MSE estimates for multitaper spectral estimation and off-grid compressive sensing

TL;DR

This paper provides analytic estimates for the MSE of multitaper spectral estimators and compressive sensing methods for multi-band signals, confirming and quantifying the convergence of Slepian functions to ideal band-pass kernels.

Contribution

It offers a rigorous proof and convergence rate analysis for the approximation of Slepian functions, validating heuristic MSE control methods in spectral estimation and compressive sensing.

Findings

Slepian functions converge to ideal band-pass kernels as their number increases.

The paper quantifies the convergence rate in the L1 norm.

Provides MSE approximation bounds for compressive acquisition methods.

Abstract

We obtain estimates for the Mean Squared Error (MSE) for the multitaper spectral estimator and certain compressive acquisition methods for multi-band signals. We confirm a fact discovered by Thomson [Spectrum estimation and harmonic analysis, Proc. IEEE, 1982]: assuming bandwidth $W$ and $N$ time domain observations, the average of the square of the first $K=2NW$ Slepian functions approaches, as $K$ grows, an ideal band-pass kernel for the interval $[-W,W]$. We provide an analytic proof of this fact and measure the corresponding rate of convergence in the $L^{1}$ norm. This validates a heuristic approximation used to control the MSE of the multitaper estimator. The estimates have also consequences for the method of compressive acquisition of multi-band signals introduced by Davenport and Wakin, giving MSE approximation bounds for the dictionary formed by modulation of the critical number of prolates.