A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity

TL;DR

This paper introduces a deterministic sparse Fourier transform algorithm that efficiently handles functions with structured frequency support, outperforming existing methods in speed and noise robustness.

Contribution

The paper presents a novel deterministic algorithm for sparse Fourier transforms that breaks the quadratic runtime barrier for structured frequency support functions.

Findings

Algorithm is faster than standard sparse Fourier transforms.

The method is robust to noise.

Numerical experiments confirm theoretical error bounds.

Abstract

In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the oft-considered set of block frequency sparse functions of the form $$f(x) = \sum^{n}_{j=1} \sum^{B-1}_{k=0} c_{\omega_j + k} e^{i(\omega_j + k)x},~~\{ \omega_1, \dots, \omega_n \} \subset \left(-\left\lceil \frac{N}{2}\right\rceil, \left\lfloor \frac{N}{2}\right\rfloor\right]\cap\mathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.