Computing the Discrete Fourier Transform of signals with spectral frequency support

TL;DR

This paper introduces an efficient $O(k \,\log k)$ algorithm for computing the DFT of signals with known spectral support, generalizing the radix-2 algorithm using recent spectral set characterizations.

Contribution

It presents a novel $O(k \log k)$ algorithm for DFT computation of signals with spectral support, extending standard radix-2 methods.

Findings

The algorithm achieves $O(k \log k)$ complexity for spectral set signals.

It generalizes the radix-2 algorithm to spectral support cases.

The approach leverages recent spectral set characterizations.

Abstract

We consider the problem of finding the Discrete Fourier Transform (DFT) of $N-$ length signals with known frequency support of size $k$. When $N$ is a power of 2 and the frequency support is a spectral set, we provide an $O(k \log k)$ algorithm to compute the DFT. Our algorithm uses some recent characterizations of spectral sets and is a generalization of the standard radix-2 algorithm.