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

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 ratios 𝜇₁=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 𝜇_𝑖=𝑅_𝑖/𝑟_𝑖, 𝑖=1,…,𝐿 are obtained. Random variables in numerator, 𝑅_𝑖, as well as random variables in denominator, 𝑟_𝑖, are exponentially correlated. To the best of the authors' knowledge, analytical expressions for the PDF of minimum and product moments of {𝜇_𝑖}^𝐿_𝑖=1 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.

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 variables 𝜇₁=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 𝑝_𝜇1𝜇2𝜇₁,𝜇₂=∬^∞₀|𝐽|𝑝_𝑅1𝑅2𝜇₁𝑟₁,𝜇₂𝑟₂𝑝_𝑟1𝑟2𝑟₁,𝑟₂𝑑𝑟₁𝑑𝑟₂, (1) where 𝑝_𝑅1𝑅2(𝑅₁,𝑅₂) and 𝑝_𝑟1𝑟2(𝑟₁,𝑟₂) are the joint PDFs of random variables in nominator and denominator, respectively, and |𝐽| is the Jacobian transformation given by |𝐽|=𝑑𝑅₁𝑑𝜇₁𝑑𝑅₁𝑑𝜇₂𝑑𝑅₂𝑑𝜇₁𝑑𝑅₂𝑑𝜇₂=𝑟₁𝑟₂. (2)

The PDF expression of maximum of ratios of random variables 𝜇_max=max{𝜇₁,𝜇₂} can be derived as 𝑝_𝜇max𝜇=^𝜇₀𝑝_𝜇1𝜇2𝜇,𝜇₂𝑑𝜇₂+^𝜇₀𝑝_𝜇1𝜇2𝜇₁,𝜇𝑑𝜇₁. (3)

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 𝑅_𝑖 and 𝑟_𝑖 (𝑖=1,2) are given by [1, Equation (6.2)] as follows: 𝑝_𝑅1𝑅2𝑅₁,𝑅₂=4𝑅₁𝑅₂Ω_𝑑1Ω_𝑑2−11−𝜌×exp𝑅1−𝜌Ω²¹_𝑑1+𝑅²₂Ω_𝑑2×𝐼₀2√𝜌𝑅₁𝑅₂1−𝜌Ω_𝑑1Ω_𝑑2,𝑝_𝑟1𝑟2𝑟₁,𝑟₂=4𝑟₁𝑟₂Ω_𝑐1Ω_𝑐2−11−𝜌×exp𝑟1−𝜌Ω²¹_𝑐1+𝑟²₂Ω_𝑐2×𝐼₀2√𝜌𝑟₁𝑟₂1−𝜌Ω_𝑐1Ω_𝑐2, (4) where 𝐼₀(⋅) is the modified Bessel function of the first kind and zero order, 𝜌 is the correlation coefficient, and Ω_𝑑𝑖 and Ω_𝑐𝑖 are the mean-square values of 𝑅_𝑖 and 𝑟_𝑖, 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 𝑝_𝜇1𝜇2𝜇₁,𝜇₂=41−𝜌^2∞_𝑘,ℓ=0𝜌^𝑘+ℓ𝜇₁𝜇₂^2𝑘+1Γ²(2+𝑘+ℓ)Ω_𝑑1Ω_𝑑2^1+𝑘Ω_𝑐1Ω_𝑐2^1+ℓ(𝑘!ℓ!)²×𝜇²₁Ω_𝑑1+1Ω_𝑐1𝜇²₂Ω_𝑑2+1Ω_𝑐2^{−(2+𝑘+ℓ)}. (5)

