Closed-Form Bounds to the Rice and Incomplete Toronto Functions and Incomplete Lipschitz-Hankel Integrals

TL;DR

This paper introduces new closed-form bounds and explicit expressions for the Rice function, incomplete Toronto function, and Lipschitz-Hankel integrals, facilitating analytical studies in wireless communications.

Contribution

It provides novel tight bounds and explicit closed-form expressions for these special functions, enhancing analytical tools for wireless communication performance analysis.

Findings

Derived tight upper and lower bounds for the Rice function.

Obtained explicit closed-form expressions for the incomplete Toronto function.

Presented bounds and expressions applicable to MIMO system analysis.

Abstract

This article provides novel analytical results for the Rice function, the incomplete Toronto function and the incomplete Lipschitz-Hankel Integrals. Firstly, upper and lower bounds are derived for the Rice function, $Ie(k,x)$. Secondly, explicit expressions are derived for the incomplete Toronto function, $T_{B}(m,n,r)$, and the incomplete Lipschitz-Hankel Integrals of the modified Bessel function of the first kind, $Ie_{\mu,n}(a,z)$, for the case that $n$ is an odd multiple of 0.5 and $m \geq n$. By exploiting these expressions, tight upper and lower bounds are subsequently proposed for both $T_{B}(m,n,r)$ function and $Ie_{\mu,n}(a,z)$ integrals. Importantly, all new representations are expressed in closed-form whilst the proposed bounds are shown to be rather tight. Based on these features, it is evident that the offered results can be utilized effectively in analytical studies related to wireless communications. Indicative applications include, among others, the performance evaluation of digital communications over fading channels and the information-theoretic analysis of multiple-input multiple-output systems.