# Statistics for Ratios of Rayleigh, Rician, Nakagami-m, and Weibull Distributed Random Variables

Authors: Dragana Č. Pavlović; Nikola M. Sekulović; Gradimir V. Milovanović; Aleksandra S. Panajotović; Mihajlo Č. Stefanović; Zoran J. Popović
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The distributions of ratios of random variables are of interest in many areas of the sciences. In this brief paper, we present the joint probability density function (PDF) and PDF of maximum of ratiosμ1=R1/r1 and μ2=R2/r2 for the cases where R1, R2, r1, and r2 are Rayleigh, Rician, Nakagami-m, and Weibull distributed random variables. Random variables R1 and R2, as well as random variables r1 and r2, are correlated. Ascertaining on the suitability of the Weibull distribution to describe fading in both indoor and outdoor environments, special attention is dedicated to the case of Weibull random variables. For this case, analytical expressions for the joint PDF, PDF of maximum, PDF of minimum, and product moments of arbitrary number of ratios μi=Ri/ri, i=1,…,L are obtained. Random variables in numerator, Ri, as well as random variables in denominator, ri, are exponentially correlated. To the best of the authors' knowledge, analytical expressions for the PDF of minimum and product moments of {μi}i=1L are novel in the open technical literature. The proposed mathematical analysis is complemented by various numerical results. An application of presented theoretical results is illustrated with respect to performance assessment of wireless systems.


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## Body

## 1. Introduction

Radio-wave propagation through wireless channels is a complicated phenomenon characterized by fading which is the result of multipath propagation. When a received signal envelope experiences fading during transmission, it fluctuates over time. Multivariate statistics is a useful mathematical tool for modeling and analyzing wireless channels. There is a very wide range of statistical models for fading channels [1] which their accuracy and veracity depend on propagation environment and communication scenario. Rayleigh, Rician, Nakagami-m, and Weibull are the most frequently applied models in the open technical literature.Fading can seriously degrade performance of wireless communications systems. Techniques that can be used to minimize the degradation effects due to fading have received a great deal of research interest. Diversity combining [1], which combines two or more replicas of the received signal, is a practical and powerful technique that can be used to alleviate the detrimental effects of fading and to improve the performance of wireless communications systems without increasing transmission power and bandwidth. Space diversity [2], achieved by using multiple antennas at the receiver, is the most common form of diversity. The most popular space diversity techniques are selection combining (SC), equal-gain combining (EGC), and maximal-ratio combining (MRC).In digital contemporary communications systems, fading channels are correlated due to insufficient antenna spacing when diversity is applied in small terminals (e.g., hand-held mobile terminals and compact base stations). Several spatial correlation models have been proposed and used for the performance analysis of various wireless systems, corresponding to specific modulation, detection, and diversity schemes [1]. Spatial exponential correlation model is one of the most frequently used in performance analysis of wireless systems with multichannel reception [3–5]. The correlation matrix of this model is described by ∑ij‍≡ρ|i-j|, i,j=1,…,L, where ρ (0≤ρ<1) is the correlation coefficient between adjacent channels [1, Equation (9.164)] and L is the number of channels. The exponential correlation model corresponds to the scenario of multichannel reception from equispaced diversity antennas in which the correlation among the pairs of combined signals decays as the spacing between the antennas increases.The distribution of the ratio of random variables is of interest in statistical analysis in biological and physical sciences, econometrics, and ranking and selection [6]. It has been studied by several authors especially when random variables are independent and come from the same family [7–10]. In [11, 12], the distribution of the ratio of correlated random variables is considered.The ratios of random variables are also of interest in analyzing wireless communications systems in fading environment [13–17]. Namely, the random variable in nominator may present desired signal envelope while the random variable in denominator may present interference signal envelope. Rayleigh, Rician, Nakagami-m, and Weibull distributions are included in our analysis. Having in mind that it is well known that the assumption of independence among the input diversity channels is not accurate for compact, hand-held, mobile terminals and indoor base stations with no sufficiently separated antennas, random variables in nominator, as well as random variables in denominator, are correlated. It shows that results presented in the paper can be efficiently used in analyzing realistic correlated fading channels.


