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Abstract—The generalized method of moments (GMM) is introduced in the framework of estimating the Nakagami-m fading parameter. This GMM approach ...
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

Generalized Method of Moments Estimation of the Nakagami-m Fading Parameter Ning Wang, Student Member, IEEE, Xuegui Song, Student Member, IEEE, and Julian Cheng, Member, IEEE

Abstract—The generalized method of moments (GMM) is introduced in the framework of estimating the Nakagami-m fading parameter. This GMM approach provides a systematic procedure for finding the moment-based m parameter estimators. Using the multivariate delta method, we present a derivation for the asymptotic variance of the GMM Nakagami m parameter estimators. Monte Carlo simulation results show that the GMM approach can lead to estimators outperforming existing momentbased m parameter estimators over a wide range of channel conditions. It is shown that the asymptotic performance of these GMM estimators are close to that of the maximum-likelihood based estimator. The proposed method can be easily applied to both noiseless and noisy environments. Index Terms—Generalized method of moments, moment-based parameter estimation, Nakagami-m fading model.

I. I NTRODUCTION MONG all the statistical models that have been proposed to characterize multipath fading in wireless communications, the Nakagami-m distribution is of practical importance. Originally deduced from experimental data, the Nakagamim distribution can provide better fits to empirical multipath fading measurements than any other fading channel model over a wide range of fading conditions [1]. In addition, with a simple exponential form, the Nakagami-m distribution often leads to closed-form analytical results when it is adopted as a fading model in performance analysis of wireless communication systems. The probability density function (PDF) of the Nakagami-m fading envelope R is given by [1]  m  2  m m 2m−1 r exp − r2 , m ≥ 1/2 (1) fR (r) = Γ(m) Ω Ω

A

where Γ(·) is the Gamma function, Ω = E[R2 ] is the scale parameter, and where the operator E[·] denotes the expectation. The parameter m in (1) defined as m=

Ω2 − Ω)2 ]

E [(R2

(2)

is the shape parameter, also known as the fading parameter. Manuscript received October 9, 2011; revised May 24, 2012; accepted June 5, 2012. The associate editor coordinating the review of this paper and approving it for publication was M. Win. This paper was presented in part at the 2011 IEEE International Conference on Communications, Kyoto, Japan, June 2011. N. Wang was with the School of Engineering, the University of British Columbia, Kelowna, BC, Canada. He is now with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8P 5C2 Canada (e-mail: [email protected]). X. Song and J. Cheng are with the School of Engineering, the University of British Columbia, Kelowna, BC, V1V 1V7 Canada (e-mail: {xuegui.song, julian.cheng}@ubc.ca). Digital Object Identifier 10.1109/TWC.2012.071612.111838

Parameter estimation of Ω and m is essential and critical in many practical wireless applications involving the Nakagamim fading model. Such knowledge can be used, for example, to design a better transmission scheme adaptive to the wireless links [2]-[4], to better characterize wireless channels for link budget analysis [5]-[7], and to facilitate the design of high performance wireless localization and ranging algorithms [8][10]. Since Ω is defined as the second-order moment of the Nakagami-m fading envelope, it can be estimated in a straightforward manner. Therefore the primary task is to determine the shape parameter m. In [11], Greenwood and Durand showed that the maximumlikelihood (ML) based m parameter estimator is the solution to a transcendental equation involving a natural logarithmic function and a digamma function ψ(x) = Γ (x)/Γ(x) as N    N  1 1  2 ln R − Ri2 (3) −ψ(m) + ln(m) = ln N i=1 i N i=1 where the random variates R1 , R2 , . . . , RN are drawn independently according to the Nakagami-m distribution characterized by (1). The most well-known ML-based m parameter estimator is the Greenwood-Durand estimator [11], which was developed in the framework of ML-based Gamma shape parameter estimation. Using the first-order and the secondorder approximations to the digamma function in (3), Cheng and Beaulieu derived two approximate ML-based estimators for m in [13]. Zhang [14] pointed out that the same design developed in [13] was reported earlier for the Gamma distribution parameter estimation in another discipline [15]. In order to avoid solving the transcendental equation resulting from the ML estimation, the method of moments (MoM) [16] can be applied to the m parameter estimation problem. Several MoM-based Nakagami m parameter estimators for both noiseless case and noisy case have been proposed and studied in the wireless communications literature. Using the definition of the parameter m, Abdi and Kaveh [17] proposed the inverse normalized variance (INV) estimator for m using the second-order and the fourth-order noiseless sample moments of the Nakagami-m distribution. It is well known that sample moments are subject to the influence of outliers [18], and this is more problematic for estimators using higherorder sample moments that can cause estimation inaccuracy. To alleviate the outlier problem, whenever possible, lowerorder sample moments should be considered. An m parameter estimator exploiting the ratio of the first-order and the thirdorder moments of the Nakagami-m distribution was proposed in [19]. This estimator was further generalized to a family of fractional moment-based m parameter estimators for noise-

