A sparse Fast Fourier Algorithm for Real Nonnegative Vectors

TL;DR

This paper introduces a fast Fourier transform algorithm tailored for real nonnegative signals with short support, significantly reducing computational complexity and sample requirements without prior sparsity knowledge.

Contribution

The paper presents a novel sparse FFT algorithm that automatically detects support length and adapts, outperforming traditional FFT in efficiency for sparse signals.

Findings

Achieves ${ m O}(m \, ext{log} \, m \, ext{log} (N/m))$ complexity for short support signals.

Requires only ${ m O}(m \, ext{log} (N/m))$ Fourier samples.

Demonstrates numerical stability through examples.

Abstract

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m)) $ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if ${\bf x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.