Fast algorithm of adaptive Fourier series

TL;DR

This paper introduces a fast Fourier series algorithm that integrates FFT to significantly reduce computational complexity, making adaptive Fourier decomposition more practical for signal processing applications.

Contribution

It formulates a 1-D adaptive Fourier decomposition algorithm incorporating FFT, reducing complexity from O(M N^2) to O(M N log N), enhancing efficiency.

Findings

Algorithm complexity reduced to O(M N log N)

High efficiency demonstrated through experiments

Enhanced applicability of adaptive Fourier decomposition

Abstract

Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. AFD merged with the greedy algorithm idea, and in particular, motivated the so-called pre-orthogonal greedy algorithm (Pre-OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper offers one formulation of the 1-D AFD algorithm by building the FFT algorithm into it. Accordingly, the algorithm complexity is reduced, from the original $\mathcal{O}(M N^2)$ to $\mathcal{O}(M N\log_2 N)$, where $N$ denotes the number of the discretization points on the unit circle and $M$ denotes the number of points in $[0,1)$. This greatly enhances the applicability of AFD. Experiments are carried out to show the high efficiency of the proposed algorithm.