Moments of the quadrivariate Rayleigh distribution with applications for diversity receivers

Abstract

Wireless channels exhibit time, frequency, and spatial correlation. Models in literature that study four-branch diversity receivers make assumptions such as independence, constant correlation, exponential correlation, or some other kind between received signals at each antenna. However, these models are not accurate in many scenarios. Addressing this issue, we provide novel results for the moments, moment-generating function, probability density function, and cumulative distribution function of the quadrivariate Rayleigh distribution with an arbitrary correlation model. To the best of our knowledge, our model is the most comprehensive and the only one that can incorporate the 3GPP suggested spatial correlation structure. We use our new results to derive an analytical expression for the moments of the output signal-to-noise ratio of the four-branch equal gain combining receiver and the four-branch maximal ratio combining receiver. We provide original insight about their output signal-to-noise ratio distributions through their higher order moments in different scenarios. Our expressions are valid for all moments.

Appendices

Appendix A: Derivation of the PDF of the quadrivariate Rayleigh distribution

The joint PDF of zero-mean equal variance (ζ) Gaussian RVs $\boldsymbol {Z} = \{{z}_{{I}_{1}}, {z}_{{Q}_{1}}, \dots , {z}_{{I}_{4}}, {z}_{{Q}_{4}}\}$ is given by (1) where (⋅)^T denotes the transpose operator, $\overline { \boldsymbol {Z}} = [{z}_{{I}_{1}}, {z}_{{Q}_{1}}, \dots , {z}_{{I}_{4}}, {z}_{{Q}_{4}}]^{T}$ and Ψ is a positive definite covariance matrix [16] (7.18a). We define $\rho _{|i-j|}={E[{z}_{{I}_{i}}{z}_{{I}_{j}}]}/{\zeta }={E[{z}_{{Q}_{i}}{z}_{{Q}_{j}}]}/{\zeta }$ for $i,j \in \{1,\dots ,4\}$. Substituting ${z}_{{I}_{k}}=r_{k} \cos \limits (\theta _{k})$, ${z}_{{Q}_{k}}=r_{k}\sin \limits (\theta _{k})$ yields the joint PDF of R = {r₁, r₂, r₃, r₄} and Θ = {θ₁, θ₂, θ₃, θ₄}:

$$ \begin{array}{@{}rcl@{}} &&f_{\boldsymbol{R},\mathbf{\varTheta}}(r_{1},\theta_{1},\dots,r_{4},\theta_{4})\\ &&=\frac{r_{1}r_{2}r_{3}r_{4}}{(2\pi)^{4}(\det \boldsymbol{\Psi})^{\left( 1/2\right)}} e^{-\left( \tfrac{\phi_{1}}{2\zeta}\left( {r_{1}^{2}}+{r_{4}^{2}}\right)+\tfrac{\phi_{2}}{2\zeta}\left( {r_{2}^{2}}+{r_{3}^{2}}\right)\right)} \\ &&\times e^{-\left( \tfrac{\phi_{3}}{\zeta}r_{1}r_{2}\cos(\theta_{1}-\theta_{2})+\tfrac{\phi_{4}}{\zeta}r_{1}r_{3}\cos(\theta_{1}-\theta_{3})\right)}\\ &&\times e^{-\left( \tfrac{\phi_{5}}{\zeta}r_{1}r_{4}\cos(\theta_{1}-\theta_{4})+\tfrac{\phi_{6}}{\zeta}r_{2}r_{3}\cos(\theta_{2}-\theta_{3})\right)}\\ &&\times e^{-\left( \tfrac{\phi_{4}}{\zeta}r_{2}r_{4}\cos(\theta_{2}-\theta_{4})+\tfrac{\phi_{3}}{\zeta}r_{3}r_{4}\cos(\theta_{3}-\theta_{4})\right)} \end{array} $$

(40)

We substitute $e^{A\cos \limits (x)}=I_{0}(A)+2{\sum }_{a=1}^{\infty }I_{a}(A)\cos \limits (ax)$ as given in [17] (9.6.34) and integrate over phases, Θ. After algebraic manipulation we reach (4).

