An Approximation of the First Order Marcum QQ-Function with Application to Network Connectivity Analysis

An exponential-type approximation of the first order Marcum $Q$-function is presented, which is robust to changes in its first argument and can easily be integrated with respect to the second argument. Such characteristics are particularly useful in network connectivity analysis. The proposed approximation is exact in the limit of small first argument of the Marcum $Q$-function, in which case the optimal parameters can be obtained analytically. For larger values of the first argument, an optimization problem is solved, and the parameters can be accurately represented using regression analysis. Numerical results indicate that the proposed methods result in approximations very close to the actual Marcum $Q$-function for small and moderate values of the first argument. We demonstrate the accuracy of the approximation by using it to analyze the connectivity properties of random ad hoc networks operating in a Rician fading environment.Comment: 6 pages, 4 figures, 1 tabl

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

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.Comment: Published in IEEE Transactions on Information Theory, August 2009. Only slight formatting modification

Design and performance analysis of quadratic-form receivers for fading channels

Ph.DDOCTOR OF PHILOSOPH

Analytic Expressions and Bounds for Special Functions and Applications in Communication Theory

This paper is devoted to the derivation of novel analytic expressions and bounds for a family of special functions that are useful in wireless communication theory. These functions are the well-known Nuttall Q-function, incomplete Toronto function, Rice Ie-function, and incomplete Lipschitz-Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and Humbert, Φ1, function as well as for specific cases of the Kampé de Fériet function. These functions can be considered as useful mathematical tools that can be employed in applications relating to the analytic performance evaluation of modern wireless communication systems, such as cognitive radio, cooperative, and free-space optical communications as well as radar, diversity, and multiantenna systems. As an example, new closed-form expressions are derived for the outage probability over nonlinear generalized fading channels, namely, α-η-μ, α-λ-μ, and α-κ-μ as well as for specific cases of the η-μ and λ-μ fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and corresponding optimum cutoff signal-to-noise ratio for single-antenna and multiantenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results