c 2012 IEEE 1536-1276/12$31.00 

WANG et al.: GENERALIZED METHOD OF MOMENTS ESTIMATION OF THE NAKAGAMI-M FADING PARAMETER

less case. The previously developed INV m estimator was shown as a special case of the proposed estimator family [19]. In [20], Chen and Beaulieu studied the m parameter estimation problem in additive white Gaussian noise (AWGN), and proposed a moment-based estimator based on moment conditions up to the fourth order. Under the same noisy channel assumption, another m parameter estimator was later derived in [21] based on the first two integer moments of the noisy fading envelope and the idea of surface matching. More recently, Tepedelenlioglu and Gao proposed a new class of integer moment-based m parameter estimators for both noiseless and AWGN environments in [22]. However, in all aforementioned work, finding the momentbased estimators for the Nakagami m parameter seems ad hoc. There is a lack of standard procedure or principle to derive moment-based parameter estimators as in the ML approach. It is also unclear how one can search for the best momentbased estimator, and how the best moment-based estimator will perform relative to the ML-based estimators. In this work, we address these questions by introducing the generalized method of moments (GMM), a statistical tool developed in the early 1980’s from the study of econometrics [23]-[25], to the Nakagami-m fading parameter estimation problem. We consider the asymptotic variance (AsVar) of the GMM based estimators and compare their performance with ML-based estimation. Both noiseless case and noisy case m parameter estimation are studied. The rest of this paper is organized as follows. In Section II we review some existing moment-based m parameter estimators. In particular, we present a new estimator as the limiting member of the fractional moment-based estimator family proposed in [19] and study its asymptotic performance. In Section III, we introduce a generic GMM algorithm for Nakagami fading parameter estimation. We then present a derivation of the asymptotic variance of the GMM estimators for m. Sections IV and V study the fading parameter estimation in noiseless and noisy environments, respectively. Finally, we make some concluding remarks in Section VI. II. R EVIEW OF M OMENT-BASED NAKAGAMI m PARAMETER E STIMATORS A. Integer Moment-Based m Parameter Estimators The kth-order moment for the Nakagami-m random variable (RV) is [1]

 Γ(m + k/2) Ω k/2 . (4) μk = E R k = Γ(m) m By taking the ratio of μ4 and μ2 , one obtains



μ4 Ω Γ(m + 2) Ω = = (m + 1) . 2 μ2 Γ(m) m m

(5)

Solving (5) for m by using the iterative property of the Gamma function Γ(m + 1) = mΓ(m), Abdi and Kaveh derived the INV m parameter estimator as [17] m ˆ IN V = where μ ˆk =

1 N

N

i=1

μ ˆ22 μ ˆ4 − μ ˆ22

Rik is the kth-order sample moment.

(6)

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Similarly, by taking the ratio of μ3 and μ1 , one can derive another estimator for m as [19] ˆ2 μ ˆ1 μ . (7) m ˆt = 2(ˆ μ3 − μ ˆ1 μ ˆ2 ) Notice from (7) that the highest order of sample moments used ˆ IN V . Therefore in m ˆ t is three, which is smaller than that of m m ˆ t is expected to suffer less from the outlier problem and has improved estimation accuracy over m ˆ IN V . B. A Family of Fractional Moment-Based m Parameter Estimators Motivated by the idea used in the derivations of m ˆ IN V and m ˆ t , a family of fractional moment-based m parameter estimators was developed in [19] μ ˆ1/p μ ˆ2 

m ˆ 1/p = (8) 2p μ ˆ2+1/p − μ ˆ1/p μ ˆ2 where p is a positive real number. It is straightforward to show ˆ t and that when p = 1 and p = 1/2, m ˆ 1/p specializes to m m ˆ IN V , respectively. The limiting member of the estimator family in (8) is the one when p approaches infinity. Denote the limiting estimator by m ˆ 0 and let k = 1/p, we have from (8) kμk μ2 lim k→0 2(μ2+k − μk μ2 ) kE[Rk ]μ2 = lim 2+k k→0 2(E[R ] − E[Rk ]μ2 )

  μ2 E[Rk ] + kE Rk ln R (9) = lim k→0 2 (E [R2+k ln R] − μ2 E [Rk ln R]) μ2 = 2 2 E [R ln R ] − μ2 E [ln R2 ] μ2 = Cov [R2 , ln R2 ] where the second equality is obtained by the L’Hˆopital’s rule. As a consequence, the limiting estimator has the form μ ˆ2 m ˆ 0 = 1 N . (10) N 2 2 2 ˆ2 N1 i=1 Ri ln Ri − μ i=1 ln Ri N Combining (8) and (10), we can then give a complete description of the fractional moment-based m parameter estimator family ⎧ kμ ˆk μ ˆ2 ⎪ , k > 0; ⎪ ⎨ 2 (ˆ μ2+k − μ ˆk μ ˆ2 ) m ˆk = μ ˆ2 ⎪ ⎪ , k = 0. ⎩ 1 N N 2 ln R2 − μ 2 R ˆ2 N1 i i=1 i i=1 ln Ri N (11) We observe from (11) that the smaller the k value is, the smaller the highest order of the sample moments is used; and therefore, a better estimation performance is expected. In the limiting case, m ˆ 0 is expected to achieve the best performance among this fractional moment-based estimator family. ˆ k in (11) has been derived The asymptotic variance σk2 of m in [26] as ⎡ ⎤ ⎧ 2 vk+2 ⎪ − v 2k+2 ⎪ v 2k v 2 ⎦ ⎨ m2 ⎣ , k > 0; + vk2 (k/2)2 vk2 (12) σk2 = ⎪ ⎪ ⎩ 2 m [1 + mψ  (m + 1)] , k=0

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Fig. 1. Normalized asymptotic variances Vk2 of m ˆ k and normalized 2 asymptotic variance VM ˆ M L versus m. L of m

Fig. 2. Normalized asymptotic variances Vk2 of m ˆ k versus k when m = 1. The horizontal dashed and dotted line is V02 = π 2 /6.

where vk = Γ(m + k/2)/Γ(m), and ψ  (z) is the firstorder derivative of the digamma function. The normalized ˆ k is plotted against m asymptotic variance Vk2 = σk2 /m2 of m in Fig. 1 for k = 0, 1, and 2. For comparison purpose, we also plot the normalized asymptotic variance of the ML estimator m ˆ ML , which is obtained using the Cram´er-Rao lower bound (CRLB) as [26] 

