EFFICIENT COMPUTATION of MULTIVARIATE RAYLEIGH and EXPONENTIAL DISTRIBUTIONS 457 Sensors Are Being Considered

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]Ð[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]Ð[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. (2π) u u z2 z2 1 2N − 1 − 2N × ··· e 2 ···e 2 g(z) dz, (7) A. Multivariate Exponential CDF l1 l2N Consider two real Gaussian random vectors X = i−1 T T where li =(li − j =1 cij zj )/cii and ui =(ui − (X1,...,XN ) and Y =(Y1,...,YN ) with zero mean i−1 )/ ≤ ≤ (z)= whose second-order moments are given in general by j =1 cij zj cii ,for1 i 2N, and g N i 2 N +i 2 −1/2 i=1 [( j =1 cij zj ) +( j =1 c(N +i)j zj ) ] . E 2 = E 2 = σ2, for 1 ≤ ≤ , Xk Yk k k N 2) Using vi =Ψ(zi ), where Ψ(z) 2 √ z − θ E X Xj = E Y Yj = ρ σ σj , for k = j , 2 θ/( 2π) k k kj k −∞ e d , yields ⎡ E Xk Yj =0, for 1 ≤ k, j ≤ N . (2) ⎛ ⎞2 N i e1 e2N ⎢⎝ −1 ⎠ Thus, ρkj is the correlation coefficient between Xk and Xj FE(a)= ··· ⎣ cij Ψ (vj ) b b or Yk and Yj . The multivariate Gaussian PDF fX,Y(·, ·) of X 1 2N i=1 j =1 and Y is ⎛ ⎞ ⎤− 1 2 2 N+i 1 1 T T −1 x −1 ⎥ fX,Y(x, y)= exp − x y Σ , (3) +⎝ Ψ ( )⎠ ⎦ v, |Σ|(2π)2N 2 y c(N +i)j vj d (8) j =1 T T where x =(x1,...,xN ) , y =(y1,...,yN ) , Σ = i−1 −1 where bi =Ψ([li − cij Ψ (vj )]/cii ) and ei = X T T j =1 E X Y , and |·|denotes determinant. i−1 −1 Y Ψ([ui − j =1 cij Ψ (vj )]/cii ),for1≤ i ≤ 2N. 458 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 8, NO. 2, APRIL 2019

=( − )/( − TABLE I 3) Applying the transformation wi vi di ei COMPARISON OF CDFS COMPUTED AT PRE-SPECIFIED POINTS (GIVEN di ) changes (8) to the following integral over a unit IN [11,TBL.1])USING PROPOSED APPROACH AND ISA, AND hyper-cube: COMPUTATIONAL COMPLEXITY OF ISA 1 1 FE(a)= ··· f (w)dw, (9) 0 0 where 2N f (w)= (ek − bk ) ⎡ k=1 ⎛ ⎞2 N i ⎢ −1 × ⎣⎝ cij Ψ (bj + wj (ej − bj ))⎠ i=1 j =1

