ (s) =Cÿ

i=1

ÊiÔ—i

expA

µ2

i s

—i

B

Q

3≠ µi

÷iÔ

—i

4, (2.45)

where Q(.) is the Gaussian Q-function defined as Q (x) = 1Ô2fi

´Œx exp

1≠u

2

2

2du.

2.4.2 Raw Moments

E [“n] =Cÿ

i=1

ÊiÔ8“fi÷i

ˆ Œ

0

“n

Ô“

exp

Q

ca≠1Ò

““ ≠ µi

22

2÷2

i

R

db d“. (2.46)

By taking the change of variables x =Ò

““ , and after some mathematical simplifications,

we get

E [“n] =Cÿ

i=1

Êi“n

ˆ Œ

0

x2n

Ô2fi÷i

expA

≠(x ≠ µi)2

2÷2

i

B

dx, (2.47)

Alternatively, we can write (2.46) as

E [“n] =Cÿ

i=1

Êi“n E

ËX2n

i

È, (2.48)

25

where Xi ≥ N (µi, ’i) is the ith Gaussian component. Using the MGF approach, (2.48) canbe expressed as

E [“n] =Cÿ

i=1

Êi“n d (2n)MX

i

(s)ds(2n)

|s=0

, (2.49)

where MXi

(s) = E{e≠sXi} is the MGF of Xi, which is given by

MXi

(s) = expA

µi

s + ÷2

i

s2

2

B

. (2.50)

Equation (2.49) is mathematically convenient for solving the first few moments.An alternative approach that yields a closed form expression can be attained by following

the same method in [50], where the vth raw moments of Xi are derived as

E [Xvi ] = ÷v

i 2v

2

�1

v2

+ 1

2

2

Ôfi

1

F1

C

≠v

2 ,12 , ≠ µ2

i

2÷2

i

D

, (2.51)

where v is an even integer (note that there is no loss in generality).By substituting (2.51) into (2.48), the nth raw moment of the MoG distribution is

derived as

E [“n] =Cÿ

i=1

Êi“n÷2n

i 2n�

1n + 1

2

2

Ôfi

1

F1

C

≠n,12 , ≠ µ2

i

2÷2

i

D

, (2.52)

2.4.3 Amount of Fading

The AoF measure was firstly introduced by Charash [51] as a measure of the severity ofthe fading channel. The AoF requires the knowledge of only the first two moments in thecorresponding fading channel, where it is defined by

AoF = E#“2

$ ≠ E [“]2

E [“]2. (2.53)

By solving (2.52) for the first two moments, we obtain

AoF =q

C

i=1

Êi

!µ4

i

+ 6µ2

i

÷2

i

+ 3÷4

i

"Ëq

C

i=1

Êi

!µ2

i

+ ÷2

i

"È2

≠ 1. (2.54)

2.4.4 Outage Probability

The outage probability is a standard performance criterion used over fading channels. It isdefined as F (“th) =

´ “th

0

f“(x) dx. By performing the following change of variables appliedto (2.27)

y =

Òx“ ≠ µi

÷i, (2.55)

26

and after some mathematical manipulations, the CDF of (2.27) can be written as

F (“th) =Cÿ

i=1

ÊiÔfi

ˆÒ

“

th

“

≠µ

i

÷

i

≠µ

i

÷

i

expA

≠y2

2

B

dy. (2.56)

Further simplifications yield

F (“th) =Cÿ

i=1

Êi

S

UQ

3≠µi

÷i

4≠ Q

Q

a

Ò“

th

“ ≠ µi

÷i

R

b

T

V , (2.57)

2.4.5 Average Channel Capacity

When only the receiver has knowledge about the channel state information (CSI), theaverage channel capacity C is expressed as

C = B

ln 2

ˆ Œ

0

ln (1 + “) f“ (“) d“. (2.58)

where B is the channel bandwidth measured in Hertz. Unfortunately, the exact solutionof (2.58) is intractable. Instead, a computationally simple and very accurate form canbe obtained by following [52], where ln(1 + “) is expanded about the mean value of theinstantaneous SNR, E{“}, using Taylor’s series, yielding

ln (1 + “) = ln (1 + E [“]) +Œÿ

w=1

(≠1)w≠1

w

(“ ≠ E [“])w

(1 + E [“])w

¥ ln (1 + E [“]) + “ ≠ E [“]1 + E [“] + (“ ≠ E [“])2

2 (1 + E [“])2

+ oË(x ≠ E [“])2

È.

(2.59)

where o (.) is one of the Landau symbols defined as f = o („) which implies that f„ æ 0.

Substituting the logarithm function approximation from (2.59) into (2.58), a secondorder approximation for the average channel capacity can be evaluated. It is noted thatthe solution to (2.58) is obtained by taking the expectation of ln (1 + “). Hence, taking theexpectation of (2.59), we get

C ¥ B

ln 2

ˆ Œ

0

ln (1 + E [“]) + “ ≠ E [“]1 + E [“] + (“ ≠ E [“])2

2 (1 + E [“])2

f“ (“) d“

¥ B

ln 2

C

ln (1 + E [“]) ≠ E#“2

$ ≠ E2 [“]2 (1 + E [“])2

D

, (2.60)

where E[“n] is evaluated using (2.52).

27

2.4.6 Symbol Error Analysis

In order to further demonstrate the significance of the MoG distribution, we study the per-formance of independent but not identically distributed (i.n.i.d.) L-branch MRC diversityreceiver over various composite and non-composite fading scenarios. The MRC scheme isthe optimal combining scheme at the expense of increased complexity, where the receiverrequires knowledge of all channel fading parameters [4]. Here the receiver sums up allreceived instantaneous SNR replicas “k as follows

“MRC

=Lÿ

k=1

“k

. (2.61)

The corresponding MGF is thus

M“MRC

(s) = E{e≠sq

L

k=1

“k } =

LŸ

k=1

M“k

(s) , (2.62)

where M“k

(s) is derived in (2.45). The SER, Ps (E), for coherent binary signals, can becomputed as follows [4]

Ps (E) = E“MRC

#Q

!2g“

MRC

"$, (2.63)

where g is some constant resembling several coherent binary signals, such as coherent binaryphase shift keying (BPSK) and coherent orthogonal binary frequency shift keying (BFSK)corresponding to g = 1 and g = 1

2

, respectively. By substituting the Q-function by itsdefinition in [4, eq. (4.2)], the SER is written as

Ps (E) = 1fi

ˆ fi

2

0

ˆ Œ

0

exp3

≠ g “MRC

sin2 (◊)

4f“

MRC

(“MRC

) d“MRC

d◊. (2.64)

The inner integral in (2.64) is the equivalent MGF derived in (2.62), yielding

Ps (E) = 1fi

ˆ fi

2

0

LŸ

k=1

M“k

3g

sin2 (◊)

4d◊, (2.65)

where M“k

(.) was derived in (2.45).Following a similar approach, and by utilizing [4, eq. (8.23)] and [4, eq. (8.12)], the

SER expressions for M -PSK and square M -QAM signaling schemes are given, respectively,by

Ps (E) = 1fi

ˆ (M≠1)fi

M

0

LŸ

k=1

M“k

Asin2

!fiM

"

sin2 (◊)

B

d◊, (2.66)

28

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Threshold SNR γth

(dB)

Outa

ge P

robabili

ty

(Analytical) m=1,ζ=1 dB

(Analytical) m=3,ζ=1 dB

Simulation

γ = 0dB

γ = 5dB

γ = 10dB

γ = 20dB

Figure 2.8: Analytical and simulated outage probability versus “th for two scenarios.

Ps (E) = 4fi

AÔM ≠ 1Ô

M

B Cˆ fi

2

0

LŸ

k=1

M“k

3gQAM

sin2 (◊)

4d◊ ≠

AÔM ≠ 1Ô

M

Bˆ fi

4

0

LŸ

k=1

M“k

3gQAM

sin2 (◊)

4d◊

D

,

(2.67)where g

QAM

= 3

2

(M ≠ 1).

2.5 Simulation Results

In this section, we present some analytical and simulation results for the outage probability,the average channel capacity, the SER of MRC scheme and the average detection probabilityfor cognitive radio.

Fig. 2.8 and 2.9 depict the outage probability, as in (2.57), versus the threshold SNR“

th

and the average SNR “, respectively. Two NL scenarios are considered in Fig. 2.8,where the multipath severity is reduced from m = 1 to m = 3 , and two Ÿ ≠ µ scenariosare considered in Fig. 2.9, where the control parameter Ÿ is increased from Ÿ = 1 to Ÿ = 3.Here one can notice how accurate the approximation is. Also, it is very noticeable how themultipath fading severity a�ects the outage probability performance.

Fig. 2.10 depicts the average channel capacity, as in (2.60), for some selected scenariosfrom the ÷ ≠ µ, Ÿ ≠ µ and Ÿ ≠ µ Shadowed fading models. As shown, the severe NLOSconfiguration of the ÷ ≠µ scenario exhibits the worse capacity. In addition, one can see that

29

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average SNR γ (dB)

Outa

ge P

robabili

ty

(Analytical) κ=1,µ=1

(Analytical) κ=3,µ=1

Monte Carlo Simulationγth

= 10dB

γth

= 0dB

γth

= 20dB

Figure 2.9: Analytical and simulated outage probability versus “ for two scenarios.

introducing the Nakagami-m shadowing to the Ÿ≠µ distribution has worsened the capacity.The term simulation refers to cross-validating the results via the recursive adaptive simpsonquadrature method performed by the aid of a mathematical package.

Fig. 2.11 illustrates the analytical SER of the BPSK signaling scheme for various NLscenarios, including mild and severe fading cases in both shadowing and multipath. Thesolid squared line represents the corresponding Monte Carlo simulation. It is quite no-ticeable how the multipath severity plays greater role in determining the SER, where asobserved, incrementing m by only 1 yields a SER performance improvement of about an or-der of magnitude at observed mid-range average SNR values. On the other hand, increasing’ from 1 to 3 dB, while fixing m, yields a very similar SER performance.

Fig. 2.12 features the analytical SER of the 16-QAM signaling scheme for variousŸ≠µ Shadowed fading scenarios, where the control parameter µ and the shadowing severityparameter m are varied. Here L corresponds to the number of antennas, and it is noticeablethat our model is still very accurate for high antenna diversity order.

2.6 Conclusions

In this chapter, the MoG distribution has been considered to characterize the amplitude andthe SNR statistics for wireless propagation. The parameters of the mixtures are evaluatedusing the EM and VB algorithms and the MSE and KL have been evaluated to challenge theproposed model’s accuracy. The proposed distribution enjoys both simplicity and accuracy

30

0 5 10 15 200

1

2

3

4

5

6

7

Average SNR γ0 (dB)

Ca

pa

city

(b

ps/

Hz)

(κ−µ) κ=1,µ=1

(κ−µ Shadowed) κ=1,µ=1,m=1

(κ−µ Shadowed) κ=1,µ=3,m=3

(η−µ) η=0.7,µ=0.4

Monte Carlo Simulation

Figure 2.10: Analytical and simulated average channel capacity, B=1

2

.

0 5 10 1510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR γ (dB)

SE

R

(Analytical) m=1,ζ=1 dB

(Analytical) m=1,ζ=2 dB

(Analytical) m=1,ζ=3 dB

(Analytical) m=2,ζ=1 dB

(Analytical) m=3,ζ=1 dB

Monte Carlo Simulation

Figure 2.11: Analytical and simulation SER of 2-branch MRC diversity receiver for BPSKsignaling scheme for RL and NL fading channels.

31

5 10 15 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR γ0 (dB)

SE

R

(Analytical) κ=1,µ=1,m=1 (L=1)

(Analytical) κ=1,µ=3,m=1 (L=1)

(Analytical) κ=1,µ=1,m=3 (L=1)

(Analytical) κ=1,µ=1,m=1 (L=2)

(Analytical) κ=1,µ=3,m=1 (L=2)

(Analytical) κ=1,µ=1,m=3 (L=2)

Monte Carlo Simulation

Figure 2.12: Analytical and simulation SER of L-branch MRC diversity receiver for 16-QAMsignaling scheme for for various Ÿ ≠ µ Shadowed fading scenarios.

and we have shown that the proposed pdf expression can accurately represent a wide rangeof both composite and non-composite channels.

We also provided a glimpse into the similarities and di�erences between the EM andVB algorithms. Given su�cient number of observations, EM performs reasonably well indetermining the parameters of the MoG. However, it does not provide an explicit methodto determine C. Therewith, the BIC was employed in conjunction with the EM to findan appropriate number of components. On the other hand, we experimented with VBframework which does determine the optimal value of C directly without the need to resortto the BIC or AIC. The two methods o�ered comparable estimates for C when the numberof observations is su�ciently large. The benefits of VB over EM are particularly evidentwhen dealing with low number of observations.