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 𝑝_𝜇max𝜇=1−𝜌^2∞_𝑘,ℓ=0𝜌^𝑘+ℓΓ²(2+𝑘+ℓ)𝜇^4𝑘+3(𝑘!ℓ!)²()Ω1+𝑘^𝑐1Ω_𝑐2Ω_𝑑1Ω_𝑑2^1+𝑘×21+Ω_𝑐1/Ω_𝑑1𝜇^22+𝑘+ℓ×₂𝐹₁Ω2+𝑘+ℓ,1+𝑘;2+𝑘;−^𝑐2Ω_𝑑2𝜇²+21+Ω_𝑐2/Ω_𝑑2𝜇^22+𝑘+ℓ×₂𝐹₁Ω2+𝑘+ℓ,1+𝑘;2+𝑘;−^𝑐1Ω_𝑑1𝜇², (6) where ₂𝐹₁(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, 𝑅_𝑖 and 𝑟_𝑖 (𝑖=1,2) are distributed according to [15, Equation (1)] as follows: 𝑝_𝑅1𝑅2𝑅₁,𝑅₂=4𝑅₁𝑅₂(𝐾+1)²Ω²_𝑑1−𝜌²−𝑅×exp²¹+𝑅²₂(𝐾+1)+2𝐾Ω_𝑑1−𝜌Ω_𝑑1−𝜌²×^∞_𝑘=0𝜀_𝑘𝐼_𝑘2𝑅₁𝑅₂𝜌(𝐾+1)Ω_𝑑1−𝜌²×𝐼_𝑘2𝑅₁1+𝜌𝐾(𝐾+1)Ω_𝑑𝐼_𝑘2𝑅₂1+𝜌𝐾(𝐾+1)Ω_𝑑,𝑝_𝑟1𝑟2𝑟₁,𝑟₂=4𝑟₁𝑟₂(𝐾+1)²Ω²_𝑐1−𝜌²−𝑟×exp²¹+𝑟²₂(𝐾+1)+2𝐾Ω_𝑐1−𝜌Ω_𝑐1−𝜌²×^∞_ℓ=0𝜀_ℓ𝐼_ℓ2𝑟₁𝑟₂𝜌(𝐾+1)Ω_𝑐1−𝜌²×𝐼_ℓ2𝑟₁1+𝜌𝐾(𝐾+1)Ω_𝑐𝐼_ℓ2𝑟₂1+𝜌𝐾(𝐾+1)Ω_𝑐, (7) where 𝐼_𝑛(⋅) is the modified Bessel function of the first kind and 𝑛th order, 𝜀₀=1, 𝜀_𝑘=2, 𝑘≥1, Ω_𝑑=𝑅²_𝑖, Ω_𝑐=𝑟²_𝑖, 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: 𝑝_𝜇1𝜇2𝜇₁,𝜇₂=exp−4𝐾/1+𝜌(𝐾+1)²×^∞_{𝑘,ℓ,𝑚,𝑛,𝑝,𝑞,𝑠,𝑤=0}𝜇₁^{2(𝑚+𝑛+𝑘)}Ω_𝑑^{2(𝑞+ℓ+1)+𝑠+𝑤}Ω_𝑐^{2(𝑚+𝑘+1)+𝑛+𝑝}1+𝜌^{2(𝑛+𝑝+𝑘+𝑠+𝑤+ℓ)}×𝜌𝑚!𝑛!𝑝!𝑞!𝑠!𝑤!