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Parameter Estimation for Wireless Fading and Turbulence Channels Wang, Ning 2010-06-30

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PARAMETER ESTIMATION FOR WIRELESSFADING AND TURBULENCE CHANNELSbyNING WANGB.Eng., Tianjin University, China, 2004A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(OKANAGAN)June 2010c© Ning Wang, 2010AbstractIn this thesis, we mainly investigate the parameter estimation problem for fading and atmo-spheric turbulence channel models for wireless communications. A generalized method ofmoments (GMM) estimation scheme is introduced to the estimation of Nakagami fading pa-rameter. Our simulation results and asymptotic performance analysis reveal that this GMMframework achieves the best performance among all method of moments estimators based onthe same moment conditions. Further improved performance can be achieved using additionalmoment conditions in the GMM. In the study of the maximum-likelihood (ML) based Nak-agami m parameter estimators, we observe that a parameter ∆, which is defined as the logarith-mic ratio of the arithmetic mean to the geometric mean of the Nakagami-m fading power, canbe used to assess the estimation performance of ML-based estimators analytically. For smallsample size, the probability density function (PDF) of ∆ is derived by the moment generatingfunction (MGF) method. For large sample size scenarios, we use a moment matching methodto approximate the PDF of ∆ by a two-parameter Gamma PDF. This approximation is vali-dated by the Kolmogorov-Smirnov (K-S) test as well as simulation results. When studying theGamma-Gamma turbulence model for free-space optical (FSO) communication, an estimationscheme for the shape parameters of the Gamma-Gamma distribution is introduced based onthe concept of fractional moments and convex optimization. A modified estimation scheme,which exploits the relationship between the Gamma-Gamma shape parameters in FSO com-munication, is also proposed. Simulation results show that this modified scheme can achievesatisfactory estimation performance over a wide range of turbulence conditions.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fading and Turbulence Models for Wireless Communication . . . . . . . . . . 21.2.1 Multipath Fading Models . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Atmospheric Turbulence Models . . . . . . . . . . . . . . . . . . . . 61.3 Moment-Based Estimation Techniques . . . . . . . . . . . . . . . . . . . . . 71.4 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 82 Moment-Based Estimation for the Nakagami-m Fading Parameter . . . . . . . . 102.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Moment-Based m Parameter Estimation with Integer and Fractional Moments . 112.2.1 Integer-Moment-Based m parameter Estimators . . . . . . . . . . . . 11iiiTable of Contents2.2.2 A Family of Fractional Moment-Based m Parameter Estimators . . . . 132.2.3 Large Sample Properties: Asymptotic Variance Analysis . . . . . . . . 152.3 Generalized Method of Moments Estimation . . . . . . . . . . . . . . . . . . 162.3.1 GMM Estimation for the Nakagami m Parameter . . . . . . . . . . . . 162.3.2 Derivation for Asymptotic Variance of the GMM Estimator . . . . . . 192.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 On Statistics of Logarithmic Ratio of Arithmetic Mean to Geometric Mean forNakagami-m Fading Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 ML-Based Nakagami-m Parameter Estimators . . . . . . . . . . . . . . . . . 283.2 Statistical Properties of ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Alternative Expression of ∆ . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Nonnegative Property of ∆ . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Moment Generating Function of ∆ . . . . . . . . . . . . . . . . . . . 313.2.4 Probability Density Function of ∆ . . . . . . . . . . . . . . . . . . . . 323.3 Gamma Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 Gamma Approximation for PDF of ∆ . . . . . . . . . . . . . . . . . . 363.3.2 Validating the Gamma Approximation . . . . . . . . . . . . . . . . . 373.4 Applications and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 413.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Moment-Based Estimation for the Gamma-Gamma Distribution with FSO Appli-cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Statistical Properties of the Gamma-Gamma Turbulence Model . . . . . . . . 474.2.1 Parameterization of the Gamma-Gamma Turbulence Model . . . . . . 474.2.2 Moments of The Gamma-Gamma Turbulence Model . . . . . . . . . . 49ivTable of Contents4.3 An MoM/CVX Estimation Scheme for Gamma-Gamma Shape Parameters . . . 514.4 A Modified MoM/CVX Estimation Scheme for the Shape Parameter α . . . . . 554.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62AppendicesA Derivation of (2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67B Exponential Family Property of the Nakagami-m Distribution . . . . . . . . . . 69vList of Tables3.1 Kolmogorov-Smirnov test for gooodness-of-fit for the Gamma approximation. . 403.2 Numerical MSE performance evaluations for ML-based Nakagami m parame-ter estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41viList of Figures2.1 Simulated MSE performance of moment-based Nakagami fading parameterestimators with sample size N = 10,000. . . . . . . . . . . . . . . . . . . . . . 252.2 Asymptotic relative estimation efficiencies of moment-based Nakagami fadingparameter estimators with respect to ML. . . . . . . . . . . . . . . . . . . . . 263.1 Comparison of empirical PDFs and analytical PDFs of ∆ for m = 0.5,1,2 withsample size N = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Comparison of empirical PDFs and analytical PDFs of ∆ for m = 0.5,1,2 withsample size N = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Comparison of empirical PDFs and Gamma approximated PDFs of ∆ for m =0.5,1,2 with sample size N = 10. . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Comparison of empirical PDFs and Gamma approximated PDFs of ∆ for m =0.5,1,2 with sample size N = 100. . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Analytical and simulated MSE performance of ML-based Nakagami m param-eter estimators with sample size N = 100. . . . . . . . . . . . . . . . . . . . . 434.1 Gamma-Gamma shape parameters α and β as functions of σR. . . . . . . . . . 484.2 Gamma-Gamma PDFs with Rytov variance σ 2R = 0.25, 2, and 11. . . . . . . . . 504.3 MSE performance of the MoM/CVX estimator and the modified MoM/CVXestimator with k = 0.5 and sample size N = 100,000. . . . . . . . . . . . . . . 544.4 Absolute derivative functions of the Gamma-Gamma shape parameters α andβ with respect to σR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56viiList of Figures4.5 Flow chart of the modified MoM/CVX estimator for the Gamma-Gamma shapeparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58viiiList of AcronymsAcronyms DefinitionsAF Amount of FadingCDF Cumulative Distribution FunctionCLT Central Limit TheoremCRLB Cramer-Rao Lower BoundFSO Free-Space OpticalGDE Greenwood-Durand EstimatorGMM Generalized Method of MomentsINV Inverse Normalized VarianceK-S Kolmogorov-SmirnovLOS Line-of-SightLS Least SquaresMGF Moment Generating FunctionFFT Fast Fourier TransformML Maximum-LikelihoodMLE Maximum-Likelihood EstimatorMoM Method of MomentsMoM/CVX Method of Moments/Convex OptimizationMSE Mean Square ErrorMVUE Minimum Variance Unbiased EstimatorPDF Probability Density FunctionixList of AcronymsRF Radio FrequencyRV Random VariableWLLN Weak Law of Large NumbersxAcknowledgmentsFirstly and foremost, I would like to take this chance to give my warm and grateful thanksto my supervisor, Dr. Julian Cheng, for providing me with the opportunity to work in thefield of wireless communication, and for his continuous support, encouragement, and guidancethroughout my master’s study.I am deeply indebted to Dr. Richard Klukas and Dr. Stephen O’Leary for their great effortand significant amount of time to serve on my M.A.Sc. committee. I would also like to expressmy thanks to Dr. Shawn Wang from UBC Okanagan Mathematics and Statistics Departmentfor both his help on my coursework and his willingness to serve as my external examiner. Ireally appreciate their valuable time and constructive comments on my thesis. In addition, Iwould like to thank Dr. Chintha Tellambura from University of Alberta for his feedback andvaluable suggestions on my research work.My thanks to all my labmates and friends at UBC Okangan for their help and friendship.Special thanks to Xian Jin, Chiun-Shen Liao, Mingbo Niu, and Xuegui Song for always beingavailable for technical discussions and for sharing their academic experiences generously.Finally, I would like to express my deepest gratitude for the constant support, understandingand unconditional love that I received from my family.xiChapter 1Introduction1.1 Background and MotivationWith the capability of providing globally interconnected voice and data communication sys-tems, as well as establishing local communication architecture for the interconnection of elec-tronic devices, wireless communication is undoubtedly one of the most vibrant areas in com-munication theory research today. Modern wireless communication dates back to the inventionof wireless telegraph system by Guglielmo Marconi over one hundred years ago. However,even though it emerged only 20 years after the invention of the telephone by Alexander Gra-ham Bell, wireless communication was not widely used in the consumer communication mar-ket until the early 1980s. For nearly one century’s time in the modern telecommunicationhistory, most of the market was dominated by wireline communication.Technologically, what impeded wireless communication from extensive application werethe undesirable features of the wireless transmission environment. Being an open transmissionmedium, wireless channels can bring much more attenuation and uncertainty to the transmit-ted signal than wireline does, thus more sophisticated technologies have to be implemented inwireless systems to combat this disadvantage. Generally speaking, there are two fundamentalchallenges that have to be addressed for wireless communication. The first challenge is therandom fluctuation of the transmitted signal. In radio frequency (RF) wireless communicationsystems, this is known as fading which is mainly due to the multipath effect and shadowing;whereas for a more recent line-of-sight (LOS) wireless communication technology, the free-space optical (FSO) communication, much smaller wavelength and directionality determine11.2. Fading and Turbulence Models for Wireless Communicationthat signal fluctuation in FSO is dominated by the effect of atmospheric turbulence but notfading. The second challenge for wireless communication is the interference from other usersor other communication systems. Unlike wireline communication which uses a bounded trans-mission medium, communication through a wireless channel is more subject to interferencebecause different users and different systems are sharing the same transmission medium. Inthis thesis, we address the first challenge and focus our study to the problem of parameterestimation for wireless fading and turbulence channel models. Knowledge of these parametervalues can be used, for example, to design a better transmission scheme adaptive to the wirelesslinks and to better characterize wireless channels for link budget analysis.1.2 Fading and Turbulence Models for WirelessCommunicationTo address the first challenge for wireless communication, the random fluctuations of the trans-mitted signal, fading and atmospheric turbulence models are proposed for different applicationscenarios and channel conditions based on statistical study of received signals. However, mod-elling the pattern of the signal fluctuations is just the first step. In order to employ a fading