A Deterministic Sub-linear Time Sparse Fourier Algorithm via Non-adaptive Compressed Sensing Methods

TL;DR

This paper introduces the first deterministic sub-linear time sparse Fourier algorithm suitable for failure-intolerant applications, building on compressed sensing techniques to reliably identify dominant frequencies without probabilistic failure.

Contribution

It develops a novel deterministic Fourier algorithm with sub-linear time complexity, extending compressed sensing methods for failure-free frequency estimation.

Findings

First deterministic sub-linear time sparse Fourier algorithm

Improved deterministic compressed sensing method with exponential decay

Applicable to failure-intolerant hardware processing signals

Abstract

We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $\textbf{A}$ of length $N \gg B$. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of $\hat{\textbf{A}}$, and estimate their coefficients, in polynomial$(B,\log N)$ time. Randomized sub-linear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) \cite{CMDetCS3,CMDetCS1,CMDetCS2} in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM's algebraic compressibility results while simultaneously maintaining their results concerning exponential decay.