_{2(𝑚+𝑞)+𝑘+ℓ}𝜇₂^{2(𝑚+𝑝+𝑘)}1−𝜌^{2𝑛+𝑝+𝑘+𝑠+𝑤+ℓ+2}×𝜀_𝑘𝜀_ℓ𝐾^{𝑛+𝑝+𝑘+𝑠+𝑤+ℓ}𝐸×Ω𝑘,ℓ,𝑚,𝑛,𝑝,𝑞,𝑠,𝑤_𝑑+Ω_𝑐𝜇²₁^{𝑚+𝑛+𝑘+𝑞+𝑠+ℓ+2}×Ω_𝑑+Ω_𝑐𝜇²₂^{𝑚+𝑝+𝑘+𝑞+𝑤+ℓ+2−1}, (8) where 𝐸(𝑘,ℓ,𝑚,𝑛,𝑝,𝑞,𝑠,𝑤)≡𝐸 is given by ×Γ()Γ()ΓΓ()Γ()𝐸=Γ𝑚+𝑛+𝑘+𝑞+𝑠+ℓ+2×Γ𝑚+𝑝+𝑘+𝑞+𝑤+ℓ+2𝑚+𝑘+1𝑛+𝑘+1𝑝+𝑘+1×Γ𝑞+ℓ+1𝑠+ℓ+1𝑤+ℓ+1⁻¹, (9) while the PDF of 𝜇_max can be written as in [17, Eq. (11)] as 𝑝_𝜇max𝜇=exp−4𝐾×1+𝜌_{∞𝑘,ℓ,𝑚,𝑛,𝑝,𝑞,𝑠,𝑤=0}𝜀_𝑘𝜀_𝑙𝐾^{𝑛+𝑝+𝑘+𝑠+𝑤+ℓ}𝜌^{2(𝑚+𝑞)+𝑘+ℓ}×1−𝜌^{2𝑛+𝑝+𝑘+𝑠+𝑤+ℓ+2}×1+𝜌^{2(𝑛+𝑝+𝑘+𝑠+𝑤+ℓ)}𝑚!𝑛!𝑝!𝑞!𝑠!𝑤!⁻¹×𝜇^{4(𝑚+𝑘)+2(𝑝+𝑛+1)}𝑆^𝑞+ℓ+1𝑆+𝜇^{22(𝑘+𝑚)+𝑛+𝑝+ℓ+𝑞+3}𝐸×𝑆𝑘,ℓ,𝑚,𝑛,𝑝,𝑞,𝑠,𝑤𝑆+𝜇^2𝑤2𝐹₁𝜇𝑚+𝑛+𝑘+1,−ℓ−𝑞−𝑠;𝑚+𝑛+𝑘+2;²𝜇²×()+𝑆𝑚+𝑛+𝑘+1⁻¹+𝑆𝑆+𝜇^2𝑠2𝐹₁𝜇𝑚+𝑝+𝑘+1,−ℓ−𝑞−𝑤;𝑚+𝑝+𝑘+2;²𝜇²×+𝑆𝑚+𝑝+𝑘+1⁻¹, (10) where 𝑆=Ω_𝑑/Ω_𝑐 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, 𝑅_𝑖 and 𝑟_𝑖 (𝑖=1,2) follow the distributions 𝑝_𝑅1𝑅2𝑅₁,𝑅₂=4𝑚^𝑚1₁⁺¹𝑅₁𝑅₂^𝑚1Γ𝑚₁Ω_𝑑1Ω_𝑑21−𝜌𝜌Ω_𝑑1Ω_𝑑2^𝑚1−1−𝑚×exp¹𝑅²₁+𝑅²₂Ω_𝑑1Ω_𝑑21−𝜌×𝐼_𝑚1−12√𝜌𝑚₁𝑅₁𝑅₂Ω_𝑑1Ω_𝑑2,𝑝1−𝜌_𝑟1𝑟2𝑟₁,𝑟₂=4𝑚^𝑚2₂⁺¹𝑟₁𝑟₂^𝑚2Γ𝑚₂Ω_𝑐1Ω_𝑐21−𝜌𝜌Ω_𝑐1Ω_𝑐2^𝑚2−1−𝑚×exp²𝑟²₁+𝑟²₂Ω_𝑐1Ω_𝑐21−𝜌×𝐼_𝑚2−12√𝜌𝑚₂𝑟₁𝑟₂Ω_𝑐1Ω_𝑐2.1−𝜌 (11)

