# American Institute of Mathematical Sciences

September  2022, 4(3): 355-393. doi: 10.3934/fods.2022010

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

 1 Department of Anesthesiology, Yale University, New Haven, Connecticut, USA 2 Department of Thoracic Medicine, Chang Gung Memorial Hospital, Linkou, College of Medicine, Chang Gung University, Taoyuan, Taiwan 3 Department of Mathematics, Duke University, Durham, North Carolina, USA 4 Department of Statistical Science, Duke University, Durham, North Carolina, USA

*Corresponding author: Hau-Tieng Wu

The authors would like to thank the anonymous reviewers for their helpful comments to improve the manuscript

Received  October 2021 Revised  February 2022 Published  September 2022 Early access  May 2022

Phase is the most fundamental physical quantity when we study an oscillatory time series. There have been many tools aiming to estimate phase, and most of them are developed based on the analytic function model. Unfortunately, these analytic function model based tools might be limited in handling modern signals with intrinsic nonstartionary structure, for example, biomedical signals composed of multiple oscillatory components, each with time-varying frequency, amplitude, and non-sinusoidal oscillation. There are several consequences of such limitation, and we specifically focus on the one that 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. 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. 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.

Citation: Aymen Alian, Yu-Lun Lo, Kirk Shelley, Hau-Tieng Wu. Reconsider phase reconstruction in signals with dynamic periodicity from the modern signal processing perspective. Foundations of Data Science, 2022, 4 (3) : 355-393. doi: 10.3934/fods.2022010
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(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 $2\pi$. 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. 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 $5\pi/4$. The phase 0 is associated with the R peaks, and the phase $5\pi/4$ is associated with the T end. In our proposed standardization, the phase is defined as a monotonically increasing function. This might cause some confusion in the illustrated phase in the red curve. Since the phase is the argument (or the angle) of the complex value, a phase value θ is not different from a phase value $\theta+2k\pi$ for any integer k. Therefore, one way to plot the phase function is plotting the residue of the phase function by finding its value modulo $2\pi$. (b) A summary of models for the definition of phase function. The traditional phase is shown on the right-hand side, and the proposed fundamental phase is shown on the left-hand side
(a) illustration of simulated signals satisfying different models. Top: the harmonic model in equation (1) $f(t) = e^{i2\pi t}$; middle: the adaptive harmonic model in equation (3) $f(t) = (1+0.3\cos⁡(t)) e^{i2\pi(t+0.02t^{2.5})}$; bottom: the wave-shape model in equation (5) with $L = 1$, $f(t) = e^{i2\pi(t+0.02t^{2.5})}+0.95e^{i[4\pi(t+0.02t^{2.5} )+3]}$. The black and red curves are the real and imaginary parts of the complex form, respectively. (b) An illustration of various respiratory signals recorded simultaneously. The morphologies of those respiratory signals are different, and this difference is mainly encoded in the multiples. (c) An illustration of photoplethymogram (PPG) and peripheral venous pressure (PVP) signals recorded simultaneously during the lower body negative pressure (LBNP) experiment. The four signals from top to bottom are the PPG signal during the baseline, the PPG signal during the -75 mmHg negative pressure, the PVP signal during the baseline, and the PVP signal during the -75 mmHg negative pressure. The morphologies of the same channels are different during different physiological status, and this difference is mainly encoded in the multiples
Demonstration of two typical algorithms. (a) An example of determining phases of various respiratory signals by the Hilbert transform followed by the band-pass filter (BPF-HT), where the band is 0.1-0.4 Hz determined by the dominant respiratory frequency. The phases (plotted with a mod of $10\pi$ to enhance the visualization), and hence their derivatives, are not consistent from one sensor to another. More importantly, while the definition of instantaneous frequency is mathematically correct, physiologically it is not sensible. The result by the continuous wavelet transform with a fixed spectral band (CWT-fixed) is similar and not shown. (b) An example of determining phases of various respiratory signals by the synchrosqueezing transform (SST). From left to right: the time-frequency representations of the pressure, flow, end-tidal CO2, and impedance signals determined by SST. (c) Top panel: the phases of four respiratory signals estimated by SST (plotted with a mod of 10π to enhance the visualization). Bottom panel: the instantaneous frequency of four respiratory signals. The respiratory phases estimated from different sensors are more consistent, except a global phase shift. Also, the derivatives, which is the instantaneous frequency by our definition, are consistent. The results by CWT with a time-varying spectral band (CWT-tvBPF) and BKD are similar and not shown