−1 1 2 2  VML = m ψ (m) − . (13) m

between population moments and their sample counterparts. Typically, the number of moment conditions is greater than the number of unknowns in GMM estimation, which formulates an over-determined estimation problem. The most well-known implementation of the GMM method is an iterative regression process proposed by Hansen in his original GMM paper [23]. This procedure is called Hansen’s two-step algorithm. In this section, we outline Hansen’s procedure to perform the Nakagami m parameter estimation using GMM. With N independent, identically distributed (i.i.d.) realizations of a Nakagami-m random variable R1 , R2 , . . . , RN and s moment conditions μk1 , μk2 , . . . , μks , the GMM estimation for the Nakagami m parameter is formulated as minimization of the orthogonal condition

From Fig. 1 we observe that smaller k value gives better estimation performance. As expected, the limiting member m ˆ0 achieves the best performance among this fractional momentbased estimator family. The normalized asymptotic variance ˆ 0 approaches that of m ˆ ML , especially when m is V02 of m large. Fig. 2 plots the normalized asymptotic variance versus index k for m = 1. In this figure, the horizontal dashed and dotted line represents V02 . When m = 1, it can be shown from (12) that V02 = 1 + ψ  (2) = π 2 /6. From Fig. 2 we observe that the normalized asymptotic variance of m ˆ k increases monotonically with k. The minimum of Vk2 is attained by m ˆ 0. In the remainder of this paper, m ˆ IN V , m ˆ t and m ˆ 0 are compared with a new class of moment-based m parameter estimators obtained using the GMM approach. III. GMM FOR THE NAKAGAMI -m PARAMETER E STIMATION A. General Idea of the GMM Approach Unlike the classical method of moments which aims to obtain explicit expressions for unknown parameters from a set of moment conditions, the GMM does not give explicit expressions for the estimators. Instead, the GMM approach performs parameter estimation by minimizing weighted distances

T (m)WgN (m) Q(m; r) = gN

(14)

where r = (R1 , R2 , . . . , RN )T is the observation vector, W is an s× s weighting matrix, and gN (m) is the moment distance vector defined by ⎧ ⎫ μ ˆ k1 − μk1 (m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ μ ˆ k2 − μk2 (m) ⎬ gN (m) = . (15) .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ μ ˆks − μks (m) In (15), μ ˆki ’s (i = 1, 2, . . . , s) are the (ki)th-order sample moments, and μki (m)’s denote the (ki)th-order population moment conditions as functions of the parameter m. Note that the moment conditions can either be noiseless moments or moments of noisy measurements in our Nakagami m parameter estimation problem. A key step in GMM is to determine an appropriate weighting matrix W. Since higher-order sample moments may deviate from the population moments more significantly (or they are less reliable than the lower-order sample moments in terms of variance), it is necessary to assign higher-order moment conditions with less weight through W.

WANG et al.: GENERALIZED METHOD OF MOMENTS ESTIMATION OF THE NAKAGAMI-M FADING PARAMETER

It will be shown in Section III-B that the optimal (in MLsense) weighting matrix W is the inverse of the covariance matrix of the moment distance vector gN (m). Therefore the main task here is to determine the covariance matrix of gN (m). The first step of Hansen’s procedure is to set W = I, an s× s identity matrix. It implies that we first give the same weights to all moment conditions and solve for an initial estimate m ˆ (0) from the following least squares (LS) problem T (m)gN (m). m ˆ (0) = argmin gN m

(16)

The solution to the LS problem in (16) can be easily found with software tools such as MATLAB. Then we use this initial estimate of m to obtain more accurate estimates by an iterative regression process. In the second step, we focus on obtaining the covariance matrix of gN (m) using an iterative regression method described in [27]. By using the initial estimate m ˆ (0) , we first compute the residue vectors

(0) 

(0)  T  k1 ˆ t = Rt − μk1 m ˆ , . . . , Rtks − μks m ˆ u (t = 1, 2, . . . , N ) for all N observations. Then the auto-covariance matrices Sj for lag length j are determined by Sj =

N 1  ˆ tu ˆ Tt−j , j = 0, 1, . . . , l u N t=j+1

(17)

where l is a prescribed maximum lag length. With l autocovariance matrices, we can estimate the covariance matrix of gN (m) using ˆ=S ˆ0 + S

l 

  ˆj + S ˆ Tj wj S

(18)

j=1

where wj ’s are the weights for auto-covariance matrices with different lag values. In this work, we employ a widely used weighting scheme proposed by Bartlett [28] with wj = 1 − ˆ −1 , an improved estimate of the m j/(l + 1). Letting W = S parameter can be obtained by solving a new LS problem T ˆ −1 gN (m). (m)S m ˆ (1) = argmin gN m

(19)

Step 2 or (19) is then iterated until the absolute difference between two consecutive estimates of m is less than a predetermined threshold , or until the number of iterations reaches a predetermined limit M . Table I summarizes the GMM algorithm for estimating the Nakagami m parameter.

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TABLE I GMM A LGORITHM FOR THE N AKAGAMI m PARAMETER E STIMATION . Initialization: Prescribe the estimation accuracy , the maximum iteration number M , and the maximum lag length l. Set the iteration counter i = 0. Hansen’s two-step procedure: Step 1: Set W = I, calculate the initial estimate T (m)g (m) m ˆ (0) = argmin gN N m

Step 2: - For each t = 1, 2, · · · , N , calculate the residue vector   (i)   (i) T ˆ t = Rk1 ˆ , . . . , Rks ˆ u t − μk1 m t − μks m - For each lag value j = 1, 2, · · · , l, calculate the auto-covariance matrix 1 N ˆtu ˆT Sj = N t−j n=j+1 u - Use wj = 1 − j/(l + 1), calculate the covariance matrix of gN   ˆj + S ˆT ˆ=S ˆ 0 + l wj S S j=1

j

ˆ −1 and i = i + 1, calculate a new - Set W = S iteration estimate T (m)S ˆ −1 gN (m) m ˆ (i) = argmin gN m (i) −m ˆ (i−1) >  and i < M , iterate Step 2; - If m ˆ otherwise output m ˆ (i) .

