IEEE Transactions on Information Theory

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 → ∞.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.