## 2. On Two Ratios of Rayleigh, Rician, and Nakagami-m Random Variables

In this section, the joint probability density function (PDF) and PDF of maximum of two ratios of random variablesμ1=R1/r1 and μ2=R2/r2 are presented. Random variables R1 and R2, as well as random variables r1 and r2, are correlated. The joint PDF of μ1 and μ2 can be obtained as
(1)pμ1μ2(μ1,μ2)=∬0∞|J|pR1R2(μ1r1,μ2r2)pr1r2(r1,r2)dr1dr2,
where pR1R2(R1,R2) and pr1r2(r1,r2) are the joint PDFs of random variables in nominator and denominator, respectively, and |J| is the Jacobian transformation given by
(2)|J|=|dR1dμ1dR1dμ2dR2dμ1dR2dμ2|=r1r2.The PDF expression of maximum of ratios of random variablesμmax=max{μ1,μ2} can be derived as
(3)pμmax(μ)=∫0μpμ1μ2(μ,μ2)dμ2+∫0μpμ1μ2(μ1,μ)dμ1.In the analysis of wireless communications systems, depending on the nature of the radio propagation environment, there are different models describing the statistical behavior of the multipath fading envelope. In the rest of this section, correlative Rayleigh, Rician, and Nakagami-m fading models are considered. The existence of correlation is the real scenario in practical multiantennas wireless systems due to insufficient antenna spacing. Since, this paper considers spatial correlation—the correlation between appropriate pair of receive antennas—all signals, regardless of their nature, received by these antennas explore the same correlation coefficient.(a) Rayleigh Case. The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight (LOS) path. The joint bivariate PDFs of correlated Rayleigh distributed random variables Ri and ri (i=1,2) are given by [1, Equation (6.2)] as follows:
(4)pR1R2(R1,R2)=4R1R2Ωd1Ωd2(1-ρ)×exp[-11-ρ(R12Ωd1+R22Ωd2)]×I0[2ρR1R2(1-ρ)Ωd1Ωd2],pr1r2(r1,r2)=4r1r2Ωc1Ωc2(1-ρ)×exp[-11-ρ(r12Ωc1+r22Ωc2)]×I0[2ρr1r2(1-ρ)Ωc1Ωc2],
where I0(·) is the modified Bessel function of the first kind and zero order, ρ is the correlation coefficient, and Ωdi and Ωci are the mean-square values of Ri and ri, respectively. Substituting (2) and (4) in (1) and using the infinite-series representation of the modified Bessel function [18, Equation (8.447/1)], integrals can be solved with the aid of [18, Equation (3.478/1)]. Analytical expression for the joint PDF of μ1 and μ2 can be finally written as
(5)pμ1μ2(μ1,μ2)=4(1-ρ)2∑k,ℓ=0∞ρk+ℓ(μ1μ2)2k+1Γ2(2+k+ℓ)(Ωd1Ωd2)1+k(Ωc1Ωc2)1+ℓ(k!ℓ!)2×((μ12Ωd1+1Ωc1)(μ22Ωd2+1Ωc2))-(2+k+ℓ).Substituting (5) in (3) and using [18, Equation (3.194/1)], the PDF of maximum of two ratios of Rayleigh random variables can be written as