being a moment-based m parameter estimator, converges to a zero-mean Gaussian random variable with variance σ 2 √ 

L as N → +∞. (20) N (m ˆ − m) → N 0, σ 2 The variance term σ 2 is the asymptotic variance of the corresponding moment-based m parameter estimator. To derive the asymptotic variance for GMM m parameter estimators, we use the assumptions made in Section III-A, i.e., N i.i.d. realizations of a Nakagami-m RV and s population moment conditions μk1 , μk2 , . . . , μks . For large sample size N , the joint distribution of the elements of T μk1 − μk1 , μ ˆk2 − μk2 , . . . , μ ˆks − μks ) approaches a gN = (ˆ multivariate Gaussian distribution N (0, Σ), where Σ is the covariance matrix of the elements of the random vector gN , the difference vector between available population moment conditions and their sample counterparts. The element of Σ at the ith row and the jth column is Σij = μki+kj − μki μkj . Thus, the joint PDF of the observed sample moment vector ˆk2 , . . . , μ ˆks )T is μ ˆ = (ˆ μk1 , μ ˆks ) f (ˆ μk1 , . . . , μ

 

−1 Σ 1 T = gN (m)  1 exp −gN (m) N s  (2π) 2  Σ  2

(21)

N

B. Asymptotic Variance of GMM Estimators With finite sample size, moment-based estimators are usually biased without optimality properties, and their analytical performance is difficult to obtain [16]. However, because of the consistency of moment-based estimators, we can derive their asymptotic variance analytically. The asymptotic variance is useful in performance comparison of moment-based estimators in large sample size scenarios. Based on the central limit theorem (CLT) and the weak law of large numbers √ (WLLN), all moment-based m parameter unbiased. estimators are N -consistent and asymptotically √ This implies that the random variable N (m ˆ − m), with m ˆ

where | · | denotes the determinant of a square matrix. The estimate m ˆ GMM in a maximum-likelihood sense can be expressed as m ˆ GMM =argmax ln f (ˆ μk1 , . . . , μ ˆks ) m  1 T =argmax − ln(|Σ|) − N gN (m)Σ−1 gN (m) + C 2 m

(22)

where C is a constant which does not depend on m. For large sample size N , the quadratic term in (22) will be the dominant term. Thus, eq. (22) can be well approximated

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by m ˆ GMM = argmin m

T gN (m)Σ−1 gN (m)

= argmin Q (m; μ ˆ)|W=Σ−1

(23)

m

where Q (m; μ ˆ)|W=Σ−1 is the orthogonal condition with weighting matrix W = Σ−1 . Equation (23) suggests that the optimal weighting matrix W in (14) is the inverse of the covariance matrix of the moment difference vector gN (m). Since the covariance matrix Σ is positive definite, in the ML sense the GMM estimator m ˆ GMM is the zero of the following function ∂Q (m; μ ˆ) h (m; μ ˆ) = ∂m ∂gT (m) −1 (24) Σ gN (m) = −2 N ∂m

∂Σ T (m)Σ−1 − gN Σ−1 gN (m) ∂m where we have used the derivative identity of matrix inverse

∂Σ ∂Σ−1 = −Σ−1 (25) Σ−1 . ∂m ∂m By the multivariate " [29], the asymptotic variance !√ delta method 2 = Var N m ˆ GMM can be obtained as σGMM



T  ∂m ˆ GMM ∂m ˆ GMM  2 σGMM = Σ  µˆ k1 = µk1 . (26)  μ ˆ μ ˆ . . . µ ˆ ks = µks

Considering h (m ˆ GMM ; μ ˆ) = 0 as an implicit function of m ˆ GMM in terms of μ ˆ, we write the derivative of the implicit function as ⎧ ∂h (m ˆ GMM ; μ ˆ) ∂ m ˆ GMM ˆ) ∂h (m ˆ GMM ; μ ⎪ + =0 ⎪ ⎪ ⎪ ∂m ˆ GMM ∂μ ˆk1 ∂μ ˆk1 ⎪ ⎨ .. (27) . ⎪ ⎪ ⎪ ⎪ ∂h (m ˆ GMM ; μ ˆ) ∂ m ˆ GMM ˆ) ˆ GMM ; μ ⎪ ⎩ ∂h (m + =0 ∂m ˆ GMM ∂μ ˆks ∂μ ˆks and have ⎛ ∂h(m ⎞ ⎛ ∂m ˆ GM M ;ˆ μ) ˆ GM M ⎞ ∂μ ˆ k1 ∂μ ˆ k1 ⎜ ⎟ 1 ⎟ ⎜ .. .. ⎜ ⎟ . (28) ⎠ = − ∂h(m ⎝ . . ⎠ ˆ GM M ;ˆ μ) ⎝ ∂m ˆ GM M ∂μ ˆ ks

∂m ˆ GM M

∂h(m ˆ GM M ;ˆ μ) ∂μ ˆ ks

Calculating the partial derivatives of h (m ˆ GMM ; μ ˆ ) in (28) and substituting (28) into (26), we can show in Appendix A that the asymptotic variance of the GMM m parameter estimator is 1 (29) Σ2GMM = η where η is defined as     ∂μk1  ∂μks  η= ,..., Σ−1 ∂m m=m ∂m m=m ˆ GM M ˆ GM M ⎞ ⎛ ∂μ  k1  (30) ˆ GM M ⎟ ⎜ ∂m m=m ⎟ ⎜ . . ×⎜ ⎟.  . ⎠ ⎝ ∂μks   ∂m m=m ˆ GM M

Because of the consistency of the GMM estimation scheme, the asymptotic variance of m ˆ GMM for large sample size can be further simplified to ⎛ ∂μk1 ⎞⎤−1 ⎡