Additionally, in this chapter, we derived closed-form expressions or approximations forseveral performance metrics used in wireless communication systems, including the MGF,the raw moments, the AoF, the outage probability, and the average channel capacity. Itshould be highlighted that the adopted approach provides a generalized distribution forwireless communication systems where all fading scenarios–be it a single fading channel ora function of fading channels–can be modeled with the same analytical expression. Severalanalytical tools essential for the evaluation of performance analysis of digital communi-

32

cations were presented. This new model can be applied to various scenarios including,diversity systems, cooperative communications, and cognitive radio networks.

33

Chapter 3

Modeling of Wireless FadingChannels using MG Distribution

3.1 Introduction

In this chapter, we present an alternative novel approach to approximate both generalizedand composite fading channels by the MG distribution. Specifically, the underlying problemis formalized as a MLE problem and is solved via the EM algorithm with the aid of theNewton-Raphson maximization algorithm. Similar to previous chapters, since our approachrelies on the channel instantaneous SNR observations only, it allows us to approximate anyfading model, regardless of its mathematical form. Hence, we can consider more channelsthat were not considered in [24], such as the Weibull and Lognormal models. Lastly, wedemonstrate the accuracy and e�ciency of the new approach by studying the system perfor-mance of a dual-hop fixed-gain amplify-and-forward (AF) relaying architecture. Unlike thechannel state information (CSI) assisted relaying strategy, which has been studied in theliterature [53, 54], the fixed-gain relaying technique has not been yet investigated in gener-alized fading models. In this chapter, we evaluate the average channel capacity and the AoFby deriving a new closed form expression for the raw moments of the end-to-end SNR ofa dual-hop fixed-gain relay scheme. Numerical simulation are incorporated to corroboratethe analytical findings.

The rest of this chapter is organized as follows: In Section 3.2, the MG distributionis introduced together with a description of the proposed approximation methodology. InSection 3.2.1, we assess the accuracy of the proposed approach. In Section 3.3, we providean analytical closed expression for the raw moments of a dual-hop fixed-gain AF relayingarchitecture, which in turn paves the way to study the average channel capacity and amountof fading for the considered system. Simulation results and numerical analysis are presentedin Section 3.3.2, while Section 3.4 concludes this chapter. It is noted that this work ispublished in [55].

34

3.2 Proposed Model and Approximation Methodology

The MG distribution was shown to accurately approximate numerous generalized and com-posite fading channels, such as the NL [24, eq. (5)], KG [24, eq. (8)], ÷≠µ [24, eq. (10)], andŸ≠µ [24, eq. (17)] . The pdf of the MG distribution consists of a convex linear combinationof Gamma distributions as

f“(x) =Kÿ

i=1

–i

“(x

“)—

i

≠1 exp(≠’ix

“), (3.1)

where K denotes the number of mixture components, –i, i = 0, .., K, is the mixing coe�cientof the ith component having the constraints 0 Æ –

i

�(—i

)

’—

i

i

Æ 1 andqK

i=1

–i

�(—i

)

’—i

i

= 1, where�(.) is the gamma function [30, eq. (8.310.1)]. The scale and shape parameters of theith component are —i and ’i, respectively, while “ = E

b

N0

= E[“] is the average SNR persymbol. As previously stated, in [24], several fading models of specific integral forms wereapproximated through the use of Gauss-quadrature based approximations. Here we extendthis distribution to more fading channels through the use of an MG based EM approximationmethodology.

Let the jth entry of an observed instantaneous SNR vector x = {x1

, ...xj , ..., xN } beregarded as incomplete data, we assume that X follows an MG distribution expressed as

fX(xj) =Kÿ

i=1

–i„(xj |—i, ’i), (3.2)

where parameter K resembles the number of components, and each ith component followsa Gamma distribution, written as

„(xj |—i, ’i) = x—i

≠1

j e≠’i

xj (3.3)

where parameter –i is the weight of the ith Gamma component and shall sum up to unity,—i and ’i are the scale and shape parameters of the ith Gamma component, respectively.Denote ◊ = ({–i, —i, ’i}K

i=1

), then ideally the target is to find the maximum likelihoodestimator (MLE) of ◊ such that the log-likelihood value is maximized, as follows

◊MLE = arg max◊œ�

L(MG)

(◊|x, K) = arg max◊œ�

ln pX

(x|◊, K) (3.4)

= arg max◊œ�

Nÿ

j=1

ln5 Kÿ

i=1

–ix—

i

≠1

j exp(≠’ixj)6. (3.5)

However as shown before, maximizing the log-likelihood probability is not tractable anddi�cult to optimize [44]. The EM algorithm, which was coined by [26], solves this issue by

35

introducing some latent (missing) random variable Z, where the joint density V = (X, Z)resembles the complete data, and Z œ {1, .., K} is a discrete random variable that defineswhich Gamma component that data X come from, namely

P (Z = i) = –i, i = 1, ..., K.

Instead, the EM algorithm solves the MLE problem by maximizing the corresponding Q-function as follows [42]

◊(t+1) = arg max◊œ�

Q1◊|◊(t)

2= arg max

◊œ�

EV |x,◊(t)

[log pV (V |◊)], (3.6)

where t is the iteration index. In the context of the MG model, the Q1◊|◊(t)

2function can

be written as

Q(◊|◊(t)) =Nÿ

j=1

EZj

|xj

,◊(t)

[ln[pV (Zj |◊) pV (xj |Zj , ◊)]

=Nÿ

j=1

EZj

|yj

,◊(t)

[ln[KŸ

i=1

–Z

j

i „Zj (xj |—i, ’i)]

=Nÿ

j=1

Kÿ

i=1

EZj

|yj

,◊(t)

ËZj

!ln –i ≠ ln „(xj |—i, ’i)

"È

=Nÿ

j=1

Kÿ

i=1

EZj

|yj

,◊(t)

ËZj

È!ln –i ≠ ln „(xj |—i, ’i)

"(3.7)

=Nÿ

j=1

Kÿ

i=1

·(t)ij

!ln –i ≠ ln „(xj |—i, ’i)

".

Here the EM process iteratively performs two main steps upon random initialization of MGmodel parameters:

E-Step: The posterior is calculated using

·(t)ij = –i„(xj |—i, ’i)q

l=1

–l„(xj |—l, ’l). (3.8)

M-Step: Given the posterior, we find the parameters that maximize 3.7, i.e.

◊(t+1) = arg max◊œ„

Q(◊|◊(t)). (3.9)

By di�erentiation of the former quantity with respect to –i, one obtains

–(t+1)

i = 1N

Nÿ

j=1

·(t)ij . (3.10)

36

As for —i, ’i coe�cients, they are approximated directly using non-linear approximationmethod, namely Newton-Raphson algorithm. The package used is the ’nlm’ in Software R,where it is based on the work of [56]. The EM technique is terminated upon reaching some|◊(t) ≠ ◊(t+1)| < ”, which is set to around 10≠5 in our upcoming approximations.

3.2.1 Proposed Model Analysis and Comparisons

In this section, the EM algorithm presented is used to approximate several scenarios of theaforementioned fading channels. The adopted criterion of accuracy is the MSE, defined as

MSE = E[f“(x) ≠ f“(x)], (3.11)

where f“(x) is the approximated instantaneous SNR distribution (MG), and f“(x) is theactual instantaneous SNR distribution. Fig. 3.1 depicts the approximation results forWeibull, Lognormal, and RL instantaneous SNR distributions, given in 2.9, 2.3, and 2.7.Note that more approximations of other scenarios are tabulated in C.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

γ

f γ(γ)

(Exact) Lognormal ζ=1 dB(Approximation) MG− MSE= 0.00021984(Exact) Weibull m=4(Approximation) MG− MSE= 0.00010115

(Exact) RL ζ=1.5 dB(Approximation) MG− MSE= 3.8656e−05

Figure 3.1: Approximation of instantaneous SNR distribution of Lognormal, Weibull, andRL for M = 0 dB.

Fig. 3.2 depicts the approximation results for several scenarios of the NL instantaneousSNR distribution defined in (10), where the Nakagami fading parameter m is varied.

As shown, the approximations are quite accurate, noting that all approximations wereachieved by only two mixture components K = 2. Further verification of accuracy via

37

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ

f γ(γ)

(Exact) NL m=1(Approximation) MG− MSE= 1.6227e−06(Exact) NL m=2(Approximation) MG− MSE= 1.8282e−06(Exact) NL m=4(Approximation) MG− MSE= 3.9112e−06

Figure 3.2: Approximation of instantaneous SNR distribution of several scenarios of NLdistributions for M = 0 dB, ’ = 1

2

dB, with varying m.

numerical means is addressed in Section 3.3.2. There are two methods for increasing theaccuracy of approximation, which are either by increasing K or by lowering the convergencecriterion ”. However, this was not needed as we have achieved su�cient accuracy foraccurate performance analysis. Our approximations of the NL scenarios were slightly lessaccurate than the approach presented in [24], but su�ciently accurate for performanceanalysis. For instance, when approximating a NL channel with m = 1, ’ = 1

2

dB, our modelachieves an MSE of 1.4(10)≠6, while the latter achieves an MSE of 2.1(10)≠7. However, thepurpose of this approach is not to increase on the accuracy but rather to provide a unifyinganalysis over both composite and non-composite models, whereby in contrast to [24], ourapproach is able to incorporate more channels, such as the Weibull and the Lognormalfading models.

3.3 Applications of Mixture Gamma

The proposed approach provides a simple and unifying framework for the evaluation of dig-ital communications systems over generalized fading channels. We point out that extensionto other diversity schemes, e.g. MIMO and cooperative communications, which have beeninvestigated extensively in the literature [57, 58, 59, 60, 61], is straightforward.

38

In [62], the performance analysis for diversity reception schemes, namely as the MRCand SC, are studied for the MG distribution. In [53, 54], the performance analysis of dual-hop AF cooperative system was evaluated. Note that all previous works assume that relayshave access to CSI. However, in fixed-gain dual-hop relaying, the relays have no access toCSI. Up to the authors’ knowledge, there has been no previous works on fixed-gain relayingscheme for the generalized fading model. In the next section, we derive a novel closed formexpression of the moments of this relaying strategy. Moreover, we utilize the derived rawmoments to study the AoF and the average channel capacity of such systems.

3.3.0.1 System Model

Figure 3.3: Fixed-gain dual-hop cooperative communication system.

Consider the dual-hop cooperative scenario, shown in Fig. 3.3, where terminal S com-municates with terminal D with the aid of an a non-regenerative relay R

1

. Further, it isassumed that R

1

has no access to CSI. Here |hSR1

|, |hR1

D| are assumed to follow the MGgeneralized fading model. The channels are assumed to be i.n.i.d. The received signal atR

1

is written as

ySR1

(t) = hSR1

(t) x(t) + nSR1

(t), (3.12)

where nSR1

is a complex Gaussian random variable with variance N01

2

as per dimension, x

is the transmitted signal with an average symbol energy of E[|x[k]|2] = E1

. The receivedsignal at D is written as

yRi

D(t) = G hR1

D(t) ySR1

(t) + nR1

D(t), (3.13)

where nR1

D is a complex Gaussian random variable with variance N02

2

as per dimension.The equivalent end-to-end SNR statistic, under the fixed-gain relaying, is thus written as[63]

“end≠to≠end = “SR1

“R1

D

“R1

D + U, (3.14)

39

where U = 1

G2N01

is a fixed-gain constant, dependent on the amplitude gain G. Anothersubclass of relays presented in [63] is the semi-blind relays, in which the relay has knowledgeof the average fading power of the channel, which changes slowly, allowing the relay not tomonitor the channel continuously. Thus, the semi-blind strategy is a less exhaustive schemebut at the expense of slight error degradation. the gain of the relay in this case is given by

G2 = E[ 1h2

SR1

+ N01

]. (3.15)

3.3.1 Raw Moments of the end-to-end SNR

The moments of the end-to-end SNR are very crucial in the performance analysis of thecooperative scheme. The order moments allow one to compute the average SNR, the AoF,and the average channel capacity. Let “

1

= “SR1

, and “2

= “R1

D. The nth moment of theend-to-end SNR is given by

E[“n] =ˆ Œ

0

ˆ Œ

0

( “1

“2

U + “2

)n f“1

(“1

) f“2

(“2

) d“1

d“2

, (3.16)

where the SNR pdf of each channel is modeled as an MG distribution, written as

f“i

(x) =Kÿ

i=1

–i

“j

( x

“j

)—i

≠1e≠’

i

x

“

j , j = 1, 2. (3.17)

where “j is the average SNR of each channel. Eq. (3.16) can be written as

E{“n} =ˆ Œ

0

“nf“1

(“1

) d“1

ˆ Œ

0

( “2

K + “2

)n f“2

(“2

) d“2

. (3.18)

