Applications of Stochastic Ordering to Wireless Communications

TL;DR

This paper introduces stochastic ordering of SNRs as a unified framework to compare wireless communication performance metrics across different fading channels, encompassing error rates, capacity, and complex systems like relay networks.

Contribution

It develops a novel approach using stochastic orders to compare diverse performance metrics and systems in wireless communications, unifying existing metrics through convex and completely monotonic functions.

Findings

Stochastic orders relate error rates and capacity to SNR in fading channels.

Performance comparisons are possible even without closed-form expressions.

Simulations confirm the theoretical results.

Abstract

Stochastic orders are binary relations defined on probability distributions which capture intuitive notions like being larger or being more variable. This paper introduces stochastic ordering of instantaneous SNRs of fading channels as a tool to compare the performance of communication systems over different channels. Stochastic orders unify existing performance metrics such as ergodic capacity, and metrics based on error rate functions for commonly used modulation schemes through their relation with convex, and completely monotonic (c.m.) functions. Toward this goal, performance metrics such as instantaneous error rates of M-QAM and M-PSK modulations are shown to be c.m. functions of the instantaneous SNR, while metrics such as the instantaneous capacity are seen to have a completely monotonic derivative (c.m.d.). It is shown that the commonly used parametric fading distributions for modeling line of sight (LoS), exhibit a monotonicity in the LoS parameter with respect to the stochastic Laplace transform order. Using stochastic orders, average performance of systems involving multiple random variables are compared over different channels, even when closed form expressions for such averages are not tractable. These include diversity combining schemes, relay networks, and signal detection over fading channels with non-Gaussian additive noise, which are investigated herein. Simulations are also provided to corroborate our results.