A simpler derivation for an integral useful in wireless communication theory

Abstract

The recent paper by Beaulieu [A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates, IEEE Trans. Comm. 54 (May 2006) 802–805] derived a closed form expression for an integral that represents the average over Rayleigh fading of the product of two Gaussian Q functions. In this note, a simpler derivation of the result of Beaulieu [A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates, IEEE Trans. Comm. 54 (May 2006) 802–805] is provided.

Introduction

An integral that occurs frequently in wireless communication theory (see [1], [2], [3], [4]) is that given by

$$I={\int }_0^{\infty }f(r)Q({\mathit{rA}}_1)Q({\mathit{rA}}_2)\mathrm{d}r\text{,}$$

where

$$f(r)=\frac{r}{{\sigma }^2}\exp \left(-\frac{r^2}{2{\sigma }^2}\right)$$

for $r⩾0$ and

$$Q(x)=\frac{1}{\sqrt{2\pi }}{\int }_x^{\infty }\exp \left(-\frac{t^2}{2}\right)\mathrm{d}t\text{.}$$

Actually, (1) represents the probability density function of a random Rayleigh fading amplitude, where $A_1$, $A_2$ and $\sigma$ are some constants. Until recently, a closed form expression for (1) has not been known. Beaulieu [5] derived the following elementary expression for (1):

$$I=\frac{1}{4}-\frac{1}{2\pi }\left[\sqrt{\frac{{\sigma }^2{A}_1^2}{1+{\sigma }^2{A}_1^2}}\arctan \left(\frac{A_1}{A_2}\sqrt{\frac{1+{\sigma }^2{A}_1^2}{{\sigma }^2{A}_1^2}}\right)\right.\left.+\sqrt{\frac{{\sigma }^2{A}_2^2}{1+{\sigma }^2{A}_2^2}}\arctan \left(\frac{A_2}{A_1}\sqrt{\frac{1+{\sigma }^2{A}_2^2}{{\sigma }^2{A}_2^2}}\right)\right]\text{.}$$

The proof of (4) in Beaulieu [5] was too long and did not use known results in the mathematics literature. Here, we provide a simpler proof.