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

Mahdi Shaghaghi, Sergiy A. Vorobyov|arXiv (Cornell University)|Feb 26, 2015

Sparse and Compressive Sensing TechniquesEngineering35 references17 citations

TL;DR

This paper derives the Cramér-Rao Bound (CRB) for parameter estimation in low-rank signal models with a small number of parameters, particularly for sparse signals not necessarily sparse in a finite basis. It shows that unbiased estimation with finite variance requires more compressed samples than the number of sources, and validates this via DOA estimation examples where CRB increases with compression (reduced Ny).

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.

Motivation & Objective

• To establish performance bounds for parameter estimation in low-rank signal models with few parameters.
• To derive a closed-form Cramér-Rao Bound (CRB) for such models under compressed sensing.
• To determine the minimum number of compressed samples required for unbiased estimation with finite variance.
• To apply the derived CRB to direction-of-arrival (DOA) and spectral estimation problems.
• To analyze the impact of compression on estimation accuracy via numerical examples.

Proposed method

• Derives the log-likelihood function for compressed measurements of a low-rank signal model with parametric structure.
• Computes the Fisher Information Matrix (FIM) using derivatives of the log-likelihood with respect to real/imaginary parts of amplitudes and unknown parameters Ω.
• Uses matrix identities and Schur complement to express the CRB in closed form, particularly for parameter vector Ω.
• Analyzes FIM singularity to determine conditions under which unbiased estimation with finite variance is impossible.
• Applies the framework to DOA estimation using steering vectors and derives simplified expressions for D(t) and the CRB.
• Uses numerical simulations with a uniform linear array to illustrate CRB behavior across varying numbers of compressed samples (Ny).

Experimental results

Research questions

• RQ1 What is the Cramér-Rao Bound (CRB) for parameter estimation in a generic low-rank signal model with a small number of unknown parameters?
• RQ2 What is the minimum number of compressed samples (Ny) required for an unbiased estimator to have finite estimation variance?
• RQ3 How does compression (reduced Ny) affect the CRB in direction-of-arrival (DOA) estimation?
• RQ4 Can the derived CRB be expressed in closed form for parametric low-rank models?
• RQ5 How does the CRB behave when Ny ≤ K (number of sources) compared to Ny > K?

Key findings

• The Cramér-Rao Bound (CRB) for parameter estimation in the low-rank model is derived in closed form, particularly for the parameter vector Ω.
• Unbiased estimation with finite variance is impossible when the number of compressed samples Ny ≤ K (number of sources), as the FIM becomes singular.
• When Ny > K, the FIM is non-singular, and the CRB is finite, indicating that unbiased estimators with finite variance can exist.
• Numerical results show that the CRB for DOA estimation increases as Ny decreases, with a sharp rise when Ny ≤ K.
• The minimum number of compressed samples required for finite-variance unbiased estimation is Ny = K + 1, as demonstrated in the DOA example with K = 11 sources.