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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 
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 timevarying frequency, amplitude, and nonsinusoidal 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 abovementioned intrinsic nonstartionary structure, we introduce the adaptive nonharmonic model (ANHM), provide a definition of phase called fundamental phase, which is a vectorvalued function describing the dynamics of all oscillatory components in the signal, and suggest a timevarying bandpass filter (tvBPF) scheme based on timefrequency analysis tools to estimate the fundamental phase. The proposed approach is validated with a simulated database and a realworld database with experts' labels, and it is applied to two realworld databases, each of which has biomedical signals recorded from different sensors, to show how to standardize the definition of phase in the realworld 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 abovementioned scientific challenge. We expect its scientific impact on a broad range of applications.
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show all references
References:
[1] 
CapnoBase IEEE TBME Respiratory Rate Benchmark, 2022, Accessed from: https://dataverse.scholarsportal.info/dataset.xhtml?persistentId=doi:10.5683/SP2/NLB8IT. 
[2] 
Matlab code for Ensemble Empirical Mode Decomposition (EEMD), 2022, Accessed from: https://github.com/benpolletta/HHTTutorial/tree/master/HuangEMD. 
[3] 
Matlab code of Blaschke decomposition (BKD), 2022, Accessed from: https://github.com/hautiengwu/BlaschkeDecomposition. 
[4] 
Matlab code used in Section 4, 2022, Accessed from: https://github.com/hautiengwu/ReconsiderPhase. 
[5] 
The TimeFrequency Toolbox, (TFTB), 2022, Accessed from: http://tftb.nongnu.org. 
[6] 
WAVELAB850, 2022, Accessed from: https://statweb.stanford.edu/ wavelab/. 
[7] 
A. A. Alian, N. J. Galante, N. S. Stachenfeld, D. G. Silverman and K. H. Shelley, Impact of central hypovolemia on photoplethysmographic waveform parameters in healthy volunteers part 2: Frequency domain analysis, J. Clinical Monitoring and Computing, 25 (2011), 387396. doi: 10.1007/s108770119317x. 
[8] 
A. A. Alian and K. H. Shelley, Photoplethysmography, Best Practice & Research Clinical Anaesthesiology, 28 (2014), 395406. doi: 10.1016/j.bpa.2014.08.006. 
[9] 
P. Ashwin, S. Coombes and R. Nicks, Mathematical frameworks for oscillatory network dynamics in neuroscience, J. Math. Neurosci., 6 (2016), 192. doi: 10.1186/s1340801500336. 
[10] 
R. P. Bartsch, A. Y. Schumann, J. W. Kantelhardt, T. Penzel and P. C. Ivanov, Phase transitions in physiologic coupling, Proceedings of the National Academy of Sciences, 109 (2012), 1018110186. doi: 10.1073/pnas.1204568109. 
[11] 
E. Bedrosian, The analytic signal representation of modulated waveforms, Proc. IRE, 50 (1962), 20712076. doi: 10.1109/JRPROC.1962.288236. 
[12] 
E. Bedrosian, A product theorem for hilbert transforms, Proceedings of the IEEE, 5 (1963), 868869. doi: 10.1109/PROC.1963.2308. 
[13] 
G. Benchetrit, Breathing pattern in humans: Diversity and individuality, Respiration Physiology, 122 (2000), 123129. doi: 10.1016/S00345687(00)001547. 
[14] 
M. Chavez, M. Besserve, C. Adam and J. Martinerie, Towards a proper estimation of phase synchronization from time series, J. Neuroscience Methods, 154 (2006), 149160. doi: 10.1016/j.jneumeth.2005.12.009. 
[15] 
Y.C. Chen, M.Y. Cheng and H.T. Wu, Nonparametric and adaptive modeling of dynamic seasonality and trend with heteroscedastic and dependent errors, J. Roy. Stat. Soc. B, 76 (2014), 651682. doi: 10.1111/rssb.12039. 
[16] 
L. Cohen, Timefrequency distributionsa review, Proceedings of the IEEE, 77 (1989), 941981. doi: 10.1109/5.30749. 
[17] 
R. R. Coifman and S. Steinerberger, Nonlinear phase unwinding of functions, J. Fourier Anal. Appl., 23 (2017), 778809. doi: 10.1007/s0004101694893. 