First, we solve the first integral, I1

=´Œ

0

“nf“1

(“1

) d“1

, as follows:

I1

=Kÿ

j=1

–1

j

“1

ˆ Œ

0

“n+—

1

j

≠1

1

exp(≠’

1

j

“1

“1

) d“1

. (3.19)

By the aid of [30, eq. 3.326.2], I1

is solved as

I1

=Kÿ

j=1

–1

“n1

�(n + —1

j

)

’n+—

1

j

1

j

, (3.20)

where �(.) is the gamma function [30, eq. 8.310.1]. As for the second integral, I2

=´Œ0

( “2

K+“2

)n f“2

(“2

) d“2

, we rely on the following Meijer’s G-function relations provided as[64]:

(x + 1)≠n = (�(n))≠1 G 1,11,1

A1

n

----- x

B

, (3.21)

40

exp(≠x) = G 1,00,1

A≠0

----- x

B

. (3.22)

Using (3.21) and (3.22), and by utilizing the theorem in [64, eq. (21)], I2

is written as

I2

=Kÿ

j

–2j

“—

2j

2

K’2j

�(n) G 1,22,1

A1,1+—

2j

n+—2j

-----“

2

K’2j

B

. (3.23)

Combining I1

and I2

, the nth order moment for the end-to-end SNR is derived as

E[“n] =Kÿ

j=1

–1j “n �(n + —

1j)’

n+—1j

1j

Kÿ

k=1

–2k

“—2k

2

U ’2k

�(n) G 1,22,1

A1,1+—

2k

n+—2k

-----“

2

U’2k

B

(3.24)

3.3.1.1 Amount of Fading (AoF) of the end-to-end SNR

The AoF measure was firstly introduced by Charash [51], as a measure of the severity of thefading channel. The AoF measure requires the computation of the first and second ordermoments, and it is calculated as follows

AoF =E[“2

end≠to≠end] ≠ (E[“end≠to≠end])2

(E[“end≠to≠end])2

, (3.25)

where 3.24 is used to calculate the first and second order statistics.

3.3.1.2 Average Channel Capacity of the end-to-end SNR

The average channel capacity is defined as the mean of the instantaneous mutual information(IMI), and thus is expressed as

C = B

ln 2E[ln(1 + “end≠to≠end)]. (3.26)

Adopting the conventional method, which is to calculate the integral inherent in (3.26) iscumbersome, since one has to find the end-to-end pdf. Similar to the approach providedin Chapter 2, ln(1 + “) is expanded about the mean value of the instantaneous SNR usingTaylor’s series yielding

ln(1 + “) = ln(1 + E[“]) +Œÿ

w=1

(≠1)w≠1(x ≠ E[“])w

w (1 + E[“])w. (3.27)

Taking the expectation of (3.27) and truncating the result to the first two moments, theaverage channel capacity can be expressed as

41

C ≥= B

ln 2[ln(1 + E[“end≠to≠end]) ≠ (3.28)

E[“2

end≠to≠end] ≠ E2[“end≠to≠end]2(1 + E[“end≠to≠end])2

],

where E[“2

end≠to≠end], and E[“end≠to≠end] are calculated from (3.24).

3.3.2 Simulation Results

We first start by defining a cooperative scenario, where the first channel |hSR1

| is assumedto follow a Weibull distribution with fading parameter m = 4, and |hR

1

D| is assumed tofollow a NL distribution with the following parameters (m = 4, ’ = 2 dB). The selectedscenarios is chosen to illustrate the flexibility of the MG model, where one may combinebetween multipath fading models and composite fading models. Fig. 3.4 depicts the averagechannel capacity of the selected cooperative scenario. The fixed-gain coe�cient U is set to0.5. Note that if one chooses U , such that it satisfies (3.15), then the scheme is called semi-blind. The term ’simulation’ refers to cross-validating our results by numerically computing(3.26) via the trapezoidal integration method on MATLAB. As seen, the solution o�ered isvery accurate for the whole operating SNR region. For the selected scenario, the AoF wascalculated to be 0.3145 using (3.24).

Average SNR (dB)0 5 10 15 20 25 30 35 40

Ca

pa

city

(b

its/s

/Hz)

0

1

2

3

4

5

6

7

(Analytical)(Simulation)

Figure 3.4: Average channel capacity for the selected scenario with U = 0.5, B = 0.5.

42

3.4 Conclusions

We have presented a new approach to representing fading channels by the MG distribu-tion using the EM technique. We have demonstrated its ability to represent more fadingscenarios than those existing in literature, and thus yielding a unifying framework for bothcomposite and non-composite fading models. Another contribution is to analyze and studythe performance analysis of dual-hop fixed-gain AF cooperative systems. New closed-formexpression of the raw moments has been derived, which facilitates the derivation of furtheranalytical tools, such as the amount of fading, and the average channel capacity. We believethat such an approach would help the research community in resolving many intractableproblems in many fields, such as cooperative communications, and cognitive radio networks.

43

Chapter 4

Energy Detection over Generalizedand Composite Fading Channels

4.1 Introduction

The need for e�cient utilization of spectrum resources has become a fundamental require-ment in modern wireless networks, mainly due to the aforementioned spectrum scarcityand the ever-increasing demand for higher data rate applications and Internet services [65].In this context, cognitive radio (CR) communications is a particularly interesting wirelesstechnology that has been proposed as an e�ective method that is capable of mitigating thespectrum scarcity by adapting their transmission parameters according to the respectiveenvironment [66]. To this end, cognitive radios have been shown to be highly e�cient inmaximizing spectrum utilization due to their inherent spectrum sensing capability. In a CRnetwork environment, users are categorized in either primary users (PUs) or secondary users(SUs). Based on this, the former are the ones who have been typically assigned licensedspectrum slots, and hence, have higher priority, whereas the latter are accessing vacantfrequency bands opportunistically.

Based on the above, numerous spectrum sensing techniques have been proposed overthe past decade and can be classified into three main categories, namely, energy detection(ED), matched filter detection and cyclostationary or feature detection. One of the earli-est methods is the likelihood ratio test (LRT) [67], which although it has been consideredoptimal, its techincal exploitation is rather limited and impractical as it requires the exactknowledge of the SNR distributions as well as the corresponding channel information [68].On the contrary, matched filter detection techniques [69], [70] typically require accuratesynchronization and exact information about the transmitted signal waveform, such as itsbandwidth and modulation type. Likewise, cyclostationary detection [71] uses the statis-tical properties of the transmitted signals to enhance the probability of detection. On thecontrary, ED based sensing practically constitutes the most common detection method and

44

has received considerable attention [72], [73] thanks to its low computational and imple-mentation complexity. In ED, the presence of a PU signal is simply detected by comparingthe output of the energy detector with a pre-determined energy threshold which dependson the a priori knowledge of the noise power level [74]. Therefore, poor knowledge of thenoise power level leads to a high probability of false-alarm and an SNR floor. Based onthis, several analyses have been proposed for resolving this issue by estimating the noisepower level, e.g. see [69], [75], [76] and the references therein. For instance, the authors in[76] proposed an iterative algorithm that optimizes the decision threshold for fulfilling thefalse-alarm probability requirement. Unlike the conventional ED based spectrum sensing,methods which rely on the statistical covariance of the received signal, do not require theknowledge of the noise power level since their operation relies on the di�erence between thatstatistical covariance matrices of the received signal and the noise. For interested readers,see e.g. [77] and references therein.

It has been also extensively shown that fading phenomena create detrimental e�ects onthe performance of conventional and emerging wireless communications, including cognitiveradio systems. In this context, the ED performance over multipath fading channels, such asRayleigh, Rician, and Nakagami-m was analyzed in [78], and [79], respectively, whereas thecorresponding performance over the more generalized Ÿ≠µ and Ÿ≠µ extreme fading channelswas investigated in [80]. However, in addition to multipath fading, in most scenarios, thereceived signal is also degraded by shadowing e�ects since it has been shown that multipathand shadowing e�ects typically occur simultaneously [81]. Therefore, it is evident that thereis an undoubted necessity to quantify and analyze the CR performance over compositemultipath/shadowing fading channels [82]. Nevertheless, it has been shown that such ananalysis is particularly tedious, since composite fading models can only be represented bycumbersome, if not intractable, infinite integrals. For example, the probability of detectionof the ED based spectrum sensing over NL fading channels was addressed in [81]; yet,the o�ered solution is semi-analytic, as it is not represented in closed form, while theimpact of fading and shadowing e�ects is evaluated numerically. Based on this, severalalternative models that characterize the composite fading channels have been shown toprovide simplified performance analysis for the CR networks. For example, in the analysesof [83, 84, 85, 86, 87], the K distribution is utilized to study the ED performance overRL channels. The energy detector performance for the MoG distribution [29] is derived in[88]. In [89], the su�ciency and optimality of cooperative wireless sensor networks that arebased on energy detection is analyzed over NLOS fading environments, where zero-meanGaussian mixtures are assumed as a viable model for NLOS fading channels. A unifiedand versatile analyses over the ED performance can be made feasible through the use ofmore recent generalized composite fading models, such as Ÿ ≠ µ/Inverse-Gaussian [90] and÷ ≠ µ/Inverse-Gaussian [91] models.

45

In the present chapter, we consider the generic and versatile MG based approach toderive new exact expressions for the average detection probability over generalized andcomposite fading channels. Specifically, a simple closed-form expression is derived for thecase of integer values of the involved scale parameter —k. It is recalled here that the MGmodel [24], [55] has been proposed as an alternative model to various generalized andcomposite fading channels, namely Lognormal [55, eq. (6)], Weibull [55, eq. (5)], NL [55,eq. (9)], KG [24, eq. (8)], ÷ ≠ µ [24, eq. (10)], Ÿ ≠ µ [24, eq. (17)], Hoyt [24, eq. (15)], andRician [24, eq. (20)] channels. This model is both accurate and flexible to represent allaforementioned fading channels and thus, it constitutes a generic unified fading model. Inthe present analysis, the derived average detection probability is also extended to the case ofdiversity reception by means of the SLC and SLS schemes. It is noted that the probabilityof detection of the MG model has been derived in [24], yet, the solution provided thereinis di�erent from the approach in this paper. In addition, the present chapter also providesnovel and useful expressions for the SLC and SLS schemes.

The reminder of the chapter is organized as follows: Section 4.2 provides a brief de-scription of the system model, while Section 4.3.1 is devoted to the derivation of averagedetection probability expressions using the MG model with and without diversity reception.Analytical and numerical simulation results are presented in Section 4.4, while closing re-marks are provided in Section 4.5. The content of the present chapter can be found in [92],[93].

4.2 System Model

In a typical opportunistic cognitive radio configuration, a secondary user, which is assumedto employ energy detection for spectrum sensing, aims to determine whether a PU utilizesits assigned frequency band or not. Here we assume that the channel gains, hj , are i.i.d.and are modeled using the generalized MG distribution, where j = 1, ..., L. Furthermore,the received signal copies at the SU node and jth antenna can have two possible hypotheses,modeled as

H0

: vj(t) = hj(t)H

1

: yj(t) = hj(t) + vj(t) ,(4.1)

where H0

and H1

represent the absence and presence of a signal, respectively, s(t) corre-sponds to the transmitted signal from the PU, with energy Es = E[|s(t)|2], and vj(t) ≥CN (0, ‡2

n) is the circularly symmetrical complex AWGN. Here, the SU utilizes an energydetector that compares the amplitudes |yj |Mj=1

of the received signal to a threshold ⁄. There-fore, the output of this process for each antenna can be represented as follows

Zj = |yj |2H

1

?H

0

⁄ , (4.2)

46

where the time index t has been omitted for the sake of notational simplicity.

For the conventional case of AWGN channels, the conditional detection and false-alarmprobabilities are determined with the aid of [94], namely

Pd = Qu(Ò

2“j ,

⁄n), (4.3)

Pf =�(u, ⁄

n

2

)�(u) , (4.4)

where u is time-bandwidth product, Qu(·, ·) is the generalized Marcum-Q function [95],�(·, ·) is the upper incomplete gamma function [30, eq. (8.35)], �(·) is the standard gammafunction [30, eq. (8.31)], ⁄n = ⁄/‡n

2 is the normalized threshold, and “j = |hj

|2Es

2‡2

n

is theinstantaneous SNR of the jth PU-SU link.

As the probability of false-alarm is based on the null hypothesis, it remains the sameregardless of the involved fading conditions. Thus, in the subsequent sections, we focus onthe derivation of the average detection probability for both the conventional single-channelcommunication and for multi-channel communications with diversity reception.

