A note on the estimation of Nakagami-m fading parameter
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IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 6, JUNE 2002 237 A Note on the Estimation of Nakagami- Fading Parameter Q. T. Zhang, Senior Member, IEEE Abstract—Recently, a number of publications address the issue of estimating the -parameter in Nakagami fading. The purpose of this note is not to derive a new estimator; but rather, to bring to the attention of the community the fact that this issue has been solved in the literature for more than four decades. Numerical re- sults are presented to demonstrate that none of the recently pro- posed estimators exceeds the performance of the classical one by Greenwood and Durand. Index Terms— -parameter estimation, Nakagami fading channel. I. HISTORIC NOTES G IVEN independent and identically distributed (i.i.d.) samples, , drawn from a Nakagami pop- ulation with fading parameter and average power , we want to estimate . Since the power of a Nakagami variable follows the Gamma distribution, estimation of in the framework of the Nakagami or Gamma distribution is equivalent. Let de- note the sample estimate of the th moment such that (1) and let denote the geometric mean of sample power such that . Define . Estimation of can be either based on or . The moment-based method em- ploys the former whereas the maximum-likelihood (ML) esti- mator relies on the latter. The standard moment method is defined by (2) Recently, Abdi et al. [1] numerically compared the accuracy of various estimators for . However, their comparison did not include the most important and pioneering work by Greenwood [3], Thom [2] and Bowman [4]. Estimation of the -parameter has long been conducted in different disciplines. It is well known that the log-likelihood function is given by (3) Manuscript received January 9, 2002. The associate editor coordinating the review of this letter and approving it for publication was Dr. N. Al-Dhahir. This work was supported, in part by CERG Grant 9040624 from the Hong Kong Government. The author is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong (e-mail: [email protected]). Publisher Item Identifier S 1089-7798(02)05791-5. where is called the digamma function. It is not easy to solve this equation for the ML estimate of due to the fact that there is no simple expansion for the digamma function. Although the use of an asymptotic expan- sion of for large enables us to represent explicitly as an infinite power series in , the resulting expression has coeffi- cients with increasing magnitudes and sign pattern , possibly leading to a divergent series. In other words, increasing the approximation order of the digamma function does not nec- essarily imply a better approximation. Therefore in 1958, Thom [2] adopted the second-order approximation to obtaining an approximate ML estimator (4) The first-order approximation is occasionally used if is large (5) The properties of these two estimators are well known. As in- dicated by Bowman [4, p. 48], the error of (4) increases from 0.001% at to 0.93% at , and to 70% at . Clearly, we face a dilemma in solving the ML equation (3). Increasing the approximation order to can lead to diver- gence. Using a lower order, on the other hand, will cause a larger approximation error. Greenwood and Durand [3] wisely devise a rational approximation to (3) with coefficients determined by polynomial fitting, so that the resulting estimator possesses the robustness of lower order approximation while having the accu- racy close to the true ML estimator. Their estimator is given by (6) According to Bowman [4], the maximum relative errors of the above two expressions are 0.0088% and 0.0054%, respectively. In 1988, Bowman suggested an alternative technique to solving the ML equation (3) in a recursive manner, as shown by [4, p. 48] (7) with the initial value determined by (4). This algorithm is quite powerful and converges rapidly, on the basis of computa- tional results. 1089-7798/02$17.00 © 2002 IEEE
Transcript of A note on the estimation of Nakagami-m fading parameter
IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 6, JUNE 2002 237
A Note on the Estimation of Nakagami-mFading Parameter
Q. T. Zhang, Senior Member, IEEE
Abstract—Recently, a number of publications address the issueof estimating the -parameter in Nakagami fading. The purposeof this note is not to derive a new estimator; but rather, to bringto the attention of the community the fact that this issue has beensolved in the literature for more than four decades. Numerical re-sults are presented to demonstrate that none of the recently pro-posed estimators exceeds the performance of the classical one byGreenwood and Durand.
Index Terms— -parameter estimation, Nakagami fadingchannel.
I. HISTORIC NOTES
G IVEN independent and identically distributed (i.i.d.)samples, , drawn from a Nakagami pop-
ulation with fading parameter and average power, we wantto estimate . Since the power of a Nakagami variable followsthe Gamma distribution, estimation of in the framework ofthe Nakagami or Gamma distribution is equivalent. Letde-note the sample estimate of theth moment such that
(1)
and let denote the geometric mean of sample power such that. Define . Estimation of
can be either based on or . The moment-based method em-ploys the former whereas the maximum-likelihood (ML) esti-mator relies on the latter.
