A simple exponential integral representation of the generalized Marcum Q-function Q<inf>M</inf> (a, b) for real-order M with applications

Abstract

This article derives a new exponential-type integral representation for the generalized M-th order Marcum Q-function, Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> (alpha, beta), when M is not necessarily an integer. Our new representation includes a well-known result due to Helstrom for the special case of positive integer M and an additional integral correction term that vanishes when M is an integer. The new form has both computational utility (numerous existing computational algorithms for Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> (alpha, beta) are limited to integer M) and analytical utility (e.g., performance analysis of selection diversity in bivariate Nakagami-m fading with arbitrary fading severity index, computation of the complementary cumulative distribution function of a noncentral chi-square random variable for both odd and even orders, unified analysis of correlated binary and quaternary modulations over generalized fading channels, development of a Markovian threshold model for packet errors in correlated Nakagami-m fading channels and so on). Our alternative representation for Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> (alpha, beta) also leads into a new, exact exponential integral formula for the cumulative distribution function (CDF) of signal-to-noise ratio (SNR) at the output of a dual-diversity selection combiner in correlated Nakagami-m fading (including the non-integer fading index case), which several researchers in the past have concluded as unobtainable. Simple yet tight upper and lower bounds for Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> (alpha, beta) (for any arbitrary real order Mges0.5) are also derived.