⎤− 1 ⎛ ⎞2 2 III. COMPUTATIONAL COMPLEXITY AND COMPARISON N+i ⎝ −1 ⎠ ⎥ We now compare our approach with the infinite series + c(N +i)j Ψ (bj + wj (ej − bj )) ⎦ . (10) j =1 approach (ISA) of [11] and [15] given that it applies to arbitrary correlation models. Consider the following correlation model for Gaussian random vectors X and Y, in which This integral is computed using the RQMC method that |k−j | uses a carefully selected, deterministic sequence of P samples ρkj = ρ and σk =1,for1≤ k, j ≤ N. For it, the s1,...,sP for the vector w, where the qth sample sq is the multivariate Rayleigh CDF of R is given in the form of the ( ,..., ) following infinite series [11], [15]: vector sq,1 sq,2N . An example of√ this is the Kronecker q,i = { i }, i ∞ sequence, which is given by s q p where p is the ρ2(i1+···+iN −1) th 2 GN i prime number and {·} denotes the remainder modulo 1. To FR(r1,...,rN )=(1− ρ ) N −1( !)2 each sample sq , a random shift u =(u1,...,u2N ) is added, i ,...,i =0 j =1 ij 1 N −1 where u1,...,u2N are independent and identically distributed 2 2 2 r1 r2 (1 + ρ ) (i.i.d.) RVs that are uniformly distributed between 0 and 1, to × γ i1 +1, γ i1 + i2 +1, shift 2(1 − ρ2) 2(1 − ρ2) get the shifted sample sq = {sq + u}. The integrand is 2 2 averaged over M such random shifts, where M is typically rN −1(1 + ρ ) ×···× γ iN −2 + iN −1 +1, between 8 and 12. To speed up convergence, f (w) is replaced 2(1 − ρ2) with (f (w) + f (1−w))/2 and w with |2w − 1| in the integrand r 2 in (9). Thus, we get × γ +1, N , iN −1 2(1 − ρ2) (13) 1 M P FE(a) ≈ f (|2{sq + ui }−1|) 2 −[i1+2i2+···+2iN −2+iN −1+N −2] 2 where GN =(1+ρ ) ,for MP i=1 q=1 N ≥ 3, and GN =1,forN = 2. +f (1 −|2{sq + ui }−1|) . (11) Table I tabulates the number of terms required by ISA and the CDFs of ISA and the proposed approach to an accuracy The RQMC method has a convergence rate of O(1/P) of 4 decimal places at pre-specified points. The pre-specified [16, Ch.√ 4], which is better than the convergence rate of points and correlation coefficients are chosen because they = = O(1/ P) of the conventional Monte Carlo method that takes were used in [11, Table 1] for N 3 and 4. For N 5, r1 is ,..., ,..., an average over any P random samples of w. The computa- set as 1 and r2 r5 are set to be the same as r1 r4 for = tional complexity of the proposed approach is O(MP). N 4. For ISA, the number of terms that need to be summed over increases exponentially as N or ρ increases. Even for N as small as 5, it exceeds 105 for ρ =0.9. The number of terms C. Multivariate Rayleigh Distribution required by the proposed approach for M = 8 and P = 1000, T Let R =(R1,...,RN ) be the vector of correlated which are sufficient for all the considered values of N and ρ, Rayleigh RVs generated from X and Y, such that Ri = is only MP = 8000. A comparison for larger N is not shown 2 2 1/2 (Xi + Yi ) ,for1≤ i ≤ N. Then, the multivariate CDF since ISA becomes computationally infeasible. T FR(r) of R evaluated at r =(r1,...,rN ) can be expressed (·) in terms of the exponential CDF FE as follows: IV. ILLUSTRATIVE APPLICATION TO N-BRANCH SC (r)=Pr( ≤ ,..., ≤ ), Consider a wireless communication system that consists of FR R1 r1 RN rN 2 2 2 2 a transmitter with one antenna and a receiver with N antennas. =PrR1 ≤ r1 ,...,R ≤ r , 2 N N Let γk = Rk ωs /N0 be the instantaneous signal-to-noise ratio 2 2 th = FE(r1 ,...,rN ). (12) (SNR) of the signal received at the k antenna, where Rk is its channel amplitude, which is a Rayleigh RV, ωs is the trans- Therefore, the multivariate Rayleigh CDF can also be easily mitted symbol energy, and N0 is the additive white Gaussian computed using the above approach. noise power spectral density. After selection combining, the ISAAC AND MEHTA: EFFICIENT COMPUTATION OF MULTIVARIATE RAYLEIGH AND EXPONENTIAL DISTRIBUTIONS 459

Fig. 2 plots Pout as a function of γth for this model when ρmin =0.5 and ρmax =0.9. The analysis and simulation results are in good agreement even for N as large as 30. Similar to Fig. 1, Pout decreases as N increases. Compared to the exponential correlation model, Pout of this model is higher for a given γth and N. Intuitively, this is because the RVs are more correlated in the arbitrary correlation model than in the exponential correlation model for the parameters used.

V. C ONCLUSION The complexity of computing the multivariate CDF with an arbitrary covariance matrix using conventional approaches increased exponentially as N increased. We presented a novel approach for computing the multivariate Rayleigh and exponential CDFs for any N and for an arbitrary correlation structure. It had a much lower computational complexity that Fig. 1. Exponential correlation model: Outage probability of SC as a function ρ of N for different ρ (γth = 3dBandωs /N0 =1). no longer increased as N or increased. We then demonstrated the utility of our approach by analyzing the outage probability of an N-branch SC receiver.

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