[18] 
R. R. Coifman, S. Steinerberger and H.T. Wu, Carrier frequencies, holomorphy, and unwinding, SIAM J. Math. Anal., 49 (2017), 48384864. doi: 10.1137/16M1081087. 
[19] 
M. A. Colominas and H.T. Wu, Decomposing nonstationary signals with timevarying waveshape functions, IEEE Trans. Signal Process., 69 (2021), 50945104. doi: 10.1109/TSP.2021.3108678. 
[20] 
I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. doi: 10.1137/1.9781611970104. 
[21] 
I. Daubechies, J. Lu and H.T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decompositionlike tool, Appl. Comput. Harmon. Anal., 30 (2011), 243261. doi: 10.1016/j.acha.2010.08.002. 
[22] 
I. Daubechies, Y. Wang and H.T. Wu, ConceFT: Concentration of frequency and time via a multitapered synchrosqueezing transform, Philos. Trans. Roy. Soc. A, 374 (2016), 20150193, 19 pp. doi: 10.1098/rsta.2015.0193. 
[23] 
K. Dragomiretskiy and D. Zosso, Variational mode decomposition, IEEE Trans. Signal Process., 62 (2014), 531544. doi: 10.1109/TSP.2013.2288675. 
[24] 
D. Dvorak and A. A. Fenton, Toward a proper estimation of phase–amplitude coupling in neural oscillations, J. Neuroscience Methods, 225 (2014), 4256. doi: 10.1016/j.jneumeth.2014.01.002. 
[25] 
M. Feldman, Timevarying vibration decomposition and analysis based on the hilbert transform, J. Sound and Vibration, 295 (2006), 518530. doi: 10.1016/j.jsv.2005.12.058. 
[26]  P. Flandrin, TimeFrequency/TimeScale Analysis, Wavelet Analysis and its Applications, 10. Academic Press, Inc., San Diego, CA, 1999. 
[27] 
D. Gabor, Theory of communication. part 1: The analysis of information, J. Institution of Electrical EngineersPart III: Radio and Communication Engineering, 93 (1946), 429441. 
[28] 
J. Garnett, Bounded Analytic Functions, volume 236., Springer, New York, 2007. 
[29] 
H. Gesche, D. Grosskurth, G. Küchler and A. Patzak, Continuous blood pressure measurement by using the pulse transit time: Comparison to a cuffbased method, European J. Applied Physiology, 112 (2012), 309315. doi: 10.1007/s0042101119833. 
[30] 
A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiology, 117 (1952), 500. 
[31] 
J. Huang, Y. Wang and L. Yang, Vakman's problem and the extension of hilbert transform, Appl. Comput. Harmon. Anal., 34 (2013), 308316. doi: 10.1016/j.acha.2012.08.009. 
[32] 
N. E. Huang, et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis, Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 903995. doi: 10.1098/rspa.1998.0193. 
[33] 
Y.C. Huang, T.Y. Lin, H.T. Wu, P.J. Chang, C.Y. Lo, T.Y. Wang, C.H. S. Kuo, S.M. Lin, F.T. Chung and H.C. Lin, et al., Cardiorespiratory coupling is associated with exercise capacity in patients with chronic obstructive pulmonary disease, BMC Pulmonary Medicine, 21 (2021), 110. doi: 10.1186/s12890021014001. 
[34]  J. Keener and J. Sneyd, Mathematical Physiology 1: Cellular Physiology, 2 edition, Springer, New York, 2009. doi: 10.1007/9780387793887. 
[35] 
D. Khodagholy, J. N. Gelinas and G. Buzsáki, Learningenhanced coupling between ripple oscillations in association cortices and hippocampus, Science, 358 (2017), 369372. doi: 10.1126/science.aan6203. 
[36] 
S.H. Kim, J.G. Song, J.H. Park, J.W. Kim, Y.S. Park and G.S. Hwang, Beattobeat tracking of systolic blood pressure using noninvasive pulse transit time during anesthesia induction in hypertensive patients, Anesthesia & Analgesia, 116 (2013), 94100. doi: 10.1213/ANE.0b013e318270a6d9. 
