A deterministic sparse FFT algorithm for vectors with small support

TL;DR

This paper introduces a deterministic, sublinear algorithm for reconstructing sparse signals with small support from their Fourier transform, achieving efficiency improvements for both exact and noisy measurements.

Contribution

It presents a novel deterministic sparse FFT algorithm tailored for signals with known small support, with optimal complexity for exact and stable for noisy data.

Findings

Exact Fourier measurements require O(m log m) operations.

Stable reconstruction from noisy data requires O(m log N) operations.

Algorithm outperforms previous methods in efficiency for sparse signals.

Abstract

In this paper we consider the special case where a discrete signal ${\bf x}$ of length N is known to vanish outside a support interval of length $m < N$. If the support length $m$ of ${\bf x}$ or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute ${\bf x}$ from its discrete Fourier transform. In case of exact Fourier measurements we require only ${\cal O}(m \log m)$ arithmetical operations. For noisy measurements, we propose a stable ${\cal O}(m \log N)$ algorithm.