On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall Q-Functions

In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and NuttallQ-functions. More precisely, a simple probabilistic method is first given to prove the monotonicity of these two functions. Then, the log-concavity of the generalized MarcumQ-function and its deformations is established with respect to each of the three parameters. Since the NuttallQ-function has similar probabilistic interpretations as the generalized MarcumQ-function, we deduce the log-concavity of the NuttallQ-function. By exploiting the log-concavity of these two functions, we propose new tight lower and upper bounds for the generalized Marcum and NuttallQ-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 to0asb ¿. 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.