Cramer-Rao Bound for Sparse Signals Fitting the Low-Rank Model with Small Number of Parameters

TL;DR

This paper derives the Cramer-Rao bound for low-rank signals with a small number of parameters, analyzing how compression affects estimation accuracy in applications like DOA and spectral estimation.

Contribution

It provides a new theoretical bound for parameter estimation in low-rank models and explores the impact of compression on estimation performance.

Findings

CRB increases with more compression, reducing the number of samples.

Number of compressed samples must exceed the number of sources for unbiased estimation.

Application to DOA and spectral estimation demonstrates practical implications.

Abstract

In this paper, we consider signals with a low-rank covariance matrix which reside in a low-dimensional subspace and can be written in terms of a finite (small) number of parameters. Although such signals do not necessarily have a sparse representation in a finite basis, they possess a sparse structure which makes it possible to recover the signal from compressed measurements. We study the statistical performance bound for parameter estimation in the low-rank signal model from compressed measurements. Specifically, we derive the Cramer-Rao bound (CRB) for a generic low-rank model and we show that the number of compressed samples needs to be larger than the number of sources for the existence of an unbiased estimator with finite estimation variance. We further consider the applications to direction-of-arrival (DOA) and spectral estimation which fit into the low-rank signal model. We also investigate the effect of compression on the CRB by considering numerical examples of the DOA estimation scenario, and show how the CRB increases by increasing the compression or equivalently reducing the number of compressed samples.