On Convexity of Error Rates in Digital Communications

Abstract

Convexity properties of error rates of a class of decoders, including the ML/min-distance one as a special case, are studied for arbitrary constellations, bit mapping and coding. Earlier results obtained for the AWGN channel are extended to a wide class of noise densities, including unimodal and sphericallyinvariant noise. Under these broad conditions, symbol and bit error rates are shown to be convex functions of the SNR in the high-SNR regime with an explicitly-determined threshold, which depends only on the constellation dimensionality and minimum distance, thus enabling an application of the powerful tools of convex optimization to such digital communication systems in a rigorous way. It is the decreasing nature of the noise power density around the decision region boundaries that insures the convexity of symbol error rates in the general case. The known high/low SNR bounds of the convexity/concavity regions are tightened and no further improvement is shown to be possible in general. The high SNR bound fits closely into the channel coding theorem: all codes, 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. Convexity properties in signal amplitude and noise power are also investigated. Some applications of the results are discussed. In particular, it is shown that fading is convexity-preserving and is never good in low dimensions under spherically-invariant noise, which may also include any linear diversity combining. Index TermsConvexity/concavity, error rate, BER, pairwise probability of error, maximum-likelihood decoding, unimodal noise, spherically-invariant noise.

Universal Bounds on the Derivatives of the Symbol Error Rate for Arbitrary Constellations

“…The subsequent analysis relies on a slightly modified version of the representation for the SER that was originally established in the proof of Theorem 2 in [5] (and also employed in the proof of Theorem 1 in [7]). For any constellation of dimensionality , the conditional SER , given that is transmitted, can be expressed equivalently to (6) as (8) where represents the vector of angles, is the distance between the origin and the boundary of the decision region at the direction , denotes the range of the angles (9) and is a PDF defined over the range as (10) where is the Gamma function [9].…”

“…For any constellation of dimensionality , the conditional SER , given that is transmitted, can be expressed equivalently to (6) as (8) where represents the vector of angles, is the distance between the origin and the boundary of the decision region at the direction , denotes the range of the angles (9) and is a PDF defined over the range as (10) where is the Gamma function [9]. This representation is obtained by first expressing (6) in hyperspherical coordinates , where and is the vector of angles, and then substituting in the resulting integral [5], [10].…”

“…Proposition 2: Consider an arbitrary constellation of dimensionality impaired by AWGN. Let the polynomial be defined as in (15) Although the convexity properties of the SER were studied in [4], [5] for arbitrary constellations (i.e., ), and the complete monotonicity of the SER was established in [6] for oneand two-dimensional constellations (i.e., ), the results presented in this paper cover those in [4]- [6] as special cases, and extend them to higher order derivatives of the SER. It should be noted that such an analysis was avoided in the referenced papers as it was considered to be rather involved.…”

“…Lower and upper bounds on the first and second derivatives of the SER in SNR are also derived. In the follow-up paper [5], the authors have provided tighter SNR thresholds to identify the concavity/convexity regions, and extended the earlier results to a more general class of decoders under unimodal and spherically invariant noise. In [6], the SER of the minimum distance detector impaired by AWGN is found to be a completely monotone (c.m.)…”

“…SYSTEM MODEL We consider the standard baseband discrete-time system model for -ary communications through AWGN, which is described as [4], [5], [7] (1)…”