4.3 Probability of Detection over Composite Fading Chan-nels

4.3.1 Single-antenna Scenario

It is recalled that the MG distribution is a generic and versatile distribution since it has beenshown capable of providing accurate representation of several generalized and compositefading models. The corresponding pdf can be expressed as [85]

f“(x) =Kÿ

k=1

–k

“0

3x

“0

4—k

≠1

exp3

≠’kx

“0

4, (4.5)

where the scale and shape parameters of the kth component are denoted by —k and ’k,respectively. Furthermore, the mixing coe�cient of the kth component is denoted by –k,having the constraints, 0 Æ –k�(—k)/’—

k

k Æ 1 andqK

k=1

–k�(—k)/’—k

k = 1. To this e�ect,the average probability of detection for the MG distribution can be written as

P d,MG =Kÿ

k=1

–k

“0

Œ

0

Qu(Ô

2x,

⁄n).( x

“0

)—

k

≠1

e≠ ’

k

x

“

0 dx . (4.6)

Here, an exact closed-form expression is derived under the assumption that —k is a positiveinteger, i.e. —k œ N. This is realized with the aid of Theorem 1 in [96, eq. (3)] and by

47

carrying out some long but basic algebraic simplifications yielding

P d,MG =Kÿ

k=1

–k�(—k)�(u, ⁄n

2

)�(u)’—

k

k

+Kÿ

k=1

—k

≠1ÿ

l=0

–k�(—k)“—

k

0

◊(⁄

n

2

)u1

F1

(l + 1, u + 1, ⁄n

/2

1+1

’k

“0

)

u!( ’k

“0

)—k

≠l(1 + ’k

“0

)l+1 exp(⁄n

2

), (4.7)

where1

F1

(., ., .) is the confluent hypergeometric function [30, eq. (9.210.1)]. It is noted herethat the above expression has a relatively simple algebraic representation which renders itconvenient to handle both analytically and numerically since the confluent hypergeometricfunction,

1

F1

(., ., .) is included as built-in function in popular software packages such asMATLAB, MAPLE and MATHEMATICA. It is worth noting that (4.7) coincides numeri-cally with the expressions for the case of Rayleigh fading channel in [78] and [79].

4.3.2 Diversity Reception

4.3.2.1 Square-Law Combining

Under SLC, the received signals from each branch are integrated, squared, and then summedup. The SLC is similar to the MRC scheme in the sense that the total instantaneous SNRat the output of the combiner is equivalent to that in MRC, i.e.

“�

=Lÿ

l=1

“l . (4.8)

Nevertheless, SLC does not require channel estimation [96]. As a result, the conditionalfalse-alarm probability would follow (4.4), with u replaced by Lu. In order to evaluate thecorresponding average detection probability, it is essential to derive the pdf of “

�

. To thisend, for the case of L = 2, the pdf of “

�

can be obtained as follows:

f (2)

“�

=ˆ “

0

f“1

(x)f(“ ≠ x)dx =Kÿ

i=1

Kÿ

j=1

–i–j

“0

—i

+—j

◊ˆ “

0

x—i

≠1e≠ ’

i

“

0

x(“ ≠ x)—j

≠1e≠ ’

j

“

0

(“≠x)

dx. (4.9)

48

In order to evaluate (4.9), we split the solution into two scenarios, namely when ’i = ’j and’i ”= ’j . In the former scenario, eq. (4.9) reduces to the following integral

f (2)

“�

|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i

“0

—i

–j

“0

—j

e≠ ’

j

“

0

“

◊ˆ “

0

x—i

≠1(“ ≠ x)—j

≠1dx . (4.10)

By performing the change of variables u = x“ and with the aid of [30, eq. (8.380)] and the

functional relation in [30, eq. (8.384)], we obtain the following closed-form solution

f (2)

“�

|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i–j

“0

—i

+—j

�(—i)�(—j)�(—i + —j) e

≠ ’

j

“

0

““—

i

+—j

≠1. (4.11)

On the contrary, for the case ’i ”= ’j , eq. (4.9) is solved with the aid of the binomial theoremin [30, eq. (1.111)] and under the assumption that —j œ N. To this e�ect, the representationin (4.9) can be equivalently re-written as follows

f (2)

“�

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B

(≠1)l –i–j

“0

—i

+—j

◊ “—j

≠l≠1

e’

j

“

0

“

ˆ “

0

x—i

+l≠1e≠ x

“

0

(’i

≠’j

)

dx. (4.12)

Evidently, the above integral can be expressed in closed-form with the aid of [30, eq. (8.350.1)]yielding

f (2)

“�

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j

“0

—j

≠l(’i ≠ ’j)—i

+l

◊ “—j

≠l≠1e≠ ’

j

“

0

““

3—i + l,

“(’i ≠ ’j)“

0

4, (4.13)

where “(a, x) ,´ x

0

ta≠1e≠tdt denotes the lower incomplete gamma function. Thus, byexpressing the “(a, x) function according to [30, eq. (9.352.6)], one obtains the followingclosed-form expression,

f (2)

“�

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j

“0

—j

≠l

◊ (’i ≠ ’j)≠—i

≠l“—j

≠l≠1e≠ ’

j

“

0

“�(—i + l)

◊ (1 ≠ e≠ “(’

i

≠’

j

)

“

0

—i

+l≠1ÿ

t=0

1“(’

i

≠’j

)

“0

2t

t! ), (4.14)

49

which is valid for —i œ N, while

f (2)

“�

= f (2)

“�

|(’

i

=’j

)

+ f (2)

“�

|(’

i

”=’j

)

. (4.15)

By following the same methodology, a similar expression can be obtained for f(3)

“�

as

f (3)

“�

|(’

i

=’j

=’k

)

=Kÿ

i=1

Kÿ

j=1

–i–j–k

“0

—i

+—j

+—k

�(—i)�(—j)�(—k)�(—i + —j + —k) e

≠ ’

k

“

0

““—

i

+—j

+—k

≠1, (4.16)

and

f (3)

“�

|(’

i

”=’j

”=’k

)

=Kÿ

i=1

Kÿ

j=1

Kÿ

k=1

—j

≠1ÿ

l=0

—k

≠1ÿ

r=0

–i–j–k

“0

—k

≠r

A—j ≠ 1

l

BA—k ≠ 1

r

B

(4.17)

◊ (≠1)—j �(l + —i)

(’i ≠ ’j)l+—i’

r+—j

≠lk

“—k

≠r≠1

e“

“

0

(’k

+’j

)

“1bj + r ≠ l, ≠ ’

k

““

0

2

≠Kÿ

i=0

Kÿ

j=0

Kÿ

k=0

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

—k

≠1ÿ

r=0

(≠1)l+r –i–j–k

“0

—k

≠r

A—j ≠ 1

l

BA—k ≠ 1

r

B

◊ �(l + —i)(’i ≠ ’j)t≠l≠—i

t!(’i ≠ ’j ≠ ’k)r+t+—j

≠l

“—k

≠r≠1

e“

“

0

(’k

+’j

)

“

3bj + r + t ≠ l,

“(’i ≠ ’j ≠ ’k)“

0

4.

It is noted here that the above methodology allows the derivation of similar expressionsfor f

(4)

“�

, f(5)

“�

and so forth.Based on this, the corresponding average detection probability is readily obtained by

P(L)

d,� =ˆ Œ

0

QLu(

2“�

,Ô

⁄) f (L)

“�

(“�

) d“�

. (4.18)

For the case of L = 2 and by inserting (4.11) and (4.14) in (4.18), it follows that

P(2)

d,�|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i–j

“0

—i

+—j

�(—i)�(—j)�(—i + —j)

◊ˆ Œ

0

Qu(Ô

2“�

,Ô

⁄)e≠ ’

j

“

0

“

“≠(—i

+—j

≠1)

d“, (4.19)

50

and

P(2)

d,�|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)“

0

—j

≠l(’i ≠ ’j)—i

+lI

1

(“0

)

≠Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

A—j ≠ 1

l

B

◊ (≠1)l–i–j�(—i + l)t!“

0

—j

≠l+t(’i ≠ ’j)—i

+l≠tI

2

(“0

), (4.20)

whereI

1

(“0

) =ˆ Œ

0

Qu(Ô

2“�

,Ô

⁄)“—j

≠l≠1

e’

j

“

0

“d“ , (4.21)

andI

2

(“0

) =ˆ Œ

0

Qu(Ô

2“�

,Ô

⁄)“—j

+t≠l≠1

e’

i

“

0

“d“ . (4.22)

Notably, the involved integrals in (4.19), (4.21), and (4.22) have the same algebraicrepresentation as (4.6). Therefore, by utilizing Theorem 1 in [96, eq. (3)] and after somealgebraic manipulations yields the closed-form expressions expressed as follows

P(2)

d,�|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i–j�(—i)�(—j)5 �(u, ⁄

2

)’

—i

+—j

j �(u)

+—

i

+—j

≠1ÿ

n=0

“0

≠n(⁄2

)u1

F1

(n + 1, u + 1,⁄

2

1+

’

j

“

0

)

u!(’j)—i

+—j

≠n(1 + ’j

“0

)n+1 exp(⁄2

)

6, (4.23)

and

P(2)

d,�|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)“

0

—j

≠l(’i ≠ ’j)—i

+l(4.24)

◊Ë�(—j ≠ l)�(u, ⁄

2

)( ’

j

“0

)—j

≠l�(u)+

—j

≠l≠1ÿ

n=0

(⁄2

)u�(—j ≠ l)u! ( ’

j

“0

)—j

≠l≠n

1

F1

(n + 1, u + 1,⁄

2

1+

’

j

“

0

)

(1 + ’j

“0

)n+1 exp(⁄2

)

È

≠Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)

t!“0

—j

≠l+t(’i ≠ ’j)—i

+l≠t

◊Ë�(—j + t ≠ l)�(u, ⁄

2

)( ’

i

“0

)—j

+t≠l�(u)+

—j

+t≠l≠1ÿ

n=0

(⁄2

)u�(—j + t ≠ l)1

F1

(n + 1, u + 1,⁄

2

1+

’

i

“

0

)

u! ( ’i

“0

)—j

+t≠l≠n(1 + ’i

“0

)n+1 exp(⁄2

)

È.

In the same context, by following a similar methodology one can obtain the averagedetection probability for higher diversity orders while the probability of false alarm remains

51

unchanged, i.e. Pf,� = Pf in (4.4).

4.3.2.2 Square-Law Selection

Under SLS, the branch with the maximum “j is selected as follows

“SLS

= maxj=1,..,L

(“j) . (4.25)

Under H0

, the false-alarm probability for the SLS scheme can be expressed as

Pf,SLS

= 1 ≠ Pr(“SLS

< ⁄n|H0

) . (4.26)

Substituting (4.25) in (4.26), we obtain

Pf,SLS

= 1 ≠ Pr(max(“1

, “2

, .., “L) < ⁄n|H0

) . (4.27)

Accordingly, this translates to [97]

Pf,SLS

= 1 ≠ [1 ≠ Pf ]L . (4.28)

Similarly, the unconditional probability of detection over the AWGN channel is obtainedby

Pd,SLS

= 1 ≠LŸ

j=1

Ë1 ≠ Qu(

Ò2“j ,

⁄n)

È. (4.29)

Hence, averaging (4.29) over (4.6) yields the unconditional probability of detection underthe SLS scheme, P d,SLS

, which is given by

Pd,SLS

= 1 ≠LŸ

j=1

[1 ≠ Pd,MG] . (4.30)

To the best of the authors’ knowledge, the o�ered analytic results have not been previ-ously reported in the open technical literature.

4.4 Numerical Results and Discussions

As already mentioned, the derived expressions are applicable to numerous generalized andcomposite fading channels, such as NL, RL, K, KG, ÷ ≠µ, Ÿ≠µ, Hoyt, and Rician channels.In this section, we present corresponding analytical and simulation results for the receiveroperating characteristic (ROC) with and without diversity over certain fading scenarios. Tothis end, Fig. 4.1 depicts the analytical and simulated average missed-detection probability,

52

0 0.2 0.4 0.6 0.8 110−6

10−5

10−4

10−3

10−2

10−1

100

Aver

age

mis

s−de

tect

ion

prob

abilit

y

False−alarm probability

(η−µ) η=3.5, µ=15(κ−µ) κ=1.0, µ=3.0(Hoyt) q=0.5(KG) k=1.0, m=2.0Simulation

Figure 4.1: Complimentary ROC for various fading channels and no diversity, with “0

=10 dB, u = 2.

