On the monotonicity, log-concavity and tight bounds of the generalized Marcum and Nuttall Q-functions

TL;DR

This paper investigates the monotonicity and log-concavity of generalized Marcum and Nuttall Q-functions, deriving tighter bounds and demonstrating their practical applications in wireless communication system analysis.

Contribution

It introduces a probabilistic method to prove monotonicity, establishes log-concavity of these functions, and proposes significantly tighter bounds with practical relevance.

Findings

Proved monotonicity of the functions using a probabilistic approach.

Established log-concavity of the generalized Marcum and Nuttall Q-functions.

Derived bounds with less than 5% relative error in most cases.

Abstract

In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and Nuttall Q-functions. More precisely, a simple probabilistic method is firstly given to prove the monotonicity of these two functions. Then, the log-concavity of the generalized Marcum Q-function and its deformations is established with respect to each of the three parameters. Since the Nuttall Q-function has similar probabilistic interpretations as the generalized Marcum Q-function, we deduce the log-concavity of the Nuttall Q-function. By exploiting the log-concavity of these two functions, we propose new tight lower and upper bounds for the generalized Marcum and Nuttall Q-functions. Our proposed bounds are much tighter than the existing bounds in the literature in most of the cases. The relative errors of our proposed bounds converge to 0 as b tends to infinity. The numerical results show that the absolute relative errors of the proposed bounds are less than 5% in most of the cases. The proposed bounds can be effectively applied to the outage probability analysis of interference-limited systems such as cognitive radio and wireless sensor network, in the study of error performance of various wireless communication systems operating over fading channels and extracting the log-likelihood ratio for differential phase-shift keying (DPSK) signals.