Reconsider phase reconstruction in signals with dynamic periodicity from the modern signal processing perspective

Phase is the most fundamental physical quantity when we study an oscillatory time series. There are many tools aiming to estimate phase, most of them are developed based on the analytic function model. Unfortunately, this approach might not be suitable for modern signals with intrinsic nonstartionary structure, including multiple oscillatory components, each with time-varying frequency, amplitude, and non-sinusoidal oscillation, e.g., biomedical signals. Specifically, due to the lack of consensus of model and algorithm, phases estimated from signals simultaneously recorded from different sensors for the same physiological system from the same subject might be different. This fact might challenge reproducibility, communication, and scientific interpretation and thus we need a standardized approach with theoretical support over a unified model. In this paper, after summarizing existing models for phase and discussing the main challenge caused by the above-mentioned intrinsic nonstartionary structure, we introduce the adaptive non-harmonic model (ANHM), provide a definition of phase called fundamental phase, which is a vector-valued function describing the dynamics of all oscillatory components in the signal, and suggest a time-varying bandpass filter (tvBPF) scheme based on time-frequency analysis tools to estimate the fundamental phase. The proposed approach is validated with a simulated database and a real-world database with experts labels, and it is applied to two real-world databases, each of which has biomedical signals recorded from different sensors, to show how to standardize the definition of phase in the real-world experimental environment. Specifically, we report that the phase describing a physiological system, if properly modeled and extracted, is immune to the selected sensor for that system, while other approaches might fail. In conclusion, the proposed approach resolves the above-mentioned scientific challenge. We expect its scientific impact on a broad range of applications.