Signal Shaping for BICM at Low SNR

TL;DR

This paper characterizes the conditions under which BICM systems achieve the Shannon limit at low SNR, showing that probabilistic shaping does not provide additional benefits beyond geometrical shaping in this regime.

Contribution

It introduces a new linear transform to analyze BICM capacity at low SNR and proves the equivalence between uniform and nonuniform input distributions for achieving the Shannon limit.

Findings

BICM systems achieve the Shannon limit if they are a zero-mean linear projection of a hypercube.

Probabilistic shaping offers no extra advantage over geometrical shaping at low SNR.

Numerical results confirm the theoretical analysis and show benefits of probabilistic shaping at medium SNR.

Abstract

The mutual information of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR), i.e., in the wideband regime. A new linear transform that depends on bits' probabilities is introduced. This transform is used to prove the asymptotical equivalence between certain BICM systems with uniform and nonuniform input distributions. Using known results for BICM systems with a uniform input distribution, we completely characterize the combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit -1.59 dB. The main conclusion is that a BICM system achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube, which is the same condition as for uniform input distributions. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR mutual information of BICM systems, in addition to what is provided by geometrical shaping. These analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping of BICM can improve the mutual information in the low and medium SNR range over any coded modulation system with a uniform input distribution.