Setting 𝑚𝑊=¹𝜇²₁Ω_𝑑1Ω_𝑑2+𝑚₂Ω_𝑐1Ω_𝑐2×𝑚₁𝜇²₂Ω_𝑑1Ω_𝑑2+𝑚₂Ω_𝑐1Ω_𝑐2, (12) the joint PDF of ratios of the Nakagami-𝑚 distributed random variables can be obtained as 𝑝_𝜇1𝜇2𝜇₁,𝜇₂=41−𝜌^𝑚1+𝑚2Γ𝑚₁Γ𝑚₂×^∞_𝑘,𝑙=0𝜌^𝑘+𝑙Γ²𝑘+𝑙+𝑚₁+𝑚₂×𝜇₁𝜇₂^{2(𝑘+𝑚1)−1}𝑚^2(𝑚1₁^+𝑘)𝑚^2(𝑚2₁^+𝑙)×𝑘!𝑙!Γ𝑘+𝑚₁Γ𝑙+𝑚₂Ω_𝑐1Ω_𝑐2^𝑙+𝑚2×Ω_𝑑1Ω_𝑑2^𝑘+𝑚1𝑊^{𝑘+𝑙+𝑚1+𝑚2−1}, (13) while the PDF of maximum of ratios of Nakagami-𝑚 distributed random variables is 𝑝_𝜇max𝜇=41−𝜌^𝑚1+𝑚2Γ𝑚₁Γ𝑚₂×^∞_𝑘,𝑙=0𝜌^𝑘+𝑙Γ²𝑘+𝑙+𝑚₁+𝑚₂×𝜇^{4(𝑘+𝑚1)−1}Ω_𝑐1Ω_𝑐2^{(𝑚1+𝑘−𝑚2−𝑙)/2}×𝑚^2(𝑚1₁^+𝑘)𝑚^{(𝑚2+𝑙−𝑚1}₂^−𝑘)×𝑘!𝑙!Γ𝑘+𝑚₁Γ𝑙+𝑚₂×𝑘+𝑚₁Ω_𝑑1Ω_𝑑2^(𝑚1+𝑘)×𝜇²𝑚₁Ω_𝑑1Ω_𝑑2+𝑚₂Ω_𝑐1Ω_𝑐2^{𝑘+𝑙+𝑚1+𝑚2−1}×₂𝐹₁𝑘+𝑙+𝑚₁+𝑚₂,𝑚₁+𝑘;𝑘+𝑚₁+1;−𝜇²𝑚₁Ω_𝑐1Ω_𝑐2/Ω_𝑑1Ω_𝑑2𝑚₂. (14)