orturbulence model in system design and performance analysis, it is also critical to determine orestimate the parameters of the model, which will fit the model to the specific channel condi-tions. This estimation process is what we will mainly discuss in this thesis. Before discussingthe parameter estimation problem further, we will first review some well-known fading andturbulence models for wireless communications.1.2.1 Multipath Fading ModelsAs a result of the random propagation (reflection, diffraction and scattering etc.) of radiowaves in the wireless transmission environment, several replicas of the transmitted signal, with21.2. Fading and Turbulence Models for Wireless Communicationdifferent amplitudes and phases, arrive at the receiver end. When these signal replicas are notresolvable, they are added up constructively or destructively at the receiver, causing multipathfading or fading in short.The delay spread, denoted by Td, is one of the most important channel properties; it char-acterizes the time domain dispersive nature of the fading channel. In brief, Td describes thearrival time span of all the available signal replicas. The reciprocal of Td is known as the coher-ent bandwidth, which is denoted by Wc. When the signal bandwidth W is much smaller thanWc, the channel is considered as frequency-nonselective or flat, which means that all the fre-quency components of the transmitted signal experience the same attenuation and phase shift.Otherwise, the channel is considered as frequency-selective fading.Another important parameter which characterizes the frequency dispersive nature of thefading channel is the Doppler spread Ds; the reciprocal of Ds is the coherent time Tc. When Tcis much larger than the delay requirement of the system, which is usually set to be the symbolduration T , the channel is considered to be a fast fading channel; otherwise the channel is saidto be slow.Therefore, based on the relative relation between properties of the transmitted signal (sym-bol duration T in time domain and signal bandwidth W in frequency domain) and propertiesof the wireless fading channel (coherent time Tc in time domain and coherent bandwidth Wcin frequency domain), we can classify fading channels into four basic types: fast frequency-selective fading, fast frequency-nonselective fading, slow frequency-selective fading, and slowfrequency-nonselective fading. In this thesis, we focus on estimation of slow frequency-nonselective fading models.Regardless of noise and interference, the slow frequency-nonselective fading channel canbe described by the following expressionsr(t) = α ·st(t) (1.1)31.2. Fading and Turbulence Models for Wireless Communicationwhere st(t) and sr(t) are transmitted and received complex signals respectively, and the com-plex random variable α represents the fading channel characteristics.The first-order statistics of the fading channel, which characterizes the fading envelopeor amplitude of the parameter α, is the most widely used approach to study fading effect.Among all the statistical models proposed for the fading envelope, the most well-known onesare Rayleigh, Rician, and Nakagami-m models.The Rayleigh and Rician fading models are derived from Clasrke’s one-ring model [1]for the electromagnetic field of the received multipath signal. The Clarke’s model assumesthat between the transmitter and the receiver, there are N unresolvable paths with randomamplitudes and phases. If no LOS path exists, for sufficiently large N, by the central limittheorem (CLT), all independent paths will have Gaussian distributed amplitudes and uniformphases. Therefore the real and imaginary parts of the sum will be independent and identically-distributed (i.i.d.) zero mean Gaussian RVs, and the corresponding fading envelope R will havea Rayleigh distribution with probability density function (PDF)fR(r) = 2rΩ e−r2Ω , r ≥ 0 (1.2)where the parameter Ω is the fading power Ω = EbracketleftbigR2bracketrightbig. When an LOS or specular path withknown amplitude exists between the transmitter and the receiver, the real part and the imaginarypart of the combined signal will be correlated, and the fading envelope will have a Riciandistribution. The PDF of the Rician distribution is given byfR(r) = 2r(K +1)Ω expbraceleftbigg−K − (K +1)r2ΩbracerightbiggI0parenleftBigg2rradicalbiggK (K +1)ΩparenrightBigg, r ≥ 0 (1.3)where I0(·) is the zeroth order modified Bessel function of the first kind, and the parameter Kis known as Rician K-factor which is defined as the ratio of the power in the specular path tothe power in the scattered paths. According to the definition, the Rician K-factor indicates the41.2. Fading and Turbulence Models for Wireless Communicationrelative strength of the LOS component: whenK = 0, the LOS component disappears, anf theRician distribution specializes to Rayleigh distribution; when K approaches infinity, the scat-tering components in the signal are negligible, and the channel becomes static or deterministic.Different from the Rayleigh and Rician distributions, the Nakagami-m distribution was notderived from any theoretical model. It was originally deduced from experimental data [2].Thus for a variety of fading conditions, the Nakagami-m model can fit the practical multipathfading measurements better than the other models. The PDF of the Nakagami-m distributedfading envelope is given byfR(r) = 2Γ(m)parenleftBigmΩparenrightBigmr2m−1 expparenleftBig−mΩr2parenrightBig, r ≥ 0, m ≥ 12 (1.4)where m is known as the fading parameter whose reciprocal quantifies the degree of fading,and Γ(·) is the Gamma function defined byΓ(z) =integraldisplay +∞0tz−1e−tdt. (1.5)The Nakagami-m distribution covers a wide range of fading conditions. It can be shown thatwhen m = 12, which corresponds to the most severe fading condition, the Nakagami-m distribu-tion becomes the one-sided Gaussian distribution. The Rayleigh distribution can also be foundas a special case of the Nakagami-m distribution by letting m = 1. Being capable of modellinga wide range of fading conditions as well as having a tractable PDF, the Nakagami-m fadingmodel is a popular and widely used fading model in wireless communication research. Esti-mation of the Nakagami-m fading model is thus of great interest in wireless communicationsresearch.51.2. Fading and Turbulence Models for Wireless Communication1.2.2 Atmospheric Turbulence ModelsAs a typical LOS communication technology, FSO differs from most RF systems which sufferfrom fading due to multipath propagation. In FSO communication, the main impairment iscaused by atmospheric turbulence-induced fluctuations [3]. Therefore, multipath fading mod-els are no longer applicable to system design and performance analysis for FSO systems. In-stead we focus on the study of atmospheric turbulence models.For weak turbulence conditions, Parry [4] and Phillips and Andrews [5] independently sug-gested a log-normal PDF to model the irradiance, which is the power density of the opticalbeam. With unit mean irradiance and scintillation index σ 2I , the log-normal PDF of the irradi-ance I is given by [3]fL(I) = 1IradicalBig2piσ 2IexpbraceleftBigg−bracketleftbiglnI + 12σ2I bracketrightbig22σ 2IbracerightBigg, I > 0. (1.6)When turbulence becomes stronger, the negative exponential distribution was introduced as alimit distribution for the irradiance. This limit distribution can only provide sufficient accuracywhen the system goes far into the saturation regime [6]. The K-distribution, which is basedon an assumed modulation process, was later introduced to model the irradiance in strongturbulence scenarios [7]. The K-distribution with unit mean irradiance is characterized by thePDF [3]fK(I) = 2αΓ(α)(αI)(α−1)/2Kα−1parenleftBig2√αIparenrightBig, I > 0 (1.7)where α is a positive shape parameter related to the effective number of discrete scatterers andKv(·) is the vth order modified Bessel function of the second kind.Being a widely accepted turbulence model for FSO communication under strong turbu-lence conditions, the K-distribution is, however, incapable of modeling the irradiance whenturbulence is weak. This is because the scintillation index given by the K distributed irradi-ance is always greater than unity, which is not valid for weak turbulence scenarios. Another61.3. Moment-Based Estimation Techniquesmodulation-based model, the Gamma-Gamma distribution, was later proposed by Al-Habashet al. [8] to model the irradiance in FSO systems. The PDF of the Gamma-Gamma distributionis given byfG(I) = 2(αβ)(α+β)/2Γ(α)Γ(β) Iα+β2 −1Kα−βparenleftBig2radicalbigαβIparenrightBig, α > 0,β > 0 (1.8)where α and β are the shape parameters. Note that by setting the shape parameter β = 1, theGamma-Gamma distribution will degenerate to the K-distribution. The Gamma-Gamma turbu-lence model is desirable because for both weak and strong turbulence scenarios, this model canprovide a good fit to the experimental measurements of irradiance [8]. Thus, the key advan-tage of using the Gamma-Gamma turbulence model is that it covers a wide-range of turbulenceconditions.1.3 Moment-Based Estimation TechniquesA number of statistical signal processing approaches have been introduced to parameter esti-mation in wireless communication research, among which the most popular ones in practicalapplications are the ML estimation and the method of moments (MoM) or the moment-basedapproach. Because of its asymptotic efficiency, the maximum likelihood estimator (MLE) isapproximately the minimum variance unbiased estimator (MVUE) and the ML approach canalso give us the Cram´er-Rao lower bound (CRLB) for the MVUE which describes the bestachievable estimation performance for unbiased estimators [9]. Therefore the ML approachis usually the more preferable one for theoretical studies. However, when the model, or morespecifically the PDF of the fading or atmospheric turbulence that will be intensively discussedin this thesis, involves transcendental functions, the ML approach will likely involve solvingan nonlinear transcendental equation or equation set, which can be undesirable in practice.As an alternative approach, the method of moments can usually lead to estimators which are71.4. Thesis Outline and Contributionseasy to determine and implement [9]. Even though there exists no optimality properties for themoment-based approach, it can usually give satisfactory estimates when the sample size is largeenough. The basic principle of the method of moments is to equate the population moments ofthe model to their sample counterparts. Several estimation techniques under the basic frame-work of MoM have been proposed by researchers. The most widely used one is the classicalmethod of moments, which solves for unknown parameters in an equation or equation set de-rived from the moment conditions of the model. The generalized method of moments (GMM)approach [10] proposed a regression estimation scheme for determined and over-determinedproblems. Using linear combinations of order statistics, the L-moment method, a more robustmethod that suffers less from sampling variability, was introduced for estimation of distri-butions [11]. In addition, combinations of the moment-based approach and other estimationapproaches are also reported [12] [13] [14]. Multiple moment-based estimation methods aswell as combinations of estimation methods will be used in this thesis.1.4 Thesis Outline and ContributionsThis thesis have been divided into five chapters. Chapter 1 reviews some background knowl-edge about fundamental challenges for wireless communication: random fluctuation of thetransmitted signal and the interference problem. To address the signal fluctuation problem,we first need to model the fluctuation pattern and then estimate corresponding characteristicparameters when applying the model in system design and performance analysis for specificapplication scenarios. This motivates researchers to find better estimators for popular fadingand atmospheric turbulence