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

摘要

This work is devoted to the derivation of novel analytic expressions andbounds for a family of special functions that are useful in wirelesscommunication theory. These functions are the well-known Nuttall$Q{-}$function, the incomplete Toronto function, the Rice $Ie$-function and theincomplete Lipschitz-Hankel integrals. Capitalizing on the offered results, useful identities are additionallyderived between the above functions and the Humbert, $\Phi_{1}$, function aswell as for specific cases of the Kamp${\it \acute{e}}$ de F${\it\acute{e}}$riet function. These functions can be considered useful mathematicaltools that can be employed in applications relating to the analytic performanceevaluation of modern wireless communication systems such as cognitive radio,cooperative and free-space optical communications as well as radar, diversityand multi-antenna systems. As an example, new closed-form expressions arederived 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 presentedfor the channel capacity for the truncated channel inversion with fixed rateand the corresponding optimum cut-off signal-to-noise ratio for single-andmulti-antenna communication systems over Rician fading channels. The accuracyand validity of the derived expressions is justified through extensivecomparisons with respective numerical results.