On the Use of Padé Approximation for Performance Evaluation of Maximal

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 Pade´ 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 Pade´ 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 fading. Also, in [14], the second-order statistics and the ca- pacity of the Weibull channel have been derived. Finally, The use of the Weibull distribution as a statistical model that we have analyzed the performance of cellular networks with better describes the actual short term fading phenomenon composite Weibull-lognormal faded links as well as the per- over outdoor as well as indoor wireless channels has been formance of MRC diversity systems in Weibull fading in proposed decades ago [1–3]. More recently, the appropri- presence of cochannel interference (CCI) in terms of outage ateness of the Weibull distribution has been further con- probability in [15, 16], respectively. firmed by experimental data collected in the cellular band The closed-form expressions provided in [7, 8], despite by two independent groups in [4, 5]. Since then, the Weibull being the first of their kind in the open literature and de- distribution has attracted much attention among the radio spite having a very elegant form, suffer from a major draw- community. In particular, the performance of receive diver- back. The expressions involve the Meijer G-function, which, sity systems over Weibull fading channels has been exten- although easy to evaluate by itself using the modern math- sively studied in [6–13]. Also, a closed-form expression for ematical packages such as Mathematica and Maple, these the moment-generating function (MGF) of the Weibull ran- packages fail to handle integrals involving this function and dom variable (RV) was obtained in [7] when the Weibull fad- lead to numerical instabilities and erroneous results when m ing parameter (which will be defined in the sequel), usually increases. This renders the expressions impractical from the denoted by m, assumes only integer values. Another expres- ease of computation point of view. Hence, it is highly desir- sion for the MGF for arbitrary values of m was also derived able to find alternative closed-form expressions for the MGF in [8]. Both expressions were given in terms of the Meijer G- of the Weibull random variable (RV) that are simple to evalu- function and were used for evaluating the performance of ate and in the same time can be used for arbitrary values of m. digital modulation schemes over the single-channel recep- Pade´ approximation (PA) is a well-known method that is tion and multichannel diversity reception assuming Weibull 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 and {bn} can be easily obtained by matching the coefficients limited number of coefficients is known [17, 18]. This tech- of like powers on both sides of (3). Specifically, taking b0 = 1, nique was recently used for outage probability calculation in without loss of generality, one can find that diversity systems in Nakagami fading in [19]. The approximation is given in terms of a simple rational function of ar- Nq − = ≤ ≤ bitrary numerator and denominator orders. In this paper, we bncNp n+j 0, 1 j Nq,(4) illustrate how this technique can be used to obtain simple-to- n=0 evaluate approximate rational expressions for the MGF of the or equivalently, Weibull RV based on the knowledge of its moments. We then use these expressions to evaluate the performance of linear Nq =− ≤ ≤ digital modulations over flat Weibull fading channels in the bncNp−n+j cNp+j ,1 j Nq. (5) case of both single-channel reception and multichannel re- n=1 ception employing maximal ratio combining (MRC). The above equations form a system of N linear equations for The rest of the paper is organized as follows. In Section 2, q the N unknown denominator coefficients. This system can we give a brief overview of the Pade´ approximation tech- q be written in matrix form as nique. In Section 3, we apply this technique to the prob- lem at hand. The performance of digital modulation sys- Cb =−c,(6) tems over the Weibull fading channel is then revisited in Section 4. Examples and numerical results as well as compar- where isons with previously published results in the literature and = ··· ··· T Monte Carlo simulations are provided in Section 5 before the b bNq bk b1 , paper is finally concluded in Section 6. T c = cN +1 ···cN +k+1 ···cN +N , ⎛ p p p q ⎞ c − c − ... c 2. PADE´ APPROXIMATION OVERVIEW ⎜ Np Nq+1 Np Nq+2 Np ⎟ ⎜ . ⎟ ⎜ . ⎟ (7) ⎜ . ⎟ Let g(z) be an unknown function given in terms of a power ⎜ ⎟ C = ⎜c − c − ... c − ⎟ , series in the variable ∈ C, the set of complex numbers, ⎜ Np Nq+k Np Nq+k+1 Np+k 1 ⎟ z ⎜ . ⎟ namely, ⎝ . ⎠ cNp cNp+1 ... cNp+Nq−1 ∞ n · T g(z) = cnz , cn ∈ R,(1)with ( ) representing the transpose operation. After solving = n 0 the matrix equation in (6), the set {an} can be obtained by where R is the set of real numbers. There are several reasons min(Nq,j) = ≤ ≤ to look for a rational approximation for the series in (1). The aj cj + bj cj−i,0 j Np. (8) = series might be divergent or converging too slowly to be of i 1 any practical use. Also, it is possible that a compact rational An important remark is now in order. It might seem that the form is needed in order to be used in later computations. choice of the values of Nq and Np is completely arbitrary. Not to mention the fact that it might be possible that only This, in fact, is not accurate. An insightful look at (6)re- ffi { } few coe cients of cn are known [17].ThePAmethodis veals that in order to be able to uniquely solve such system capable of dealing with all the problems mentioned above. of equations, it is necessary to have |C| = 0, where |·|is In particular, it can capture the limiting behavior of a power the determinant. In [17], using what we refer to as the rank- series in a rational form. order plots, it is shown that there exists a value of Nq above [N /N ] The one-point PA of order [Np/Nq], P p q (z), is defined which the matrix C becomes rank deficient. This clearly rep- from the series g(z) as a rational function by [17, 18] resents an upper bound on the permissible values of Nq. Also, as mentioned in [20], N is chosen to be equal to N − 1as p q Np n this guarantees the convergence of the PA. In this paper, we = anz P[Np/Nq](z) = n 0 ,(2) Nq take Nq such that it guarantees the uniqueness of the solution = b zn n 0 n of (6)andNp = Nq − 1. ffi { } { } where the coe cients an and bn are defined such that 3. APPLICATION TO THE WEIBULL MGF + Np n Np Nq The MGF of an RV X>0isdefinedas n=0 anz + +1 = c zn + O zNp Nq ,(3) ∞ Nq n n = −sX −sx n=0 bnz n 0 MX (s) = E e = e fX (x)dx,(9) 0 N +N +1 with O(z p q ) representing the terms of order higher than where E(·) is the expectation operator and fX (x) is the prob- Np + Nq. It is straightforward to see that the coefficients {an} ability density function (PDF) of X. The PDF of the Weibull M. H. Ismail and M. M. Matalgah 3 RV is given by the Weibull slow flat-fading channel has been conducted. It is well known that, in general, the performance of any com- mxm−1 xm munication system, in terms of bit error rate (BER), symbol fX (x) = exp − , x ≥ 0, (10) error rate (SER), or signal outage, will depend on the statis- γ γ tics of the signal-to-noise ratio (SNR). Assuming that both the average signal and noise powers are unity, then the SNR where m>0 is usually referred to as the Weibull distribu- will be equal to the squared channel amplitude, X2.Oneof tion fading parameter and γ>0 is a parameter related to the interesting properties of the Weibull RV with parame- the moments and the fading parameter of the distribution.

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