models. It is pointed out that the method of moments is sometimesa preferable approach to ML when the model takes an intractable form.In Chapter 2, we provide a detailed discussion on moment-based estimation for the Nakagami-m fading model. Firstly a family of classical moment-based m parameter estimators is re-viewed, both integer moments scenario and fractional moments scenario are discussed. Then81.4. Thesis Outline and Contributionsthe GMM method which exploits information resides in all moment conditions in a determinedor over-determined estimation problem is introduced to fading parameter estimation for the firsttime. At last, a systematic performance comparison for moment-based m parameter estimatorsis conducted by both the simulated mean square error approach and the analytical asymptoticvariance approach.In Chapter 3, we use moment-based method to study performance of ML-based Nakagamim parameter estimators. By examining the derivation of the ML-based m parameter estimationproblem, it is found that a parameter ∆, which is defined as the logarithmic ratio of arith-metic mean to geometric mean for Nakagami-m fading power, is critical to the ML-based mparameter estimation. Closed-form expressions are derived for the moment generating func-tion (MGF) and the PDF of ∆. For large sample size, we use a moment matching method toapproximate the PDF of ∆ by a two-parameter Gamma PDF. This approximation is validatedby the Kolmogorov-Smirnov (K-S) test. As an application, the approximate PDF is used tostudy the performance of three well known ML-based Nakagami m parameter estimators, theGreenwood-Durand estimator [15] and the first and second order Cheng-Beaulieu estimators[16].Chapter 4 studies the parameter estimation of the Gamma-Gamma turbulence model forFSO communication. A novel estimation scheme for the shape parameters of the Gamma-Gamma distribution is proposed based on a combination of fractional MoM estimation andconvex optimization. Then a modified estimation scheme, which turns out to be an improvedone, is proposed by considering relationship between the Gamma-Gamma shape parameters forFSO applications. Revealed by computer simulation results, the modified scheme can achieveimproved performance over a wide range of turbulence conditions.Chapter 5 summarizes contents and contributions of this thesis, and suggests some possiblefuture works in related topics.9Chapter 2Moment-Based Estimation for theNakagami-m Fading Parameter2.1 Background and MotivationThe Nakagami-m fading model is important in wireless communications research because itfits the empirical multipath fading measurements better than the other fading models for avariety of fading conditions [2]. The Nakagami-m model is also desirable because error rateperformance analysis with Nakagami fading often leads to closed-form analytical results.The PDF of the Nakagami-m fading envelope R has a two-parameter form, which is givenby [2]fR(r) = 2Γ(m)parenleftBigmΩparenrightBigmr2m−1 expparenleftBig−mΩr2parenrightBig, r ≥ 0, m ≥ 12 (2.1)where Ω=E[R2], and the fading parameter m is defined as [2]m = Ω2E[(R2−Ω)2], m ≥12. (2.2)Given N independent realizations of the Nakagami-m random variable R1, R2, . . . , RN, sincethe parameter Ω is defined as the second-order moment of the Nakagami-m fading envelope, itis straightforward to use the second order sample moment of the fading envelope to estimateΩ,which is ˆΩ = ˆµ2 = 1N ∑Ni=1 R2i . Thus in order to characterize wireless communication chan-nels using the Nakagami model, it is critical to determine or estimate the value of m, which is102.2. Moment-Based m Parameter Estimation with Integer and Fractional Momentsalso known as reciprocal of the amount of fading (AF), from N independent observations. No-tice that the squared value of a Nakagami-m distributed random variable is a Gamma randomvariable, Nakagami m parameter estimation is sometimes related to estimation for the Gammadistribution.Greenwood and Durand demonstrated that the ML-based m parameter estimation problemleads to solving a non-linear transcendental equation involving a natural logarithmic functionand a digamma function [15]. The most well-known ML-based m parameter estimators, theGreenwood-Durand estimator [15] and the Cheng-Beaulieu estimators [16] are actually ap-proximate solutions to the ML Nakagami m parameter estimation problem. This undesirablefeature of the ML approach has, in part, motivated researchers to use a moment-based approachto find alternative Nakagami m parameter estimators.2.2 Moment-Based m Parameter Estimation with Integerand Fractional MomentsIn this section, we review some moment-based Nakagami m fading parameter estimators de-rived from analytical moment expressions of the Nakagami-m distribution.2.2.1 Integer-Moment-Based m parameter EstimatorsThe kth moment expression for the Nakagami-m distribution is given byµk =EbracketleftBigRkbracketrightBig= Γ(m+k/2)Γ(m)parenleftbiggΩmparenrightbiggk/2. (2.3)To avoid transcendental functions in deriving moment-based m parameter estimators, first weneed to find a way to cancel the Gamma functions in (2.3).Recall the iterative property of the Gamma function Γ(z+1) = z·Γ(z), it is straightforward112.2. Moment-Based m Parameter Estimation with Integer and Fractional Momentsto show that for even k values, Γ(m +k/2) can be written as product of Γ(m) and a polyno-mial function of m. Gamma functions in the moment expression can then be canceled, and agenerally preferred algebraic equation is obtained. This is the basic idea of the moment-basedNakagami m parameter estimator proposed by Abdi and Kaveh [17]. For finite sample size,higher order sample moments may deviate from the value of the true moments significantly[18] (which is known as the outlier problem), smaller k values are preferred in this moment-based m parameter estimation scheme. However, the second order moment of the Nakagami-mdistribution is simply the parameter Ω, which does not have parameter m in it. Therefore Abdiand Kaveh derived a moment-based m parameter estimator based on the fourth order momentexpression of the Nakagami-m distribution. Substituting Ω in the fourth order moment expres-sion by the second order sample moment, and then solve for m, a moment-based Nakagami mparameter estimator was found as [17]ˆmINV = ˆµ22ˆµ4 − ˆµ22 (2.4)where ˆµk = 1N ∑Ni=1 Rki is the kth order sample moment. This estimator was named the inversenormalized variance (INV) estimator, because it can be obtained by replacing the moments inthe definition of m in (2.2) with the sample moments.An alternative way to cancel the Gamma functions in (2.3) is to take the ratio of two dif-ferent moments of the Nakagami-m distribution. Again by the iterative property of the Gammafunction, we observe that the ratio of the kprimeth and kth moments, where kprime−k = a is a non-zeroeven integer, also formulates an algebraic equation which can be easily solved. The simplestcase of this approach is to use kprime = 3 and k = 1µ3µ1 =Γ(m+3/2)Γ(m+1/2)parenleftbiggΩmparenrightbigg=parenleftbiggm+ 12parenrightbiggparenleftbiggΩmparenrightbigg. (2.5)Solving (2.5) for m with Ω substituted by its estimator, Cheng and Beaulieu derived another m122.2. Moment-Based m Parameter Estimation with Integer and Fractional Momentsparameter estimator based on integer moments [19]ˆmt = ˆµ1 ˆµ22( ˆµ3 − ˆµ1 ˆµ2). (2.6)Notice that the highest order of sample moments used in estimator ˆmt is 3, which is smallerthan that of ˆmINV , therefore ˆmt is expected to suffer from the outlier problem less than theINV estimator, which suggests a better estimation performance. This intuitive result will beconfirmed by the asymptotic variance analysis in Section 2.2.3.2.2.2 A Family of Fractional Moment-Based m Parameter EstimatorsCheng and Beaulieu observed that the INV estimator can also be derived by taking the ratioof the fourth and second moments of the Nakagami-m distribution and solving for parameterm [19]. This suggests that both estimators discussed in Section 2.2.1 belong to a family ofmoment-based Nakagami m parameter estimators derived by the ratio-of-moments approach.Notice that the order index k in the Nakagami-m moment expression (2.3) is not restrictedto positive integers. This estimator family can thus be expanded into a family of fractionalmoment-based m parameter estimators. As briefly discussed in Section 2.2.1, because of theoutlier problem, smaller moment order indices are preferred in this estimator family. There-fore admitting the use of fractional moments can actually result in better m parameter estima-tors. This novel idea was first introduced to Nakagami m parameter estimation by Cheng andBeaulieu in [19].Based on the framework of the ratio-of-moments approach discussed in Section 2.2.1, andassuming that k = 1/p and kprime = 2+1/p, where p is a positive real number, the ratio of momentscan then be expressed asµ2+1/pµ1/p =Γ(m+1+1/2p)Γ(m+1/2p)parenleftbiggΩmparenrightbigg=parenleftbiggm+ 12pparenrightbiggparenleftbiggΩmparenrightbigg. (2.7)132.2. Moment-Based m Parameter Estimation with Integer and Fractional MomentsSolving (2.7) and replacing the population moments with their sample counterparts, a generalexpression of this estimator family was found as [19]ˆm1/p = ˆµ1/p ˆµ22pparenleftbigˆµ2+1/p − ˆµ1/p ˆµ2parenrightbig. (2.8)It is straightforward to show that when p = 1, ˆm1/p is actually ˆmt, and ˆmINV corresponds to ap value of 0.5.However, when p approaches +∞, (2.8) is found to have a 00 indeterminate form after somealgebraic manipulations. Therefore we need to go back to the population moment expressionof (2.8) to find the expression for the limiting case.Assuming k = 1/p and recognizing that the limiting value of k is 0, we denote the limitingestimator as ˆm0. Then apply L’Hˆopital’s rule to the population moment expression of theestimator family, we havelimk→0kµkµ22(µ2+k −µkµ2) = limk→0kE[Rk]µ22(E[R2+k]−E[Rk]µ2)= limk→0µ2parenleftbigE[Rk]+kEbracketleftbigRk lnRbracketrightbigparenrightbig2parenleftbigEbracketleftbigR2+k lnRbracketrightbig−µ2EbracketleftbigRk lnRbracketrightbigparenrightbig= µ22(E[R2 lnR]−µ2E[lnR])= µ2E[R2 lnR2]−µ2E[lnR2].(2.9)Replacing the population moments and expected value expressions in (2.9) by their samplecounterparts, the limiting estimator was found to be [20]ˆm0 = ˆµ21N ∑Ni=1 R2i lnR2i − ˆµ21N ∑Ni=1 lnR2i. (2.10)Combining (2.8) and (2.10) together and using the order index k consistently, we can rewrite142.2. Moment-Based m Parameter Estimation with Integer and Fractional Momentsthe fractional moment-based Nakagami m parameter estimator family as [19]ˆmk =k ˆµk ˆµ22( ˆµ2+k − ˆµk ˆµ2), k > 0ˆµ21N ∑Ni=1 R2i lnR2i − ˆµ21N ∑Ni=1 lnR2i, k = 0.