(6)pμmax(μ)=(1-ρ)2∑k,ℓ=0∞ρk+ℓΓ2(2+k+ℓ)μ4k+3(k!ℓ!)2(1+k)(Ωc1Ωc2Ωd1Ωd2)1+k×{2(1+((Ωc1/Ωd1)μ2))2+k+ℓ×2F1(2+k+ℓ,1+k;2+k;-Ωc2Ωd2μ2)+2(1+((Ωc2/Ωd2)μ2))2+k+ℓ×F12(2+k+ℓ,1+k;2+k;-Ωc1Ωd1μ2)2(1+((Ωc1/Ωd1)μ2))2+k+ℓ},
where F12(a,b;c;d) is the Gauss hypergeometric function.(b) Rician Case. The Rician distribution is often used to model propagation paths consisting of one strong direct LOS component and many random weaker components. In this case, Ri and ri (i=1,2) are distributed according to [15, Equation (1)] as follows:
(7)pR1R2(R1,R2)=4R1R2(K+1)2Ωd2(1-ρ2)×exp(-(R12+R22)(K+1)+2KΩd(1-ρ)Ωd(1-ρ2))×∑k=0∞‍ɛkIk(2R1R2ρ(K+1)Ωd(1-ρ2))×Ik(2R11+ρK(K+1)Ωd)Ik(2R21+ρK(K+1)Ωd),pr1r2(r1,r2)=4r1r2(K+1)2Ωc2(1-ρ2)×exp(-(r12+r22)(K+1)+2KΩc(1-ρ)Ωc(1-ρ2))×∑ℓ=0∞‍ɛℓIℓ(2r1r2ρ(K+1)Ωc(1-ρ2))×Iℓ(2r11+ρK(K+1)Ωc)Iℓ(2r21+ρK(K+1)Ωc),
where In(·) is the modified Bessel function of the first kind and nth order, ɛ0=1, ɛk=2, k≥1, Ωd=Ri2¯, Ωc=ri2¯, and K is the Rician factor. The Rician distribution spans the range from Rayleigh fading (K→0) to no fading (K→∞). Following the same procedure as for the Rayleigh case, the joint PDF of μ1 and μ2 becomes [17, Equation (6)] as follows:
(8)pμ1μ2(μ1,μ2)=exp(-4K/(1+ρ))(K+1)2×∑k,ℓ,m,n,p,q,s,w=0∞μ12(m+n+k)Ωd2(q+ℓ+1)+s+wΩc2(m+k+1)+n+p(1+ρ)2(n+p+k+s+w+ℓ)m!n!p!q!s!w!×(ρ2(m+q)+k+ℓμ22(m+p+k)(1-ρ2)n+p+k+s+w+ℓ+2×ɛkɛℓKn+p+k+s+w+ℓE(k,ℓ,m,n,p,q,s,w)×((Ωd+Ωcμ12)m+n+k+q+s+ℓ+2×(Ωd+Ωcμ22)m+p+k+q+w+ℓ+2)-1ρ2(m+q)+k+ℓμ22(m+p+k)),
where E(k,ℓ,m,n,p,q,s,w)≡E is given by
(9)E=Γ(m+n+k+q+s+ℓ+2)×Γ(m+p+k+q+w+ℓ+2)×(Γ(m+k+1)Γ(n+k+1)Γ(p+k+1)×Γ(q+ℓ+1)Γ(s+ℓ+1)Γ(w+ℓ+1))-1,
while the PDF of μmax can be written as in [17, Eq. (11)] as
(10)pμmax(μ)=exp(-4K1+ρ)×∑k,ℓ,m,n,p,q,s,w=0∞(×((1+ρ)2(n+p+k+s+w+ℓ)m!n!p!q!s!w!)-1ɛkɛlKn+p+k+s+w+ℓρ2(m+q)+k+ℓ×(1-ρ2)n+p+k+s+w+ℓ+2×((1+ρ)2(n+p+k+s+w+ℓ)m!n!p!q!s!w!)-1)×μ4(m+k)+2(p+n+1)Sq+ℓ+1(S+μ2)2(k+m)+n+p+ℓ+q+3E(k,ℓ,m,n,p,q,s,w)×[(SS+μ2)w(F12(m+n+k+2;μ2μ2+Sm+n+k+1,-ℓ-q-s;m+n+k+2;μ2μ2+S)×(m+n+k+1)-1F12(m+n+k+2;μ2μ2+Sm+n+k+1,-ℓ-q-s;)+(SS+μ2)s(F12(m+p+k+2;μ2μ2+S)m+p+k+1,-ℓ-q-w;m+p+k+2;μ2μ2+S)×(m+p+k+1)-1F12(m+p+k+2;μ2μ2+S)m+p+k+1,-ℓ-q-w;))(SS+μ2)w],
where S=Ωd/Ωc and E is defined before.(c) Nakagami-
m
Case. The Nakagami-m distribution has gained widespread application in the modeling of physical fading radio channels [19]. The primary justification of the use of Nakagami-m fading model is its good fit to empirical fading data. It is versatile and through its parameter m, we can model signal fading conditions that range from severe to moderate, to light fading, or no fading. It includes the one-sided Gaussian distribution (m=0.5) and the Rayleigh distribution (m=1) as special cases. In Nakagami-m fading environment, Ri and ri (i=1,2) follow the distributions
