456 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 8, NO. 2, APRIL 2019 Efficient Computation of Multivariate Rayleigh and Exponential Distributions Reneeta Sara Isaac and Neelesh B. Mehta , Senior Member, IEEE Abstract—We propose an efficient approach for the computa- covariance matrices, but severely limits the number of corre- tion of cumulative distribution functions of N correlated Rayleigh lated random variables (RVs) N. For example, infinite series or exponential random variables (RVs) for arbitrary covariance representations are derived for N = 2in[5] and N = 3 and 4 matrices, which arise in the design and analysis of many wireless in [6]–[8] for the arbitrary correlation model. systems. Compared to the approaches in the literature, it employs a fast and accurate randomized quasi-Monte Carlo method that A second class of approaches limits itself to special forms markedly reduces the computational complexity by several orders of the covariance matrix. For example, for the constant cor- of magnitude as N or the correlation among the RVs increases. relation model, the CDF FE(a) of N exponential RVs E = Numerical results show that an order of magnitude larger values (E1,...,EN ) evaluated at a =(a1,...,aN ) is written in of N can now be computed for. Its application to the performance terms of an infinite series as follows [9]: analysis of selection combining is also shown. Index Terms—Multivariate, Rayleigh, exponential, correlated 1 − ρ ∞ ρ n fading, cumulative distribution function. (a)= FE 1+( − 1)ρ 1+( − 1)ρ N n=0 N N × n γ ai , +1 , I. INTRODUCTION li (1) l1,...,lN c(1 − ρ) (l ,...,l ) i=1 ULTIVARIATE or correlated Rayleigh and exponen- 1 N l1≥0,...,lN ≥0 M tial distributions arise in the design and analysis of l1+···+lN =n many wireless systems. An example of this is a multiple input multiple output (MIMO) system when the transmit or receive 2 where ρ is the correlation coefficient between Ei and Ej ,for antennas are spaced close together, which leads to the chan- i = j , c is the mean of Ei ,for1≤ i ≤ N, and γ(x, k) nel gains of the different transmit-receive antenna pairs being is the lower incomplete gamma function [10, eq. (6.5.2)]. correlated [1, Ch. 6], [2, Ch. 13]. Another example is a sensor Computing the nth term of this series requires enumerating network, in which the measurements of different sensor nodes N + n − 1 all combinations of the vector of non-negative are correlated. Alternately, when the sensor nodes or the relay N − 1 nodes in a cooperative relay system are close to each other N integers (l1,...,lN ) that satisfy li = n. This entails a or when there is a common scatterer in the propagation envi- i=1 computational complexity of O(nN −1), which increases expo- ronment, the channel gains seen by the different nodes are nentially with N. This significant complexity challenge also correlated [3], [4]. These distributions also arise in orthogonal arises while evaluating the CDF of correlated Nakagami-m frequency division multiplexing (OFDM) systems, where the RVs that is derived in [11]. subchannel gains of a user are correlated [2, Ch. 19]. When the inverse of the covariance matrix of the underly- Performance analyses of communication techniques ing Gaussian RVs has a tridiagonal structure, the multivariate employed in these systems often lead to expressions that Rayleigh CDF simplifies [12]. However, it is still in the form involve the multivariate cumulative distributive function of an infinite series, evaluating which entails O(nN −1) com- (CDF) of such correlated channel gains. For example, the plexity. The one exception is the specialized correlation model outage probability of an N-branch selection combiner (SC) in of [13] in which the multivariate PDF and CDF are given a multi-antenna system [1, Ch. 9] or of an N-relay cooperative in single-integral forms. These can be numerically integrated system [4] is written in terms of the multivariate CDF of the using Gauss-Laguerre quadrature [10, eq. (25.4.45)]. channel gains. A third class of approaches considers arbitrary covari- Considerable effort has been made in the literature to ance matrices and applies to any N.In[14], the multivariate derive tractable mathematical representations for the multi- Nakagami-m CDF for an arbitrary correlation matrix is given variate probability density functions (PDFs) and CDFs of as a multi-dimensional integration of the PDF, which itself is these distributions. One class of approaches considers arbitrary in the form of an infinite series. In [15], the Green’s approach Manuscript received July 20, 2018; accepted October 7, 2018. Date of is used to approximate