On the BICM Capacity

TL;DR

This paper analyzes the capacity of bit-interleaved coded modulation (BICM), focusing on optimal labelings and input distributions, especially at low SNR, and introduces first-order asymptotics and conditions for optimal constellations.

Contribution

It develops first-order asymptotics for BICM capacity and characterizes FOO constellations using the Hadamard transform, providing new insights into optimal input distributions and labelings.

Findings

Proper input distributions can nearly eliminate the 1 dB gap in BICM capacity.

Only 72 classes of labelings differ in asymptotic behavior for 8-PAM, reduced to 26 for 8-PSK.

FOO constellations are linear projections of hypercubes, with specific conditions for PAM, QAM, and PSK.

Abstract

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit $-1.59 \tr{dB}$. It is shown that the $\Eb/N_0$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.