Exact Performance Analysis of the Oracle Receiver for Compressed Sensing Reconstruction

TL;DR

This paper derives an exact, closed-form expression for the mean square error of the oracle receiver in compressed sensing, applicable to noisy, correlated measurements, and confirms its accuracy through simulations.

Contribution

It provides the first exact, closed-form performance analysis of the oracle receiver in compressed sensing, independent of sensing matrix realization and noise correlation.

Findings

Exact MSE expression matches simulations

Performance is robust to noise correlation

Results improve understanding of oracle receiver behavior

Abstract

A sparse or compressible signal can be recovered from a certain number of noisy random projections, smaller than what dictated by classic Shannon/Nyquist theory. In this paper, we derive the closed-form expression of the mean square error performance of the oracle receiver, knowing the sparsity pattern of the signal. With respect to existing bounds, our result is exact and does not depend on a particular realization of the sensing matrix. Moreover, our result holds irrespective of whether the noise affecting the measurements is white or correlated. Numerical results show a perfect match between equations and simulations, confirming the validity of the result.