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

TL;DR

This paper derives new analytic expressions and bounds for special functions used in wireless communication theory, enabling improved performance analysis of various communication systems with complex fading channels.

Contribution

It introduces novel analytic formulas and bounds for special functions, facilitating advanced performance evaluation in modern wireless communication systems.

Findings

Derived closed-form outage probability expressions for generalized fading channels

Provided simple channel capacity formulas for Rician fading channels

Validated accuracy of expressions through extensive numerical comparisons

Abstract

This work 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, the incomplete Toronto function, the Rice $Ie$-function and the incomplete Lipschitz-Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and the Humbert, $\Phi_{1}$, function as well as for specific cases of the Kamp${\it \acute{e}}$ de F${\it \acute{e}}$riet function. These functions can be considered 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 multi-antenna systems. As an example, new closed-form expressions are derived for the outage probability over non-linear generalized fading channels, namely, $\alpha{-}\eta{-}\mu$, $\alpha{-}\lambda{-}\mu$ and $\alpha{-}\kappa{-}\mu$ as well as for specific cases of the $\eta{-}\mu$ and $\lambda{-}\mu$ fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and the corresponding optimum cut-off signal-to-noise ratio for single-and multi-antenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results.