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, 𝜇_𝑖 denotes signal-to-interference ratio, Ω_𝑑𝑖 and Ω_𝑐𝑖 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 𝜇_max 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 𝑅_𝑖 and 𝑟_𝑖 (𝑖=1,2) are given by [23, Equation (11)] as follows: 𝑝_𝑅1𝑅2𝑅₁,𝑅₂=𝛽₁𝛽₂𝑅^𝛽1₁⁻¹𝑅^𝛽2₂⁻¹Ω_𝑑1Ω_𝑑2−11−𝜌×exp𝑅1−𝜌Ω^𝛽11_𝑑1+𝑅^𝛽2₂Ω_𝑑2×𝐼₀2√𝜌𝑅^𝛽1₁^/2𝑅^𝛽2₂^/21−𝜌Ω_𝑑1Ω_𝑑2,𝑝_𝑟1𝑟2𝑟₁,𝑟₂=𝛽₁𝛽₂𝑟^𝛽1₁⁻¹𝑟^𝛽2₂⁻¹Ω_𝑐1Ω_𝑐2−11−𝜌×exp𝑟1−𝜌Ω^𝛽11_𝑐1+𝑟^𝛽2₂Ω_𝑐2×𝐼₀2√𝜌𝑟^𝛽1₁^/2𝑟^𝛽2₂^/21−𝜌Ω_𝑐1Ω_𝑐2, (15) while the joint multivariate PDFs of Weibull distributed random variables 𝑅_𝑖 and 𝑟_𝑖, 𝑖=1,…,𝐿, with exponential correlation can be expressed as [23, Equation (22)] as follows: 𝑝_{𝑅1⋯𝑅𝐿}𝑅₁,…,𝑅_𝐿=11−𝜌^𝐿𝐿−1_𝑖=1𝛽_𝑖𝑅^𝛽𝑖_𝑖⁻¹Ω_𝑑𝑖−1×exp𝑅1−𝜌Ω^𝛽11_𝑑1+𝑅^𝛽𝐿_𝐿Ω_𝑑𝐿+1+𝜌^𝐿−1_𝑖=2𝑅^𝛽𝑖_𝑖Ω_𝑑𝑖×^𝐿−1_𝑖=1𝐼₀2√𝜌𝑅^𝛽𝑖_𝑖^/2𝑅^𝛽𝑖+1/2_𝑖+11−𝜌^𝐿Ω_𝑑1⋯Ω_𝑑𝐿,𝑝_{𝑟1⋯𝑟𝐿}𝑟₁,…,𝑟_𝐿=11−𝜌^𝐿𝐿−1_𝑖=1𝛽_𝑖𝑟^𝛽𝑖_𝑖⁻¹Ω_𝑐𝑖−1×exp𝑟1−𝜌Ω^𝛽11_𝑐1+𝑟^𝛽𝐿_𝐿Ω_𝑐𝐿+1+𝜌^𝐿−1_𝑖=2𝑟^𝛽𝑖_𝑖Ω_𝑐𝑖×^𝐿−1_𝑖=1𝐼₀2√𝜌𝑟^𝛽𝑖_𝑖^/2𝑟^𝛽𝑖+1/2_𝑖+11−𝜌^𝐿Ω_𝑐1⋯Ω_𝑐𝐿, (16) where 𝛽_𝑖 is the Weibull parameter. Weibull parameter expresses the fading severity (𝛽_𝑖>0). As it increases, the severity of fading decreases, while for 𝛽_𝑖=2, the Weibull distribution reduces to the Rayleigh distribution. The joint PDF of ratios of Weibull random variables, 𝜇_𝑖=𝑅_𝑖/𝑟_𝑖, 𝑖=1,…,𝐿, can be obtained as 𝑝_{𝜇1⋯𝜇𝐿}𝜇₁,…,𝜇_𝐿=^∞₀⋯^∞_0𝐿-fold|𝐽|𝑝_{𝑅1⋯𝑅𝐿}𝜇₁𝑟₁,…,𝜇_𝐿𝑟_𝐿×𝑝_{𝑟1⋯𝑟𝐿}𝑟₁,…,𝑟_𝐿𝑑𝑟₁⋯𝑑𝑟_𝐿, (17) where |𝐽| is the Jacobian transformation given by |𝐽|=𝑑𝑅₁𝑑𝜇₁…𝑑𝑅₁𝑑𝜇_𝐿⋮𝑑𝑅_𝐿𝑑𝜇₁𝑑𝑅_𝐿𝑑𝜇_𝐿=^𝐿_𝑖=1𝑟_𝑖. (18)

