The Fast Slepian Transform

TL;DR

This paper introduces fast algorithms for approximate projection onto the Slepian basis, enabling efficient signal processing tasks with complexity comparable to FFT, and provides new eigenvalue distribution bounds.

Contribution

The paper develops fast, approximate algorithms for Slepian basis projections, reducing computational complexity and establishing new eigenvalue bounds for time-frequency localization operators.

Findings

Algorithms achieve FFT-like complexity for Slepian projections

Simulations show comparable accuracy to exact Slepian methods

New bounds on eigenvalue distribution of localization operators

Abstract

The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). We show that there exist fast constructions for computing approximate projections onto the leading Slepian basis elements. The complexity of the resulting algorithms is comparable to the FFT, and scales favorably as the quality of the desired approximation is increased. In the process of bounding the complexity of these algorithms, we also establish new nonasymptotic results on the eigenvalue distribution of discrete time-frequency localization operators. We then demonstrate how these algorithms allow us to efficiently compute the solution to certain least-squares problems that arise in signal processing. We also provide simulations comparing these fast, approximate Slepian methods to exact Slepian methods as well as the traditional FFT based methods.