∂m ∂μks ⎜ ⎟⎥ ⎢ ∂μk1 2 σGMM ,..., =⎣ Σ−1 ⎝ ... ⎠⎦ . (31) ∂m ∂m ∂μ ks

∂m

IV. GMM E STIMATION OF m WITHOUT N OISE In this section, we implement the GMM scheme for the Nakagami m parameter estimation in a noiseless environment. Performance of the GMM estimators is evaluated and compared with that of several classical moment-based m parameter estimators. Performance of ML-based m parameter estimation is also provided as a benchmark comparison. The kth order moment of the Nakagami-m distribution is given by (4). By using population moment conditions of orders k1, · · · , ks and their sample counterparts obtained from N noiseless Nakagami-m fading envelope samples, we calculate the moment distance vector gN (m) specified in (14). The GMM algorithm described in Section III-A can then be implemented to obtain estimates of m. Asymptotic performance evaluation as well as mean square error (MSE) analysis based on Monte Carlo simulation were conducted for the GMM m parameter estimators. The MSE performance is the mean value of the squared estimation error for a series of Monte Carlo simulations conducted under the same environment settings. Performance of the fractional moment-based m parameter estimator family introduced in [19] was chosen for comparison. Specifically, from the estimator family we choose the INV estimator m ˆ IN V that uses the first two even moment conditions in (6), the estimator m ˆ t with the first three integer moments in (7), as well as the limiting estimator m ˆ 0 given in (10), which is shown to be the best performing estimator in the fractional moment estimator family. To have a meaningful comparison, we consider GMM estimators using the same moment conditions. Therefore we examine ˆ GMM1,2,3 , and GMM m parameter estimators m ˆ GMM2,4 , m m ˆ GMM1,2 . Since moment-based estimators are asymptotically unbiased, we also use performance of the ML estimation as a benchmark for comparison purpose. More specifically, the normalized asymptotic variance of ML estimation for m, which is given by (13), is plotted in the normalized asymptotic variance comparison. The CRLB for m parameter estimation is used to be compared with the MSE performance of momentbased m estimators. Fig. 3 shows the MSE performance comparison with sample size N = 10, 000. We observe that for all the m values considered here, the limiting estimator m ˆ 0 and the GMM estimator m ˆ GMM1,2,3 with the first three integer moment conditions achieve better MSE performance than the other moment-based m parameter estimators, and their MSE performances are very close to the CRLB. In addition, the MSE performance ˆ GMM2,4 are of m ˆ IN V and that of the GMM estimator m almost the same, whereas a noticeable MSE performance gap exists between the GMM estimator m ˆ GMM1,2,3 and m ˆ t . For example, a 20% MSE gap is observed in Fig. 3 between m ˆt and m ˆ GMM1,2,3 at m = 2. Similar observation can be made in

WANG et al.: GENERALIZED METHOD OF MOMENTS ESTIMATION OF THE NAKAGAMI-M FADING PARAMETER

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Fig. 5. Asymptotic relative efficiencies of the fractional moment-based estimator family and GMM estimators with respect to ML estimation.

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Fig. 4. Normalized MSE comparison of the fractional moment-based estimator family and GMM estimators for the Nakagami m parameter with sample size N = 10, 000.

the normalized MSE performance comparison shown in Fig. 4. For asymptotic performance comparison, we compute the asymptotic relative efficiency (ARE) [29] of different momentbased m parameter estimators with respect to the ML-based estimation. The relative efficiency eZ1 Z0 of estimator Zˆ1 with respect to Zˆ0 is defined as eZ1 Z0 =

Var(Zˆ0 ) . Var(Zˆ1 )

(32)

According to the above definition, the ARE of the MLbased estimator with respect to itself is one. The ARE of

We also compare the asymptotic variance between m ˆ l,k proposed in [22] and m ˆ GMMl,k , which use the lth-order and kth-order moments of the Nakagami-m distribution, for different l and k values. From the comparison result shown in Fig. 6 we observe that, as expected, the smaller the order of the sample moments that are used, the better estimation performance is for both estimators. It is also interesting to ˆ GMMl,k observe from Fig. 6 that, the estimators m ˆ l,k and m have the same asymptotic variance performance. However, as we increase the number of moment conditions used, the GMM estimator is expected to outperform m ˆ l,k . In addition, to confirm our observation from Figs. 3-5 that the performance of m ˆ IN V and that of the GMM estimator m ˆ GMM2,4 are virtually the same, we present a detailed derivation of the asymptotic variance of m ˆ IN V and that of m ˆ GMM2,4 in Appendix B. From Appendix B we conclude ˆ GMM2,4 that the asymptotic variance of m ˆ IN V and that of m are in fact identical.

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of the noisy Nakagami-m fading envelope is given by  μ ˜k = E Zik

m m k 2 k 2 +1 = (2σ ) Γ 2 γ+m

γ k × F m, + 1; 1; 2 γ+m

8 7 6

Ω where γ = 2σ 2 is the average signal-to-noise ratio (ASNR) at the receiver, and F (·, ·; ·; ·) is the hypergeometric function [30, eq. (9.100)]. Knowing the population moments given by (35) and their sample counterparts obtained from noisy fading envelope measurements zi ’s, we implement the GMM algorithm described in Section III-A for m parameter estimation with noise.

m ˆ 1,2 4

m ˆ 1,3



AsVar

5

m ˆ 2,3

3

m ˆ 2,4 GMM1,2

2

GMM1,3 GMM2,3

1

(35)

GMM2,4 0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

B. Numerical Results and Discussions

5

m

Fig. 6. Asymptotic variance (AsVar) comparison between m ˆ l,k [22] and m ˆ GM Ml,k for different l and k values.

