(l1,l2)-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements

TL;DR

This paper evaluates the effectiveness of the Projected Back-Projection method for reconstructing sparse signals from phase-only measurements, extending one-bit compressive sensing to complex signals and analyzing its theoretical and empirical performance.

Contribution

It introduces an analysis of PBP for phase-only measurements, establishing RIP-like properties for complex Gaussian matrices and providing error bounds.

Findings

PBP performs well in phase-only measurement scenarios.

Complex Gaussian matrices satisfy a RIP variant with high probability.

Simulation results show competitive accuracy compared to classical CS.

Abstract

This letter analyzes the performances of a simple reconstruction method, namely the Projected Back-Projection (PBP), for estimating the direction of a sparse signal from its phase-only (or amplitude-less) complex Gaussian random measurements, i.e., an extension of one-bit compressive sensing to the complex field. To study the performances of this algorithm, we show that complex Gaussian random matrices respect, with high probability, a variant of the Restricted Isometry Property (RIP) relating to the l1 -norm of the sparse signal measurements to their l2 -norm. This property allows us to upper-bound the reconstruction error of PBP in the presence of phase noise. Monte Carlo simulations are performed to highlight the performance of our approach in this phase-only acquisition model when compared to error achieved by PBP in classical compressive sensing.