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\ura ¿. 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.

In this paper, we study the generalized MarcumQ-functionQnu(a,b), wherea, nu > 0 andbges 0. Our aim is to extend the results of Corazza and Ferrari (IEEETrans.Inf.Theory, vol. 48, pp. 3003-3008, 2002) to the generalized MarcumQ-function in order to deduce some new tight lower and upper bounds. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind and some classical inequalities, i.e., the Cauchy-Buniakowski-Schwarz and Chebyshev integral inequalities. These bounds are shown to be very tight for largeb, i.e., the relative errors of our bounds converge to zero asbincreases. Both theoretical analysis and numerical results are provided to show the tightness of our bounds.