![]() Since the phase is the argument (or the angle) of the complex value, a phase value θ is not different from a phase value θ + 2 kπ for any integer k. This might cause some confusion in the illustrated phase in the red curve. In our proposed standardization, the phase is defined as a monotonically increasing function. The phase 0 is associated with the R peaks, and the phase 5π/ 1 is associated with the T end. The blue arrows below the ECG signal indicate the status associated with the phase 0, and the red arrows above the ECG signal indicate the status associated with the phase 5π/ 1. The cycle indicated by the magenta box is zoomed in to illustrate the idea of phase with a clock-like circling plot, where the time flow is indicated by the gray arrows. (a) The red curve on the top is the phase function associated with the ECG shown in the black curve, which ranges from 0 to 1 with the unit 2π. This list of scientific merits of phase information is far from exhaustive. ![]() For example, the pulse transit time reflects the phase relationship between electrocardiogram (ECG) and photoplethysmogram (PPG), and it is known to be related to the blood pressure the coupling between the heart and lung, known as the cardiopulmonary coupling (CPC), is quantified by evaluating the phase relationship of the heart rate and respiration the coupling between different frequency bands of the electroencephalogram (EEG) or the amplitude-phase coupling, has been well explored in the neuroscience society, among others. In biomedicine, the phase contains crucial physiological information. It is closely related to the notion of frequency and period. Roughly speaking, the phase of an oscillatory time series is a value that describes the signal’s status within the span of a single oscillatory period ( Figure 1a). In this paper, we focus on discussing the phase function of an oscillatory (or periodic) time series (or signal). In the flow-volume loop example, the parameters of phase are the pressure and volume, which jointly describe the respiratory dynamics. The flow-volume loop or pressure-volume diagram are common examples that physicians commonly use in clinics for decision making. In a very general sense, we may collect all possible statuses of the system and call the collection the phase space, which might be a high dimensional space. It has been studied over centuries and is a standard material covered in almost all scientific fields. Phase is the most fundamental physical quantity when we study the dynamics of a system of interest. We expect its scientific impact on a broad range of applications. In conclusion, the proposed approach resolves the above-mentioned scientific challenge. 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. 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. 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. This fact might challenge reproducibility, communication, and scientific interpretation and thus we need a standardized approach with theoretical support over a unified model. 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. 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. There are many tools aiming to estimate phase, most of them are developed based on the analytic function model. Phase is the most fundamental physical quantity when we study an oscillatory time series.
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