Thomson's Multitaper Method Revisited

TL;DR

This paper revisits Thomson's multitaper spectral estimation method, providing new linear algebra insights, nonasymptotic bounds, and an efficient approximation technique that improves spectral leakage protection and computational efficiency.

Contribution

It offers a linear algebra perspective on the multitaper method, establishes nonasymptotic bounds, and introduces an $ ext{O}( ext{log}(NW) ext{log}(1/ ext{epsilon}))$ FFT-based approximation.

Findings

Using more tapers ($2NW - O( ext{log}(NW))$) better reduces spectral leakage.

The new approximation method significantly reduces computational complexity.

The approach provides nonasymptotic bounds similar to existing asymptotic results.

Abstract

Thomson's multitaper method estimates the power spectrum of a signal from $N$ equally spaced samples by averaging $K$ tapered periodograms. Discrete prolate spheroidal sequences (DPSS) are used as tapers since they provide excellent protection against spectral leakage. Thomson's multitaper method is widely used in applications, but most of the existing theory is qualitative or asymptotic. Furthermore, many practitioners use a DPSS bandwidth $W$ and number of tapers that are smaller than what the theory suggests is optimal because the computational requirements increase with the number of tapers. We revisit Thomson's multitaper method from a linear algebra perspective involving subspace projections. This provides additional insight and helps us establish nonasymptotic bounds on some statistical properties of the multitaper spectral estimate, which are similar to existing asymptotic results. We show using $K=2NW-O(\log(NW))$ tapers instead of the traditional $2NW-O(1)$ tapers better protects against spectral leakage, especially when the power spectrum has a high dynamic range. Our perspective also allows us to derive an $\epsilon$-approximation to the multitaper spectral estimate which can be evaluated on a grid of frequencies using $O(\log(NW)\log\tfrac{1}{\epsilon})$ FFTs instead of $K=O(NW)$ FFTs. This is useful in problems where many samples are taken, and thus, using many tapers is desirable.