1 ≠ Pd, versus the false-alarm probability for di�erent fading conditions with no diversity.It is clearly shown that the analytical and simulated curves are in tight agreement thanksto the arbitrarily accurate representation of the MG distribution. It is also shown that thepresented ÷ ≠ µ scenario exhibits the best ROC, which is expected since it represents arather light fading scenario with ÷ = 3.5 and µ = 15.

Fig. 4.2 depicts the analytical and simulated ROC curves over several scenarios of thecomposite NL fading channel with SLC diversity scheme with L = 2. As expected, changingthe multipath severity parameter, m, has more prominent influence on the detection perfor-mance than changing the shadowing parameter, ’. For example, at Pf = 0.48, increasingm from 3 to 4 resulted in the ratio, 1≠P

d

|m=4

1≠Pd

|m=3

= 4.47, while reducing ’ from 4 to 1 improvedthe ROC curve by only 1≠P

d

|’=1 dB

1≠Pd

|’=4 dB

= 1.40. Fig. 4.3 depicts the analytical and simulatedROC curves over one scenario of the composite NL fading channel with SLS scheme withvarying L. Comparing Figs. 4.2 and 4.3 exhibits that SLC performs better than SLS forL = 2; yet, the improvement is not significant for this particular case. For instance, atPf,SLS = Pf,� = 0.48, the corresponding improvement ratio was 1≠P

d,�

1≠P d,SLS = 1.80. Also,one can observe how e�ective is the spatial diversity in combating the severity of the mul-tipath fading and shadowing e�ects.

MG

53

0 0.2 0.4 0.6 0.8 110−6

10−5

10−4

10−3

10−2

10−1

100

Aver

age

mis

s−de

tect

ion

prob

abilit

y

False−alarm probability

(NL) m=4, ζ=1 dB(NL) m=3, ζ=1 dB(NL) m=1, ζ=1 dB(NL) m=1, ζ=4 dBSimulation

Figure 4.2: Complimentary ROC for composite NL fading scenarios and SLC scheme, withL = 2, “

0

= 10 dB, u = 2.

0 0.2 0.4 0.6 0.8 110−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Aver

age

mis

s−de

tect

ion

prob

abilit

y

False−alarm probability

(NL) m=3, ζ=1 dB, L=1(NL) m=3, ζ=1 dB, L=2(NL) m=3, ζ=1 dB, L=4(NL) m=3, ζ=1 dB, L=6Simulation

Figure 4.3: Complimentary ROC for selected NL fading scenario and SLS scheme, withvarying L and “

0

= 10 dB, u = 2.

54

4.5 Conclusions

We proposed a unified framework for the performance analysis of an energy detector ingeneralized and composite MG-based fading channels. Novel analytical expressions for theaverage detection probability have been derived for both the single-antenna case and themultiple-antenna case with SLC and SLS schemes. The derived expressions have been shownto be both generalized in terms of fading characterization and algebraically versatile.

55

Chapter 5

Analysis of Wireless Systems overGeneralized and Composite FadingChannels with Impulsive Noise

5.1 Introduction

Diversity techniques, that exploit multiple copies of the transmitted signal, have been widelyinvestigated to overcome detrimental e�ects of wireless channels. Most of the contributionsin the literature assume AWGN in each diversity branch. Although this assumption in-corporates the e�ect of background thermal noise, it ignores the impact of the impulsivenoise caused by atmospheric, man-made partial discharge, switching e�ect, and electromag-netic interference, etc., [98, 99]. The MCA noise model [100] is one of the most accuratestatistical-physical models for narrowband impulsive noise. There are many works in theliterature that employed MCA model to characterize impulsive noise in wireless commu-nication systems, c.f., [98, 99], [101, 102, 103, 104, 105]. Expressions for the BEP in SCand MRC receivers over Rayleigh fading channel with MCA noise were derived in [106].In [98], the same channel model was considered to derive the BEP for di�erent combiningschemes, including SC and MRC. The BEP over Rician channel in the presence of MCAnoise was derived for both SC and equal gain combining (EGC) in [102]. Moreover, sincethe MCA model contains an infinite number of noise states, a relatively tractable model,widely known as ‘-mixture noise model, with two terms and two tunable parameters is con-sidered in [104, 107]. Although there have been considerable research e�orts on diversityanalysis over conventional multipath fading channels, such as Rayleigh and Nakagami-m,to the best of our knowledge, there exist no reported results that incorporates generalizedor composite fading channels along with MCA or ‘-mixture noise.

In this chapter, we aim to fill this research gap and investigate the performance of SIMOsystem over composite fading channels with MCA and ‘-mixture noise. Following [24] and

56

[55], we express numerous generalized and composite fading channels, such as the NL, KG,÷ ≠ µ, Ÿ ≠ µ, Lognormal, and Weibull, by the MG distribution. The contributions in thischapter are multifold, and can be summarized as follows:

• We propose a simple and e�ective information theoretic approach to determine theoptimal number of components for the MG distribution based on the BIC.

• We derive analytical pairwise error probability (PEP) expressions that involve finitesingle-fold integrals assuming MCA and ‘-mixture noise with MRC or SC. Further-more, more tractable PEP expressions are derived for the case of integer values of theinvolved parameter —k.

• We derive analytical tractable expressions for the average channel capacity assumingMCA and ‘-mixture noise with MRC or SC.

The remainder of the chapter is organized as follows: In Section 5.1, we provide a briefintroduction to the two impulsive noise models. In Section 5.3, we propose the BIC as anapproach to determine the optimal number of components for numerous fading models. TheSIMO system model with MCA and ‘-mixture noise, followed by the derivation of exactPEP, and approximate average ergodic capacity expressions are introduced in Section 5.4.Numerical results and Monte Carlo simulations are presented in Section 5.5, followed bysome concluding remarks in Section 5.6.

5.2 Middleton’s Class-A and ‘-Mixture Impulsive Noise Mod-els

In this chapter, following [98], we assume that the noise at the receiver is modeled as, n =ng +ni, where ng, ni are the background Gaussian noise with variance ‡2

g , and the impulsivenoise with variance ‡2

i , respectively. When the number of independent active sources, T , islarge enough, the occurences of interference would follow a Poisson distribution, i.e.

Pr(T = k) = e≠AAk

k! , (5.1)

where A is the impulsive index that describes the average number of impulses during someinterference time [98], and it is typically in the range of 10≠4 to 0.5. As such, the pdf of theMCA noise can be expressed as [108]

fn(x) =qŒ

k=0

Pr(T = k)fn|T =k(x|k) =Œÿ

k=0

e≠AAk

k!Ò

2fi‡2

k

exp(≠ x2

2‡2

k

), (5.2)

In (5.2), ‡2

k = kA≠1

+�

1+�

‡2, where ‡2 = N0

2

is the power of n, ⁄ = ‡2

g

‡2

i

is the Gaussian Factor,which resembles the ratio of the variances of the background Gaussian component the

57

impulsive component, and it is typically in the range of 10≠5 to 1. Although this distributionincludes an infinite summation, it is completely described by three parameters, ‡2, ⁄, and A.As A and ⁄ tends to zero, the noise becomes more impulsive. In the analysis hereafter,we truncate the MCA model to C components. In [100], Zabin and Poor proposed anexpectation-maximization (EM) based method to estimate ⁄ and A. The method relies onthe iterative maximization of the log-likelihood function of the envelope of the MCA noise,where the envelope is expressed as an infinite summation of weighted Rayleigh distributions.

Another popular impulsive noise model is the ‘-mixture [108], which resembles a Gaussian-Gaussian mixture model, where the real part of the pdf is expressed as

fn(x) = 1 ≠ ‘Ò

2fi‡2

g

exp(≠ x2

2‡2

g

) + ‘Ò

2fi‡2

i

exp(≠ x2

2‡2

i

). (5.3)

In (5.3), ‘ denotes the fraction of time for which the impulsive noise occurs, with 0 < ‘ < 1.The ratio of the variances of the impulsive component to the Gaussian component is givenby › = ‡2

i

‡2

g

. Here the variance of n is given by

‡2 = N0

2 = (1 ≠ ‘)‡2

g + ‘‡2

i . (5.4)

In this context, Gaussian mixture based EM estimation technique can be utilized for theestimation of ‘, ‡2

g , and ‡2

i . It is worth noting that although a truncated two-term MCAmodel has been shown to accurately represent the infinite MCA noise model when A and �are small enough, the truncated two-term MCA model remains as a subset of the ‘-mixturenoise model, as shown in [108].

5.3 Approximating Fading Models with the MG Distribution

Recall that the pdf of the MG distribution consists of a convex linear combination of Gammadistributions as

f“(x) =Kÿ

j=1

–j

“(x

“)—

j

≠1 exp(≠’jx

“), (5.5)

where K denotes the number of mixture components. –j , j = 0, .., K, is the mixing coef-ficient of the jth component having the constraints 0 Æ –

j

�(—j

)

’—j

j

Æ 1 andqK

j=1

–j

�(—j

)

’—j

j

= 1,

where �(.) is the gamma function [30, eq. (8.310.1)]. The scale and shape parameters ofthe jth component are —j and ’j , respectively, while “ = E

b

N0

= E[“] is the average SNR perbit.

In this chapter, we determine the appropriate number of components for the MG dis-tribution. In [24], the number of components is selected manually, such that the MSE orKL divergence between the actual true distribution and the MG to be below a predefined

58

threshold. This method requires determination of a suitable target threshold empirically orby means of Monte Carlo simulations, that may be a tedious task to accomplish. Choosing asmall number of components would yield an inaccurate representation, while a large numberwould unnecessarily increase the complexity of the distribution and may cause overfitting.Instead, we propose the use of a simple yet e�ective unsupervised criterion, named the BIC,which was introduced by Gideon Schwarz in [46].

Let x = {x1

, ..., xz, ..xN } correspond to N i.i.d. samples, drawn from any of the actualaforementioned SNR fading models, then the approximation methods used in [24] and [55]rely on the maximization of the likelihood function, Pr(x|◊, K), where ◊ = ({–i, —i, ’i}K

i=1

)are the estimated parameters and K is the corresponding number of components. The BIC isan asymptotic approximation to the transformation of the Bayesian a posteriori probability,Pr(◊|x, K). As such, in a large-sample setting, the number of components determined bythe BIC is optimal from the perspective of the Bayesian posterior probability. In addition,the BIC does not rely on the specification of prior distributions of ◊, but only on the log-likelihood function, which can be readily achieved. For a candidate model with complexitylabelled by K components, the log-likelihood function is expressed as

L(MG)

(◊|x, K) =Nÿ

z=1

log5 Kÿ

i=1

–ixˆ—

i

≠1

z exp(≠’ixz)6. (5.6)

The corresponding BIC estimate can be computed as

BICK = ≠2L(MG)

(◊|x, K) + K ln N. (5.7)

It can be seen that the BIC penalizes the model complexity by adding the regularization co-e�cient, K ln N . Here we select the candidate model satisfying the minimum BIC estimateor equivalently the asymptotically maximum Bayesian posterior probability as

Kopt = arg minKœN

BICK . (5.8)

Fig. 5.1 depicts the BIC versus the number of components of various MG distributions.The corresponding optimal number of components, Kopt, indicated in the legend, will beadopted in the simulations and numerical results hereafter, denoted by K.

Another possible criterion is the Akaike information criterion (AIC) [109], which can becomputed as

AICK = ≠2L(MG)

(◊|x, K) + 2K. (5.9)

The AIC serves as an asymptotic unbiased estimator of the KL divergence between theactual distribution and the MG distribution. We point out that the BIC tends to be morepenurious as compared to the AIC, i.e. the BIC favors model candidates with smaller K,since K ln N > 2K for N > e2.

59

5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

3

4

Number of Components, K

Baye

sian Info

rmatio

n C

rite

rion

(NL) (m=5.0,λ=2.0 dB) Kopt

=4

(KG

) (m=5.0,k=2.0) Kopt

=8

(η−µ) (η=3.5,µ=15.0) Kopt

=13

(κ−µ) (κ=1.0,µ=3.0) Kopt

=7

Figure 5.1: The BIC versus number of components for various MG based fading channelsusing the BIC. The considered BIC is normalized to have zero mean and unit variance.

5.4 Performance Analysis

5.4.1 System Model

Consider a point-to-point communication scenario with a multi-link channel having L i.i.d.slowly varying and flat fading channels hl, l = 1, .., L . The received signal copy via lth

branch is given byrl = hl s + nl, (5.10)

where s œ S is the transmitted symbol belongs to the constellation S, with an averagesignal energy Eb = E[|s|2] = 1, nl is the noise impaired in lth receiver branch, following theMCA noise model, i.e., nl ≥ qŒ

k=0

e≠AAk

k!