any arbitrary covariance matrix to a publication October 15, 2018; date of current version April 9, 2019. This matrix whose inverse has a tridiagonal form. However, the work was supported by the Department of Telecommunications, Ministry multivariate Rayleigh CDF again involves (N − 1) nested of Communications, India, as part of the Indigenous 5G Test Bed Project infinite series, which entails a computational complexity of and the Qualcomm Innovation Fellowship. The associate editor coordinating N −1 the review of this paper and approving it for publication was A. Kammoun. O(n ) if the series is truncated after n terms. Therefore, (Corresponding author: Neelesh B. Mehta.) results for N > 5 are seldom available in the literature for The authors are with the Department of Electrical Communication general correlation models. However, larger values of N are Engineering, Indian Institute of Science, Bengaluru 560012, India (e-mail: [email protected]; [email protected]). of interest in practice. For example, more antennas for next Digital Object Identifier 10.1109/LWC.2018.2875999 generation systems and wireless systems with more relays and 2162-2345 c 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. ISAAC AND MEHTA: EFFICIENT COMPUTATION OF MULTIVARIATE RAYLEIGH AND EXPONENTIAL DISTRIBUTIONS 457 sensors are being considered. In many systems, the covariance Converting to polar coordinates, let matrix need not take a specialized form. 2 2 −1 Ei = X + Y and φi = tan (Yi /Xi ). (4) Contributions: In this letter, we first propose a novel i i 2 approach to numerically compute the multivariate CDF of It can be seen that Ei is an exponential RV with mean 2σi and N Rayleigh or exponential RVs with an arbitrary covariance φi is uniformly distributed between 0 and 2π. Furthermore, Ei matrix. Its key ideas are as follows. It employs a series and φj are mutually independent for all i and j (including i = j) 2 2 2 of transformations to first convert the multivariate PDF of and (E[Ei Ej ] − 4σi σj )/( var(Ei )var(Ej )) = ρkj . Rayleigh or exponential RVs to the multivariate PDF of cor- T ThemultivariateCDFFE(·) of E =(E1,...,EN ) related Gaussian RVs. The CDF expression is then accurately T evaluated at a =(a1,...,aN ) is then and quickly computed using the randomized quasi-Monte 2π 2π a a Carlo (RQMC) method [16, Ch. 4]. In contrast to the above 1 N FE(a)= ··· ··· fE,Φ(e, θ) de dθ, (5) approaches, we show that it can compute the CDF easily even 0 0 0 0 for N = 50, which is much larger than the values considered Φ =(φ ,...,φ )T (·) in [5]–[9], [11], [12], [14], and [15]. Notably, it can do so where 1 N and fE,Φ is the joint CDF of E Φ even when the RVs are highly correlated. This is unlike the and . above approaches, which require more terms to be computed Using the rules governing transformation of variables, we as the RVs become more correlated since the rate of conver- get the following from (3) and (4): u u gence of the series decreases. We then illustrate the utility of 1 2N fX,Y(x, y) (a)= ··· x y, our approach by presenting results for the performance anal- FE d d (6) l1 l2N 2 + 2 ··· 2 + 2 ysis of an N-branch SC, which is a popular receive diversity x1 y1 xN yN combining technique [1, Ch. 9], for a range of values of N. √ √ where l1 = − a1,..., lN = − aN , lN +1 = Outline: In Section II, we present the proposed approach. √ − − 2,..., = − − 2 = ,..., = In Section III, we compare its accuracy and computational a1 x1 l2N aN xN , u1 a1 uN √ complexity with existing approaches. In Section IV, we ana- 2 2 aN , uN +1 = a1 − x1 ,...,u2N = aN − xN . lyze the outage probability of SC. Our conclusions follow in Next, we show how to efficiently compute the integral in (6) Section V. using the RQMC method of [16, Ch. 4]. Notations: The probability of an event A is denoted by Pr(A). The expectation of an RV X is denoted by E[X ] and B. RQMC Method its variance is denoted by var(X). The joint CDF of a random vector X =(X1,...,XN ) evaluated at x =(x1,...,xN ) is It transforms the integral in (6) into an integral over a unit denoted by FX(x)=Pr(X1 ≤ x1,...,XN ≤ xN ). The joint hyper-cube. This is then numerically computed using quasi- (·) Monte Carlo techniques. It is as follows. PDF of X is denoted by fX . The transpose of a vector x is T denoted by xT . 1) Take the Cholesky decomposition of Σ = CC , where C =[cij ] is a lower triangular matrix with cii > 0. T Using the variable transformation z = C−1 xT yT II. COMPUTATION OF MULTIVARIATE EXPONENTIAL changes (6)to AND RAYLEIGH CDF 1 We first consider the multivariate exponential distribution (a)= FE 2N and then the multivariate Rayleigh distribution.