Belief-propagation reconstruction for compressed sensing: Quantization vs. Gaussian approximation

This work considers the compressed sensing (CS) of i.i.d. signals with sparse measurement matrices and belief-propagation (BP) reconstruction. In general, BP reconstruction for CS requires the passing of messages that are distributions over the real numbers. To implement this in practice, one typically uses either quantized distributions or a Gaussian approximation. In this work, we use density evolution to compare the reconstruction performance of these two methods. Since the reconstruction performance depends on the signal realization, this analysis makes use of a novel change of variables to analyze the performance for a typical signal. Simulation results are provided to support the results.