CN (0, ‡2

k), with ‡2

k = kA≠1

+�

1+�

‡2, where ‡2 = N0

isthe total noise variance1. Here the power gain, “l = “h2

l , of the lth channel is modeled byMG distribution as discussed in Section 5.3, where “ = E

b

N0

. In the following subsections,we derive expressions for the PEP and ergodic capacity for both MRC and SC schemes.

1

Although, not explicitly derived, extending the herein analysis to ‘-mixture noise is straightforward,

where nl

≥ (1 ≠ ‘) CN (0, ‡2

1

) + ‘ CN (0, ‡2

2

) as in Section II, with ‡2

= N0

= (1 ≠ ‘)‡2

g

+ ‘‡2

i

.

60

5.4.2 Pairwise Error Probability

Although the L2

-norm detection scheme is suboptimal for the case of impulsive noise, it isconsidered in our analysis due to its versatile application and low computational complexity.For a single branch, the L

2

-norm detector selects the most likely symbol using

s = minsœS

||rl ≠ hl s||2. (5.11)

5.4.2.1 Pairwise Error Probability for MRC

It follows that we can express the conditional PEP between any two codewords s and s œ Sfor the lth single branch as

P(l)P EP (d|“l) =

˙Cÿ

k=0

e≠AAk

k! Pr{||r ≠ hs||2‡2

k

<||r ≠ hs||2

‡2

k

}, (5.12)

where d denotes the Euclidean distance between s and s. From (5.12), the conditional PEPof the lth branch is thus written as

P(l)P EP (d|“l) =

˙Cÿ

k=0

e≠AAk

k!Ò

2fi‡2

k

/2

ˆ ŒÒ

“

l

d

2

2

exp(≠ x2

2‡2

k

/2

) dx, (5.13)

which can be solved as

P(l)P EP (d|“l) =

˙Cÿ

k=0

e≠AAk

k! Q(Û

“ld2

‡2

k

), (5.14)

where Q(.) is the Gaussian Q-function [4, eq. (4.1)].The unconditional pairwise error probability can be derived by averaging P

(l)P EP (d|“l)

over the lth fading channel, i.e. P(l)P EP (d) = E“

l

[PP EP (d|“l)], yielding

P(l)P EP (d) =

˙Cÿ

k=0

Kÿ

j=1

e≠AAk–j

k! “—j

ˆ Œ

0

x—j

≠1 exp(≠’jx

“)Q(

Ûxd2

‡2

k

) dx. (5.15)

Using [64, eq. (11)] and [64, eq. (12)], we can re-write (5.15) as

P(l)P EP (d) =

˙Cÿ

k=0

Kÿ

j=1

–j

2Ô

fi

e≠AAk

k!“—j

ˆ Œ

0

x—j

≠1G 1,00,1 ( ≠,≠

0, | ’jx

“)G 2,0

1,2 ( ≠,1

0, 1

2

| xd2

2‡2

k

)dx, (5.16)

where G(.) is the Meijer’s G-function [64, eq. (18)]. It is noted that (5.16) is the Mellintransformation of the product of the two Meijer’s G-functions. Therefore, using [64, eq.

61

(21)] and after some manipulations, we obtain

P(l)P EP (d) =

˙Cÿ

k=0

Kÿ

j=1

–je≠AAk

Ôfi(k!) ( ‡2

k

“d2

)—j 22—

j

≠1G 1,22,2 ( 1≠—

j

, 1

2

≠—j

0,≠—j

| 2’j‡2

k

“d2

), (5.17)

where by using [64, eq. (17)], (5.17) can be re-written as

P(l)P EP (d) =

˙Cÿ

k=0

Kÿ

j=1

–jAk

eAk! ( ‡2

k

“d2

)—j

�(2—j)�(1 + —j) 2

F1

(—j ,12 + —j , 1 + —j ; ≠2’j‡2

k

d2“), (5.18)

where2

F1

(., ., .; .) is the Gauss hypergeometric function [30, eq. (9.113)].For the MRC scheme, the received signal copies are coherently weighted and summed

up in order to maximize the instantaneous output SNR, where the total instantaneous SNRat the output of the MRC method is given by

“�

=Lÿ

l=1

“l. (5.19)

In order to evaluate the corresponding PEP, one needs to derive the pdf of “�

2

. To thisend, for the case of L = 2, the pdf of “

�

2

can be obtained by

f“�

2

=ˆ “

0

f“1

(x)f(“ ≠ x)dx

=Kÿ

i=1

Kÿ

j=1

–i

“—i

–j

“—j

ˆ “

0

x—i

≠1e≠ ’

i

“

x(“ ≠ x)—j

≠1e≠ ’

j

“

(“≠x)

dx. (5.20)

In order to solve (5.20), we split the solution into two scenarios, namely when ’i = ’j and’i ”= ’j . In the former scenario, eq. (5.20) reduces to the following integral

f“�

2

|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i

“—i

–j

“—j

e≠ ’

j

fl

“ˆ “

0

x—i

≠1(“ ≠ x)—j

≠1dx. (5.21)

By the change of variables u = x“ and with the aid of [30, eq. (8.380)] and the functional

relation in [30, eq. (8.384)], we obtain the following closed-form solution

f“�

2

|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

–i–j

“—i

+—j

�(—i)�(—j)�(—i + —j) e

≠ ’

j

fl

““—

i

+—j

≠1. (5.22)

Likewise, for the case ’i ”= ’j , eq. (5.21) is solved with the aid of the binomial theorem in[30, eq. (1.111)] and under the assumption that —j œ N. To this e�ect, the representation

62

in (5.20) can be equivalently re-written as follows

f“�

2

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B

(≠1)l –i

“—i

–j

“—j

“—j

≠l≠1e≠ ’

j

“

“ˆ “

0

x—i

+l≠1e≠ x

“

(’i

≠’j

)

dx.

(5.23)

Evidently, the above integral can be expressed in closed-form with the aid of [30, eq.(8.350.1)] yielding

f“�

2

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j

“—j

≠l(’i ≠ ’j)—i

+l“—

j

≠l≠1e≠ ’

j

“

““

3—i + l,

“(’i ≠ ’j)“

4,

(5.24)

where “(a, x) ,´ x

0

ta≠1e≠tdt denotes the lower incomplete gamma function. Thus, byexpressing the “(a, x) function according to [30, eq. (8.352.6)] one obtains the followingclosed-form expression,

f“�

2

|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j

“—j

≠l(’i ≠ ’j)—i

+l“—

j

≠l≠1

◊e≠ ’

j

“

“�(—i + l)

Q

ca1 ≠ e≠ “(’

i

≠’

j

)

“

—i

+l≠1ÿ

t=0

1“(’

i

≠’j

)

“

2t

t!

R

db , (5.25)

which is valid for —i œ N, while

f“�

2

= f“�

2

|(’

i

=’j

)

+ f“�

2

|(’

i

”=’j

)

. (5.26)

By following the same methodology, a similar expression can be obtained for f“�

3

, namely

f“�

3

|(’

i

=’j

=’k

)

=Kÿ

i=1

Kÿ

j=1

–i–j–k

“—i

+—j

+—k

�(—i)�(—j)�(—k)�(—i + —j + —k) e

≠ ’

k

“

““—

i

+—j

+—k

≠1, (5.27)

63

f“�

3

|(’

i

”=’j

”=’k

)

=Kÿ

i=1

Kÿ

j=1

Kÿ

k=1

—j

≠1ÿ

l=0

—k

≠1ÿ

r=0

–i–j–k

“—k

≠r

A—j ≠ 1

l

BA—k ≠ 1

r

B(≠1)—

j �(l + —i)(’i ≠ ’j)l+—

i’r+—

j

≠lk

◊ “—k

≠r≠1

e“

“

(’k

+’j

)

“

3bj + r ≠ l, ≠’k“

“

4≠

Cÿ

i=0

Cÿ

j=0

Cÿ

k=0

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

—k

≠1ÿ

r=0

(≠1)l+r –i–j–k

“—k

≠r

A—j ≠ 1

l

B

◊A

—k ≠ 1r

B�(l + —i)(’i ≠ ’j)t≠l≠—

i

t!(’i ≠ ’j ≠ ’k)r+t+—j

≠l“—

k

≠r≠1e≠ “

“

(’k

+’j

)

“

3bj + r + t ≠ l,

“(’i ≠ ’j ≠ ’k)“

4.

(5.28)

It is noted here that the above methodology allows for the derivation of similar expressionsfor f“

�

4

, f“�

5

and so forth. Based on the above, the PEP for MRC is readily obtained by

PP EP,�(d) =˙Cÿ

k=0

e≠AAk

k!

ˆ Œ

0

f (L)

“�

(“�

) Q(Û

“�

d2

2‡2

k

) d“�

. (5.29)

For the case of L = 2 and by inserting (5.22) and (5.25) in (5.29), it follows that

PP EP,�2

(d)|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

–i–j

“—i

+—j

�(—i)�(—j)�(—i + —j)

e≠AAk

k!

ˆ Œ

0

e≠ ’

j

“

““—

i

+—j

≠1Q(Û

“d2

‡2

k

) d“,

(5.30)and

PP EP,�2

(d)|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)“—

j

≠l(’i ≠ ’j)—i

+l

ˆ Œ

0

“—j

≠l≠1e≠ ’

j

“

“Q(

Û“d2

‡2

k

) d“

≠Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)(’i ≠ ’j)t

“t+—j

≠l(’i ≠ ’j)—i

+lt!

◊ˆ Œ

0

“t+—j

≠l≠1e≠ ’

i

“

“Q(

Û“d2

‡2

k

) d“. (5.31)

Notably, the three integrals in (5.30) and (5.31) have the same algebraic representation as(5.15). Therefore, following a similar methodology presented in (5.16) and (5.17) and aftersome algebraic manipulations yields the following expressions

PP EP,�2

(d)|(’

i

=’j

)

=Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

–i–j�(—i)�(—j)�(—i + —j)

�(2—j + 2—i)�(1 + —j + —i)

e≠AAk

k! (5.32)

◊ ( ‡2

k

2“d2

)—i

+—j

2

F1

(—i + —j ,12 + —i + —j , 1 + —i + —j ; ≠2’j‡2

k

d2“),

64

PP EP,�2

(d)|(’

i

”=’j

)

=Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

—j

≠1ÿ

l=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)

(’i ≠ ’j)—i

+l

�(2(—j ≠ l))�(1 + —j ≠ l) (5.33)

◊ ( ‡2

k

2“d2

)—j

≠l2

F1

(—j ≠ l,12 + —j ≠ l, 1 + —j ≠ l; ≠2’j‡2

k

d2“) ≠

˙Cÿ

k=0

Kÿ

i=1

◊Kÿ

j=1

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)

(’i ≠ ’j)—i

+l≠tt!�(2(t + —j ≠ l))�(1 + t + —j ≠ l)

◊ ( ‡2

k

2“d2

)t+—j

≠l2

F1

(t + —j ≠ l,12 + t + —j ≠ l, 1 + t + —j ≠ l; ≠2’i‡

2

k

d2“),

For the case —j œ R, the PEP for the MRC scheme can be realized via the moment generatingfunction (MGF) approach. By utilizing [4, eq. (4.2)] and [30, eq. (3.381.4)] and realizingthat M“

�

(s) = E“l

[exp(qL

l=1

“l)] = �Ll=1

M“l

(s), where M“l

(s) is the MGF of the lth branch,one obtains the following single-fold integral

PP EP,�(d) =˙Cÿ

k=0

e≠AAk

k!fi

ˆ fi

2

0

[Kÿ

j=1

–i�(—j)( “d2

2‡2

k

sin

2 „+ ’j)—

j

]L

¸ ˚˙ ˝M

“

�

(

“d

2

2‡

2

k

sin

2

„

)

d„. (5.34)

5.4.2.2 Pairwise Error Probability for Selection Combining

The equivalent SNR for the SC scheme is given by

“SC = maxl={1,...,L}

“l. (5.35)

Accordingly, we can obtain F“SC

(x) = �Ll=1

F“l

(x), where

F“1

(x) =Kÿ

i=1

–i’≠—

i

i “(—i,’ix

“) (5.36)

is the cumulative distribution function (CDF) of “l. For the case of two independent andidentically distributed (i.i.d.) branches, di�erentiating F“

SC

(x) w.r.t. to x yields

f“SC

2

(x) =Kÿ

i=1

Kÿ

j=1

2–i–j

“’—

j

j

(x

“)—

i

≠1 exp(≠’ix

“)“(—j ,

’jx

“). (5.37)

A rather tractable expression valid for —j œ N can be obtained as follows. By expressing“(—j ,

’j

x“ ) according to [30, eq. (8.352.6)], eq. (5.37) can be re-written as

f“SC

2

(x) =Kÿ

i=1

Kÿ

j=1

2–i–j

“’—

j

j

(x

“)—

i

≠1 exp(≠’ix

“)�(—j)

11 ≠ e

≠ ’

j

x

“

—j

≠1ÿ

t=0

( ’j

x“ )t

t!2. (5.38)

65

Based on the above pdf, the PEP for SC can be obtained by

PP EP,SC(d) =˙Cÿ

k=0

e≠AAk

k!