Substituting the adequate joint PDFs of 𝑅_𝑖 and 𝑟_𝑖 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)]: 𝑝_𝜇1𝜇2𝜇₁,𝜇₂=1−𝜌²𝛽₁𝛽₂×^∞_𝑘,ℓ=0𝜌^𝑘+ℓ𝜇^{(1+𝑘)𝛽1}₁⁻¹𝜇^{(1+𝑘)𝛽2}₂⁻¹Γ²(2+𝑘+ℓ)Ω_𝑑1Ω_𝑑2^1+𝑘Ω_𝑐1Ω_𝑐2^1+ℓ(𝑘!ℓ!)²×𝜇^𝛽1₁Ω_𝑑1+1Ω_𝑐1𝜇^𝛽2₂Ω_𝑑2+1Ω_𝑐2^{−(2+𝑘+ℓ)}, (19) while the joint multivariate PDF of {𝜇_𝑖}^𝐿_𝑖=1 can be written as in [16, Equation (4)] as 𝑝_{𝜇1⋯𝜇𝐿}𝜇₁,…,𝜇_𝐿=1−𝜌²×^𝐿_𝑖=1𝛽_𝑖^∞_{𝑘1,…,𝑘𝐿−1,ℓ1,…,ℓ𝐿−1=0}𝜌^{∑𝐿−1𝑖=1(𝑘𝑖+ℓ𝑖)}1+𝜌^{∑𝐿−1𝑖=2(2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1)}×𝜇^{(1+𝑘1)𝛽1}₁⁻¹𝜇^{(1+𝑘𝐿−1)𝛽𝐿}_𝐿⁻¹Γ2+𝑘₁+ℓ₁×Γ2+𝑘_𝐿−1+ℓ_𝐿−1×^𝐿_𝑖=1Ω^{1+(2/𝐿)∑𝐿−1𝑗=1𝑘𝑗}_𝑑𝑖Ω^{1+(2/𝐿)∑𝐿−1𝑗=1ℓ𝑗}_𝑐𝑖×𝜇^𝛽1₁Ω_𝑑1+1Ω_𝑐1^2+𝑘1+ℓ1×𝜇^𝛽𝐿_𝐿Ω_𝑑𝐿+1Ω_𝑐𝐿^{2+𝑘𝐿−1+ℓ𝐿−1𝐿−1}_𝑖=1𝑘_𝑖!ℓ_𝑖!²⁻¹×^𝐿−1_𝑖=2𝜇^{𝛽𝑖(1+𝑘𝑖+𝑘𝑖−1}_𝑖⁾⁻¹𝜇^𝛽𝑖_𝑖Ω_𝑑𝑖+1Ω_𝑐𝑖^{−(2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1)}×Γ2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1. (20)

(b) Product Moments. Product moments are also important in wireless communications systems because they are a useful statistical tool to characterize a distribution. The (∑^𝐿_𝑖=1𝑛_𝑖)th-order moment of the product of {𝜇_𝑖}^𝐿_𝑖=1 can be derived as 𝜀^𝐿_𝑖=1𝜇^𝑛𝑖_𝑖=^∞₀⋯^∞_0𝐿-fold^𝐿_𝑖=1𝜇^𝑛𝑖_𝑖𝑝_{𝜇1⋯𝜇𝐿}𝜇₁,…,𝜇_𝐿𝑑𝜇₁⋯𝑑𝜇_𝐿. (21)

Substituting the adequate joint PDF of ratios of Weibull distributed random variables in (21) and using [18, Equation (3.241/4)], for 𝑛_𝑖/𝛽_𝑖<1, 𝑖=1,…,𝐿, product moments of two and multiple ratios of Weibull random variables are 𝜀𝜇^𝑛1₁𝜇^𝑛2₂=𝑈^∞_𝑘,ℓ=0𝜌^𝑘+ℓ(𝑘!ℓ!)²Γ𝑛1+𝑘+¹𝛽₁Γ𝑛1+𝑘+²𝛽₂𝑛×Γ1+ℓ−¹𝛽₁Γ𝑛1+ℓ−²𝛽₂, (22) where 𝑈=(1−𝜌)²(Ω_𝑑1/Ω_𝑐1)^𝑛1/𝛽1(Ω_𝑑2/Ω_𝑐2)^𝑛2/𝛽2 and 𝜀^𝐿_𝑖=1𝜇^𝑛𝑖_𝑖=1−𝜌²×^∞_{𝑘1,…,𝑘𝐿−1,ℓ1,…,ℓ𝐿−1=0}𝜌^{∑𝐿−1𝑖=1(𝑘𝑖+ℓ𝑖)}Ω^{𝑘1−(2/𝐿)∑𝐿−1𝑖=1𝑘𝑖+(𝑛1/𝛽1)}_𝑑1×Ω^{ℓ1−(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖−(𝑛1/𝛽1)}_𝑐1×1−𝜌^{∑𝐿−1𝑖=2(2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1)−1}×Ω^{𝑘𝐿−1−(2/𝐿)∑𝐿−1𝑖=1𝑘𝑖+(𝑛𝐿/𝛽𝐿)}_𝑑𝐿Ω^{ℓ𝐿−1−(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖−(𝑛𝐿/𝛽𝐿)}_𝑐𝐿∏^𝐿−1_𝑖=1𝑘_𝑖!ℓ_𝑖!²×Γ1+𝑘₁+𝑛₁𝛽₁Γ1+𝑘_𝐿−1+𝑛_𝐿𝛽_𝐿×Γ1+ℓ₁−𝑛₁𝛽₁Γ1+ℓ_𝐿−1−𝑛_𝐿𝛽_𝐿×^𝐿−1_𝑖=2Ω^{𝑘𝑖+𝑘𝑖−1−(2/𝐿)∑𝐿−1𝑗=1𝑘𝑗+(𝑛𝑖/𝛽𝑖)}_𝑑𝑖Ω^{ℓ𝑖+ℓ𝑖−1−(2/𝐿)∑𝐿−1𝑗=1ℓ𝑗−(𝑛𝑖/𝛽𝑖)}_𝑐𝑖×Γ1+𝑘_𝑖+𝑘_𝑖−1+𝑛_𝑖𝛽_𝑖×Γ1+ℓ_𝑖+ℓ_𝑖−1−𝑛_𝑖𝛽_𝑖, (23) respectively.