An illustration of the fundamental phase estimation by the proposed tvBPF with SST in a simulated database with ground truth, and the results of other methods, including the Hilbert transform with fixed band-pass filter (HT), the Blaschke decomposition (BKD) and the variational mode decomposition (VMD). (a) one realization of the clean signal $f = f_1+f_2$ contaminated by noise, denoted as Y; (b) the time-frequency representation (TFR) of Y; (c) the extracted instantaneous frequency (IF) of $f_1$ is superimposed as the red dashed curve; (d) the reconstructed fundamental component of $f_1$ (blue) superimposed on the true fundamental component of $f_1$ (gray); (e) the reconstructed $f_1$, denoted as $\tilde{f}_1$ by apply the shape adaptive mode decomposition (red) superimposed on the true $f_1$ (gray); (f) $Y-\tilde{f}_1$, which is an estimate of $f_2$; (g) the TFR of $Y-\tilde{f}_1$; (h) the extracted IF of $f_2$ is superimposed as the red dashed curve; (i) the reconstructed fundamental component of $f_2$; (j) the difference of the estimated fundamental phase of $f_1$ and the ground true; (k) the difference of the estimated fundamental phase of $f_2$ and the ground true; (l)(n)(p): the estimated fundamental component of $f_1$ by VDM, BKD and HT (blue) superimposed on the true fundamental component of $f_1$ (gray); (m)(o)(q): the estimated fundamental component of $f_2$ by VDM, BKD and HT (blue) superimposed on the true fundamental component of $f_2$ (gray)
A comparison of different phase reconstruction methods from different respiratory sensors. (a) Three different respiratory signals. (b) The time-frequency representation (TFR) of the flow, impedance and end-tidal CO2 (EtCO2) signals, which were determined by CWT (top and middle) and SST (bottom) respectively. The fixed spectral band for CWT (CWT-fixed) reconstruction is shown in the top subfigure as red lines. The time-varying spectral bands for CWT and SST are plotted as dashed red curves above and below the solid red curve indicating the instantaneous frequency in the middle and bottom plots. (c) The reconstructed fundamental component by different methods. Each column is associated with one signal, and each row is associated with one method. (d) The phase estimated by CWT-fixed mod by π is shown in the top row, where the phases are different. The phase difference between two phases estimated from two sensors is shown in the bottom row. (e) Same as (d), but the phase is estimated by SST. It is obvious that the fluctuation is less compared with the CWT-fixed method. The phase estimated by CWT with the time-varying bandpass filter is similar (not shown)
An illustration of the potential problem the Hilbert transform might encounter. The first signal, the real part of $f_1 (t) = e^{i2\pi(t+0.02t^{2.5})}+0.95e^{i[4\pi(t+0.02t^{2.5})+3]}$, is shown on the left top panel, and the second signal, the real part of $f_2 (t) = e^{i2\pi(t+0.02t^{2.5})}+1.01e^{i[4\pi(t+0.02t^{2.5} )+3]}$ is shown on the left bottom panel. Note that the smaller trough of the real part of $f_2 (t)$ crosses zero since the first harmonic has a larger amplitude than the fundamental component. In the left middle columns, the red lines are the difference of the phase of the fundamental component, $\phi(t) = 2\pi(t+0.02t^{2.5} )$, and the estimated phase $\tilde{\phi}(t)$ by directly applying the Hilbert transform. It is clear that for $f_1 (t)$, there is an obvious zig-zag difference between the estimated phase function $\tilde{\phi}(t)$ and the proposed phase function $\phi(t)$. However, for $f_2 (t)$, there is an obvious cumulative phase estimation error as time evolves. This comes from the winding number issue due to the larger amplitude of the first harmonic. Due to the frequency modulation, the bandpass filter or continuous wavelet transform does not work as well (results not shown). In the right middle columns, the time-frequency representations of $f_1 (t)$ and $f_2 (t)$ determined by the synchrosqueezing transform (SST) are shown. In the right column, the blue lines are the difference of the estimated phase by SST and the proposed phase function $\phi(t)$. SST provides a consistent phase estimation
Demonstration of decomposed components for the standard phase estimation by different algorithms. (a) is the end-tidal CO2 (EtCO2) signal, (b) is the flow signal (AWF), and (c) is the impedance (Impe) signal
