Finite-Length and Asymptotic Analysis of Correlogram for Undersampled Data

TL;DR

This paper introduces a spectrum estimation method for undersampled data called correlogram, analyzing its bias, variance, asymptotic behavior, and demonstrating its effectiveness through numerical simulations.

Contribution

The paper provides a novel spectrum estimation approach for undersampled signals, including theoretical analysis and proof of consistency without requiring sparsity.

Findings

Estimator is unbiased and consistent

Variance depends on parameters and decreases with data length

Spectral segment estimates become uncorrelated asymptotically

Abstract

This paper studies a spectrum estimation method for the case that the samples are obtained at a rate lower than the Nyquist rate. The method is referred to as the correlogram for undersampled data. The algorithm partitions the spectrum into a number of segments and estimates the average power within each spectral segment. This method is able to estimate the power spectrum density of a signal from undersampled data without essentially requiring the signal to be sparse. We derive the bias and the variance of the spectrum estimator, and show that there is a tradeoff between the accuracy of the estimation, the frequency resolution, and the complexity of the estimator. A closed-form approximation of the estimation variance is also derived, which clearly shows how the variance is related to different parameters. The asymptotic behavior of the estimator is also investigated, and it is proved that this spectrum estimator is consistent. Moreover, the estimation made for different spectral segments becomes uncorrelated as the signal length tends to infinity. Finally, numerical examples and simulation results are provided, which approve the theoretical conclusions.