V. GMM E STIMATION OF m W ITH N OISY C HANNEL S AMPLES The GMM technique can be easily extended to the Nakagami m parameter estimation problem when the signal envelope is corrupted by noisy channel samples without resorting to complicated surface matching method [20]. In this section, we adopt the noisy system model considered in [20] and compare the performance of the GMM estimator m ˆ GMM1,2 with that of the estimators proposed in [20] and [21]. A. System Model According to [20], in the ith symbol duration, the received signal in a noisy system can be expressed as Xi (t) = Ri ejΘi Si (t) + Wi (t)

(33)

where Ri is the Nakagammi-m fading amplitude in the ith symbol duration, Si (t) is the ith transmitted signal, Θi is the fading phase in the ith symbol duration and Wi (t) is the complex AWGN with known power spectra density N0 . The fading phase Θi can be modeled as a uniform RV on [0, 2π]. At the output of the correlator, the norm of the complex output normalized by the transmitted signal power ES , i.e.,   Zi = Ri ejΘi + Ni  (34) is the sample of the noisy Nakagami-m fading envelope in the ith symbol duration, where Ni is the normalized AWGN term with zero mean and variance σ 2 = N0 /2ES . In this section, the noisy samples Zi are used to estimate the Nakagami fading parameter m. To adopt the GMM technique to the noisy case of Nakagami fading parameter estimation, we need to know the expression of the population moments of the noisy Nakagami-m fading envelope. It has been shown in [20] that the kth order moment

The performance of the GMM Nakagami m parameter estimator m ˆ GMM1,2 with the first two integer moment conditions is evaluated and compared with that of the estimators m ˆ1 proposed in [21] and m ˆ 2 proposed in [20]. To avoid notational confusion, we rename the two estimators from [21] and [20] respectively as m ˆ N 1 and m ˆ N 2 . In our simulations, the values of m vary from 0.5 to 10, and the scale parameter Ω is set to be 20. Figure 7 compares the MSE performance of m ˆ GMM1,2 , ˆ N 2 with sample size N = 1, 000, and ASNR set at m ˆ N 1 , and m 20 dB and 25 dB. We observe that under the ASNR conditions considered, the GMM estimator m ˆ GMM1,2 outperforms the ˆ N 2 over a wide range of m values. estimators m ˆ N 1 and m When ASNR increases, the advantage of the GMM estimation scheme becomes greater over the estimators m ˆ N 1 and m ˆ N 2. As expected, we observe that when the ASNR becomes large ˆ N2 (25 dB), the GMM estimator m ˆ GMM1,2 and the estimator m have lower MSE compared with the smaller ASNR case (20 dB). However, the MSE performance of m ˆ N 1 deteriorates for ASNR = 25 dB. For instance, at m = 5, the MSE of m ˆ N1 when ASNR = 25 dB is more than three times of that for the ASNR = 20 dB case. This is because the surface-matching method used in [21] becomes less accurate in the large ASNR regime, which in turn gives less accurate estimates for the parameter m. Moreover, since m ˆ N 1 is an approximate solution to a moment-based equation involving the parameter m, it does not have asymptotic properties (e.g. consistency) of GMMbased estimators. ˆ N 1 , and Figure 8 plots the normalized MSE of m ˆ GMM1,2 , m m ˆ N 2 with sample size N = 1, 000 and two different ASNR values (20 dB and 25 dB). From Fig. 8 we observe that for m values less than unity, the two estimators m ˆ N 1 and m ˆ N2 have large normalized MSE, which is undesirable. When m = 0.5, the normalized MSE of m ˆ N 1 and m ˆ N 2 are almost 0.03, which corresponds to root mean squared error of over 15% for both ASNR conditions. That is one order of magnitude larger than that of the GMM estimator m ˆ GMM1,2 . This observation ˆ N2 suggests that, under severe fading conditions, m ˆ N 1 and m may give unreliable estimates of m. Furthermore, for ASNR = 25 dB, the normalized MSE of m ˆ N 1 increases with m. On the other hand, we observe from Fig. 8 that the GMM estimator m ˆ GMM1,2 has a flat normalized MSE over a wide range of

WANG et al.: GENERALIZED METHOD OF MOMENTS ESTIMATION OF THE NAKAGAMI-M FADING PARAMETER

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estimators over a wide range of channel conditions. By combining available moment conditions optimally through an iterative process, this GMM scheme can achieve the best asymptotic performance among all possible consistent moment-based estimators using the same moment conditions.

0

10

−1

MSE

10

A PPENDIX A By taking partial derivatives of h (m ˆ GMM ; μ ˆ) in (28) and setting μ ˆk1 = μk1 , . . . , μ ˆks = μks , we have  ∂h (m ˆ GMM ; μ ˆ)   µˆk1 = µk1 ∂m ˆ GMM .

−2

10

m ˆ N 1 ASNR=25dB m ˆ N 2 ASNR=25dB GMM1,2 ASNR=25dB

−3

10

m ˆ N 1 ASNR=20dB m ˆ N 2 ASNR=20dB GMM1,2 ASNR=20dB



−4

10

1

2

3

4

5

6

7

8

9

10

m

Fig. 7. Comparison of MSE of the GMM estimator m ˆ GM M1,2 and m ˆ N1 [21] and m ˆ N2 [20] for the Nakagami m parameter in noisy fading channels with sample size N = 1, 000.

−1

10

m ˆ N 1 ASNR=25dB

m ˆ N 1 ASNR=20dB m ˆ N 2 ASNR=20dB GMM1,2 ASNR=20dB

2

   ∂μk1  ∂μks  = −2 ,..., Σ−1  ∂m m=m ∂m ˆ GM M m=m ˆ GM M ⎞ ⎛ ∂μ  k1  ˆ GM M ⎟ ⎜ ∂m m=m ⎟ ⎜ .. ×⎜ ⎟ ⎠ ⎝  . ∂μks  ∂m m=m ˆ GM M (A.1a)

m ˆ N 2 ASNR=25dB GMM1,2 ASNR=25dB

MSE Normalized by m

. . µ ˆ ks = µks

 ∂h (m ˆ GMM ; μ ˆ) ˆ)  ∂h (m ˆ GMM ; μ ,...,  ∂μ ˆk1 ∂μ ˆks 

 ∂μk1  = −2 ∂m m=m ˆ GM M ⎛ 1 0 ··· 0 ⎜ 0 1 ··· 0 ⎜ ×⎜ . . . . . ... ⎝ .. ..

