Measurement Design for Detecting Sparse Signals

TL;DR

This paper investigates optimal measurement designs for detecting sparse signals in noise, proposing a lexicographic optimization approach to maximize SNR under measurement constraints, and establishing connections to tight frames.

Contribution

It introduces a novel lexicographic optimization method for measurement design in sparse signal detection, linking optimal solutions to tight frame constructions.

Findings

Optimal measurement matrix is a Grassmannian line packing for worst-case SNR.

Optimal measurement matrix is a uniform tight frame with minimum sum-coherence for average SNR.

Measurement design improves detection performance under measurement budget constraints.

Abstract

We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level $k$ is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k-1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most---and a uniform tight frame for all---sparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most---and a tight frame for all---sparse signals.