A Fast Algorithm for Computing the Fourier Spectrum of a Fractional Period

TL;DR

This paper introduces a novel, efficient algorithm for directly computing the Fourier spectrum at fractional periods in time series, significantly reducing computational costs and aiding high-resolution analysis in biological and other data.

Contribution

The paper presents a new fast algorithm for fractional period spectrum computation that is more efficient than traditional Fourier transform methods.

Findings

The algorithm reduces computational complexity compared to traditional Fourier transform.

It effectively computes fractional periods like 3.6 in protein sequences.

The method is applicable across various research fields involving time series analysis.

Abstract

The Fourier spectrum at a fractional period is often examined when extracting features from biological sequences and time series. It reflects the inner information structure of the sequences. A fractional period is not uncommon in time series. A typical example is the 3.6 period in protein sequences, which determines the $\alpha$-helix secondary structure. Computing the spectrum of a fractional period offers a high-resolution insight into a time series. It has thus become an important approach in genomic analysis. However, computing Fourier spectra of fractional periods by the traditional Fourier transform is computationally expensive. In this paper, we present a novel, fast algorithm for directly computing the fractional period spectrum (FPS) of time series. The algorithm is based on the periodic distribution of signal strength at periodic positions of the time series. We provide theoretical analysis, deduction, and special techniques for reducing the computational costs of the algorithm. The analysis of the computational complexity of the algorithm shows that the algorithm is much faster than traditional Fourier transform. Our algorithm can be applied directly in computing fractional periods in time series from a broad of research fields. The computer programs of the FPS algorithm are available at https://github.com/cyinbox/FPS.