(c) PDF of Maximum. The PDF expression of maximum of ratios of random variables, 𝜇_max=max{𝜇₁,𝜇₂,…,𝜇_𝐿}, can be derived as 𝑝_𝜇max𝜇=^𝜇₀⋯^𝜇_{0()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇,𝜇₂,…,𝜇_𝐿𝑑𝜇₂𝑑𝜇₃⋯𝑑𝜇_𝐿+^𝜇₀⋯^𝜇_{0()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇₁,𝜇,…,𝜇_𝐿𝑑𝜇₁𝑑𝜇₃⋯𝑑𝜇_𝐿⋮+^𝜇₀⋯^𝜇_{0()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇₁,𝜇₂,…,𝜇𝑑𝜇₁𝑑𝜇₂⋯𝑑𝜇_𝐿−1. (24)

Substituting the joint PDF of {𝜇_𝑖}^𝐿_𝑖=1 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: 𝑝_𝜇max𝜇=1−𝜌^2∞_𝑘,ℓ=0𝜌^𝑘+ℓΓ²(2+𝑘+ℓ)𝜇^{(1+𝑘)(𝛽1+𝛽2)−1}(1+𝑘)(𝑘!ℓ!)²Ω_𝑐1Ω_𝑐2Ω_𝑑1Ω_𝑑2^1+𝑘×𝛽₁1+Ω_𝑐1/Ω_𝑑1𝜇^{𝛽12+𝑘+ℓ}×₂𝐹₁Ω2+𝑘+ℓ,1+𝑘,2+𝑘;−^𝑐2Ω_𝑑2𝜇^𝛽2+𝛽₂1+Ω_𝑐2/Ω_𝑑2𝜇^{𝛽22+𝑘+ℓ}×₂𝐹₁Ω2+𝑘+ℓ,1+𝑘,2+𝑘;−^𝑐1Ω_𝑑1𝜇^𝛽1,𝑝_𝜇max𝜇=1−𝜌²×^∞_{𝑘1,𝑘2,…,𝑘𝐿−1,ℓ1,ℓ2,…,ℓ𝐿−1=0}Ω^{1+𝑘1+ℓ1−(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖}_𝑐1Ω^{1+𝑘𝐿−1+ℓ𝐿−1−(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖}_𝑐𝐿1+𝜌^{∑𝐿−1𝑖=2(2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1)}×Γ2+𝑘₁+ℓ₁Γ2+𝑘_𝐿−1+ℓ_𝐿−1∏^𝐿_𝑖=1Ω_𝑑𝑖^{1+(2/𝐿)∑𝐿−1𝑖=1𝑘𝑖}∏^𝐿−1_𝑖=0𝑘_𝑖!ℓ_𝑖!²×𝜌^{∑𝐿−1𝑖=1(𝑘𝑖+ℓ𝑖)}𝜇^{(1+𝑘1)𝛽1+(1+𝑘𝐿−1)𝛽𝐿+∑𝐿−1𝑖=2(1+𝑘𝑖+𝑘𝑖−1)𝛽𝑖−1}×^𝐿−1_𝑖=2Ω^{1+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1−(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖}_𝑐𝑖×Γ2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1𝑍𝜇, (25)(26) respectively, where 𝑍𝜇=𝛽₁1+𝑘_𝐿−11+Ω_𝑐1/Ω_𝑑1𝜇^{𝛽12+𝑘1+ℓ1}×₂𝐹₁2+𝑘_𝐿−1+ℓ_𝐿−1,1+𝑘_𝐿−1;2+𝑘_𝐿−1Ω;−^𝑐𝐿Ω_𝑑𝐿𝜇^𝛽𝐿+𝛽_𝐿1+𝑘₁1+Ω_𝑐𝐿/Ω_𝑑𝐿𝜇^{𝛽𝐿2+𝑘𝐿−1+ℓ𝐿−1}×₂𝐹₁2+𝑘₁+ℓ₁,1+𝑘₁;2+𝑘₁Ω;−^𝑐1Ω_𝑑1𝜇^𝛽1×^𝐿−1_𝑖=22𝐹₁2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1,1+𝑘_𝑖+𝑘_𝑖−1;2+𝑘_𝑖+𝑘_𝑖−1Ω;−^𝑐𝑖Ω_𝑑𝑖𝜇^𝛽𝑖×1+𝑘_𝑖+𝑘_𝑖−1⁻¹+₂𝐹₁2+𝑘₁+ℓ₁,1+𝑘₁;2+𝑘₁Ω;−^𝑐1Ω_𝑑1𝜇^𝛽1×₂𝐹₁2+𝑘_𝐿−1+ℓ_𝐿−1,1+𝑘_𝐿−1;2+𝑘_𝐿−1;−Ω_𝑐𝐿Ω_𝑑𝐿𝜇^𝛽𝐿×1+𝑘₁1+𝑘_𝐿−1⁻¹×^𝐿−1_𝑗=2𝛽_𝑗1+Ω_𝑐𝑗/Ω_𝑑𝑗𝜇^{𝛽𝑗2+𝑘𝑗+𝑘𝑗−1+ℓ𝑗+ℓ𝑗−1}×^𝐿−1_{𝑖=2;𝑖≠𝑗2}𝐹₁2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1,1+𝑘_𝑖+𝑘_𝑖−1;2+𝑘_𝑖+𝑘_𝑖−1Ω;−^𝑐𝑖Ω_𝑑𝑖𝜇^𝛽𝑖×1+𝑘_𝑖+𝑘_𝑖−1⁻¹. (27)