[37] 
R. Klabunde, Cardiovascular Physiology Concepts, Lippincott Williams & Wilkins, 2011. 
[38] 
J.P. Lachaux, E. Rodriguez, J. Martinerie, C. Adam, D. Hasboun and F. J. Varela, A quantitative study of gammaband activity in human intracranial recordings triggered by visual stimuli, European J. Neuroscience, 12 (2000), 26082622. doi: 10.1046/j.14609568.2000.00163.x. 
[39] 
M. Le Van Quyen, J. Foucher, J.P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie and F. J. Varela, Comparison of hilbert transform and wavelet methods for the analysis of neuronal synchrony, J. Neuroscience Methods, 111 (2001), 8398. 
[40] 
C.Y. Lin, L. Su and H.T. Wu, Waveshape function analysis–when cepstrum meets timefrequency analysis, J. Fourier Anal. Appl., 24 (2018), 451505. doi: 10.1007/s0004101795230. 
[41] 
Y.T. Lin, J. Malik and H.T. Wu, Waveshape oscillatory model for nonstationary periodic time series analysis, Foundations of Data Science, 3 (2021), 99131. doi: 10.3934/fods.2021009. 
[42] 
Y.T. Lin, H.T. Wu, J. Tsao, H.W. Yien and S.S. Hseu, Timevarying spectral analysis revealing differential effects of sevoflurane anaesthesia: Nonrhythmictorhythmic ratio, Acta Anaesthesiol. Scand., 58 (2014), 157167. doi: 10.1111/aas.12251. 
[43] 
S. Luo, W. J. Tompkins and J. G. Webster, Cardiogenic artifact cancellation in apnea monitoring, In Proceedings of 16th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, IEEE, 2 (1994), 968–969. 
[44] 
S. Meignen, D.H. Pham and S. McLaughlin, On demodulation, ridge detection, and synchrosqueezing for multicomponent signals, IEEE Trans. Signal Process., 65 (2017), 20932103. doi: 10.1109/TSP.2017.2656838. 
[45] 
M. R. Miller, J. Hankinson, V. Brusasco, F. Burgos, R. Casaburi, A. Coates, R. Crapo, P. Enright, C. Van Der Grinten and P. Gustafsson, et al. Standardisation of spirometry, European Respiratory Journal, 26 (2005), 319338. doi: 10.1183/09031936.05.00034805. 
[46]  J. D. Murray, Mathematical Biology: I. An Introduction. Interdisciplinary Applied Mathematics, SpringerVerlag, New York, 2002. 
[47] 
M. R. Nahon, Phase Evaluation and Segmentation, Yale University, 2000. 
[48] 
R. Nevanlinna, The first main theorem in the theory of meromorphic functions, Analytic Functions, 162 (1970), 162180. doi: 10.1007/9783642855900_7. 
[49] 
A. H. Nuttall, On the quadrature approximation to the hilbert transform of modulated signals, Proc. IEEE, 54 (1966), 14581459. doi: 10.1109/PROC.1966.5138. 
[50] 
A. V. Oppenhein, R. W. Schafer and J. R. Buck, Discretetime signal processing, Prince Hall, Sec, 11 (1999). 
[51] 
M. Peltola, Role of editing of rr intervals in the analysis of heart rate variability, Frontiers in Physiology, 3 (2012), 148. 
[52] 
B. Picinbono, On instantaneous amplitude and phase of signals, IEEE Transactions on Signal Processing, 45 (1997), 552560. doi: 10.1109/78.558469. 
[53] 
A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov and J. Kurths, Phase synchronization of chaotic oscillators by external driving, Physica D: Nonlinear Phenomena, 104 (1997), 219238. doi: 10.1016/S01672789(96)003016. 
[54] 
T. Qian, Intrinsic monocomponent decomposition of functions: An advance of Fourier theory, Math. Methods Appl. Sci., 33 (2010), 880891. doi: 10.1002/mma.1214. 
[55] 
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