ˆ Œ

0

f“SC

(“SC) Q(Û

“SCd2

‡2

k

) d“SC . (5.39)

By following the methodology presented in (5.17) and (5.18) and after some algebraic ma-nipulations, one obtains

PP EP,SC2

(d) =Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

2e≠AAk

k!–i–j�(—j)

’—

j

j

�(2—i)�(1 + —i)

( ‡2

k

2“d2

)—i

◊2

F1

(—i,12 + —i, 1 + —i;

≠2’i‡2

k

d2“) ≠

Kÿ

i=1

Kÿ

j=1

˙Cÿ

k=0

—j

≠1ÿ

t=0

2e≠AAk

k!–i–j�(—j)

’—

j

≠tj t!

◊ �(2(t + —i))�(1 + t + —i)

( ‡2

k

2“d2

)t+—i

2

F1

(t + —i,12 + t + —i, 1 + t + —i;

≠2(’i + ’j)‡2

k

d2“).(5.40)

Similar to (5.34), for the case —i, —j œ R, a corresponding PEP expression for the SC schemecan be similarly expressed via the MGF approach, where M“

sc

(s) is derived using [30, eq.(6.45.2)], yielding

PP EP,SC2

(d) =˙Cÿ

k=0

2Ake≠A

fi(k!)

ˆ fi

2

0

M“

SC

(

s/sin

2

◊)

˙ ˝¸ ˚Kÿ

i,j=1

–i–j�(—ji)—i(’ji ≠ s/sin

2 ◊)—ji

◊2

F1

(1, —ji; —i + 1; ’i

(’ji ≠ s/sin

2 ◊))d◊

¸ ˚˙ ˝, (5.41)

where s = “d2

2‡2

k

, —ji = —j + —i, and ’ji = ’j + ’i.

5.4.2.3 Symbol Error Probability and Bit Error Probability

Having the PEP expressions derived, at large “, the SEP of various M -ary linear signalingschemes can be expressed as [103]

Ps = 2÷M PP EP (dM ), (5.42)

where dM is the minimum average Euclidean distance of S, and ÷M is a parameter dependenton the signaling scheme, as summarized in [103, Table. 1]. Assuming gray coding, the BEPcan be determined by Pb = P

s/log

2

M,where M is the size of S. For the case of binary phaseshift keying (BPSK), the exact SEP is obtained when ÷M = 1

2

, and dM =Ô

2. The BPSKsignaling is considered in our simulations and analyses thereafter.

66

5.4.3 Average Channel Capacity

Assuming that the channel state information is known, then the average conditional channelcapacity of a two-term mixture distribution, which can be written as

fN (x) = p0

fB(x) + p1

fI(x), (5.43)

was derived as [110]C = p

0

C(0) + p1

C(1), (5.44)

where C(0) and C(1) are the conditional capacities of fB(x) and fI(x). Extending on thisapproach, we can express the average unconditional capacity under the MCA noise as aweighted summation of the individual capacities of each Gaussian component as

C =˙Cÿ

k=0

e≠AAkB

k!ln2 E“{ln(1 + “

‡2

k

“)}. (5.45)

Denote the e�ective average SNR per kth noise component as “k = “‡2

k

, then following [24],the average channel capacity for the single-branch scenario, which is valid for —i œ N, canbe expressed in closed-form as

C =˙Cÿ

k=0

Kÿ

i=1

—iÿ

k=1

–ie≠AAk

k! “—i

k

B

ln 2�(—i)e’

i

“

k

�(k ≠ —i,’

i

“k

)( ’

i

“k

)k. (5.46)

For the MRC scheme, since f“�

(x) has the same algebraic representation as the MG distri-bution, the average channel capacity for L = 2 is readily obtained as

C�

2

|(’

i

=’j

)

=˙Cÿ

k=0

Kÿ

i=1

Kÿ

j=1

—i

+—jÿ

k=1

e≠AAk–i–j

k! “—

i

+—j

k

B

ln 2�(—i)�(—j) e’

j

“

k

�(k ≠ —i + —j ,’

j

“k

)

( ’j

“k

)k, (5.47)

C�

2

|(’

i

”=’j

)

=˙Cÿ

k=0

Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

—j

≠lÿ

k=1

B

ln 2

A—j ≠ 1

l

B(≠1)l–i–j�(—i + l)“

—j

≠lk (’i ≠ ’j)—

i

+l�[—j ≠ l] e

’

j

“

k

�(k ≠ —j ≠ l,’

j

“k

)

( ’j

“k

)k

≠˙Cÿ

k=0

Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

l=0

—i

+l≠1ÿ

t=0

—j

+t≠lÿ

k=1

e≠AAk

k!B

ln 2

A—j ≠ 1

l

B

◊ (≠1)l–i–j�(—i + l)“

—j

+t≠lk (’i ≠ ’j)—

i

≠t+lt!�[—j + t ≠ l] e

’

i

“

k

�(k ≠ —j + t ≠ l, ’i

“k

)( ’

i

“k

)k, (5.48)

whileC

�

2

= C�

2

|(’

i

=’j

)

+ C�

2

|(’

i

”=’j

)

. (5.49)

67

Similar expressions for the average channel capacity for higher diversity orders can beobtained by following the same methodology.

Likewise, for the SC scheme, an expression for the average channel capacity for L = 2can be derived from (5.38) as follows,

CSC2

=˙Cÿ

k=0

Kÿ

i=1

Kÿ

j=1

—iÿ

k=1

e≠AAk

k!B

ln 22–i–j�(—j)

“—i

k ’—

j

j

(—i ≠ 1)! e’

i

“

k

�(k ≠ —i,’

i

“k

)( ’

i

“k

)k(5.50)

≠˙Cÿ

k=0

Kÿ

i=1

Kÿ

j=1

—j

≠1ÿ

t=0

e≠AAk

k!B

ln 22–i–j�(—j)“—

i

+tk ’

—j

≠tj t!

(t + —i ≠ 1)! e’

i

+’

j

“

k

t+—iÿ

k=1

�(k ≠ t + —i,’

i

+’j

“k

)

( ’i

+’j

“k

)k,

which is valid for —i, —j œ N. In order to get expressions valid for all values of the involvedparameter —, we obtain tight approximate expressions for the average channel capacity asfollows. By utilizing the Taylor’s series, ln(1 + “k“)} can be expanded about “k, yielding

ln(1 + “k“)} = ln(1 + E[ “

‡2

i

]) +Œÿ

w=1

(≠1)w≠1

w

(x ≠ E[ “‡2

k

])w

(1 + E[ “‡2

k

])w. (5.51)

By taking the expectation of this expansion and truncating the result to contain the firsttwo moments, i.e. w = 2, we obtain an approximate expression

C¥˙Cÿ

k=0

e≠AAkB

k!filn2 [ln(1 + E[ “

‡2

k

]) ≠E[( “

‡2

k

)2] ≠ E2[ “‡2

k

]2(1 + E[ “

‡2

k

])2

]. (5.52)

Similarly, for the case of ‘-mixture noise, the capacity can be expressed as

C ¥ B

ln21(1 ≠ ‘)[ln(1 + E[ “

‡2

g

]) ≠E[( “

‡2

g

)2] ≠ E2[ “‡2

g

]2(1 + E[ “

‡2

g

])2

] +

‘[ln(1 + E[ “

‡2

i

]) ≠E[( “

‡2

i

)2] ≠ E2[ “‡2

i

]2(1 + E[ “

‡2

i

])2

]2. (5.53)

Here we derive the first two moments E[ “‡2

k

] and E[( “‡2

k

)2] by di�erentiating the correspondingMGF, as follows

E[( “

‡2

k

)n] =d(n)M“( s

‡2

k

)

ds(n)

|s=0

. (5.54)

Di�erentiating M“MRC

(s) in (5.34) using (5.54), and after some mathematical simplifica-tions, we obtain

E[“MRC

‡2

k

] = ≠L(“MRC

‡2

k

)�L≠1� (5.55)

68

E[(“MRC

‡2

k

)2] = L(“MRC

‡2

k

)2

Ë(L ≠ 1)�L≠2�2 + �L≠1�

È, (5.56)

where � =qK

i=1

–i

�(—i

)

’—

i

i

, � =qK

j=1

—j

–j

�(—j

)

’—

j

+1

j

, and � =qK

j=1

—j

–j

�(—j

)(—j

+1)

’—

j

+2

j

.

Similarly, di�erentiating M“SC

(s) in (5.41) using (5.54), and after considerable mathe-matical manipulation, we arrive at

E[“SC

‡2

k

] =Kÿ

i,j=1

�1

(i, j)C

Õ1

(i, j) + ’jÕ2

(i, j)’ij(—j + 1)

D

(5.57)

E[(“SC

‡2

k

)2] =Kÿ

i,j=1

(—ij + 1)�2

(i, j)’ij

C

Õ1

(i, j) (5.58)

+ 2’jÕ2

(i, j)’ij(—j + 1) +

2’2

j Õ3

(i, j)(—j + 1)’2

ij(—j + 2)

D

,

where �y(i, j) = (“SC

‡2

k

)y 2—ij

–i

–j

�(—ij

)

—i

’—

ij

+1

ij

, Õy(i, j) =2

F1

1y, —ij + y ≠ 1; —j + y; ’

j

’ij

2.

5.5 Simulations and Discussion

In order to evaluate the e�ectiveness of the BIC described in Section 5.3 and to validatethe derivations of Section 5.4, Fig. 5.2 depicts the analytical and simulated SEP for BPSKsignaling under MCA noise with ⁄, A = 0.1 for both MRC and SC schemes with L =4, and 2, respectively, for various MG based fading channels, namely NL, KG, Ÿ ≠ µ, and÷ ≠ µ. The number of components are chosen according to the BIC, as indicated in Section5.3. As shown, the curves are very accurate over the whole operating average SNR region.

In Fig. 5.3, we analyze the SEP for the MRC scheme, when L = 1 and L = 4, withthe fading channel following the NL composite model. We consider several scenarios ofthe MCA noise with ⁄ = 0.1, and A = (0.1, 0.3, 0.9). Recall that as A æ 0, the noisebecomes more impulsive. Therefore, as one would expect, at medium and large “, the SEPcurve becomes more flat, as A æ 0. However, interestingly, we notice that this is not thecase at very small “, where there exists a threshold, marked by an arrow, for which theerror performance associated with small A (more impulsive case) is better. Increasing thediversity order shifts this threshold to the left. When L = 1, the threshold was at “ = 5dB, whereas when L = 4, the threshold shifted to “ = 0 dB.

Lastly, in Fig. 5.4, we plot the capacity versus ‘ for the NL and ÷ ≠ µ fading channelsassuming MRC diversity scheme and ‘-mixture noise with impulsive index › = ‡2

i

/‡2

g

= 75.First, we notice that as the diversity order increases, the whole capacity curve shifts upwards.In addition, when ‘ is either small or large, one Gaussian component dominates, resulting

69

in low capacity measures, whereas the skewness of the curve reflects the fact that at large‘, the noise becomes more impulsive.

0 2 4 6 8 10 12 14 16 18 2010

−10

10−8

10−6

10−4

10−2

100

Sym

bo

l Err

or

Ra

te P

s

γ0 [dB]

(MRC) NL (m=5,λ=2.0) K=4

(MRC) KG

(k=2.0,m=5.0) K=8

(MRC) κ−µ (κ=1.0,µ=3.0) K=7

(MRC) η−µ (η=3.5,µ=15.0) K=13

(SC) NL (m=5.0,λ=2.0) K=4

(SC) KG

(k=2.0,m=5.0) K=8

(SC) κ−µ (κ=1.0,µ=3.0) K=7

(SC) η−µ (η=3.5,µ=15.0) K=13

Monte Carlo

Figure 5.2: Analytical and simulated SEP of BPSK with 4-MRC and 2-SC schemes forvarious MG based fading channels with MCA Noise of ⁄, A = 0.1, and C = 10.

5.6 Conclusions

In this chapter, we have proposed a unified and versatile approach to the performanceanalysis of SIMO over generalized and composite fading channels with impulsive noise.Specifically, we have proposed an e�ective-information theoretic approach to determine theoptimal number of components for the MG-based fading models, based on the BIC. Wealso have derived novel exact analytical PEP and average channel capacity expressions forthe performance of SIMO systems with MRC and SC schemes over MG-based generalizedand composite fading channels with impulsive noise, modeled by MCA and ‘-mixture noisemodels. Our derived expressions have been shown to be both generalized to many fadingchannels and noise environments and algebraically versatile.

