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

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 l̄, 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 l̄ → ∞ 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.