(11)pR1R2(R1,R2)=4m1m1+1(R1R2)m1Γ(m1)Ωd1Ωd2(1-ρ)(ρΩd1Ωd2)m1-1×exp(-m1(R12+R22)Ωd1Ωd2(1-ρ))×Im1-1(2ρm1R1R2Ωd1Ωd2(1-ρ)),pr1r2(r1,r2)=4m2m2+1(r1r2)m2Γ(m2)Ωc1Ωc2(1-ρ)(ρΩc1Ωc2)m2-1×exp(-m2(r12+r22)Ωc1Ωc2(1-ρ))×Im2-1(2ρm2r1r2Ωc1Ωc2(1-ρ)).Setting(12)W=(m1μ12Ωd1Ωd2+m2Ωc1Ωc2)×(m1μ22Ωd1Ωd2+m2Ωc1Ωc2),
the joint PDF of ratios of the Nakagami-m distributed random variables can be obtained as
(13)pμ1μ2(μ1,μ2)=4(1-ρ)m1+m2Γ(m1)Γ(m2)×∑k,l=0∞((Ωd1Ωd2)k+m1Wk+l+m1+m2)-1ρk+lΓ2(k+l+m1+m2)×(μ1μ2)2(k+m1)-1m12(m1+k)m12(m2+l)×(k!l!Γ(k+m1)Γ(l+m2)(Ωc1Ωc2)l+m2×(Ωd1Ωd2)k+m1Wk+l+m1+m2)-1),
while the PDF of maximum of ratios of Nakagami-m distributed random variables is
(14)pμmax(μ)=4(1-ρ)m1+m2Γ(m1)Γ(m2)×∑k,l=0∞(×(μ2m1Ωd1Ωd2+m2Ωc1Ωc2)k+l+m1+m2)-1ρk+lΓ2(k+l+m1+m2)×μ4(k+m1)-1(Ωc1Ωc2)(m1+k-m2-l)/2×m12(m1+k)m2(m2+l-m1-k)×(((μ2m1/Ωd1Ωd2)+(m2/Ωc1Ωc2))k+l+m1+m2(μ2m1Ωd1Ωd2+m2Ωc1Ωc2)k+l+m1+m2k!l!Γ(k+m1)Γ(l+m2)×(k+m1)(Ωd1Ωd2)(m1+k)×(μ2m1Ωd1Ωd2+m2Ωc1Ωc2)k+l+m1+m2)-1)×F12(-μ2m1Ωc1Ωc2/Ωd1Ωd2m2k+l+m1+m2,m1+k;k+m1+1;k+l+m1+m2,m1+k;k+m1+1;-μ2m1Ωc1Ωc2/Ωd1Ωd2m2k+l+m1+m2,m1+k;k+m1+1;).The above presented results can be efficiently used for analyzing wireless communications systems in fading environment. Namely, the joint bivariate PDF of ratios of random variables is necessary for performance analysis of dual diversity combining schemes. In that case,μi denotes signal-to-interference ratio, Ωdi and Ωci are the average powers of desired and interference signals at ith diversity branch, respectively, and ρ is the correlation between branches. Varying the Rician and Nakagami-m parameter, it is possible to simulate fading with different severity degrees. Having in mind that in interference-limited environment SC diversity receiver selects and outputs the branch with the highest signal-to-interference ratio, the PDF of μmax presents the PDF of signal-to-interference ratio at the SC output. This expression can be used to study important performance measures such as the average bit error probability and channel capacity.


## 3. On the Ratios of Weibull Distributed Random Variables

The Weibull distribution exhibits an excellent fit to experimental fading channel measurements, for both indoor [20] and outdoor [21, 22] environments. It is the reason why Weibull distribution paved its way to wireless communications applications. The fact that the diversity receiver with larger number of branches shows better