70

−10 −5 0 5 10 15

10−10

10−8

10−6

10−4

10−2

100

Sym

bol E

rror

Pro

babili

ty

γ dB

L=1, A=0.1L=1, A=0.3L=1, A=0.9 (MRC) L=4, A=0.1(MRC) L=4, A=0.3(MRC) L=4, A=0.9Monte Carlo

Figure 5.3: Analytical and simulated SEP of BPSK with MRC scheme for NL fading con-taminated with MCA Noise of ⁄ = 0.1, A = (0.1, 0.3, 0.9), and C = 10.

0 0.2 0.4 0.6 0.8 13

4

5

6

7

8

9

ε

Capacity [bps/Hz]

(η−µ) (η=3.5,µ=15.0) L=1

(η−µ) (η=3.5,µ=15.0) L=4

(NL) (m=5,λ=2.0 dB) L=1

(NL) (m=5,λ=2.0 dB) L=4

Figure 5.4: Average channel capacity with and without MRC diversity for NL and ÷ ≠ µfading channels contaminated with ‘-mixture noise with “ = 10 dB and › = 75.

71

Chapter 6

Conclusions and Future Work

In this thesis, we proposed to approximate wireless fading channels in general and general-ized and composite fading channels in particular using the MoG distribution in Chapter 2and the MG distribution in Chapter 3.

In Chapter 2, the parameters of the MoG distribution were evaluated using the EMalgorithm, while the number of components were determined using the BIC. Through severalexperiments and derivations, it has been shown that the MoG distribution is both simple andaccurate in approximating many generalized and composite fading models. Furthermore,we proposed the VB approach as a viable alternative to EM. It has been shown that thebayesian treatment enjoys many advantages over its counterpart. First and foremost, VBallows for the determination of the number of components implicitely without resortingto statistical or information theoritic approaches, such as cross validation, AIC, or BIC.Moreover, in contrast to EM, VB performs better when the number of observations issmaller and is not prone to overfitting. Several analytical tools essential for the analysis ofdigital communication systems, such as the MGF, raw moments, outage probability havebeen derived.

In Chapter 3, we proposed to approximate fading channels by the MG distributionusing the EM algorithm. In contrast to the MoG based EM approach, here there is noiterative EM treatment with closed-form estimates in the maximization step. To resolvethis issue, Newton-Raphson maximization algorithm was adopted to maximize the completelog-likelihood function numerically. Likewise, the MG distribution was shown to enjoy bothaccuracy and algeabric versatility.

In Chapters 4 and 5, further performance analysis applications on generalized and com-posite fading channels were performed. More specifically, the performance analysis of anenergy detector in generalized and composite MG-based fading channels were studied inChapter 4. Novel analytical expressions for the average detection probability have beenderived for both the single-antenna case and the multiple-antenna case with square-lawcombining and square-law selection schemes. In Chapter 5, we proposed a unified and

72

versatile approach to the performance analysis of SIMO over generalized and compositefading channels with impulsive noise. We have derived novel exact analytical pairwise errorprobability and average channel capacity expressions for the performance of SIMO systemswith MRC and SC schemes over MG based generalized and composite fading channels withimpulsive noise, modeled by MCA and ‘-mixture noise models. Our derived expressionshave been shown to be both generalized to many fading channels and noise environmentsand algebraically versatile.

There is plenty of potential future work that can be performed. With regard to the MoGand MG distributions, several analytical tools and statistics are yet to be derived. Thanksto the arbitrarily accurate yet simple representation of the MoG and MG distributions,further analytical derivation of intricate communication scenarios is now feasible. Moreover,It should be noted that this approach is not only applicable to approximating single fadingchannels, but rather it could be applied to approximating a function of random variables,such as which occurs in MIMO and cooperative communications.

73

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81

Appendix A

Mixture of Gaussian Parameters

The following tables provide the approximation parameters for all scenarios presented inChapter 2 using the MoG distribution.

Table A.1: MoG parameters for RL fading channel with ’ = 3 dB and Copt = 7

i wi µi ÷i

1 0.24621 0.76637 0.18882 0.28164 1.0954 0.274823 0.15143 1.5254 0.394554 0.077823 0.26911 0.0888155 0.025355 2.0441 0.594716 0.19662 0.48573 0.140667 0.020909 0.11846 0.050183

Table A.2: MoG parameters for NL fading channel with m = 2, ’ = 1 dB and Copt = 4

i wi µi ÷i

1 0.11009 0.48576 0.144832 0.14047 1.3455 0.342673 0.37935 1.0845 0.250994 0.37009 0.77746 0.19545

Table A.3: MoG parameters for NL fading channel with m = 4, ’ = 1 dB and Copt = 4

i wi µi ÷i

1 0.31126 0.88351 0.149272 0.13366 1.2306 0.258033 0.39008 1.0756 0.194794 0.165 0.67541 0.1405

82

Table A.4: MoG parameters for Ÿ ≠ µ fading channel with Ÿ = 1, µ = 0.5 and Copt = 10

i wi µi ÷i i wi µi ÷i

1 0.005263 0.0052273 0.0032608 6 0.1401 0.39636 0.11622642 0.050502 0.11458 0.038823 7 0.013249 0.020352 0.00898783 0.18712 1.4475 0.3582 8 0.22111 0.6682 0.183414 0.21379 1.0393 0.24872 9 0.026951 0.052855 0.0197395 0.054249 1.8749 0.48797 10 0.087676 0.22199 0.069679

Table A.5: MoG parameters for Ÿ ≠ µ fading channel with Ÿ = 3, µ = 1 and Copt = 5

i wi µi ÷i

1 0.248 1.2295 0.290682 0.23197 0.59067 0.177273 0.017972 0.24425 0.0996984 0.2658 1.0976 0.207365 0.23626 0.86379 0.16438

Table A.6: MoG parameters for ÷ ≠ µ fading channel with ÷ = 0.5, µ = 0.2 and Copt = 14

i wi µi ÷i i wi µi ÷i

1 0.092351 0.21051 0.061239 8 0.17858 0.58555 0.159842 0.030907 0.034416 0.012126 9 0.0052011 0.00092585 0.00068073 0.063814 0.12078 0.036005 10 0.091546 0.0047069 0.0022764 0.092301 1.8163 0.41813 11 0.11756 1.352 0.286775 0.00030556 4.5372 0.47284 12 0.019634 0.014034 0.0055606 0.1304 0.35591 0.10026 13 0.029039 2.5233 0.619227 0.18817 0.92188 0.22362 14 0.042591 0.067048 0.021155

Table A.7: MoG parameters for ÷ ≠ µ fading channel with ÷ = 5, µ = 10 and Copt = 3

i wi µi ÷i

1 0.32677 1.063 0.136022 0.32789 0.89548 0.0998553 0.34533 1.0116 0.10407

Table A.8: MoG parameters for Ÿ ≠ µ Shadowed fading channel with Ÿ = 1, µ = 3, m = 3and Copt = 4

i wi µi ÷i

1 0.40871 0.86626 0.183192 0.40942 1.111 0.229123 0.049081 1.4037 0.259154 0.13279 0.61749 0.15039

83

Appendix B

Sample Code for Section 2.3.3

The following sample code performs the following steps:

1. Generating a random variable following a Nakagami-m distribution.

2. Determining the optimal number of components according to the BIC.

3. Approximating the fading channel using the MoG distribution.

%%%%%%%%%%%%%%%%%%%% Written by Omar Alhussein, 28th July 2015.%%% Generation and Approximation of Fading channels using MoG model%%% References =%%%%%%%%%%%%%%%%%%% Global Variablessnr_dB = 0 ; % in dBsnr = 10.^(snr_dB./10); %linear%% Generate a NL random variate and Validate its PDF.N = 1e6 ; % # of samplesm = 3; % fading parametertheta = 1./m ;data1 = sqrt(gamrnd(1,theta,1,N)); % Since it is sqrt(). this is amplitude.[rpdf,h1] = hist(data1,100);rpdf = rpdf/(h1(2)≠h1(1))/length(data1) ; %normalized amplitude pdf.

datax = data1.^2 ; % This is the instantaneous snr RV.[rpdfx,hx] = hist(datax,100);rpdfx = rpdfx/(hx(2)≠hx(1))/length(datax) ; %normalized snr pdf.

figure;plot(h1,rpdf,'b*≠'); hold on ;plot(hx,rpdfx,'ko≠');legend('empirical amplitude PDF','empirical SNR PDF');% AoF = var(datax)./(mean(datax))^2%% Determination of the optimal number of components

ticN = 1e4;data = data1 ;C = 10 ; % upper number of components

84

e = 1e≠6 ; % error of stopping criterionnum_iter = 1e4; % number of iterations.options = statset('MaxIter',num_iter,'TolFun',e,'Display','final');BIC = zeros(1,C);obj = cell(1,C);for k = 1:C

obj{k} = gmdistribution.fit(data',k,'Options',options);BIC(k)= obj{k}.BIC; %one can also employ AIC as well.

end[minBIC,numComponents] = min(BIC);disp('minimum number of components is: ') ;numComponents

toc;figure;semilogy(1:C,BIC≠min(BIC)+1);title('BIC Behavior');xlabel('Number of Components');ylabel('BIC');%% Fitting MoG parameters using EM code

ticN=1e6;data = data1 ;C = numComponents; %number of componentse = 1e≠6 ; %tolerance of stopping criterionoptions = statset('MaxIter',num_iter,'TolFun',e,'Display','final');obj = gmdistribution.fit(data',C,'Options',options);BIC = obj.BIC;

w_est = obj.PComponents ;mu_est = obj.mu;for i =1:Csigma_est(i) = sqrt(obj.Sigma(:,:,i));endtoc;

x = h1 ;%.% Building the GMM Modelfhat = zeros(size(x));for i = 1:Cfhat = fhat + w_est(i) * normpdf(x,mu_est(i),sigma_est(i));endx = hx ;fhat_snr = zeros(size(x));for i = 1:Cfhat_snr = fhat_snr + w_est(i) ./ (sqrt(8*pi*snr.*x)*sigma_est(i)) ...

.* exp(≠(sqrt(x./snr)≠mu_est(i)).^2 ./(2*sigma_est(i)^2)) ;end

%Check Amplitude Distributionfig = figure;plot(h1,fhat,'k');hold on ;plot(h1,rpdf,'r*');legend('MoG amplitude PDF','Empirical amplitude PDF');title(['EM Based approximation with' num2str(C) ' components']);

%Visualize SNR Distribution

85

fig = figure;plot(hx,fhat_snr,'k');hold on ;plot(hx,rpdfx,'b*');legend('MoG SNR PDF','Empirical SNR PDF');title(['EM Based approximation with' num2str(C) ' components']);

%writing MoG parameters to a file.dlmwrite('NEW_MoG_KappaMu_w_Mu_Sig.txt',...

[w_est', mu_est,sigma_est'],'delimiter', '\t');

MSE = mean((fhat_snr≠rpdfx).^2) ; %Mean square error for SNR PDF.

%% If the delay bothered you, consider using Variational Bayes method%% for determining the number of components as well as getting parameters%% simulataneously. refer to Chapter~3.

86

Appendix C

Mixture Gamma Parameters

Table C.1 lists the MG distribution parameters of various fading channels. The second col-umn (Parameter) contains parameters dependent on the corresponding fading distributionwith m, and ’2 defined in (2.7).

Table C.1: MG parameters for various fading channels

Channel Parameter – — 1/’ MSELognormal ’ = 1 0.8795306 21.47391625 0.04634621 2.19E-04

- 0.1204694 19.29669267 0.06526572Weibull m = 4 0.4163066 2.547627 0.276866 8.14E-05

- 0.5836934 6.410809 0.1887993Nakagami-m mm

�(m)

m 1

m -Ray/logn - 0.2889985 0.9667361 3.908869 1.18E-06

’ = 1 0.7110015 0.972492 1.62796Ray/logn - 0.7229848 0.9223047 0.8351385

’ = 1.5 0.2770152 0.8499458 3.0324419Ray/logn m = 1 0.3491298 0.9117919 0.6304104 1.43E-6

’ = 0.5 0.6508702 1.225578 1.010205Nak/logn m = 2 0.6569638 1.9707105 0.4273802 1.55E-6

’ = 0.5 0.3430362 2.5034565 0.5267421Nak/logn m = 4 0.7775037 4.0356456 0.2247047 3.76E-6

’ = 0.5 0.2224963 5.2938705 0.2556295

87