Asymptotic Comparison of ML and MAP Detectors for Multidimensional Constellations

TL;DR

This paper derives exact asymptotic expressions for SEP and BEP of multidimensional constellations with nonequally likely symbols, comparing MAP and ML detectors and establishing conditions for their asymptotic equivalence.

Contribution

It provides closed-form asymptotic formulas for SEP and BEP, proves the union bound's tightness, and analyzes the asymptotic relationship between MAP and ML detectors.

Findings

Asymptotic expressions for SEP and BEP are derived.

Union bound is asymptotically tight under general conditions.

The ratio of MAP to ML detector error probabilities approaches a constant.

Abstract

A classical problem in digital communications is to evaluate the symbol error probability (SEP) and bit error probability (BEP) of a multidimensional constellation over an additive white Gaussian noise channel. In this paper, we revisit this problem for nonequally likely symbols and study the asymptotic behavior of the optimal maximum a posteriori (MAP) detector. Exact closed-form asymptotic expressions for SEP and BEP for arbitrary constellations and input distributions are presented. The well-known union bound is proven to be asymptotically tight under general conditions. The performance of the practically relevant maximum likelihood (ML) detector is also analyzed. Although the decision regions with MAP detection converge to the ML regions at high signal-to-noise ratios, the ratio between the MAP and ML detector in terms of both SEP and BEP approach a constant, which depends on the constellation and a priori probabilities. Necessary and sufficient conditions for asymptotic equivalence between the MAP and ML detectors are also presented.