The standard moment method is defined by
(2)
Recently, Abdiet al. [1] numerically compared the accuracy ofvarious estimators for . However, their comparison did notinclude the most important and pioneering work by Greenwood[3], Thom [2] and Bowman [4]. Estimation of the-parameterhas long been conducted in different disciplines. It is well knownthat the log-likelihood function is given by
(3)
Manuscript received January 9, 2002. The associate editor coordinating thereview of this letter and approving it for publication was Dr. N. Al-Dhahir. Thiswork was supported, in part by CERG Grant 9040624 from the Hong KongGovernment.
The author is with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon Tong, Kowloon, Hong Kong (e-mail:[email protected]).
Publisher Item Identifier S 1089-7798(02)05791-5.
where is called the digamma function.It is not easy to solve this equation for the ML estimate of
due to the fact that there is no simple expansion for thedigamma function. Although the use of an asymptotic expan-sion of for large enables us to representexplicitly asan infinite power series in, the resulting expression has coeffi-cients with increasing magnitudes and sign pattern ,possibly leading to a divergent series. In other words, increasingthe approximation order of the digamma function does not nec-essarily imply a better approximation. Therefore in 1958, Thom[2] adopted the second-order approximation to obtainingan approximate ML estimator
(4)
The first-order approximation is occasionally used ifis large
(5)
The properties of these two estimators are well known. As in-dicated by Bowman [4, p. 48], the error of (4) increases from0.001% at to 0.93% at , and to 70% at .
Clearly, we face a dilemma in solving the ML equation (3).Increasing the approximation order to can lead to diver-gence. Using a lower order, on the other hand, will cause a largerapproximation error. Greenwood and Durand [3] wisely devisea rational approximation to (3) with coefficients determined bypolynomial fitting, so that the resulting estimator possesses therobustness of lower order approximation while having the accu-racy close to the true ML estimator. Their estimator is given by
(6)
According to Bowman [4], the maximum relative errors of theabove two expressions are 0.0088% and 0.0054%, respectively.In 1988, Bowman suggested an alternative technique to solvingthe ML equation (3) in a recursive manner, as shown by [4, p.48]
(7)
with the initial value determined by (4). This algorithm isquite powerful and converges rapidly, on the basis of computa-tional results.
1089-7798/02$17.00 © 2002 IEEE
238 IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 6, JUNE 2002
Fig. 1. Root mean-square error versus the sample sizeN ;m = 0:5.
Two approximate ML estimators have recently been sug-gested in [5]. More recently, Cheng and Beaulieu [6] furthersuggest a novel moment-based estimator defined by
(8)
where is a positive number. Smaller values ofusually leadto a better estimation. This moment-based estimator is new inmethodology. The two ML estimators proposed in [5], however,are respectively identical to (5) and (4) derived in the originalpaper by Thom. This situation is certainly caused by the factthat developments in other disciplines are not popular in ourcommunity.
II. COMPARISON
We use the root mean square (rms) error as a metric for thecomparison of various estimators for the fading parameter sinceit takes both mean value and bias into consideration. The rmserror for each sample size is obtained by averaging over 2500independent estimates. The results shown in Figs. 1–3 are for
, and respectively. In these figures, the methodsbased on (8), (5), (4), (6) and (7) are respectively labeled by CBnew moment, CB ML1, CB ML2, Greenwood and Bowman.For the CB new moment method, we used to generatethe curves. By observation, the estimators based on the standardmoment and on (5) represent the worst cases. On the other hand,the classical method by Greenwood is consistently superior tothe other estimators.
III. CONCLUSION
For i.i.d. samples and among the known estimators in the lit-erature, the best approximation to the ML estimator for the Nak-agami -fading parameter is the classical technique by Green-wood and Durand. The CB new moment method approaches the
Fig. 2. Root mean square error versus the sample sizeN ;m = 1.
Fig. 3. Root mean-square error versus the sample sizeN ;m = 4:5.
accuracy of the Greenwood estimator, particularly whenislarger than one, but at the cost of increased computational com-plexity.
REFERENCES
[1] A. Abdi and M. Kaveh, “Performance comparison of three different es-timators for the Nakagamim parameter using Monte Carlo simulation,”IEEE Commun. Lett., vol. 4, pp. 119–121, Apr. 2000.
[2] H. C. Thom, “A note on the Gamma distribution,”Monthly Weather Rev.,vol. 86, no. 4, pp. 117–122, 1958.
[3] J. A. Greenwood and D. Durand, “Aids for fitting the Gamma distribu-tion by maximum likelihood,”Technometrics, vol. 2, pp. 55–65, 1960.
[4] K. O. Bowman and L. R. Shenton,Properties of Estimators for theGamma Distribution. New York: Marcel Dekker, 1988, p. 46.
[5] J. Cheng and N. Beaulieu, “Maximum-likelihood based estimation ofthe Nakagamim parameter,”IEEE Commun. Lett., vol. 5, pp. 101–103,Mar. 2001.
[6] , “Generalized moment estimators for the Nakagami fading param-eters,”IEEE Commun. Lett., vol. 6, pp. 144–146, Apr. 2002.
