Applications of Tauberian Theorem for High-SNR Analysis of Performance over Fading Channels

TL;DR

This paper uses Tauberian theorems and regular variation theory to derive simple, general high-SNR error rate asymptotics over fading channels, linking them directly to outage probability and channel distribution properties.

Contribution

It introduces a novel application of Tauberian theorems to relate high-SNR error rates to outage probability for a broad class of fading channels, simplifying existing analyses.

Findings

High-SNR error rates are asymptotically proportional to outage probability.

Diversity order and the channel gain distribution's variation exponent are equivalent under certain conditions.

Numerical results confirm the theoretical predictions across various channel models.

Abstract

This paper derives high-SNR asymptotic average error rates over fading channels by relating them to the outage probability, under mild assumptions. The analysis is based on the Tauberian theorem for Laplace-Stieltjes transforms which is grounded on the notion of regular variation, and applies to a wider range of channel distributions than existing approaches. The theory of regular variation is argued to be the proper mathematical framework for finding sufficient and necessary conditions for outage events to dominate high-SNR error rate performance. It is proved that the diversity order being $d$ and the cumulative distribution function (CDF) of the channel power gain having variation exponent $d$ at 0 imply each other, provided that the instantaneous error rate is upper-bounded by an exponential function of the instantaneous SNR. High-SNR asymptotic average error rates are derived for specific instantaneous error rates. Compared to existing approaches in the literature, the asymptotic expressions are related to the channel distribution in a much simpler manner herein, and related with outage more intuitively. The high-SNR asymptotic error rate is also characterized under diversity combining schemes with the channel power gain of each branch having a regularly varying CDF. Numerical results are shown to corroborate our theoretical analysis.