performance gives an idea to investigate the statistics of arbitrary number of ratios. In this section, the joint PDF, product moments, and PDF of maximum and minimum of arbitrary number of ratios of Weibull distributed random variables are presented. To the best of the authors’ knowledge, analytical expressions for product moments and PDF of minimum of ratios of Weibull random variables are novel in the open technical literature.(a) The Joint PDF. The joint bivariate PDFs of correlated Weibull distributed random variables Ri and ri (i=1,2) are given by [23, Equation (11)] as follows:
(15)pR1R2(R1,R2)=β1β2R1β1-1R2β2-1Ωd1Ωd2(1-ρ)×exp[-11-ρ(R1β1Ωd1+R2β2Ωd2)]×I0[2ρR1β1/2R2β2/2(1-ρ)Ωd1Ωd2],pr1r2(r1,r2)=β1β2r1β1-1r2β2-1Ωc1Ωc2(1-ρ)×exp[-11-ρ(r1β1Ωc1+r2β2Ωc2)]×I0[2ρr1β1/2r2β2/2(1-ρ)Ωc1Ωc2],
while the joint multivariate PDFs of Weibull distributed random variables Ri and ri, i=1,…,L, with exponential correlation can be expressed as [23, Equation (22)] as follows:
(16)pR1⋯RL(R1,…,RL)=1(1-ρ)L-1∏i=1L‍βiRiβi-1Ωdi×exp{-11-ρ[R1β1Ωd1+RLβLΩdL+(1+ρ)∑i=2L-1RiβiΩdi]}×∏i=1L-1‍I0[2ρRiβi/2Ri+1βi+1/2(1-ρ)Ωd1⋯ΩdLL],pr1⋯rL(r1,…,rL)=1(1-ρ)L-1∏i=1L‍βiriβi-1Ωci×exp{-11-ρ[r1β1Ωc1+rLβLΩcL+(1+ρ)∑i=2L-1riβiΩci]}×∏i=1L-1I0[2ρriβi/2ri+1βi+1/2(1-ρ)Ωc1⋯ΩcLL],
where βi is the Weibull parameter. Weibull parameter expresses the fading severity (βi>0). As it increases, the severity of fading decreases, while for βi=2, the Weibull distribution reduces to the Rayleigh distribution. The joint PDF of ratios of Weibull random variables, μi=Ri/ri, i=1,…,L, can be obtained as
(17)pμ1⋯μL(μ1,…,μL)=∫0∞‍⋯∫0∞‍︸L-fold|J|pR1⋯RL(μ1r1,…,μLrL)×pr1⋯rL(r1,…,rL)dr1⋯drL,
where |J| is the Jacobian transformation given by
(18)|J|=|dR1dμ1…dR1dμL⋮dRLdμ1dRLdμL|=∏i=1Lri.Substituting the adequate joint PDFs ofRi and ri in (17) and using the infinite-series representation of the modified Bessel function, after integrations, the joint bivariate PDF of μ1 and μ2 can be written in the following form [14, Equation (7)]:
(19)pμ1μ2(μ1,μ2)=(1-ρ)2β1β2×∑k,ℓ=0∞‍ρk+ℓμ1(1+k)β1-1μ2(1+k)β2-1Γ2(2+k+ℓ)(Ωd1Ωd2)1+k(Ωc1Ωc2)1+ℓ(k!ℓ!)2×((μ1β1Ωd1+1Ωc1)(μ2β2Ωd2+1Ωc2))-(2+k+ℓ),
while the joint multivariate PDF of {μi}i=1L can be written as in [16, Equation (4)] as