(2.11)We observe for the estimator family (2.11), the smaller k is, the smaller the order samplemoments that are used, and therefore better estimation performance is expected. In the limitingcase, ˆm0 should intuitively achieve the best performance among this fractional moment-basedm parameter estimator family. Theoretical estimation performance analysis of this m parameterestimator family using the idea of asymptotic variance will be discussed in Section 2.2.3.2.2.3 Large Sample Properties: Asymptotic Variance AnalysisFor finite sample size, moment-based estimators are usually biased, do not have optimalityproperties, and their analytical performance are difficult to obtain. However, because of theconsistency of moment-based estimators, we can derive their asymptotic variance analytically,which can be of great importance to performance analysis and comparison for large samplesize scenarios.The idea of asymptotic variance analysis of moment-based estimators is based on the cen-tral limit theorem and the weak law of large numbers (WLLN). The moment-based m parameterestimators discussed in this section are √N-consistent and asymptotically unbiased, thus therandom variable √N ( ˆm−m), with ˆm a moment-based m parameter estimator, converges in lawto a zero mean Gaussian random variable with variance σ 2√N ( ˆm−m) L→ N parenleftbig0,σ 2parenrightbig as N →+∞. (2.12)The variance term σ 2 is the asymptotic variance of the corresponding moment-based m param-eter estimator.152.3. Generalized Method of Moments EstimationIn the derivation of the asymptotic variance, we use an approach known as the multivariatedelta method [21]. Take the moment-based estimator family with k > 0 as an example. Forlarge sample size N, by the CLT, the vector √N ( ˆµ −µ) follows a trivariate Gaussian distribu-tion N (0,Σk). Here µ = (µ2,µk,µk+2) is the population moment vector, ˆµ = ( ˆµ2, ˆµk, ˆµk+2)is the corresponding sample moment vector, and Σk is the covariance matrix of ˆµ. Since theestimator ˆmk in (2.11) is a function of the sample moments, the multivariate delta method saysthe asymptotic variance of ˆmk can be derived from Σk using the Jacobian method [21].The asymptotic variance σ 2k of the fractional moment-based m parameter estimator family(2.11) has been derived by Cheng [20] asσ 2k =m2v2kv2k +v2k+2 − v2k+2v2(k/2)2v2k, k > 0m2bracketleftbig1+mψprime(m+1)bracketrightbig, k = 0(2.13)where vk = Γ(m+k/2)/Γ(m), and ψ(z)= d [lnΓ(z)]/dz = Γprime(z)/Γ(z) is the digamma function.2.3 Generalized Method of Moments Estimation2.3.1 GMM Estimation for the Nakagami m ParameterThe basic idea used in the moment-based m parameter estimators reviewed in Section 2.2 isconsidered the classical method of moments, which aims to find a closed-form solution to atheoretical equation or equation set involving the moments of a distribution. The keystoneof classical method of moments is to find a tractable equation set with moment conditions.However, desirable equations of moments like the algebraic equations derived in Section 2.2for the Nakagami-m distribution are not always easy to find, even though closed-form analyticalmoment expressions are available.The generalized method of moments estimation was first introduced by Hansen [10] in162.3. Generalized Method of Moments Estimationeconometrics literature and it is already a widely used method in this research area. However,to the author’s best knowledge, this powerful method has not been applied to communicationsresearch. The GMM gives an alternative way to exploit moment conditions in estimation prob-lems. It performs parameter estimation by minimizing weighted distances between populationmoments and their sample counterparts. Usually, more moment conditions than the numberof unknown parameters are available in GMM estimation. The GMM provides a frameworkwhich combines all available moment conditions optimally for over-determined problems.The most widely used implementation of the GMM method is an iterative regression pro-cess proposed by Hansen in his original GMM paper [10], namely Hansen’s two-step GMMprocedure. In this section, we follow Hansen’s recipe to perform the Nakagami m parameterestimation with GMM.With N i.i.d. realizations of a Nakagami-m random variable R1, R2, . . . , RN and s > 1 pop-ulation moment conditions µk1, µk2, . . . , µks, the GMM estimation for Nakagami m parameteris formulated as minimizing the orthogonal criterion functionQ(m;r) = gTN(m)WgN(m) (2.14)where r = (R1,R2,...,RN)T is the observation vector, W is a weighting matrix, and gN(m) isthe distance vector defined asgN(m) =ˆµk1 −µk1(m)ˆµk2 −µk2(m)...ˆµks −µks(m). (2.15)In (2.15), ˆµki’s (i = 1, 2, . . . , s) are the (ki)th-order sample moments, and µki(m)’s denotethe (ki)th-order population moment conditions as functions of the unknown parameter m. Asdiscussed in Section 2.2, higher order sample moments may deviate from the population mo-172.3. Generalized Method of Moments Estimationments significantly, or we can say they are less accurate than lower order moment conditions.Therefore it is intuitively necessary to give higher order moment conditions less weight in theGMM framework. This is the purpose of introducing the weighting matrix W. The accuracy ofmoment conditions can be measured by the variance covariance matrix of the sample momentstatistics.The first step of Hansen’s recipe is to set W = I, the identity matrix. It means we first givethe same weights to all moment conditions and solve for an initial estimate ˆm(0), which can beexpressed asˆm(0) = argminmgTN(m)gN(m). (2.16)The solution to the least squares (LS) problem in (2.16) can easily be found with software toolslike MATLAB. Then we can use this initial estimate of the m parameter to obtain more preciseestimates by an iterative regression process.In the second step, we first compute the residue ˆut =bracketleftBigRk1t −µk1parenleftBigˆm(0)parenrightBig,Rk2t −µk2parenleftBigˆm(0)parenrightBig,..., Rkst −µksparenleftBigˆm(0)parenrightBigbracketrightBigT(t = 1,2,...,N) for all N observations. Then the autocovariance matri-ces Sj for lag length j is estimated bySj = 1NN∑n=j+1ˆut ˆuTt−j, j = 0,1,...,l (2.17)where l is the selected maximum lag length. With all l autocovariance matrices, we can esti-mate the long-run covariance matrix byˆS = ˆS0 + l∑j=1wjparenleftbigˆSj + ˆSTjparenrightbig (2.18)where wj’s are weights for autocovariance matrices with different lag values. Generally speak-ing, giving more distant lags less weight can improve estimation accuracy. A widely usedweighting scheme is that of Bartlett [22], which is given by wj = 1− j/(l +1). Then selecting182.3. Generalized Method of Moments EstimationW = ˆS−1, the second step estimate of the m parameter can be obtained asˆm(1) = argminmgTN(m)ˆS−1gN(m). (2.19)Step 2 is then iterated until the absolute difference between two consecutive estimates is lessthan a predetermined threshold (estimation accuracy requirement) ε.2.3.2 Derivation for Asymptotic Variance of the GMM EstimatorIn Section 2.2.3 we have introduced the basic concept of asymptotic variance analysis andshowed that the asymptotic variance of the fractional moment-based m parameter estimatorfamily has been derived by Cheng [20]. However, for the GMM m parameter estimator, theasymptotic variance has not been derived in the engineering literature; besides, it is also un-clear what is the best achievable performance among all possible m parameter estimators basedon certain available moment conditions. Because the GMM provides a framework to opti-mally exploit all available moment conditions in its iteration process, it is natural to ask if theGMM attains the best asymptotic performance among all moment-based estimators using thesame moment conditions. In this section, we derive the asymptotic variance of the GMM mparameter estimator introduced in Section 2.3.1.Using the assumptions made in Section 2.3.1, we have N i.i.d. realizations of a Nakagami-m RV and s > 1 population moment conditions µk1, µk2, . . . , µks. For large sample size N,the joint distribution of the elements of d = ( ˆµk1 −µk1, ˆµk2 −µk2,..., ˆµks −µks)T approachesa multi-variate Gaussian distribution N (0,Σ), where Σ is the covariance matrix of elementsof random vector d, the difference vector between available population conditions and theirsample counterparts. The element of Σ at the ith row and the jth column is Σi j = µki+k j −192.3. Generalized Method of Moments Estimationµkiµk j. Thus, the joint PDF of the observed sample moment vector ˆµ = ( ˆµk1, ˆµk2,..., ˆµks)T isf ( ˆµk1,..., ˆµks)= 1(2pi) s2 bracketleftbigdetparenleftbigΣNparenrightbigbracketrightbig12exp−( ˆµk1 −µk1,..., ˆµks −µks)parenleftbiggΣNparenrightbigg−1ˆµk1 −µk1...ˆµks −µks(2.20)where µi’s are functions of m and Ω, and det(·) denotes the determinant of a square matrix.The estimate ˆmGMM in a maximum-likelihood sense can be expressed asˆmGMM = argmaxmln f ( ˆµk1,..., ˆµks)= argmaxm−12 ln[det(Σ)]−N ( ˆµk1 −µk1,..., ˆµks −µks)Σ−1ˆµk1 −µk1...ˆµks −µks+C(2.21)where C is a constant which does not depend on m.For large sample size N, the quadratic term in (2.21) will be the dominant term. Thus,(2.21) can be well approximated byˆmGMM = argminm( ˆµk1 −µk1,..., ˆµks −µks)Σ−1ˆµk1 −µk1...ˆµks −µks= argminmgTN(m)Σ−1gN(m)= argminmQ(m; ˆµk1,..., ˆµks)(2.22)where Q(m; ˆµk1,..., ˆµks) is the orthogonal criterion function (2.14). In the ML sense, the202.3. Generalized Method of Moments Estimationestimate ˆmGMM is the zero of the following functionh(m; ˆµk1,..., ˆµks)= ∂Q(m; ˆµk1,..., ˆµks)∂m=−2parenleftbigg∂ µk1∂m ,...,∂ µks∂mparenrightbiggΣ−1ˆµk1 −µk1...ˆµks −µks−( ˆµk1 −µk1,..., ˆµks −µks)Σ−1parenleftbigg∂Σ∂mparenrightbiggΣ−1ˆµk1 −µk1...ˆµks −µks(2.23)in which we used the derivative identity of matrix inverse∂Σ−1∂m =−Σ−1parenleftbigg∂Σ∂mparenrightbiggΣ−1. (2.24)By the multivariate delta method [21], the asymptotic variance σ 2GMM = Varbracketleftbig√N ˆmGMMbracketrightbig canbe obtained asσ 2GMM =parenleftbigg∂ ˆmGMM∂ ˆµk1 ,...,∂ ˆmGMM∂ ˆµksparenrightbiggΣ∂ ˆmGMM∂ ˆµk1...∂ ˆmGMM∂ ˆµksvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingleˆµk1=µk1,..., ˆµks=µks. (2.25)Consider h( ˆmGMM; ˆµk1,..., ˆµks)= 0 as an implicit function of ˆmGMM in terms of ( ˆµk1,..., ˆµks).212.3. Generalized Method of Moments EstimationWe write the derivative of the implicit function as∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆmGMM∂ ˆmGMM∂ ˆµk1 +∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆµk1 = 0...∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆmGMM∂ ˆmGMM∂ ˆµks +∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆµks = 0(2.26)and have ∂ ˆmGMM∂ ˆµk1...∂ ˆmGMM∂ ˆµks=− 1∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆmGMM∂h( ˆmGMM; ˆµk1,...,ˆµks)∂ ˆµk1...∂h( ˆmGMM; ˆµk1,...,ˆµks)∂ ˆµks. (2.27)Calculating the partial derivatives of h( ˆmGMM; ˆµk1,..., ˆµks) in (2.27) and substitute (2.27)into (2.25), the asymptotic variance of the GMM m parameter estimator can be written asσ 2GMM = 1η (2.28)where η is defined asη =parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM. (2.29)A detailed derivation of (2.28) is given in Appendix A. Because of the consistency of theGMM estimation scheme, for large sample size N, the asymptotic variance of ˆmGMM can befurther simplified asσ 2GMM =parenleftbigg∂ µk1∂m ,...,∂ µks∂mparenrightbiggΣ−1∂ µk1∂m...∂ µks∂m−1. (2.30)222.4. Numerical Results and Discussion2.4 Numerical Results and DiscussionIn this section, we present performance comparisons of several moment-based m parameter es-timators. Both mean square error (MSE) analysis via Monte Carlo simulation and the asymp-totic variance analysis are conducted for the classical moment-based estimators reviewed inSection 2.2 and the GMM estimator introduced in Section 2.3.Since estimators discussed in this chapter are considered asymptotically unbiased, we useperformance of the ML approach as a benchmark in the comparison. The MSE performanceof the moment-based estimators are compared with the Cram´er-Rao lower bound. For asymp-totic variance analysis, we compute the asymptotic relative efficiency (ARE) [21] of differentmoment-based m parameter estimators with respect to the ML-based estimator. The relativeefficiency eZ1Z0 of estimator ˆZ1 to ˆZ0 is defined aseZ1Z0 = Var( ˆZ0)Var( ˆZ1). (2.31)The ARE of the ML-based estimator with respect to itself is thus 1. The ARE of the moment-based estimators with respect to ML should be less than 1 because moment statistics are notthe sufficient statistics.Fig. 2.1 and Fig. 2.2 show the simulated MSE performance and the asymptotic relativeefficiency of the moment-based Nakagami m parameter estimators respectively. We observethat for all values of m, the limiting estimator of the fractional moment-based estimator family(2.11) and the GMM estimator with first, second and third order moment conditions are moreefficient than the other moment-based m parameter estimators. Specifically, when m < 1, thelimiting fractional moment-based estimator achieves the best estimation performance; whereaswhen m > 1, the GMM estimator based on the first three integer moments outperforms thelimiting estimator. The performance difference between the GMM estimator based on thefirst two integer moments and the GMM estimator based on