Compressive Spectral Estimation for Nonstationary Random Processes

TL;DR

This paper introduces a compressive spectral estimator for nonstationary, underspread, and approximately sparse processes, reducing measurement requirements while maintaining estimation accuracy through a combination of unbiased estimation and compressed sensing.

Contribution

It proposes a novel compressive estimator for the Rihaczek spectrum that leverages time-frequency sparsity and provides theoretical error bounds.

Findings

Significantly fewer measurements needed for accurate spectral estimation.

The estimator performs well in simulations for underspread processes.

Applicable to other spectra like Wigner-Ville due to spectral similarity.

Abstract

Estimating the spectral characteristics of a nonstationary random process is an important but challenging task, which can be facilitated by exploiting structural properties of the process. In certain applications, the observed processes are underspread, i.e., their time and frequency correlations exhibit a reasonably fast decay, and approximately time-frequency sparse, i.e., a reasonably large percentage of the spectral values is small. For this class of processes, we propose a compressive estimator of the discrete Rihaczek spectrum (RS). This estimator combines a minimum variance unbiased estimator of the RS (which is a smoothed Rihaczek distribution using an appropriately designed smoothing kernel) with a compressed sensing technique that exploits the approximate time-frequency sparsity. As a result of the compression stage, the number of measurements required for good estimation performance can be significantly reduced. The measurements are values of the discrete ambiguity function of the observed signal at randomly chosen time and frequency lag positions. We provide bounds on the mean-square estimation error of both the minimum variance unbiased RS estimator and the compressive RS estimator, and we demonstrate the performance of the compressive estimator by means of simulation results. The proposed compressive RS estimator can also be used for estimating other time-dependent spectra (e.g., the Wigner-Ville spectrum) since for an underspread process most spectra are almost equal.