Proof of Threshold Saturation for Spatially Coupled Sparse Superposition Codes

TL;DR

This paper proves that spatially coupled sparse superposition codes achieve capacity by demonstrating threshold saturation of the state evolution, extending ideas from LDPC codes and compressive sensing to this new coding scheme.

Contribution

It provides a rigorous proof that spatial coupling causes the potential threshold to saturate, approaching the optimal threshold for sparse superposition codes.

Findings

State evolution saturates the potential threshold.

Threshold approaches the optimal capacity limit.

Supports capacity-achieving performance without power allocation.

Abstract

Recently, a new class of codes, called sparse superposition or sparse regression codes, has been proposed for communication over the AWGN channel. It has been proven that they achieve capacity using power allocation and various forms of iterative decoding. Empirical evidence has also strongly suggested that the codes achieve capacity when spatial coupling and approximate message passing decoding are used, without need of power allocation. In this note we prove that state evolution (which tracks message passing) indeed saturates the potential threshold of the underlying code ensemble, which approaches in a proper limit the optimal threshold. Our proof uses ideas developed in the theory of low-density parity-check codes and compressive sensing.