−2

10

0 0

···

µ ˆ k1 = µk1 . . . µ ˆ ks = µks

  ∂μks  ,..., Σ−1 ∂m m=m ˆ GM M ⎞ ⎟ ⎟ ⎟ ⎠

1

. ks×ks

(A.1b)

−3

10

1

2

3

4

5

6

7

8

9

10

m

Fig. 8. Comparison of normalized MSE of the GMM estimator m ˆ GM M1,2 and m ˆ N1 [21] and m ˆ N2 [20] for the Nakagami m parameter in noisy fading channels with sample size N = 1, 000.

m values for both ASNR conditions. Therefore we conclude the GMM estimation scheme is preferred over a wide range of fading and noise conditions. VI. C ONCLUSION In this work, we have introduced the generalized method of moments to the Nakagami-m fading parameter estimation. This GMM estimation scheme provides a systematic approach for finding good moment-based m parameter estimators in both noiseless and noisy channel conditions. With the aid of the multivariate delta method, we have presented a derivation for the asymptotic variance of GMM based m parameter estimators. Numerical results have shown that the GMM estimation scheme outperforms the existing moment-based

Denoting     ∂μk1  ∂μks  η= ,..., Σ−1  ∂m m=m ∂m ˆ GM M m=m ˆ GM M ⎞ ⎛ ∂μ  k1  (A.2) ˆ GM M ⎟ ⎜ ∂m m=m ⎟ ⎜ . . ×⎜ ⎟  . ⎠ ⎝ ∂μks  ∂m  m=m ˆ GM M

and substituting (A.1) and (28) into (26), we arrive at     1 ∂μks  ∂μk1  2 σGMM = 2 ,..., η ∂m m=m ∂m m=m ˆ GM M ˆ GM M ⎛ ∂μ  ⎞ k1  ˆ GM M ⎟ ⎜ ∂m m=m ⎜ ⎟ ... × Σ−1 ΣΣ−1 ⎜ ⎟  ⎝ ⎠ ∂μks   ∂m 1 = . η

m=m ˆ GM M

(A.3)

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A PPENDIX B

R EFERENCES

From (12) we can obtain the asymptotic variance of the INV estimator as   v6 v42 2 2 v4 + 2− 3 σ2 = m v2 v2 v2  2 Γ(m + 3)/Γ(m) Γ(m + 2)/Γ(m) = m2 2 + 2 (Γ(m + 1)/Γ(m)) (Γ(m + 1)/Γ(m))  (Γ(m + 2)/Γ(m))2 − 3 (Γ(m + 1)/Γ(m)) = 2m(m + 1). (B.1) We know from (31) that the asymptotic variance of m ˆ GMM2,4 is given by 2 = σGMM 2,4



∂μ2 ∂m

∂μ4 ∂m



Σ−1



∂μ2 ∂m ∂μ4 ∂m

−1 .

(B.2)

Using the kth-order moment of the Nakagami-m distribution, it is straightforward to show that ⎧ ∂μ2 ∂Ω ⎪ ⎪ = = 0, ⎨ ∂m ∂m (B.3) (m+1)Ω2 ⎪ Ω2 ⎪ ⎩ ∂μ4 = ∂ m = − 2. ∂m ∂m m From the definition of Σ given in Section III-B, elements of Σ are given by ⎧ 2Ω3 (m + 1) ⎪ ⎪ ⎪ Σ12 = μ6 − μ2 μ4 = , ⎪ ⎪ m2 ⎪ ⎨ 3 2Ω (m + 1) (B.4) Σ21 = Σ12 = , ⎪ m2 ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎩ Σ22 = μ8 − μ24 = 2Ω (m + 1)(2m + 3) . m3 Substituting (B.3) and (B.4) into (B.2), we obtain the the asymptotic variance of m ˆ GMM2,4 as 2 σGMM 2,4     2 −Ω = 0 m2

m(2m+3) Ω2 −m2 Ω3

−m2 Ω3 m3 4 2Ω (m+1)



0

−1

−Ω2 m2

=2m(m + 1). (B.5) From (B.1) and (B.5), we conclude that the asymptotic variance of the INV estimator is identical to that of m ˆ GMM2,4 . ACKNOWLEDGMENT This work was supported by a Natural Sciences Engineering Research Council of Canada (NSERC) Discovery Grant. Julian Cheng would also like to thank Dr. Zheng Du from Huawei Technologies Co., Ltd. for his discussions and comments on an early draft of the manuscript.