the first three integer moments232.4. Numerical Results and Discussionsuggests that the GMM approach can achieve better estimation performance by adding moremoment conditions. It is interesting to notice that the INV estimator which uses the second andfourth order moments achieves the same MSE and asymptotic variance as the GMM estimatorbased on the same moment conditions. This implies that the INV estimator has achieved thebest asymptotic performance among all MoM estimators based on the second and fourth ordermoment conditions. However, we can observe that there is a huge performance gap betweenthe GMM estimator with the first three integer moment conditions and the classical moment-based estimator ˆmt. This observation suggests that based on the same moment conditions, it isstill possible to design a moment-based estimator with better performance than that of ˆmt.242.4. Numerical Results and Discussion0.5 1 1.5 2 2.5 3 3.5 4 4.5 510−410−310−2mMSE  CRLBˆmINVˆmtˆm0.01GMM2,4GMM1,2,3GMM1,2Figure 2.1: Simulated MSE performance of moment-based Nakagami fading parameter esti-mators with sample size N = 10,000.252.4. Numerical Results and Discussion0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.20.30.40.50.60.70.80.91mRelative Efficiency  Maximum Likelihoodˆm0ˆmINVˆmtGMM2,4GMM1,2,3GMM1,2Figure 2.2: Asymptotic relative estimation efficiencies of moment-based Nakagami fadingparameter estimators with respect to ML.262.5. Summary2.5 SummaryIn this chapter, we have provided a detailed discussion on moment-based estimation for theNakagami-m fading model. A family of classical moment-based m parameter estimators basedon both integer and fractional moments has been reviewed. It has been shown that some com-monly used moment-based m parameter estimators are special cases of this estimator family.The GMM estimation approach which exploits all available moment conditions in a determinedor over-determined estimation problem has been introduced to fading parameter estimation forthe first time. Systematic performance comparison for moment-based m parameter estimatorshas been conducted by both Monte Carlo simulation and asymptotic variance analysis.27Chapter 3On Statistics of Logarithmic Ratio ofArithmetic Mean to Geometric Mean forNakagami-m Fading PowerChaper 2 discusses several moment-based Nakagami m parameter estimators. In this chapter,we focus on performance analysis of maximum-likelihood based m parameter estimators.3.1 ML-Based Nakagami-m Parameter EstimatorsRecall that the PDF of the Nakagami-m fading envelope R is given byfR(r) = 2Γ(m)parenleftBigmΩparenrightBigmr2m−1 expparenleftBig−mΩr2parenrightBig, r ≥ 0, m ≥ 12 (3.1)where Ω =E[R2] is the scale parameter, and the shape parameter m is defined asm = Ω2E[(R2−Ω)2], m ≥12. (3.2)As briefly discussed in Chapter 1, in order to characterize wireless channels using theNakagami-m distribution, it is crucial to determine or estimate the value of m from N randomsamples R1,R2,...,RN drawn independently according to (3.1). Several methods for estimat-ing the m parameter have been reported in literature. The Greenwood-Durand estimator (GDE)283.1. ML-Based Nakagami-m Parameter Estimators[15], a ML-based Gamma-shape parameter estimator, is given byˆmGDE =f1(∆) ∆ < 0.5772f2(∆) 0.5772 ≤ ∆ ≤ 17(3.3)wheref1(∆) = 0.5000876+0.1648852∆−0.0544274∆2∆ (3.4a)f2(∆) = 8.898919+9.059950∆−0.9775373∆2(17.79728+11.968477∆+∆2)∆ (3.4b)and∆ = lnbracketleftBigg1NN∑i=1R2ibracketrightBigg− 1NN∑i=1lnR2i =−ψ( ˆm)+ln( ˆm) (3.5)in which ψ(·) is the digamma function defined as ψ(x) = Γprime(x)/Γ(x). More recently, Chengand Beaulieu [16] proposed to use the first-order and second-order approximations to ψ(·) inML-based m parameter estimation and derived two approximate ML estimators for m asˆm1 = 12∆ (3.6)andˆm2 = 6+√36+48∆24∆ . (3.7)It was pointed out by Zhang [23] that estimators similar to (3.6) and (3.7) were reported earlierby Thom [24] in the estimation problem for the Gamma distribution in another discipline.The ML-based estimators presented in (3.3), (3.6), and (3.7) are all functions of the param-eter ∆. This immediately implies that if we know the PDF of the parameter ∆, we can assessthe performance of ML-based estimators for the Nakagami m parameter without performingintensive computer simulations.293.2. Statistical Properties of ∆3.2 Statistical Properties of ∆3.2.1 Alternative Expression of ∆The expression of parameter ∆ in (3.5) can be written as∆ = lnparenleftBigg1NN∑i=1R2iparenrightBigg−lnparenleftBigg N∏i=1R2iparenrightBigg1N= lnbracketleftBigg 1N ∑Ni=1 R2i(∏Ni=1 R2i ) 1NbracketrightBigg. (3.8)We observe from (3.8) that the parameter ∆ is just the logarithmic ratio of the arithmetic meanto the geometric mean of N samples of the Nakagami-m fading power.It can also be shown that the Nakagami-m distribution is a member of a two-parameterexponential family, and the parameter ∆ is a function of the joint complete sufficient statistics.The detailed proof is given in Appendix B. In addition, by properties of sufficient statistics[25], the ML-based estimator of the unknown parameter is a function of the sufficient statistic.3.2.2 Nonnegative Property of ∆According to the well-known Arithmetic-Geometric inequality [26], we have1NN∑i=1R2i ≥parenleftBigg N∏i=1R2iparenrightBigg1N(3.9)and therefore we must have ∆ > 0. By recognizing the fact that when m approaches +∞ theNakagami PDF becomes an impulse function located at √Ω, we arrive atlimm→+∞∆ = limm→+∞lnbracketleftBigg 1N ∑Ni=1 R2i(∏Ni=1 R2i ) 1NbracketrightBigg= lnbracketleftBigg 1N ∑Ni=1 Ω(∏Ni=1 Ω) 1NbracketrightBigg= 0. (3.10)303.2. Statistical Properties of ∆3.2.3 Moment Generating Function of ∆To derive the MGF of ∆, denoted by Φ∆(s), we start with the definition and haveΦ∆(s) =EbracketleftBiges∆bracketrightBig=integraldisplay +∞0···integraldisplay +∞0bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightNbracketleftBigg parenleftbig∑Ni=1 R2iparenrightbigsNsparenleftbig∏Ni=1 R2iparenrightbigsNbracketrightBigg×bracketleftbigg 2Γ(m)parenleftBigmΩparenrightBigmR2m−11 e−mΩ R21bracketrightbigg×···×bracketleftbigg 2Γ(m)parenleftBigmΩparenrightBigmR2m−1N e−mΩ R2NbracketrightbiggdR1···dRN=bracketleftBig2Γ(m)parenleftbigmΩparenrightbigmbracketrightBigNNsintegraldisplay +∞0···integraldisplay +∞0bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightNN∏i=1R2m−2sN −1i ·parenleftBigg N∑i=1R2iparenrightBiggsexpparenleftBigg−mΩN∑i=1R2iparenrightBiggdR1···dRN.(3.11)If we let d = m−s/N, after a change of variable (R2i = xi), we obtainΦ∆(s) =bracketleftBig1Γ(m)parenleftbigmΩparenrightbigmbracketrightBigNNsintegraldisplay +∞0···integraldisplay +∞0bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightNN∏i=1xd−1i ·parenleftBigg N∑i=1xiparenrightBiggs·expparenleftBigg−mΩN∑i=1xiparenrightBiggdx1···dxN.(3.12)The multiple N integrals in (3.12) can be reduced to a single integral by invoking the followinguseful integral identity [27]integraldisplay +∞0···integraldisplay +∞0bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightnxα1−11 xα2−12 ···xαn−1n f(n∑i=1xi)dx1···dxn= Γ(α1)Γ(α2)···Γ(αn)Γ(α1 +α2 +···+αn)integraldisplay +∞0uα1+α2+···+αn−1 f(u)du.(3.13)313.2. Statistical Properties of ∆Letting α1 = α2 = ... = αN = d and f(x) = xs expparenleftbig−mΩxparenrightbig, we obtain a compact form for theMGF of ∆ asΦ∆(s) =bracketleftBig1Γ(m)parenleftbigmΩparenrightbigmbracketrightBigNNs ·[Γ(d)]NΓ(Nd) ·integraldisplay +∞0uNd−1us expparenleftBig−mΩuparenrightBigdu= Γ(mN)[Γ(m−s/N)]NNs[Γ(m)]NΓ(mN −s)(3.14)where in obtaining the last step we have used the definition of the Gamma function.3.2.4 Probability Density Function of ∆The PDF of ∆ can be obtained from its MGF by applying an inverse Laplace transform asf∆(δ) = 12pi jintegraldisplay c+j∞c−j∞Φ∆(−s)esδ ds= Γ(mN)[Γ(m)]N · 12pi jintegraldisplay c+j∞c−j∞Ns[Γ(m+s/N)]NΓ(mN +s) esδ ds(3.15)where j2 =−1 and c is a suitably chosen positive constant which ensures that the contour pathis in the region of convergence. The integration is taken along the vertical line ℜ{s}= c in thecomplex plane such that c is greater than the real part of any singularity of Φ∆(−s).If we now let y = s/N, the PDF becomesf∆(δ) = Γ(mN)[Γ(m)]N · N2pi jintegraldisplay cprime+j∞cprime−j∞NNy[Γ(m+y)]NΓ[N(m+y))] eNyδ dy (3.16)where cprime = c/N is another positive constant. With the aid of the Gauss multiplication theorem[28]Γ(nx) = (2pi)1−n2 nnx−12n−1∏k=0Γparenleftbiggx+ knparenrightbigg(3.17)323.2. Statistical Properties of ∆we arrive atf∆(δ) = N ·Γ(mN)[Γ(m)]N · 12pi j ·integraldisplay cprime+∞cprime−j∞NNy[Γ(m+y)]N(2pi)1−N2 NN(m+y)−12 ∏N−1k=1 Γbracketleftbigm+y+ kNbracketrightbig(eNδ)ydy= N · Γ(mN)(2pi)1−N2 NNm−12 [Γ(m)]N· 12pi j ·integraldisplay cprime+j∞cprime−j∞∏Nk=1 Γ[1−(1−m)+y]∏N−1k=0 Γbracketleftbig1−(1−m− kN)+ybracketrightbig(eNδ)ydy(3.18)Now applying the definition of Meijer’s G-function [29]Gm,np,qzvextendsinglevextendsinglevextendsinglevextendsingle a1 ··· apb1 ··· bq= 12pi jintegraldisplay ∏mj=1 Γ(b j −s)∏ j=1 nΓ(1−a j +s)∏qj=m+1 Γ(1−b j +s)∏ j=n+1 pΓ(a j −s) ·zsds (3.19)to (3.18), we can simply write the PDF of ∆ asf∆(δ) = Nξ ·G0,NN,NeNδvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle1−m ··· 1−m1−m ··· 1−m− N−1N (3.20)whereξ = Γ(mN)(2pi)1−N2 NNm−12 [Γ(m)]N. (3.21)Computer simulations were carried out to generate empirical PDFs of ∆ for different mand N values, and to compare them with the analytical PDFs obtained from (3.20). Fig. 3.1shows the analytical and empirical PDFs of ∆ for m = 0.5, 1, 2 when N = 5. Fig. 3.2 showsthe analytical and empirical PDFs of ∆ for m = 0.5, 1, 2 when N = 10. It is shown that theanalytical PDFs of ∆ have excellent agreement with the empirical ones.When the sample size N becomes large, the latest version of commercial software such asMAPLE and MATHEMATICA are incapable of evaluating our analytical PDF expression in(3.20).333.2. Statistical Properties of ∆0 1 2 3 4 500.511.522.533.5Probability Density Function of ∆Independent Variable δN=5  m=0.5 Simulationm=1 Simulationm=2 Simulationm=0.5 Analytical PDFm=1 Analytical PDFm=2 Analytical PDFm=2m=1m=0.5Figure 3.1: Comparison of empirical PDFs and analytical PDFs of ∆ for m = 0.5,1,2 withsample size N = 5.343.2. Statistical Properties of ∆0 0.5 1 1.5 2 2.5 3 3.500.511.522.533.54Probability Density Function of ∆Independent Variable δN=10  m=0.5 Simulationm=1 Simulationm=2 Simulationm=0.5 Analytical PDFm=1 Analytical PDFm=2 Analytical PDFm=1m=2m=0.5Figure 3.2: Comparison of empirical PDFs and analytical PDFs of ∆ for m = 0.5,1,2 withsample size N = 10.353.3. Gamma Approximation3.3 Gamma ApproximationTo avoid the high computational complexity associated with the Meijer’s G-function for largeN, we are motivated to approximate the PDF of ∆ using another PDF which can be easilyevaluated and is analytically tractable.3.3.1 Gamma Approximation for PDF of ∆From the nonnegative property discussed in Section 3.2, we know that ∆ is defined on [0,+∞). We propose to use a two-parameter Gamma PDF, which is also defined on [0, +∞), toapproximate the PDF of ∆. To determine the parameters θ and k in the two-parameter GammaPDFfX(x) = xk−1e−x/θθ kΓ(k) , x ≥ 0; θ,k > 0 (3.22)we can simply match the mean and variance of the two-parameter Gamma distribution to themean and variance of ∆.From the MGF of ∆ in (3.14), the first two moments of ∆ can be obtained by taking the firstand the second derivatives of the MGF with respect to s and evaluating the results at s = 0. Itis straightforward to show that the first two moments of ∆ are given byµ1 =−ψ(m)−ln(N)+ψ(mN) (3.23a)µ2 = [ψ(m)]2 +[ln(N)]2 +[ψ(mN)]2 +2ψ(m)ln(N)−2ψ(m)ψ(mN)−2ψ(mN)ln(N)+ 1N ψprime(m)−ψprime(mN).