An extended Hilbert transform method for reconstructing the phase from an oscillatory signal

TL;DR

This paper introduces an extended Hilbert transform method that improves phase reconstruction accuracy for diverse oscillatory signals, aiding the analysis of synchronization in biological systems.

Contribution

The paper develops a novel extended Hilbert transform approach that overcomes limitations of the traditional method, enabling accurate phase reconstruction from various signals.

Findings

Systematic improvement over conventional Hilbert transform

Validated with synthetic data showing higher accuracy

Potential for detecting phase shifts in observed signals

Abstract

Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.