[1] M. Nakagami, “The m-distribution, a general formula of intensity distribution of rapid fading,” Statistical Methods in Radio Wave Propagation, W. G. Hoffman, editor. Pergamon, 1960. [2] Y.-C. Ko and M.-S. Alouini, “Estimation of Nakagami-m fading channel parameters with application to optimized transmitter diversity systems,” IEEE Trans. Wireless Commun., vol. 2, pp. 250–259, Mar. 2003. [3] W. M. Gifford, M. Z. Win, and M. Chiani, “Diversity with practical channel estimation,” IEEE Trans. Wireless Commun., vol. 4, pp. 1935– 1947, July 2005. [4] A. Conti, M. Z. Win, and M. Chiani, “On the inverse symbol-error probability for diversity reception,” IEEE Trans. Commun., vol. 51, pp. 753–756, May 2003. [5] Y.-W. Lin and W.-C. Liu, “Cross-layer goodput analysis for rate adaptive IEEE 802.11a WLAN in the generalized Nakagami fading channel,” in Proc. 2004 IEEE International Conf. on Commn. [6] A. Conti, M. Z. Win, and M. Chiani, “Slow adaptive M-QAM with diversity in fast fading and shadowing,” IEEE Trans. Commun., vol. 55, pp. 895–905, May 2007. [7] W. W.-L. Li, Y. J. Zhang, A. M.-C. So, and M. Z. Win, “Slow adaptive OFDMA systems through chance constrained programming,” IEEE Trans. Signal Process., vol. 58, pp. 3858–3869, July 2010. [8] Y. Shen and M. Z. Win, “Fundamental limits of wideband localization— part I: a general framework,” IEEE Trans. Inf. Theory, vol. 56, pp. 4956–4980, Oct. 2010. [9] D. Dardari, A. Conti, U. Ferner, A. Giorgetti, and M. Z. Win, “Ranging with ultrawide bandwidth signals in multipath environments,” Proc. IEEE, vol. 97, pp. 404–426, Feb. 2009. [10] M. Guerra, D. Dardari, N. Decarli, and M. Z. Win, “Network experimentation for cooperative localization,” IEEE J. Sel. Areas Commun., vol. 30, pp. 467–475, Feb. 2012. [11] J. A. Greenwood and D. Durand, “Aids for fitting the Gamma distribution by maximum likelihood,” Technometrics, vol. 2, pp. 55–64, Feb. 1960. [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edition. Academic Press, 2007. [13] J. Cheng and N. C. Beaulieu, “Maximum-Likelihood based estimation of the Nakagami m parameter,” IEEE Commun. Lett., vol. 5, pp. 101– 103, Mar. 2001. [14] Q. T. Zhang, “A note on the estimation of Nakagami-m fading parameter,” IEEE Commun. Lett., vol. 6, pp. 237–238, June 2002. [15] H. C. Thom, “A note on the Gamma distribution,” Monthly Weather Review, vol. 86, pp. 117–122, Apr. 1958. [16] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice-Hall, 1993. [17] A. Abdi and M. Kaveh, “Performance comparison of three different estimators for the Nakagami m parameter using Monte Carlo simulation,” IEEE Commun. Lett., vol. 4, pp. 119–121, Apr. 2000. [18] V. Barnett and T. Lewis, Outliers in Statistical Data, 3rd edition. John Wiley & Sons, 1994. [19] J. Cheng and N. Beaulieu, “Generalized moment estimators for the Nakagami fading parameters,” IEEE Commun. Lett., vol. 6, pp. 144– 146, Apr. 2002. [20] Y. Chen and N. Beaulieu, “Estimation of Ricean and Nakagami distribution parameters using noisy samples,” in Proc. 2004 IEEE International Conf. on Commn. [21] Y. Chen and N. Beaulieu, “Novel Nakagami-m parameter estimator for noisy channel samples,” IEEE Commun. Lett., vol. 9, pp. 417–419, May 2005. [22] C. Tepedelenlioglu and P. Gao, “Estimators of the Nakagami-m parameter and performance analysis,” IEEE Trans. Wireless Commun., vol. 4, pp. 519–527, Mar. 2005. [23] L. P. Hansen, “Large sample properties of generalized method of moments estimators,” Econometrica, vol. 50, pp. 1029–1054, July 1982. [24] F. Hayashi, Econometrics. Princeton University Press, 2000. [25] A. R. Hall, Generalized Moment of Moments. Oxford University Press, 2005. [26] J. Cheng, “Performance analysis of digital communications systems with fading and interference,” Ph.D. thesis, University of Alberta, Edmonton, AB, Canada, Dec. 2002. [27] J. D. Hamilton, Time Series Analysis. Princeton University Press, 1994. [28] M. S. Bartlett, An Introduction to Stochastic Processes, 3rd edition. Cambridge University Press, 1978. [29] E. L. Lehmann, Elements of Large-Sample Theory. Springer-Verlag, 1999. [30] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th edition. Academic Press, 2000.

WANG et al.: GENERALIZED METHOD OF MOMENTS ESTIMATION OF THE NAKAGAMI-M FADING PARAMETER

Ning Wang received the B.E. degree in communication engineering from Tianjin University in 2004, and the M.A.Sc. degree in electrical engineering from the University of British Columbia in 2010. From 2004 to 2008, he was with China Information Technology Design and Consulting Institute as a mobile communication system engineer, specialized in large-scale mobile communication system design, traffic analysis, and network optimization. He is currently a Ph.D. student in electrical engineering at the University of Victoria, Victoria, BC, Canada. His research interests include channel modeling for wireless communications, statistical signal processing, and cooperative wireless communication system design. Xuegui Song (S’08) received the B. Eng. degree in electrical and information engineering from the Northwestern Polytechnical University, Xi’an, Shaanxi, China in 2003, the M. A. Sc. degree in electrical engineering from the University of British Columbia, Kelowna, BC, Canada in 2009. He is currently working toward the Ph.D. degree in the School of Engineering, the University of British Columbia, Kelowna, BC, Canada. His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, and optical wireless communications.

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Julian Cheng (S’96-M’04) received the B. Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995, the M.Sc. (Eng.) degree in mathematics and engineering from Queen’s University, Kingston, ON, Canada in 1997, and the Ph.D. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada in 2003. He is currently an Associate Professor in the School of Engineering, the University of British Columbia, Kelowna, BC, Canada. In 2005-2006, he was an Assistant Professor with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada. Previously, he worked for Bell Northern Research (BNR) and Northern Telecom (now NORTEL Networks). His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, spread spectrum communications, statistical signal processing for wireless applications, and optical wireless communications. Dr. Cheng was the recipient of numerous scholarships during his undergraduate and graduate studies, which included a President Scholarship from the University of Victoria and a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC). He was also a winner of the 2002 NSERC Postdoctoral Fellowship competition. Dr. Cheng is a registered Professional Engineer in the province of British Columbia, Canada.