Figure 1 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 for 𝜌=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, 𝜇_min=min{𝜇₁,𝜇₂,…,𝜇_𝐿}, can be obtained by 𝑝_𝜇min𝜇=^∞_𝜇⋯^∞_{𝜇()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇,𝜇₂,…,𝜇_𝐿𝑑𝜇₂𝑑𝜇₃⋯𝑑𝜇_𝐿+^∞_𝜇⋯^∞_{𝜇()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇₁,𝜇,…,𝜇_𝐿𝑑𝜇₁𝑑𝜇₃⋯𝜇_𝐿⋮+^∞_𝜇⋯^∞_{𝜇()𝐿−1-fold}𝑝_{𝜇1𝜇2⋯𝜇𝐿}𝜇₁,𝜇₂,…,𝜇𝑑𝜇₁𝑑𝜇₂⋯𝑑𝜇_𝐿−1. (28)

Substituting the adequate joint PDF in (28) and using [18, Equation (3.194/2)], with (𝜇^𝛽𝑖/Ω_𝑑𝑖)+(1/Ω_𝑐𝑖)=Φ_𝑖, Ω_𝑑𝑖/Ω_𝑐𝑖𝜇^𝛽𝑖=𝜑_𝑖, 𝑖=1,2,…,𝐿, the PDF of minimum of two ratios of random variables becomes 𝑝_𝜇min𝜇=1−𝜌²×^∞_𝑘,ℓ=0𝜌^𝑘+ℓΓ²(2+𝑘+ℓ)Ω_𝑑1Ω_𝑑2^1+𝑘Ω_𝑐1Ω_𝑐2^1+ℓ(𝑘!ℓ!)²()×𝛽1+ℓ₁Ω^2+𝑘+ℓ_𝑑2𝜇^{(1+𝑘)𝛽1−(1+ℓ)𝛽2−1}Φ₁^2+𝑘+ℓ×₂𝐹₁2+𝑘+ℓ,1+ℓ;2+ℓ;−𝜑₂+𝛽₂Ω^2+𝑘+ℓ_𝑑1𝜇^{(1+𝑘)𝛽2−(1+ℓ)𝛽1−1}Φ₂^2+𝑘+ℓ×₂𝐹₁2+𝑘+ℓ,1+ℓ;2+ℓ;−𝜑₁, (29) and finally, the PDF of minimum of multiple ratios of random variables can be expressed in a more complicated form as follows: 𝑝_𝜇min𝜇=1−𝜌²×^∞_{𝑘1,𝑘2,…,𝑘𝐿−1,ℓ1,ℓ2,…,ℓ𝐿−1=0}𝜌^{∑𝐿−1𝑖=1(𝑘𝑖+ℓ𝑖)}1+𝜌^{−∑𝐿−1𝑖=2(2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1)}×^𝐿_𝑖=1Ω_𝑑𝑖^{1+(2/𝐿)∑𝐿−1𝑖=1𝑘𝑖}×^𝐿_𝑖=1Ω_𝑐𝑖^{1+(2/𝐿)∑𝐿−1𝑖=1ℓ𝑖−1}×Γ2+𝑘₁+ℓ₁Γ2+𝑘_𝐿−1+ℓ_𝐿−1×^𝐿−1_𝑖=2Γ2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1×^𝐿−1_𝑖=1𝑘_𝑖!ℓ_𝑖!²⁻¹×^𝐿−1_𝑖=2Ω^{2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1}_𝑑𝑖1+ℓ_𝑖+ℓ²_𝑖−1𝐹₁2+𝑘_𝑖+𝑘_𝑖−1+ℓ_𝑖+ℓ_𝑖−1,1+ℓ_𝑖+ℓ_𝑖−1;2+ℓ_𝑖+ℓ_𝑖−1;−𝜑_𝑖×𝛽₁Ω^{2+𝑘𝐿−1+ℓ𝐿−1}_𝑑𝐿𝜇^{(1+𝑘1)𝛽1−∑𝐿−1𝑖=2(1+ℓ𝑖+ℓ𝑖−1)𝛽𝑖−(1+ℓ𝐿−1)𝛽𝐿−1}1+ℓ_𝐿−1Φ^2+𝑘1+ℓ1₁×₂𝐹₁2+𝑘_𝐿−1+ℓ_𝐿−1,1+ℓ_𝐿−1;2+ℓ_𝐿−1;−𝜑_𝐿+𝛽_𝐿Ω^2+𝑘1+ℓ1_𝑑1𝜇^{(1+𝑘𝐿−1)𝛽𝐿−∑𝐿−1𝑖=2(1+ℓ𝑖+ℓ𝑖−1)𝛽𝑖−(1+ℓ1)𝛽1−1}1+ℓ₁Φ^{2+𝑘𝐿−1+ℓ𝐿−1}_𝐿×₂𝐹₁2+𝑘₁+ℓ₁,1+ℓ₁;2+ℓ₁;−𝜑₁+₂𝐹₁2+𝑘₁+ℓ₁,1+ℓ₁;2+ℓ₁;−𝜑₁×₂𝐹₁2+𝑘_𝐿−1+ℓ_𝐿−1;1+ℓ_𝐿−1;2+ℓ_𝐿−1;−𝜑_𝐿×Ω^2+𝑘1+ℓ1_𝑑1Ω^{2+𝑘𝐿−1+ℓ𝐿−1}_𝑑𝐿1+ℓ₁1+ℓ_𝐿−1×∑^𝐿−1_𝑖=2𝛽_𝑖Φ^{2+𝑘𝑖+𝑘𝑖−1+ℓ𝑖+ℓ𝑖−1}_𝑖×^𝐿−1_{𝑗=2,𝑗≠𝑖}Ω^{2+𝑘𝑗+𝑘𝑗−1+ℓ𝑗+ℓ𝑗−1}_𝑑𝑗1+ℓ_𝑗+ℓ_𝑗−1×₂𝐹₁2+𝑘_𝑗+𝑘_𝑗−1+ℓ_𝑗+ℓ_𝑗−1,1+ℓ_𝑗+ℓ_𝑗−1;2+ℓ_𝑗+ℓ_𝑗−1;−𝜑_𝑗×𝜇^{(1+𝑘𝑖+𝑘𝑖−1)𝛽𝑖−(1+ℓ1)𝛽1−(1+ℓ𝐿−1)𝛽𝐿−∑𝐿−1𝑗=2,𝑗≠𝑖(1+ℓ𝑗+ℓ𝑗−1)𝛽𝑗−1}. (30)

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, 𝜇_𝑖 is signal-to-interference ratio, Ω_𝑑𝑖 and Ω_𝑐𝑖 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 for 𝜌=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 Table 1. 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 for Ω_𝑑𝑖=0.8, Ω_𝑐𝑖=0.2, 𝛽_𝑖=2.5, 𝑖=1,…,𝐿.

4. Conclusion

The distributions of ratios of random variables, 𝜇_𝑖=𝑅_𝑖/𝑟_𝑖, 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.

Acknowledgments

The authors were supported in part by the Serbian Ministry of Education, Science and Technological Development. The authors are also indebted to the referees for the careful reading of the paper and their suggestions which have improved the paper.