(20)pμ1⋯μL(μ1,…,μL)=(1-ρ)2×∏i=1Lβi∑k1,…,kL-1,ℓ1,…,ℓL-1=0∞ρ∑i=1L-1(ki+ℓi)(1+ρ)∑i=2L-1(2+ki+ki-1+ℓi+ℓi-1)×(×(μLβLΩdL+1ΩcL)2+kL-1+ℓL-1∏i=1L-1(ki!ℓi!)2)-1μ1(1+k1)β1-1μL(1+kL-1)βL-1Γ(2+k1+ℓ1)×Γ(2+kL-1+ℓL-1)×(×(μLβLΩdL+1ΩcL)2+kL-1+ℓL-1∏i=1L-1(ki!ℓi!)2)-1∏i=1L‍(Ωdi1+(2/L)∑j=1L-1‍kjΩci1+(2/L)∑j=1L-1‍ℓj)×(μ1β1Ωd1+1Ωc1)2+k1+ℓ1×(μLβLΩdL+1ΩcL)2+kL-1+ℓL-1∏i=1L-1(ki!ℓi!)2)-1)×∏i=2L-1[μiβi(1+ki+ki-1)-1(μiβiΩdi+1Ωci)-(2+ki+ki-1+ℓi+ℓi-1)×Γ(2+ki+ki-1+ℓi+ℓi-1)(μiβiΩdi+1Ωci)-(2+ki+ki-1+ℓi+ℓi-1)].(b) Product Moments. Product moments are also important in wireless communications systems because they are a useful statistical tool to characterize a distribution. The (∑i=1L‍ni)th-order moment of the product of {μi}i=1L can be derived as
(21)ɛ〈∏i=1L‍μini〉=∫0∞‍⋯∫0∞‍︸L-fold(∏i=1L‍μini)pμ1⋯μL(μ1,…,μL)dμ1⋯dμL.Substituting the adequate joint PDF of ratios of Weibull distributed random variables in (21) and using [18, Equation (3.241/4)], for ni/βi<1, i=1,…,L, product moments of two and multiple ratios of Weibull random variables are
(22)ɛ〈μ1n1μ2n2〉=U∑k,ℓ=0∞‍ρk+ℓ(k!ℓ!)2Γ(1+k+n1β1)Γ(1+k+n2β2)×Γ(1+ℓ-n1β1)Γ(1+ℓ-n2β2),
where U=(1-ρ)2(Ωd1/Ωc1)n1/β1(Ωd2/Ωc2)n2/β2 and
(23)ɛ〈∏i=1L‍μini〉=(1-ρ)2×∑k1,…,kL-1,ℓ1,…,ℓL-1=0∞(ρ∑i=1L-1‍(ki+ℓi)Ωd1k1-(2/L)∑i=1L-1‍ki+(n1/β1)×Ωc1ℓ1-(2/L)∑i=1L-1‍ℓi-(n1/β1)×((1-ρ)∑i=2L-1‍(2+ki+ki-1+ℓi+ℓi-1))-1)×ΩdLkL-1-(2/L)∑i=1L-1‍ki+(nL/βL)ΩcLℓL-1-(2/L)∑i=1L-1‍ℓi-(nL/βL)∏i=1L-1‍(ki!ℓi!)2×Γ(1+k1+n1β1)Γ(1+kL-1+nLβL)×Γ(1+ℓ1-n1β1)Γ(1+ℓL-1-nLβL)×∏i=2L-1‍{Γ(1+ℓi+ℓi-1-niβi)Ωdiki+ki-1-(2/L)∑j=1L-1‍kj+(ni/βi)Ωdiki+ki-1-(2/L)∑j=1L-1‍kj+(ni/βi)Ωciℓi+ℓi-1-(2/L)∑j=1L-1‍ℓj-(ni/βi)×Γ(1+ki+ki-1+niβi)Ωdiki+ki-1-2/L∑j=1L-1‍kj+ni/βi×Γ(1+ℓi+ℓi-1-niβi)Ωdiki+ki-1-(2/L)∑j=1L-1‍kj+(ni/βi)},
respectively.(c) PDF of Maximum. The PDF expression of maximum of ratios of random variables, μmax=max{μ1,μ2,…,μL}, can be derived as
(24)pμmax(μ)=∬0μ⋯∫0μ︸(L-1)-foldpμ1μ2⋯μL(μ,μ2,…,μL)dμ2dμ3⋯dμL+∬0μ⋯∫0μ︸(L-1)-foldpμ1μ2⋯μL(μ1,μ,…,μL)dμ1dμ3⋯dμL⋮+∬0μ⋯∫0μ︸(L-1)-foldpμ1μ2⋯μL(μ1,μ2,…,μ)dμ1dμ2⋯dμL-1.Substituting the joint PDF of{μi}i=1L in the previous expression and after integrations, the PDFs of maximum of two and multiple ratios of Weibull random variables can be written, respectively, as in [14, Equation(11)] and [16, Equation(12)] as follows:
(25)pμmax(μ)=(1-ρ)2∑k,ℓ=0∞ρk+ℓΓ2(2+k+ℓ)μ(1+k)(β1+β2)-1(1+k)(k!ℓ!)2(Ωc1Ωc2Ωd1Ωd2)1+k×{β1(1+((Ωc1/Ωd1)μβ1))2+k+ℓ×F21(2+k+ℓ,1+k,2+k;-Ωc2Ωd2μβ2)+β2(1+((Ωc2/Ωd2)μβ2))2+k+ℓ×F21(2+k+ℓ,1+k,2+k;-Ωc1Ωd1μβ1)β1(1+((Ωc1/Ωd1)μβ1))2+k+ℓ},(26)pμmax(μ)=(1-ρ)2×∑k1,k2,…,kL-1,ℓ1,ℓ2,…,ℓL-1=0∞Ωc11+k1+ℓ1-(2/L)∑i=1L-1‍ℓiΩcL1+kL-1+ℓL-1-(2/L)∑i=1L-1‍ℓi(1+ρ)∑i=2L-1‍(2+ki+ki-1+ℓi+ℓi-1)×Γ(2+k1+ℓ1)Γ(2+kL-1+ℓL-1)(∏i=1L‍Ωdi)1+(2/L)∑i=1L-1‍ki∏i=0L-1‍(ki!ℓi!)2×ρ∑i=1L-1‍(ki+ℓi)μ(1+k1)β1+(1+kL-1)βL+∑i=2L-1‍(1+ki+ki-1)βi-1×∏i=2L-1{Ωci1+ki+ki-1+ℓi+ℓi-1-(2/L)∑i=1L-1‍ℓi×Γ(2+ki+ki-1+ℓi+ℓi-1)Ωci1+ki+ki-1+ℓi+ℓi-1-(2/L)∑i=1L-1‍ℓi}Z(μ),
respectively, where