(3.23b)Setting the mean and variance of the two-parameter Gamma distribution equal to the mean and363.3. Gamma Approximationvariance of ∆µ1 = kθ (3.24a)µ2 −(µ1)2 = kθ 2 (3.24b)we arrive atθ =1N ψprime(m)−ψprime(mN)−ψ(m)−ln(N)+ψ(mN) (3.25a)k =1N ψprime(m)−ψprime(mN)[−ψ(m)−ln(N)+ψ(mN)]2. (3.25b)The two-parameter Gamma approximation is desirable since this PDF has a simple expo-nential form, which can be easily evaluated and manipulated in practice.3.3.2 Validating the Gamma ApproximationComputer simulations were also carried out to compare the two-parameter Gamma approxi-mate PDFs with the empirical PDFs of ∆.Fig. 3.3 shows the comparison between the empirical PDFs and the Gamma approximatedPDFs of ∆ for m = 0.5, 1, and 2 with N = 10. Fig. 3.4 presents the comparison between theempirical PDFs and the corresponding Gamma PDFs for m = 0.5, 1, and 2 with N = 100. BothFigs. 3.3 and 3.4 demonstrate that the two-parameter Gamma PDF is a good candidate forapproximating the PDF of ∆.To numerically validate the feasibility of approximating ∆ as a Gamma RV, we use theKolmogorov-Smirnov (K-S) test for goodness-of-fit. The basic idea of the K-S test is to com-pare the empirical cumulative distribution function (CDF) with the CDF of the hypothesizeddistribution. The test statistic Dn for test sample volume n is defined as the supremum of the373.3. Gamma Approximation0 0.5 1 1.5 2 2.5 3 3.500.511.522.533.54Probability Density Function of ∆Independent Variable δN=10  m=0.5 Simulationm=1 Simulationm=2 Simulationm=0.5 Gamma Approx.m=1 Gamma Approx.m=2 Gamma Approx.m=2m=1m=0.5Figure 3.3: Comparison of empirical PDFs and Gamma approximated PDFs of ∆ for m =0.5,1,2 with sample size N = 10.383.3. Gamma Approximation0 0.5 1 1.5 2024681012Probability Density Function of ∆Independent Variable δN=100  m=0.5 Simulationm=1 Simulationm=2 Simulationm=0.5 Gamma Approx.m=1 Gamma Approx.m=2 Gamma Approx.m=2m=1m=0.5Figure 3.4: Comparison of empirical PDFs and Gamma approximated PDFs of ∆ for m =0.5,1,2 with sample size N = 100.393.3. Gamma ApproximationTable 3.1: Kolmogorov-Smirnov test for gooodness-of-fit for the Gamma approximation.n = 100 n = 1,000Dmax Davg Acpt. % Dmax Davg Acpt. %N = 10m = 0.5 0.226 0.085 99.03% 0.067 0.027 99.05%m = 1 0.218 0.085 98.87% 0.069 0.027 98.83%m = 2 0.226 0.085 99.01% 0.072 0.027 99.00%m = 5 0.232 0.085 99.11% 0.078 0.027 99.01%N = 100m = 0.5 0.212 0.085 98.95% 0.075 0.027 99.00%m = 1 0.222 0.085 99.11% 0.066 0.027 98.91%m = 2 0.211 0.085 99.00% 0.074 0.027 99.03%m = 5 0.232 0.085 99.01% 0.068 0.027 98.88%absolute difference between the theoretical CDF F(x) and the empirical CDF Fn(x)Dn ≡ supx∈[0,+∞)|F(x)−Fn(x)|. (3.26)If the test statistic Dn is less than a critical value Dαn , which is determined by both the testsample volume n (degree of freedom) and a prescribed significance level α, the theoreticaldistribution is acceptable at a confidence level of 1−α.Case studies were conducted using test sample volume n = 100 and 1,000 for m = 0.5, 1, 2,and 5 with N = 10 and 100. The significance level α was chosen to be 0.01, giving a 99.00%confidence level for the K-S test.Table 3.1 shows the maximum test statistics Dmax and the average test statistics Davg ob-tained from 10,000 experiments conducted in our study. According to [30], the critical valuesfor test sample volume n = 100 and 1,000 at significance level 0.01 are D0.01100 = 1.63/√100 =0.163 and D0.011,000 = 1.63/√1000 = 0.0515 respectively. We observe that in each case of ourcase studies, about 99% of the experiments accepted the hypothesis that the random variable∆ can be modelled as a Gamma random variable at a confidence level of 99.00%. Table 3.1also shows that Dmax values, under which the hypothesis is rejected, are slightly greater than403.4. Applications and Numerical ResultsTable 3.2: Numerical MSE performance evaluations for ML-based Nakagami m parameterestimators.N = 100E[·] Var[·] Biasˆm1m = 0.5 0.40413731 0.00311112 -0.09586269m = 1 0.89122741 0.01615511 -0.10877259m = 2 1.90478583 0.07568833 -0.09521418m = 5 4.98725469 0.52291930 -0.01274531ˆm2m = 0.5 0.530691584 0.00348994 0.03069158m = 1 1.034506216 0.01677922 0.03450622m = 2 2.058782448 0.07652349 0.05878245m = 5 5.148600611 0.52392564 0.14860061ˆmGDEm = 0.5 0.500496858 0.004216106 0.000496858m = 1 0.980884952 0.018350252 -0.019115048m = 2 2.032113782 0.078701255 0.032113782m = 5 5.141084032 0.526581576 0.141084032the critical values; and the average test statistic Davg values are significantly below the corre-sponding values. In summary, the K-S test concludes that the two-parameter Gamma PDF canbe used to accurately approximate the PDF of ∆.3.4 Applications and Numerical ResultsIn this section, we use the Gamma approximate PDF to numerically evaluate the performanceof ML-based Nakagami m parameter estimators discussed in Section 3.1 for large N scenarios.Table 3.2 shows the mean, variance and bias of three ML-based Nakagami m parameterestimators ˆm1, ˆm2, and ˆmGDE for m = 0.5, 1, 2, and 5 with sample size N = 100. The numericalresults were calculated by using the Gamma approximate PDF of ∆ derived in Section 3.3.413.4. Applications and Numerical ResultsBy using the relationshipMSE( ˆm) = Var[ ˆm]+bias2( ˆm) (3.27)we can evaluate the MSE performance of ML-based m parameter estimators numerically withthe data in Table 3.2. Fig. 3.5 shows the simulated MSE of ML-based m parameter esti-mators discussed in this chapter and the MSE of these m estimators calculated by using theapproximate PDF of ∆. The plots show that the calculated MSE values give excellent fit to thesimulated MSE curves, which also validates the proposed Gamma approximation. It can beobserved in Fig. 3.5 that the Greenwood-Durand estimator ˆmGDE and the second order Cheng-Beaulieu estimator ˆm2 achieve very close MSE performance for a variety of fading conditions.In addition, for small and moderate m values (m < 3), ˆmGDE and ˆm2 perform better than the firstorder Cheng-Beaulieu estimator ˆm1 in terms of MSE; however, for m > 3, which correspondsto less severe fading scenarios, ˆm1 outperforms ˆmGDE and ˆm2.423.4. Applications and Numerical Results0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.7mMSEN=100  MSE GDE SimulationMSE MLE1 SimulationMSE MLE2 SimulationMSE GDE AnalyticalMSE MLE1 AnalyticalMSE MLE2 AnalyticalFigure 3.5: Analytical and simulated MSE performance of ML-based Nakagami m parameterestimators with sample size N = 100.433.5. Summary3.5 SummaryIn this chapter, we have studied statistical properties of a parameter ∆, which is defined as thelogarithmic ratio of the arithmetic mean to the geometric mean for the Nakagami-m fadingpower. This parameter is useful in studying the ML-based estimators of the Nakagami mfading parameter. Closed-form expressions have been derived for both the MGF and the PDFof the parameter ∆. For large sample size, it has been found that the PDF of ∆ can be wellapproximated by a two-parameter Gamma PDF. This approximation has been validated bythe Kolmogorov-Smirnov test. As an application, we have applied our results to study theperformance of three widely used ML-based Nakagami m parameter estimators.44Chapter 4Moment-Based Estimation for theGamma-Gamma Distribution with FSOApplications4.1 IntroductionBeing capable of establishing full-duplex high-speed wireless communication links over a dis-tance of several kilometers using license-free spectrums, free-space optical communication hasattracted much attention in the past decade. Because of ease and low cost of implementation,FSO system is considered as an alternative to optical fiber for the ’last mile’ problem whenfiber optic links are unavailable or too expensive to implement.As discussed in Chapter 1, in FSO communications the main impairment is caused by atmo-spheric turbulence-induced irradiance fluctuations. Therefore, when conducting system designand performance analysis for FSO systems, we need to study the atmospheric turbulence mod-els. We reviewed in Chapter 1 that the log-normal distribution [4][5] and the K-distribution [7]were proposed to model the irradiance for weak and strong turbulence conditions respectively.Another turbulence model, the Gamma-Gamma distribution, was later found to be capable ofproviding good fit to experimental measurements of irradiance for both weak and strong turbu-lence scenarios [8]. This desirable feature of the Gamma-Gamma distribution enables it to beused in a wide-range of turbulence conditions. The PDF of the Gamma-Gamma distribution is454.1. Introductiongiven byfG(I) = 2(αβ)(α+β)/2λΓ(α)Γ(β)parenleftbiggIλparenrightbiggα+β2 −1Kα−βparenleftBig2radicalbigαβI/λparenrightBig, α > 0,β > 0,λ > 0 (4.1)where λ is a scale parameter, α and β are the shape parameters, and Kv(·) is the vth ordermodified Bessel function of the second kind.To apply turbulence models to the analyses of practical FSO systems, we are often requiredto estimate the corresponding unknown parameters. Parameter estimation methods for thelog-normal distribution and the K-distribution have been well studied in [31] [32] [12] [33].However, to our best knowledge, estimator for the parameters of the Gamma-Gamma PDFhas not been reported in literature. The parameter estimation problem for the Gamma-Gammadistribution is challenging because a maximum-likelihood approach will involve derivativesof Kv(·), with respect to both its argument and the order index. For the same reason, theCram´er-Rao lower bound of the estimators can not be easily derived. Current method fordetermining the shape parameters of the Gamma-Gamma turbulence model has focused oncalculating the Rytov variance, which requires the knowledge of link distance and refractive-index structure parameter [31]. However, this requirement is not always desirable for practicalFSO systems, especially when terminals have some degrees of portability which can change thelink parameters frequently. For FSO systems with slant propagation path, the refractive-indexstructure parameter can not even be measured accurately because it is a function of altitude,which will change along the slant path.The remainder of this chapter is organized as follows. Section 4.2 reviews some importantstatistical properties of the Gamma-Gamma distribution which are useful for our estimationproblem. In Section 4.3 we propose an estimation scheme for the Gamma-Gamma turbulencemodel based on the concept of fractional moments and convex optimization. Then a modifiedestimator which makes use of the relationship between the Gamma-Gamma shape parametersin FSO applications is proposed in Section 4.4. Simulation results show that significant perfor-464.2. Statistical Properties of the Gamma-Gamma Turbulence Modelmance improvement in terms of MSE can be achieved by the modified estimation scheme.4.2 Statistical Properties of the Gamma-GammaTurbulence Model4.2.1 Parameterization of the Gamma-Gamma Turbulence ModelSimilar to the K-distribution, the Gamma-Gamma turbulence model is developed based on amodulation process, in which small scale irradiance fluctuation is modulated by large scaleirradiance fluctuation. In the Gamma-Gamma PDF specified in (4.1), the parameter α repre-sents the effective number of large-scale cells of the scattering process, and the parameter βrepresents the effective number of small-scale cells [3]. We also emphasize that parameters αand β can not be arbitrarily chosen in FSO applications, they are related through a parametercalled Rytov variance, which is a measure of optical turbulence strength. Under an assumptionof plane wave and negligible inner scale, which corresponds to long propagation distance andsmall detector area, the shape parameters of the Gamma-Gamma model satisfy the followingrelationships [3]α = g(σR) =exp 0.49σ 2RparenleftBig1+1.11σ 12/5RparenrightBig7/6−1−1(4.2a)β = h(σR) =exp 0.51σ 2RparenleftBig1+0.69σ 12/5RparenrightBig5/6−1−1(4.2b)where σ 2R is the Rytov variance. Though the relationships described in (4.2a) and (4.2b) canchange when spherical wave and a finite inner scale are taken into account [8], our estimationapproach can be similarly applied to the other scenarios considered in [8]. It can be shown474.2. Statistical Properties of the Gamma-Gamma Turbulence Model0 1 2 3 4100101102103Root of the Rytov variance,  σRValues of α and β   α=g(σR) β=h(σR) α=g(σR) β=h(σR)Figure 4.1: Gamma-Gamma shape parameters α and β