Achieving AWGN Channel Capacity With Lattice Gaussian Coding

TL;DR

This paper introduces a simplified lattice Gaussian coding scheme that achieves the AWGN channel capacity using only one lattice and no shaping lattice or dither, with comparable error performance to previous complex schemes.

Contribution

It presents a novel lattice Gaussian coding scheme that simplifies capacity-achieving coding for AWGN channels by eliminating the need for multiple lattices and shaping, while maintaining optimal performance.

Findings

Achieves AWGN capacity with a single lattice and no shaping lattice.

Error probability close to previous complex lattice schemes.

Introduces the concept of good constellations for near-Gaussian mutual information.

Abstract

We propose a new coding scheme using only one lattice that achieves the $\frac{1}{2}\log(1+\SNR)$ capacity of the additive white Gaussian noise (AWGN) channel with lattice decoding, when the signal-to-noise ratio $\SNR>e-1$. The scheme applies a discrete Gaussian distribution over an AWGN-good lattice, but otherwise does not require a shaping lattice or dither. Thus, it significantly simplifies the default lattice coding scheme of Erez and Zamir which involves a quantization-good lattice as well as an AWGN-good lattice. Using the flatness factor, we show that the error probability of the proposed scheme under minimum mean-square error (MMSE) lattice decoding is almost the same as that of Erez and Zamir, for any rate up to the AWGN channel capacity. We introduce the notion of good constellations, which carry almost the same mutual information as that of continuous Gaussian inputs. We also address the implementation of Gaussian shaping for the proposed lattice Gaussian coding scheme.