Analysis of Belief Propagation for Non-Linear Problems: The Example of CDMA (or: How to Prove Tanaka's Formula)

TL;DR

This paper analyzes belief propagation for CDMA multi-user detection with sparse signatures, proving its optimality in the large system limit and providing a rigorous justification for Tanaka's formula.

Contribution

It introduces a sparse signature scheme for CDMA, proves BP detection matches optimal detection asymptotically, and rigorously justifies Tanaka's capacity formula.

Findings

BP detection performance matches optimal detection in large systems

System capacity converges to Tanaka's formula in the limit

Sparse signature scheme enables efficient detection and optimization

Abstract

We consider the CDMA (code-division multiple-access) multi-user detection problem for binary signals and additive white gaussian noise. We propose a spreading sequences scheme based on random sparse signatures, and a detection algorithm based on belief propagation (BP) with linear time complexity. In the new scheme, each user conveys its power onto a finite number of chips l, in the large system limit. We analyze the performances of BP detection and prove that they coincide with the ones of optimal (symbol MAP) detection in the l->\infty limit. In the same limit, we prove that the information capacity of the system converges to Tanaka's formula for random `dense' signatures, thus providing the first rigorous justification of this formula. Apart from being computationally convenient, the new scheme allows for optimization in close analogy with irregular low density parity check code ensembles.