On the Monotonicity of the Generalized Marcum and Nuttall Q-Functions

TL;DR

This paper establishes monotonicity criteria and derives closed-form expressions for generalized Marcum and Nuttall Q-functions, providing bounds and extending results for real order indices, with applications in digital communications and probability theory.

Contribution

It introduces new monotonicity criteria, closed-form formulas for specific parameter cases, and extended bounds for the generalized Q-functions, advancing their theoretical understanding and practical computation.

Findings

Monotonicity criteria established for Q-functions.

Closed-form expressions derived for specific parameters.

New bounds proposed and extended for real order indices.

Abstract

Monotonicity criteria are established for the generalized Marcum Q-function, $\emph{Q}_{M}$, the standard Nuttall Q-function, $\emph{Q}_{M,N}$, and the normalized Nuttall Q-function, $\mathcal{Q}_{M,N}$, with respect to their real order indices M,N. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall Q-functions for the case when M,N are odd multiples of 0.5 and $M\geq N$. By exploiting these results, novel upper and lower bounds for $\emph{Q}_{M,N}$ and $\mathcal{Q}_{M,N}$ are proposed. Furthermore, specific tight upper and lower bounds for $\emph{Q}_{M}$, previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others.