Bit Error Rate is Convex at High SNR

TL;DR

This paper proves that the bit error rate of maximum likelihood detectors in AWGN channels is convex at high SNR, providing theoretical insights into its behavior across different noise regimes.

Contribution

It establishes the convexity of bit error rate in high SNR regimes for arbitrary constellations and coding, with explicit boundary conditions.

Findings

Bit error rate is convex at high SNR.

Bit error rate is convex in low noise/high SNR regimes.

Convexity properties hold for arbitrary constellations and coding schemes.

Abstract

Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which may also include coding under maximum-likelihood decoding. Under this generic setting, the pairwise probability of error and bit error rate are shown to be convex functions of the SNR in the high SNR regime with explicitly-determined boundary. The bit error rate is also shown to be a convex function of the noise power in the low noise/high SNR regime.