Noise Folding in Compressed Sensing

TL;DR

This paper investigates the impact of pre-measurement noise in compressed sensing, revealing that it causes noise folding, significantly reducing the effective signal-to-noise ratio in most measurement schemes.

Contribution

It introduces the concept of noise folding in compressed sensing, showing how pre-measurement noise affects the SNR and analyzing its implications across common measurement schemes.

Findings

Pre-measurement noise effectively reduces SNR by a factor proportional to p/n.

Most measurement schemes in compressed sensing are susceptible to noise folding.

Noise folding can severely impact the quality of signal recovery.

Abstract

The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show that, for the vast majority of measurement schemes employed in compressed sensing, the two models are equivalent with the important difference that the signal-to-noise ratio is divided by a factor proportional to p/n, where p is the dimension of the signal and n is the number of observations. Since p/n is often large, this leads to noise folding which can have a severe impact on the SNR.