as functions of σR.that α = g(σR) in (4.2a) is a convex function of σR on (0, ∞), and β = h(σR) in (4.2b) is amonotonically decreasing function on (0, ∞). In addition, the relationship α > β always holds,and the smaller shape parameter β is lower bounded above 0.91398 as σR approaches infinity.Fig. 4.1 plots α and β as functions of σR.As a measure of optical turbulence strength, the Rytov variance can also be used to char-acterize different turbulence levels [34]: the weak-turbulence regime refers to σ 2R ≤ 0.3; themoderate-turbulence regime has 0.3 < σ 2R ≤ 5; and the strong-turbulence regime correspondsto σ 2R > 5. However, the definition for fluctuation regimes by the Rytov variance is not strict as484.2. Statistical Properties of the Gamma-Gamma Turbulence Modelother classification schemes have also been used in literature. For example, in [35] Voelz andXiao used Rytov variance values between [1,10) to define the moderate turbulence regime forplane wave scenario. Gamma-Gamma PDFs for weak, moderate, and strong turbulence scenar-ios are plotted in Fig. 4.2, where the corresponding Rytov variance values are σ 2R = 0.25, 2, and11, and the scale parameter λ is set to unity. When the Rytov variance σ 2R approaches infin-ity, which corresponds to very severe turbulence condition or the saturation regime, the shapeparameter α approaches infinity, the shape parameter β approaches a finite constant 0.91398,and the Gamma-Gamma PDF (4.1) will approach a negative exponential PDF. We can observethis trend in Fig. 4.2.4.2.2 Moments of The Gamma-Gamma Turbulence ModelThe kth order moment of the Gamma-Gamma PDF is given by [36]µk =EbracketleftBigIkbracketrightBig= Γ(α +k)Γ(β +k)Γ(α)Γ(β)parenleftbigg λαβparenrightbiggk. (4.3)In this work, we normalize the first moment by setting λ = 1.The closed-form expression in (4.3) can be derived by applying the following integral prop-erty of the modified Bessel function of the second kind (6.561-16, [37]) in the definition of thekth order moment of the Gamma-Gamma distributionintegraldisplay ∞0xuKv(ax)dx = 2u−1a−u−1Γparenleftbigg1+u+v2parenrightbiggΓparenleftbigg1+u−v2parenrightbigg, [ℜ{u+1±v}> 0,ℜ{a}> 0].(4.4)Note that from the condition of the integral property (4.4), we requireα +k > 0β +k > 0(4.5)494.2. Statistical Properties of the Gamma-Gamma Turbulence Model0 0.5 1 1.5 2 2.5 3 3.5 400.20.40.60.81Intensity of irradiance, IProbability density function  σR2=0.25σR2=2σR2=11Figure 4.2: Gamma-Gamma PDFs with Rytov variance σ 2R = 0.25, 2, and 11.504.3. An MoM/CVX Estimation Scheme for Gamma-Gamma Shape Parameterswhich means the closed-form moment expression in (4.3) for the Gamma-Gamma distributionis valid only for moments of order greater than max{−α,−β}. From Section 4.2.1, we knowthat the minimum of α and β is the limit of β, which is 0.91398. Therefore we conclude theclosed-form expression for moments of the Gamma-Gamma distribution in (4.3) is valid formoments of order k >−0.91398. Note that the order index k is not restricted to integers, it canalso have non-integer values.4.3 An MoM/CVX Estimation Scheme for Gamma-GammaShape ParametersTaking the ratio of the (k+1)th and the kth order moments of the Gamma-Gamma distribution,we obtainµk+1µk = 1+kα +kβ +k2αβ . (4.6)From (4.3), we also find that the second-order moment of the Gamma-Gamma distribution isµ2 = 1+ 1α + 1β + 1αβ . (4.7)Using (4.6) and (4.7), a nonlinear equation set involving variables α and β is formulated as1α +1β = c1α ·1β = d(4.8)wherec =k2µ2 − µk+1µk −(k2 −1)k2−k (4.9a)d =kµ2 − µk+1µk −(k−1)k−k2 . (4.9b)514.3. An MoM/CVX Estimation Scheme for Gamma-Gamma Shape ParametersAfter some algebraic manipulations to (4.8), α and β values can be found as the roots ofthe following quadratic equationx2 − cd x+ 1d = 0. (4.10)For FSO applications, since the shape parameter α is always greater than the shape parameterβ, we designate the larger root of (4.10) to be α, and the smaller one to be β. A moment-basedshape parameter estimator for the Gamma-Gamma turbulence model can thus be expressed asˆα = ˆc2 ˆd + 12radicalBiggˆc2ˆd2 −4ˆd (4.11a)ˆβ = ˆc2 ˆd −12radicalBiggˆc2ˆd2 −4ˆd (4.11b)where ˆc and ˆd are c and d values in (4.9) calculated using sample moments.It is known that moment-based estimators with higher order moments may suffer from out-lier samples. The outlier problem can be alleviated by choosing smaller k values. To achievebetter performance, we propose to use fractional moments (0 < k < 1) instead of positive inte-ger moments in our moment-based shape parameter estimators. The application of fractionalmoments in the study of atmospheric laser scintillation has been discussed by Consortini andRigal [38]. It has been shown that using fractional moments of orders less than two can signif-icantly reduce the fitting error of moments. Even with the presence of noise and backgroundwhich can not be removed directly from fractional moments, the fitting accuracy can be guar-anteed as along as we have small enough width of the noise of the experimental setup.Although the denominators of the expressions in (4.9a) and (4.9b) become zero when k = 0,it can be shown that the equalities hold for k = 0 by applying L’Hˆopital’s rule aslimk→ 0c = limk→ 0k2µ2 − µk+1µk −(k2 −1)k2−k = limk→ 02kµ2 −parenleftBig1α +1β +2kαβparenrightBig−2k2k−1 =1α +1β (4.12a)524.3. An MoM/CVX Estimation Scheme for Gamma-Gamma Shape Parameterslimk→ 0d = limk→ 0kµ2 − µk+1µk −(k−1)k−k2 = limk→ 0µ2 −parenleftBig1α +1β +2kαβparenrightBig−11−2k =1α ·1β . (4.12b)In order to obtain real-valued roots, eqn. (4.10) must have a positive discriminant∆ =parenleftBigcdparenrightBig2− 4d > 0. (4.13)However, the discriminant ∆ may be negative, especially when the Rytov variance becomessmall (σR < 1), which corresponds to weak turbulence scenarios. In that case, the moment-based estimator in (4.11) will not give meaningful real-valued estimates for α and β.To address the above shortcoming, we observe that the left-hand side of (4.10) is a convexfunction. First, define a function f(x) = x2 − ˆcˆd x + 1ˆd . Then, a suboptimal solution to theestimation problem can be formulated as a convex optimization problemminimizeα, β[f(α)−0]2 +[f(β)−0]2subject to α > 0, β > 0.(4.14)The minimizer for the convex optimization problem described by (4.14) can be found asˆα = ˆβ = ˆc2 ˆd. (4.15)From Fig. 4.1, it can be seen that when σR < 1, α and β values are close to each other. Thusit is intuitively correct to have suboptimal estimates with ˆα = ˆβ. By combining the fractionalmoment-based estimator (4.11) and the convex optimization estimator (4.15), we arrive at arobust estimation scheme for the shape parameters α and β. We name this estimation schemethe method-of-moments/convex-optimization (MoM/CVX) approach.We use MSE as the metric for assessing the estimation performance. Monte Carlo simula-534.3. An MoM/CVX Estimation Scheme for Gamma-Gamma Shape Parameters0 0.5 1 1.5 2 2.5 3 3.5 4 4.510−410−310−210−1100101Root of Rytov Variance, σRMSE  MSE of ˆαMSE of ˆβMSE of ˆαimpvFigure 4.3: MSE performance of the MoM/CVX estimator and the modified MoM/CVX estima-tor with k = 0.5 and sample size N = 100,000.tions were carried out for the MoM/CVX estimator with k=0.5 and σR value from 0.5 to 4.5,the data sample size was chosen to be N=100,000.From the simulation results shown in Fig. 4.3, we observe that the MoM/CVX estimatorfor β can provide good estimates over a wide range of σR values. However, the estimationperformance of the MoM/CVX estimator for α is poorer. For σR = 0.5, the MSE of ˆα can beas large as 2.97, which corresponds to an average relative error of 17.8%. Therefore, we aremotivated to further improve the estimation performance for parameter α.544.4. A Modified MoM/CVX Estimation Scheme for the Shape Parameter α4.4 A Modified MoM/CVX Estimation Scheme for theShape Parameter αAn alternative method for estimating the shape parameter α, which turns out to be an improvedscheme, is to use ˆβ to estimate σR viaˆσR = h−1parenleftBigˆβparenrightBig(4.16)where h−1(·) denotes the inverse function of h(·) in (4.2b). Replacing σR in (4.2a) with itsestimates in (4.16), a new estimator for α can be obtained asˆαimpv = gparenleftBigh−1( ˆβ)parenrightBig. (4.17)The analytical expression of h−1(·) is cumbersome; however, the built-in function solve inMATLAB can be used to find numerical results for h−1(·).We observe in Fig. 4.3 that the MSE performance of the estimates of α is significantly im-proved by the modified method (dashed line). For our sample points, the largest improvementis achieved at σR = 1.5, where the MSE is reduced by 99.85%.The change in improvement achieved by the modified scheme can actually be predicted.For example, from a plot of |gprime(σR)| versus σR shown in Fig. 4.4, one obtains |gprime(σR)| = 0when σR = 1.402567471. This suggests that the modified estimator ˆαimpv is least sensitive tothe estimation error of ˆβ in the neighborhood of this point and the largest improvement forestimates of α can be achieved.Fig. 4.5 summarizes the estimation process of the modified MoM/CVX estimator. In themodified estimation scheme, we first calculate sample moments ˆµk, ˆµk+1 and ˆµ2 of the Gamma-Gamma turbulence model from the observed optical irradiance sample values. Parameters ˆcand ˆd in (4.9) can then be determined by using the sample moments. If the discriminant ∆ of554.4. A Modified MoM/CVX Estimation Scheme for the Shape Parameter α0 1 2 3 405101520253035404550Root of the Rytov variance,  σRAbsolute value of derivative functions  vextendsinglevextendsinglevextendsingle dαdσRvextendsinglevextendsinglevextendsingle = |gprime(σR)|vextendsinglevextendsinglevextendsingle dβdσRvextendsinglevextendsinglevextendsingle = |hprime(σR)|| h’(σR)|| g’(σR)|Figure 4.4: Absolute derivative functions of the Gamma-Gamma shape parameters α and βwith respect to σR.564.5. Summarythe quadratic equation in (4.10) is greater than zero, we use the quadratic solution in (4.11) toobtain the estimate of parameter β; otherwise, an estimate of β will be given by the convexoptimization solution (4.15). With an estimate of β, we can finally find an improved estimatorˆαimpv via (4.17).4.5 SummaryIn this chapter, we have studied the parameter estimation problem for the Gamma-Gammaturbulence model for free-space optical communications. An estimation scheme for the shapeparameters of the Gamma-Gamma distribution has been proposed based on the concept of frac-tional moments and convex optimization. With the proposed method, estimates of the shapeparameters can be directly obtained from observed samples, which is more straightforwardthan the current method which depends on measurements of some physical quantities. To im-prove the estimation performance, we have also proposed a modified scheme which exploitsthe relationship between the Gamma-Gamma shape parameters in FSO communications. Sim-ulation results have revealed that the modified estimation scheme can achieve MSE below 0.5and average relative estimation error below 15% for a wide range of turbulence conditions andsystem setups.574.5. SummaryFigure 4.5: Flow chart of the modified MoM/CVX estimator for the Gamma-Gamma shapeparameters.58Chapter 5ConclusionsThis chapter concludes the thesis with some general comments on applications of the method ofmoments on parameter estimation for fading and atmospheric models in wireless communica-tion, followed by a discussion of possible future work for investigation of alternative applicableestimation methods and applications.5.1 Summary of ContributionsIn this thesis, we have investigated applications of several moment-based methods in parameterestimation of fading and atmospheric turbulence models. The contributions of this thesis canbe summarized as follows.1. A detailed discussion on moment-based estimation for the Nakagami-m fading modelhas been given. A family of classical moment-based m parameter estimators has beenreviewed. Both the integer moments scenario and the fractional moments