On the Use of Padé Approximation for Performance Evaluation of Maximal Ratio Combining Diversity over Weibull Fading Channels

Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2006, Article ID 58501, Pages 1–7 DOI 10.1155/WCN/2006/58501 On the Use of Padé Approximation for Performance Evaluation of Maximal Ratio Combining Diversity over Weibull Fading Channels Mahmoud H. Ismail and Mustafa M. Matalgah Department of Electrical Engineering, Center for Wireless Communications, The University of Mississippi, University, MS 38677-1848, USA Received 1 April 2005; Revised 18 August 2005; Accepted 11 October 2005 Recommended for Publication by Peter Djuric We use the Padé approximation (PA) technique to obtain closed-form approximate expressions for the moment-generating function (MGF) of the Weibull random variable. Unlike previously obtained closed-form exact expressions for the MGF, which are relatively complicated as being given in terms of the Meijer G-function, PA can be used to obtain simple rational expressions for the MGF, which can be easily used in further computations. We illustrate the accuracy of the PA technique by comparing its results to either the existing exact MGF or to that obtained via Monte Carlo simulations. Using the approximate expressions, we analyze the performance of digital modulation schemes over the single channel and the multichannels employing maximal ratio combining (MRC) under the Weibull fading assumption. Our results show excellent agreement with previously published results as well as with simulations. Copyright © 2006 M. H. Ismail and M. M. Matalgah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION The use of the Weibull distribution as a statistical model that better describes the actual short term fading phenomenon over outdoor as well as indoor wireless channels has been proposed decades ago [1–3]. More recently, the appropriateness of the Weibull distribution has been further confirmed by experimental data collected in the cellular band by two independent groups in [4, 5]. Since then, the Weibull distribution has attracted much attention among the radio community. In particular, the performance of receive diversity systems over Weibull fading channels has been extensively studied in [6–13]. Also, a closed-form expression for the moment-generating function (MGF) of the Weibull random variable (RV) was obtained in [7] when the Weibull fading parameter (which will be defined in the sequel), usually denoted by m, assumes only integer values. Another expression for the MGF for arbitrary values of m was also derived in [8]. Both expressions were given in terms of the Meijer Gfunction and were used for evaluating the performance of digital modulation schemes over the single-channel reception and multichannel diversity reception assuming Weibull fading. Also, in [14], the second-order statistics and the capacity of the Weibull channel have been derived. Finally, we have analyzed the performance of cellular networks with composite Weibull-lognormal faded links as well as the performance of MRC diversity systems in Weibull fading in presence of cochannel interference (CCI) in terms of outage probability in [15, 16], respectively. The closed-form expressions provided in [7, 8], despite being the first of their kind in the open literature and despite having a very elegant form, suffer from a major drawback. The expressions involve the Meijer G-function, which, although easy to evaluate by itself using the modern mathematical packages such as Mathematica and Maple, these packages fail to handle integrals involving this function and lead to numerical instabilities and erroneous results when m increases. This renders the expressions impractical from the ease of computation point of view. Hence, it is highly desirable to find alternative closed-form expressions for the MGF of the Weibull random variable (RV) that are simple to evaluate and in the same time can be used for arbitrary values of m. Padé approximation (PA) is a well-known method that is used to approximate infinite power series that are either not 2 EURASIP Journal on Wireless Communications and Networking guaranteed to converge, converge very slowly or for which a limited number of coefficients is known [17, 18]. This technique was recently used for outage probability calculation in diversity systems in Nakagami fading in [19]. The approximation is given in terms of a simple rational function of arbitrary numerator and denominator orders. In this paper, we illustrate how this technique can be used to obtain simple-toevaluate approximate rational expressions for the MGF of the Weibull RV based on the knowledge of its moments. We then use these expressions to evaluate the performance of linear digital modulations over flat Weibull fading channels in the case of both single-channel reception and multichannel reception employing maximal ratio combining (MRC). The rest of the paper is organized as follows. In Section 2, we give a brief overview of the Padé approximation technique. In Section 3, we apply this technique to the problem at hand. The performance of digital modulation systems over the Weibull fading channel is then revisited in Section 4. Examples and numerical results as well as comparisons with previously published results in the literature and Monte Carlo simulations are provided in Section 5 before the paper is finally concluded in Section 6. and {bn } can be easily obtained by matching the coefficients of like powers on both sides of (3). Specifically, taking b0 = 1, without loss of generality, one can find that Nq  bn cN p −n+ j = 0, Nq  bn cN p −n+ j = −cN p + j , cn z n , cn ∈ R, (1) n=0 where R is the set of real numbers. There are several reasons to look for a rational approximation for the series in (1). The series might be divergent or converging too slowly to be of any practical use. Also, it is possible that a compact rational form is needed in order to be used in later computations. Not to mention the fact that it might be possible that only few coefficients of {cn } are known [17]. The PA method is capable of dealing with all the problems mentioned above. In particular, it can capture the limiting behavior of a power series in a rational form. The one-point PA of order [N p /Nq ], P [N p /Nq ] (z), is defined from the series g(z) as a rational function by [17, 18] N p a zn P [N p /Nq ] (z) = nN=q 0 n , n n=0 bn z (2) where the coefficients {an } and {bn } are defined such that N p N p +Nq n    n=0 an z = cn zn + O zN p +Nq +1 , N q n n=0 n=0 bn z (5) The above equations form a system of Nq linear equations for the Nq unknown denominator coefficients. This system can be written in matrix form as Cb = −c, (6) where   b = bNq · · · bk · · · b1 T , c = cN p +1 · · · cN p +k+1 · · · cN p +Nq T , cN p −Nq +1 ⎜ .. ⎜ ⎜ . cN p −Nq + (...truncated)