Detection of large-scale noisy multi-periodic patterns with discrete double Fourier transform

TL;DR

This paper introduces the discrete double Fourier transform (DDFT), a novel spectral analysis method designed to detect large-scale noisy multi-periodic patterns in signals with long-range correlated noise, demonstrated through simulations and real-world applications.

Contribution

The paper presents DDFT, a new spectral analysis technique that enhances detection of multi-periodic patterns in noisy, long-range correlated data, extending cepstrum analysis.

Findings

DDFT effectively detects large-scale multi-periodic patterns in noisy signals.

DDFT outperforms traditional cepstrum transform in pattern detection.

Applications include solar wind analysis and DNA sequence pattern detection.

Abstract

In many processes, the variations in underlying characteristics can be approximated by noisy multi-periodic patterns. If large-scale patterns are superimposed by a noise with long-range correlations, the detection of multi-periodic patterns becomes especially challenging. To solve this problem, we developed a discrete double Fourier transform (DDFT). DDFT is based on the equidistance property of harmonics generated by multi-periodic patterns in the discrete Fourier transform (DFT) spectra. As the large-scale patterns generate long enough equidistant series, they can be detected by the iteration of the primary DFT. DDFT is defined as Fourier transform of intensity spectral harmonics or of their functions. It comprises widely used cepstrum transform as a particular case. We present also the relevant analytical criteria for the assessment of statistical significance of peak harmonics in DDFT spectra in the presence of noise. DDFT technique was tested by extensive numerical simulations. The practical applications of DDFT technique are illustrated by the analysis of variations in solar wind speed related to solar rotation and by the study of large-scale multi-periodic patterns in DNA sequences. The latter application can be considered as generic example for the general spectral analysis of symbolic sequences. The results are compared with those obtained by the cepstrum transform. The mutual combination of DFT and DDFT provides an efficient technique to search for noisy large-scale multi-periodic patterns.