Spectral Estimation from Undersampled Data: Correlogram and Model-Based Least Squares

TL;DR

This paper compares two spectrum estimation methods for undersampled data, deriving their statistical properties, and introduces a new model-based algorithm for reconstructing sparse signals with high resolution and near-optimal accuracy.

Contribution

It presents a comprehensive analysis of the correlogram method's bias, variance, and consistency, and proposes a novel sparse spectrum reconstruction algorithm that approaches the Cramer-Rao bound.

Findings

Correlogram estimator is consistent with a bias-variance tradeoff.

The new sparse spectrum algorithm achieves high-resolution estimates.

Proposed method approaches the Cramer-Rao bound at high SNR.

Abstract

This paper studies two spectrum estimation methods for the case that the samples are obtained at a rate lower than the Nyquist rate. The first method is the correlogram method for undersampled data. The algorithm partitions the spectrum into a number of segments and estimates the average power within each spectral segment. We derive the bias and the variance of the spectrum estimator, and show that there is a tradeoff between the accuracy of the estimation and the frequency resolution. The asymptotic behavior of the estimator is also investigated, and it is proved that this spectrum estimator is consistent. A new algorithm for reconstructing signals with sparse spectrum from noisy compressive measurements is also introduced. Such model-based algorithm takes the signal structure into account for estimating the unknown parameters which are the frequencies and the amplitudes of linearly combined sinusoidal signals. A high-resolution spectral estimation method is used to recover the frequencies of the signal elements, while the amplitudes of the signal components are estimated by minimizing the squared norm of the compressed estimation error using the least squares technique. The Cramer-Rao bound for the given system model is also derived. It is shown that the proposed algorithm approaches the bound at high signal to noise ratios.