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

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.

Bounds for the symmetric difference of generalized Marcum Q-functions

“…This leads to the desired inequalities due to [5,Theorem 3(b)], which claims the log-concavity of ν → Q ν (a, b) on [1, ∞). Thus we have that…”

“…where Q(·) is the Gaussian Q-function, expressed by Motivated by the above result, in the next sections we present some lower and upper bounds based on the main results in [5], [6]. The advantage of this approach is that we are able to establish bounds also in the case of real valued parameter ν.…”

“…This is the case for example when in [4] the authors characterize the bit error rate performance (BER) of digital communication systems with quadratic-form receivers. Since this function and its symmetric difference has a rather complicated nature, many papers (see [5], [6]) have studied its behavior due to monotonicity or log-concavity properties, closed-form expressions, aiming to derive lower and upper bounds.…”

“…To do this, the key idea is to consider a closed-form expression of Q ν (a, b) that is simpler to work with. These kind of forms are proposed in [5], [7] for the case when ν + 1 2 ∈ N. For instance, we consider the following closed-form expression of Q ν (a, b) for a, b > 0 and ν + 1 2 ∈ N:…”

Asymptotically Exact Approximations for the Symmetric Difference of Generalized Marcum <inline-formula> <tex-math notation="TeX">$Q$</tex-math></inline-formula>-Functions

“…However, it does not allow to gain insight on system performance in a simple way due to its complicated nature. For this reason, many authors have focused on the derivation of simple lower and upper bounds for this function [8][9][10][11][12]. In some scenarios, the Marcum Q-function appears in the following particular form…”