Appendix B: Derivation of the CDF of the quadrivariate Rayleigh distribution

We expand $I_{l}(x)=\left (\tfrac {x}{2}\right )^{l} {\sum }_{k=0}^{\infty }\tfrac {\left (x^{2}/4\right )^{k}}{k! {\Gamma }(l+k+1)}$ as given in [17] (9.6.10) where Γ(⋅) denotes the gamma function and integrate (4). We start with r₁:

$$ {\int}_{0}^{r_{1}^{\prime}}e^{-\tfrac{\phi_{1}{r_{1}^{2}}}{2\zeta}}{r}_{1}^{\nu_{1}+1} dr_{1}\\ $$

(41)

We make substitution $u=\tfrac {\phi _{1}{r_{1}^{2}}}{2\zeta }$.

$$ \begin{array}{@{}rcl@{}} &&=\frac{1}{2}\left( \frac{2\zeta}{\phi_{1}}\right)^{\tfrac{\nu_{1}+2}{2}} {\int}_{0}^{\tfrac{\phi_{1}\left( {r}_{1}^{\prime}\right)^{2}}{2\zeta}}e^{-u}u^{\tfrac{\nu_{1}}{2}} du \\ &&=\frac{1}{2}\left( \frac{2\zeta}{\phi_{1}}\right)^{\tfrac{\nu_{1}+2}{2}} \gamma\left( \tfrac{\nu_{1}+2}{2},\tfrac{\phi_{1}\left( {r}_{1}^{\prime}\right)^{2}}{2\zeta}\right) \end{array} $$

(42)

where ν₁ = |l| + |j| + |j + l| + 2b + 2h + 2f. We rely on the integral representation of lower incomplete gamma function as given in [17] (6.5.2) reaching the final expression in (42). We repeat the same integration process for r₂, r₃, r₄. After algebraic manipulation we reach (5).

Appendix C: Derivation of the MGF of the quadrivariate Rayleigh distribution

By definition,

$$ \begin{array}{@{}rcl@{}} &&M_{\boldsymbol R}(s_{1},s_{2},s_{3},s_{4})=E[e^{s_{1}r_{1}+s_{2}r_{2}+s_{3}r_{3}+s_{4}r_{4}}]\\ &&= {\int}_{0}^{\infty} {\int}_{0}^{\infty} {\int}_{0}^{\infty} {\int}_{0}^{\infty}e^{s_{1}r_{1}+s_{2}r_{2}+s_{3}r_{3}+s_{4}r_{4}} \\ &&\times f_{\boldsymbol{R}}(r_{1},r_{2},r_{3},r_{4})dr_{1}dr_{2}dr_{3}dr_{4} \end{array} $$

(43)

Using (4) and substituting modified Bessel function of the first kind with its infinite series representation, as given in [17] (9.6.10), we proceed:

$$ \begin{array}{@{}rcl@{}} &&{\int}_{0}^{\infty}e^{-\tfrac{\phi_{1}{r_{1}^{2}}}{2\zeta}+s_{1}r_{1}}{r}_{1}^{\nu+1} dr_{1} \\ &&=\left( \tfrac{\phi_{1}}{\zeta}\right)^{-\tfrac{\nu_{1}+2}{2}}{\Gamma}(\nu_{1}+2)e^{\left( \tfrac{{s_{1}^{2}}\zeta}{4\phi_{1}}\right)}D_{-(\nu_{1}+2)}\left( \tfrac{-s_{1}}{\sqrt{\phi_{1}/\zeta}}\right) \end{array} $$

(44)

where ν₁ = |l| + |j| + |j + l| + 2b + 2h + 2f and D_k(⋅) denotes the parabolic cylinder function [18] (9.240). We rely on the equation given in [18] (3.462.1) reaching the final expression in (44). We repeat the same integration process for r₂, r₃, r₄. After algebraic manipulation we reach (7).