Analysis for sensing resource reduction via state evolution

Abstract

This thesis focuses on the approximate message passing (AMP) based algorithms for solving compressed sensing problems and provides corresponding modifications and state evolution analyses based on the following situations.

We consider the correlated distributed compressed sensing (C-DCS) model, in which multiple measurement instances are included. This model allows correlation between measurement matrices and signals across different measurement instances. We modified the AMP algorithm for the C-DCS model such that it can handle correlated matrices and correlated signals. Correctness justification is provided for our proposed algorithm for two special cases: distributed compressed sensing (DCS) and multiple measurement vectors (MMV) models. Simulations show that the empirical results almost perfectly match the theoretical predictions achieved by state evolution.

We consider a practical signal transmission/receiving application with fixed energy budget and assume that the thermal noise is the dominant noise source. Under such conditions, we observe that the overall signal-to-noise ratio (SNR) per measurement decreases quadratically with the increase of the number of measurements. By applying the AMP algorithm and state evolution analysis, we are able to provide an optimal number of measurements to minimize the mean squared error of the estimate which is different from the common wisdom where more measurements often mean a better performance. Numerical results justify the correctness of our analysis.

The performance of AMP may severely deteriorate when the measurement matrix is not a standard Gaussian random matrix. We propose an improved AMP (IAMP) algorithm that works better for non i.i.d. Gaussian random matrices when the correlations between elements of the measurement matrix deviate from those of the standard Gaussian. The derivation is based on a modification of the message passing mechanism that removes the conditional independence assumption. Examples are provided to demonstrate the performance improvement of IAMP where both a particularly designed matrix and a matrix from real applications are used.