Error rates of capacity-achieving codes are convex

Abstract

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 also includes 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 and noise power in the high SNR/low noise regime with explicitlydetermined boundary. Any code, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem) satisfies this high SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates.

On Convexity of Error Rates in Digital Communications

“…Unless otherwise indicated, a non-fading channel, an arbitrary constellation and a decoder with center-convex decision regions are assumed. Theorem 5, ( 24), (25) AWGN: BER/PEP are convex at high SNR See [14], [16], [17] and Theorem 9…”

Radio and computing resource management in SDR clouds

“…If Φ(y) is a convex function, 1 − Φ(y) is concave and the conditions of Theorem 6.1 are satisfied (concavity implies log-concavity). The convexity of bit and symbol error probabilities have been recently proven in the high SNR regime for arbitrary constellations and bit mappings with coding and maximum-likelihood decoding [126]. Convexity in the low SNR regime exists for many common modulations.…”