(27)Z(μ)=(β1(1+kL-1)(1+((Ωc1/Ωd1)μβ1))2+k1+ℓ1×2F1(2+kL-1+ℓL-1,1+kL-1;2+kL-1;-ΩcLΩdLμβL)+βL(1+k1)(1+((ΩcL/ΩdL)μβL))2+kL-1+ℓL-1×2F1(2+k1+ℓ1,1+k1;2+k1;-Ωc1Ωd1μβ1)β1(1+kL-1)(1+((Ωc1/Ωd1)μβ1))2+k1+ℓ1)×∏i=2L-1‍(F21(ΩciΩdiμβi2+ki+ki-1+ℓi+ℓi-1,1+ki+ki-1;2+ki+ki-1;-ΩciΩdiμβi)×(1+ki+ki-1)-1F21(ΩciΩdiμβi2+ki+ki-1+ℓi+ℓi-1,1+ki+ki-1;)+(F12(2+k1+ℓ1,1+k1;2+k1;-Ωc1Ωd1μβ1)×F12(-ΩcLΩdLμβL2+kL-1+ℓL-1,1+kL-1;2+kL-1;-ΩcLΩdLμβL)×((1+k1)(1+kL-1))-1Ωc1Ωd1)×∑j=2L-1‍βj(1+((Ωcj/Ωdj)μβj))2+kj+kj-1+ℓj+ℓj-1×∏i=2;i≠jL-1[ΩciΩdi2F1(ΩciΩdiμβi2+ki+ki-1+ℓi+ℓi-1,1+ki+ki-1;2+ki+ki-1;-ΩciΩdiμβi)×(1+ki+ki-1)-12F1(ΩciΩdiμβi2+ki+ki-1+ℓi+ℓi-1,1+ki+ki-1;].Figure1 illustrates the PDF of maximum of two, three, and four ratios of random variables.Figure 1
Probability density function of maximum of ratios of Weibull random variables forρ=0.2, Ωdi=0.8, Ωci=0.2, and βi=2.5,i=1,…,L.(d) PDF of Minimum. The minimum of ratios of Weibull random variables, μmin=min{μ1,μ2,…,μL}, can be obtained by
(28)pμmin(μ)=∬μ∞⋯∫μ∞︸(L-1)-foldpμ1μ2⋯μL(μ,μ2,…,μL)dμ2dμ3⋯dμL+∬μ∞⋯∫μ∞︸(L-1)-foldpμ1μ2⋯μL(μ1,μ,…,μL)dμ1dμ3⋯μL⋮+∬μ∞⋯∫μ∞︸(L-1)-foldpμ1μ2⋯μL(μ1,μ2,…,μ)dμ1dμ2⋯dμL-1.Substituting the adequate joint PDF in (28) and using [18, Equation (3.194/2)], with (μβi/Ωdi)+(1/Ωci)=Φi, Ωdi/Ωciμβi=φi, i=1,2,…,L, the PDF of minimum of two ratios of random variables becomes
(29)pμmin(μ)=(1-ρ)2×∑k,ℓ=0∞ρk+ℓΓ2(2+k+ℓ)(Ωd1Ωd2)1+k(Ωc1Ωc2)1+ℓ(k!ℓ!)2(1+ℓ)×{β1Ωd22+k+ℓμ(1+k)β1-(1+ℓ)β2-1Φ12+k+ℓ×2F1(2+k+ℓ,1+ℓ;2+ℓ;-φ2)+β2Ωd12+k+ℓμ(1+k)β2-(1+ℓ)β1-1Φ22+k+ℓ×2F1(2+k+ℓ,1+ℓ;2+ℓ;-φ1)μ(1+k)β1-(1+ℓ)β2-1Φ12+k+ℓ},
and finally, the PDF of minimum of multiple ratios of random variables can be expressed in a more complicated form as follows:
(30)pμmin(μ)=(1-ρ)2×∑k1,k2,…,kL-1,ℓ1,ℓ2,…,ℓL-1=0∞(×(∏i=1LΩci)1+(2/L)∑i=1L-1‍ℓi)-1ρ∑i=1L-1‍(ki+ℓi)(1+ρ)-∑i=2L-1‍(2+ki+ki-1+ℓi+ℓi-1)×((∏i=1LΩdi)1+(2/L)∑i=1L-1‍ki×(∏i=1LΩci)1+(2/L)∑i=1L-1‍ℓi)-1)×((∏i=1L-1(ki!ℓi!)2)-1Γ(2+k1+ℓ1)Γ(2+kL-1+ℓL-1)×∏i=2L-1‍Γ(2+ki+ki-1+ℓi+ℓi-1)×(∏i=1L-1(ki!ℓi!)2)-1)×{∏i=2L-1(Ωdi2+ki+ki-1+ℓi+ℓi-11+ℓi+ℓi-1F12(2+ki+ki-1+ℓi+ℓi-1,1+ℓi+ℓi-1;2+ℓi+ℓi-1;-φi)Ωdi2+ki+ki-1+ℓi+ℓi-11+ℓi+ℓi-1)×(β1ΩdL2+kL-1+ℓL-1μ(1+k1)β1-∑i=2L-1‍(1+ℓi+ℓi-1)βi-(1+ℓL-1)βL-1(1+ℓL-1)Φ12+k1+ℓ1×F12(2+kL-1+ℓL-1,1+ℓL-1;2+ℓL-1;-φL)+βLΩd12+k1+ℓ1μ(1+kL-1)βL-∑i=2L-1‍(1+ℓi+ℓi-1)βi-(1+ℓ1)β1-1(1+ℓ1)ΦL2+kL-1+ℓL-1×2F1(2+k1+ℓ1,1+ℓ1;2+ℓ1;-φ1)β1ΩdL2+kL-1+ℓL-1μ(1+k1)β1-∑i=2L-1‍(1+ℓi+ℓi-1)βi-(1+ℓL-1)βL-1(1+ℓL-1)Φ12+k1+ℓ1)+F12(2+k1+ℓ1,1+ℓ1;2+ℓ1;-φ1)×F21(2+kL-1+ℓL-1;1+ℓL-1;2+ℓL-1;-φL)×Ωd12+k1+ℓ1ΩdL2+kL-1+ℓL-1(1+ℓ1)(1+ℓL-1)×∑i=2L-1‍[μ(1+ki+ki-1)βi-(1+ℓ1)β1-(1+ℓL-1)βL-∑j=2,j≠iL-1‍(1+ℓj+ℓj-1)βj-1βiΦi2+ki+ki-1+ℓi+ℓi-1×∏j=2,j≠iL-1‍(Ωdj2+kj+kj-1+ℓj+ℓj-11+ℓj+ℓj-1×F21(2+kj+kj-1+ℓj+ℓj-1,1+ℓj+ℓj-1;2+ℓj+ℓj-1;-φj)Ωdj2+kj+kj-1+ℓj+ℓj-11+ℓj+ℓj-1)×μ(1+ki+ki-1)βi-(1+ℓ1)β1-(1+ℓL-1)βL-∑j=2,j≠iL-1‍(1+ℓj+ℓj-1)βj-1βiΦi2+ki+ki-1+ℓi+ℓi-1]}.The