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

Abstract

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, $\Phi _{1}$ , function as well as for specific cases of the Kampede Feriet 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, $\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 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.