An illustration of the fundamental phase estimation by the proposed tvBPF with SST in a photoplethysmogram (PPG) signal (shown in (a)) with experts' labels on the simultaneously recorded electrocardiogram (ECG) and end-tidal CO2 (EtCO2) signals. The experts' labels lead to the ground truth phases of the respiratory and cardiac dynamics. The results of other methods, including the Hilbert transform with fixed band-pass filter (HT), the Blaschke decomposition (BKD) and the variational mode decomposition (VMD) are also shown. (b, d) the time-frequency representation (TFR) of the PPG signal shown in (a), where the extracted instantaneous frequency (IF) of the respiratory component is superimposed in (d) as a red curve. (c, e) the time-frequency representation (TFR) of the PPG signal subtracted by the reconstructed respiratory component, where the extracted instantaneous frequency (IF) of the cardiac component is superimposed in (e) as a red curve. (f, g) the reconstructed fundamental components of the respiratory and cardiac components are shown in blue curves, with the simultaneously recorded ECG and EtCO2 superimposed as gray curves respectively. (h) the phase differences between the estimated fundamental phases and the ground truth phases for the respiratory and cardiac components are shown in blue and red curves respectively. (i-k) the same as (f-h) but generated by HT. (l-n) the same as (f-h) but generated by BKD. (o-q) the same as (f-h) but generated by VMD
(a) An example of estimating phases of photoplethymogram (PPG) and peripheral venous pressure (PVP) signals by the Hilbert transform followed by a band-pass filter (BPF-HT) with different spectral bands, where the derivatives of the estimated phases, the instantaneous frequencies, are shown for a comparison. In the top panel, the spectral band is 0.5-2 Hz, while in the bottom panel, the spectral band is 0.5-4 Hz. The lower spectral bound 0.5 Hz is chosen to avoid the respiratory component impact, and the upper spectral bound 2 or 4 Hz is chosen to include sufficient cardiac information. However, in both cases, the phases, and hence their derivatives, are not consistent between the PPG and PVP signals. The difference is clear over 1300-1600 seconds. This comes from the fact that the heart rate increases dramatically during 1300-1600 seconds during the lower body negative pressure experiment, and the chosen spectral bands, if fixed, may include insufficient fundamental component information or may be contaminated by its multiples. (b) An example of the time-frequency representations (TFRs) determined by the synchrosqueezing transform (SST). The TFR of the photoplethymogram (PPG) is shown on the left and the TFR of the peripheral venous pressure (PVP) is shown on the right. (c) the phases and the associated instantaneous frequency of the cardiac component in the PPG and PVP signals. The phases associated with the PPG and PVP signals are shown on the top, and the associated instantaneous frequency of the hemodynamic oscillation determined by differentiating the estimated phases are shown in the bottom. The phases of PPG and PVP signals are consistent. Also, the derivatives, which is the instantaneous frequency by our definition, are consistent, except for some unwanted spikes in the PVP signal. This comes from the well-known fact that the PVP signal is usually too noisy to analyze with traditional tools
The median and median absolute error (MAD) of phase difference of all subjects among eight algorithms and three respiratory signals. The p value was evaluated by the Mann-Whitney Wilcoxon test when comparing with results obtained by SST. *: statistically significant with the level 0.05 corrected by the Bonferroni correction. Abbreviation: Impe: Impedance; ETCO2: End tidal CO2; BPF-HT: Hilbert transform followed by a bandpass filter; the Blaschke decomposition (BKD), the empirical mode decomposition (EMD), the ensemble empirical mode decomposition (EEMD), the variational mode decomposition (VMD), CWT-fixed: Continuous wavelet transform with a fixed spectral band; CWT-tvBPF: Continuous wavelet transform with a time varying bandpass filter; SST: Synchrosqueezing transform
Summary of Bland-Altman (BA) plot of the cardiopulmonary coupling index determined from different phase reconstruction methods, including the Hilbert transform followed by a bandpass filter (BPF-HT), the Blaschke decomposition (BKD), the empirical mode decomposition (EMD), the ensemble empirical mode decomposition (EEMD), the variational mode decomposition (VMD), the continuous wavelet transform (CWT) with a fixed spectral band (CWT-fixed), the CWT with the proposed time-varying bandpass filter (CWT-tvBPF) and the synchrosqueezing transform (SST), and different respiratory signals, including flow, impedance (Impe) and End tidal CO2 (ETCO2). The comparison suggests that the agreements of respiratory phases determined by CWT-tvBPF and SST are higher than other methods. CI means confidence interval
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