PDF of minimum of ratios of random variables is important for analyzing multihop relayed communication systems in which the source terminal communicates with the destination terminal through a number of relay terminals [24]. In that case, μi is signal-to-interference ratio, Ωdi and Ωci are the average signal desired and interference powers at ith terminal input, respectively, and ρ is the correlation between two successive terminals (in the case of independent terminals, which is the real scenario in practice, correlation coefficient tends to be zero). Figure 2 illustrates the PDF of minimum of two, three, and four ratios of random variables.Figure 2
Probability density function of minimum of ratios of Weibull random variables forρ=0.2, Ωdi=0.8, Ωci=0.2, and βi=2.5, i=1,…,L.The main problem in the infinite-series expressions can be their convergence. Expressions presented in the paper converge rapidly, and thus they can be efficiently used. As an illustrative example, the number of terms needed to be summed in expressions for PDF of minimum of ratios of Weibull random variables is shown in Table1. It is evident that the number of terms depends strongly on the correlation coefficient.Table 1
Number of terms needed to be summed to achieve accuracy at the 4th digit of expressions for PDF of minimum of ratios of Weibull random variables forΩdi=0.8, Ωci=0.2, βi=2.5, i=1,…,L.
μ
=
0.5
μ
=
1
μ
=
2
L
=
2, ρ=0.2
7
5
5
L
=
2, ρ=0.6
17
15
13
L
=
3, ρ=0.2
8
6
5
L
=
3, ρ=0.6
18
16
16


## 4. Conclusion

The distributions of ratios of random variables,μi=Ri/ri, i=1,…,L, are of interest in many areas of science. In this paper, expressions for the joint PDF and PDF of maximum of two ratios of Rayleigh, Rician, and Nakagami-m distributed random variables have been presented. Motivated by the fact that Weibull distribution exhibits an excellent fit to experimental fading channel measurements, special attention is dedicated to the case of ratios of Weibull random variables. For this case, expressions for the joint PDF, product moments, PDF of maximum, and PDF of minimum of arbitrary number of ratios of Weibull distributed random variables have been obtained. An application of these results for the wireless communications community has also been described.



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*Source: 252804-2013-12-24.xml*