scenario arediscussed. The generalized method of moment estimation which exploits informationthat resides in all available moment conditions in a determined or over-determined es-timation problem, has been introduced to fading parameter estimation for the first time.Systematic performance comparison for moment-based m parameter estimators has beenconducted by both simulation and an analytical asymptotic variance analysis.2. By investigating the statistical properties of the parameter ∆, which is defined as thelogarithmic ratio of the arithmetic mean to geometric mean for the Nakagami-m fading595.2. Future workpower, the MGF and the exact PDF of ∆ have been derived. A Gamma approximation ofthe PDF of ∆, which avoids computational complexity of the exact PDF for large samplesize, has also been proposed by using a moment matching method. The validity of usingthe two-parameter Gamma PDF to approximate the PDF of ∆ has been established usingthe Kolmogorov-Smirnov test. With the assistance of our results, numerical evaluationof the performance of ML-based Nakagami m parameter estimators is feasible withoutperforming intensive Monte Carlo simulations.3. Based on the concepts of fractional moments and convex optimization, we have pro-posed a composite estimation scheme for the shape parameters of the Gamma-Gammaatmospheric turbulence model. Our estimation technique can be used to characterizethis atmospheric turbulence model over a wide range of turbulence conditions in FSOapplications.5.2 Future workThe GMM approach introduced in Section 2.3 provides a general framework for an iterativeestimation scheme based on moment conditions. It has been shown that the GMM approachcan achieve very good estimation performance for Nakagami m parameter estimation, andunder this framework, using more moment conditions can further improve the estimation per-formance. Therefore, when the computation load is affordable and the delay requirement is notstrict, it is preferable to use the GMM in any estimation problem in wireless communicationresearch like the Gamma-Gamma estimation problem discussed in Chapter 4 and channel esti-mation etc. to improve moment-based estimation accuracy. Particularly, this GMM approachhas large potential application in wireless communication problems where the traditional ML-based estimation approach does not work.Another possible future research topic is the application of the L-moment method men-605.2. Future worktioned in Section 1.3. Most moment-based estimation schemes require the sample size to bevery large to guarantee the convergence of the sample moments. However, for many real-timeapplications, this requirement can not be satisfied. The L-moments, being linear combinationsof data, are less influenced by outliers and suffer less from sampling variability. Therefore, theL-moment method may be preferred in such real-time scenarios to the ML approach when MLhas high computational complexity.61Bibliography[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge: Cam-bridge University Press, 2005.[2] M. Nakagami, “The m-distributionca general formula of intensity distribution of rapidfading,” Statistical Methods in Radio Wave Propagation, vol. 40, pp. 757–768, Nov.1962.[3] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applica-tions. Bellingham, WA: SPIE Press, 2001.[4] G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in alaser beam,” Optica Acta, vol. 28, no. 5, pp. 715–728, May 1981.[5] R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in at-mospheric turbulence,” Journal of the Optical Society of America, vol. 71, no. 12, pp.1440–1445, Dec. 1981.[6] J. H. Churnside and R. G. Frehlich, “Experimental evaluation of lognormally modulatedRician and IK models of optical scintillation in the atmosphere,” Journal of the OpticalSociety of America. Series A, vol. 6, no. 11, pp. 1760–1766, Nov. 1989.[7] E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experi-ments,” Physics Review Letters, vol. 40, no. 9, pp. 546–550, Sept. 1978.62Chapter 5. Bibliography[8] M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradi-ance probability density function of a laser beam propagating through turbulent media,”Optical Engineering, vol. 40, no. 8, pp. 1554–1562, Aug. 2001.[9] S. M. Kay, Funcdamentals of Statistical Signal Processing: Estimation Theory. UpperSaddle River, NJ: Prentice Hall, 1993.[10] L. P. Hansen, “Large sample properties of generalized method of moments estimators,”Econometrica, vol. 50, no. 4, pp. 1029–1054, July 1982.[11] J. R. M. Hosking, “L-moments: Analysis and estimation of distributions using linearcombinations of order statistics,” Journal of the Royal Statistical Society. Series B, vol.52, no. 1, pp. 105–124, Jan. 1990.[12] D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parametersof the K-distribution,” IEEE Transactions on Signal Processing, vol. 47, no. 4, pp. 1147–1151, Apr. 1999.[13] M. Hadˇziali´c, M. Miliˇsi´e, N. Hadˇziahmetovi´e, and A. Sarajli´o, “Moment-based and max-imum likelihood-based quotiential estimation of the Nakagami-m fading parameter,” inProc. IEEE Vehicular Technology Conference, VTC2007-Spring, Dublin, Ireland, Apr.22-25, 2007, pp. 549–553.[14] G. Yin, “Bayesian generalized method of moments,” Bayesian Analysis, vol. 4, no. 2, pp.191–208, Feb. 2009.[15] J. A. Greenwood and D. Durand, “Aids for fitting the gamma distribution by maximumlikelihood,” Technometrics, vol. 2, no. 1, pp. 55–64, Feb. 1960.[16] J. Cheng and N. Beaulieu, “Maximum-likelihood based estimation of the Nakagami mparameter,” IEEE Communications Letters, vol. 5, pp. 101–103, Mar. 2001.63Chapter 5. Bibliography[17] A. Abdi and M. Kaveh, “Performance comparison of three different estimators for theNakagami m parameter using monte carlo simulation,” IEEE Communications Letters,vol. 4, no. 4, pp. 119–121, Apr. 2000.[18] V. Barnett and T. Lewis, Outliers in Statistical Data, 3rd ed. Chichester, NY: John Wiley& Sons, 1994.[19] J. Cheng and N. Beaulieu, “Generalized moment estimators for the Nakagami fadingparameters,” IEEE Communications Letters, vol. 6, pp. 144–146, Apr. 2002.[20] J. Cheng, Performance Analysis of Digital Communications Systems with Fading andInterference, Ph.D. thesis, University of Alberta, Edmonton, A.B., Canada, Dec. 2002.[21] E. L. Lehmann, Elements of Large-Sample Theory. New York: Springer-Verlag, 1999.[22] M. S. Bartlett, An Introduction to Stochastic Processes, 3rd ed. Cambridge: CambridgeUniversity Press, 1978.[23] Q. T. Zhang, “A note on the estimation of Nakagami-m fading parameter,” IEEE Com-munications Letters, vol. 6, no. 6, pp. 237–238, June 2002.[24] H. C. Thom, “A note on the Gamma distribution,” Monthly Weather Review, vol. 86, no.4, pp. 117–122, Apr. 1958.[25] R. V. Hogg and A. T. Craig, Introduction to Mathematical Statistics, 5th ed. Upper SaddleRiver, NJ: Prentice Hall, 1995.[26] P. Bullen, Handbook of Means and Their Inequalities, 2nd ed. New York: Springer, 1987.[27] G. A. Gibson, Advanced Calculus. London: Macmillan, 1931.[28] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. Cambridge:Cambridge University Press, 1962.64Chapter 5. Bibliography[29] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions With Applica-tions In Statistics And Physical Sciences. New York: Springer-Verlag, 1973.[30] A. H.-S. Ang and W. H. Tang, Probability Concepts in Engineering, 2nd ed. Hoboken,NJ: John Wiley & Sons, 2007.[31] A. C. Cohen Jr., “Estimating parameters of logarithmic-normal distributions by maximumlikelihood,” Journal of the American Statistical Association, vol. 46, no. 254, pp. 206–212, June 1951.[32] A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small sam-ples from a three-parameter lognormal distribution,” Journal of the American StatisticalAssociation, vol. 65, no. 329, pp. 212–225, Mar. 1970.[33] I. R. Joughin, D. B. Percival, and D. Winebrenner, “Maximum likelihood estimation ofK distribution parameters for SAR data,” IEEE Transactions on Geoscience and RemoteSensing, vol. 31, no. 5, pp. 989–999, Sept. 1993.[34] A. Prokeˇs, “Modeling of atmospheric turbulence effect on terrestrial FSO link,” RadioEngineering, vol. 18, no. 1, pp. 42–47, Apr. 2009.[35] D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beamsfor propagation through turbulence,” Optical Engineering, vol. 48, no. 3, pp. 036001–036001–7, Mar. 2009.[36] W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels inturbulent atmosphere with gamma-gamma distribution,” Electronics Letters, vol. 43, no.16, pp. 880–882, Aug. 2007.[37] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed.London: Academic Press, 2007.65Chapter 5. Bibliography[38] A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laserscintillation,” Pure and Applied Optics, vol. 7, no. 5, pp. 1013–1032, May 1998.[39] H. V. Poor, An Introduction to Signal Detection an Estimation, 2nd ed. New York:Springer, 1994.66Appendix ADerivation of (2.28)By taking partial derivatives of h( ˆmGMM; ˆµk1,..., ˆµks) in (2.27) and setting ˆµk1 = µk1,..., ˆµks =µks, we have∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆmGMMvextendsinglevextendsinglevextendsinglevextendsingleˆµk1=µk1,...,ˆµks=µks=−2parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM(A.1a)parenleftbigg∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆµk1 ,...,∂h( ˆmGMM; ˆµk1,..., ˆµks)∂ ˆµksparenrightbiggvextendsinglevextendsinglevextendsinglevextendsingleˆµk1=µk1,...,ˆµks=µks=−2parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−11 0 ··· 00 1 ··· 0... ... ... ...0 0 ··· 1ks×ks.(A.1b)Denotingη =parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM (A.2)67Appendix A. Derivation of (2.28)and substituting (A.1) and (2.27) into (2.25), we arrive atσ 2GMM = 1η2parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1ΣΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM= 1η2parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM=parenleftBigg∂ µk1∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMM,..., ∂ µks∂mvextendsinglevextendsinglevextendsinglevextendsinglem= ˆmGMMparenrightBiggΣ−1∂ µk1∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM...∂ µks∂mvextendsinglevextendsinglevextendsinglem= ˆmGMM−1= 1η .(A.3)68Appendix BExponential Family Property of theNakagami-m DistributionAn s-parameter exponential family is defined as a family of distributions parameterized by ans-dimensional vector θ = [θ1,θ2,...,θs]T with PDF in the form offX(x;θ) = C(θ)expbraceleftBigg s∑i=1ηi(θ)Ti(x)bracerightBiggh(x) (B.1)where C, ηi’s are real-valued functions of θ, and Tis and h are real-valued functions of x[39]. If N i.i.d. random samples X1,X2,...,XN are drawn according to an exponential familydistribution with PDF in the form of (B.1), thenparenleftbig∑Ni=1 Tj(Xi), j = 1,2,...,sparenrightbigis a joint completesufficient statistic.We rewrite the PDF of the Nakagami-m distribution (3.1) asfR(r;m,Ω) = 2Γ(m)parenleftBigmΩparenrightBigmr2m−1 expbraceleftBig−mΩr2bracerightBig= 2Γ(m)parenleftBigmΩparenrightBigmexpbraceleftbiglnr2mbracerightbig1r expbraceleftBig−mΩr2bracerightBig= 2Γ(m)parenleftBigmΩparenrightBigmexpbraceleftBig−mΩr2 +mlnr2bracerightBig1r.(B.2)If we denote C(m,Ω) = 2Γ(m)parenleftbigmΩparenrightbigm, η1(m,Ω) = −m/Ω, η2(m,Ω) = m, T1(r) = r2, T2(r) =lnr2, and h(r) = 1/r, then the Nakagami-m PDF can fit in the form of (B.1), which suggeststhat the Nakagami-m distribution is a member of a two-parameter exponential family.Thusparenleftbig∑Ni=1 R2i ,∑Ni=1 lnR2iparenrightbig=parenleftbig∑Ni=1 T1(Ri),∑Ni=1 T2(Ri)parenrightbigis a joint complete sufficient statis-69Appendix B. Exponential Family Property of the Nakagami-m Distributiontics of the Nakagami-m distribution. Therefore, the parameter ∆, which can be written as∆ = lnparenleftBigg1NN∑i=1R2iparenrightBigg−lnparenleftBigg N∏i=1R2iparenrightBigg1N= lnparenleftBigg1NN∑i=1T1(Ri)parenrightBigg− 1NN∑i=1T2(Ri)(B.3)is a function of the joint complete sufficient statistics of the Nakagami-m distribution.70

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