On the SNR Variability in Noisy Compressed Sensing

TL;DR

This paper investigates how the distribution of sensing matrix elements affects the variability of output SNR in noisy compressed sensing, revealing that fewer measurements lead to greater SNR spread and performance inconsistency.

Contribution

It provides analytic expressions for SNR spread in different sensing matrices and highlights the impact of measurement count on SNR variability in noisy CS.

Findings

SNR spread increases as the number of measurements decreases.

High compression rates exacerbate SNR variability.

Analytic expressions quantify SNR spread for common sensing matrices.

Abstract

Compressed sensing (CS) is a sampling paradigm that allows to simultaneously measure and compress signals that are sparse or compressible in some domain. The choice of a sensing matrix that carries out the measurement has a defining impact on the system performance and it is often advocated to draw its elements randomly. It has been noted that in the presence of input (signal) noise, the application of the sensing matrix causes SNR degradation due to the noise folding effect. In fact, it might also result in the variations of the output SNR in compressive measurements over the support of the input signal, potentially resulting in unexpected non-uniform system performance. In this work, we study the impact of a distribution from which the elements of a sensing matrix are drawn on the spread of the output SNR. We derive analytic expressions for several common types of sensing matrices and show that the SNR spread grows with the decrease of the number of measurements. This makes its negative effect especially pronounced for high compression rates that are often of interest in CS.