Abstract
Cardiovascular signals are intrinsically non-stationary and interact through dynamic mechanisms to maintain blood pressure homeostasis in response to internal and external perturbations. The assessment of changes in cardiovascular signals and in their interactions provides valuable information regarding the cardiovascular function. Time-frequency analysis is a useful tool to study the time-varying nature of the cardiovascular system because it provides a joint representation of a signal in the temporal and spectral domain that allows to track the instantaneous frequency, amplitude and phase of non-stationary processes. The time-frequency distributions described in this chapter belong to the Cohen’s class, and can be derived from the Wigner-Ville distribution, which represents the fundamental basis of this unified framework. Time-frequency analysis can be extended to the study of the dynamic interactions between two or more non-stationary processes. Time-frequency coherence, phase-delay, phase-locking and partial-spectra are estimators that assess changes in the coupling and phase shift of signals generated by a complex system.
This chapter introduces the reader to multivariate time-frequency analysis and covers both theoretical and practical aspects. The application of these methodologies in the study of the dynamic interactions between the most important variables of the cardiovascular function is discussed. In the introduction, classical spectral analysis of cardiovascular signals is reviewed along with its physiological interpretation. The limitations of this framework provides a motivation for implementing non-stationary tools. In the first section, time-frequency representations based on the Wigner-Ville distribution are introduced and important aspects, such as the interference cross-terms and their elimination, the time and frequency resolution and the estimation of time-frequency spectra, are described. The second section describes algorithms to assess the dynamic interactions between non-stationary signals, including time-frequency coherence, phase delay and partial spectra, while the third section provides examples of multivariate time-frequency analysis of cardiovascular data recorded during a standard test to induce an autonomic response.
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References
Borgnat, P., Flandrin, P., Honeine, P., Richard, C., Xiao, J.: Testing stationarity with surrogates: A time-frequency approach. IEEE Trans. Signal Process. 58(7), 3459–3470 (2010)
Orini, M., Laguna, P., Mainardi, L.T., Bailón, R.: Assessment of the dynamic interactions between heart rate and arterial pressure by the cross time-frequency analysis. Physiol. Meas. 33, 315–331 (2012)
Mainardi, L.T., Bianchi, A.M., Baselli, G., Cerutti, S.: Pole-tracking algorithms for the extraction of time-variant heart rate variability spectral parameters. IEEE Trans. Biomed. Eng. 42(3), 250–259 (1995). ID: 1
Mainardi, L.T., Bianchi, A.M., Furlan, R., Piazza, S., Barbieri, R., di Virgilio, V., Malliani, A., Cerutti, S.: Multivariate time-variant identification of cardiovascular variability signals: a beat-to-beat spectral parameter estimation in vasovagal syncope. IEEE Trans. Biomed. Eng. 44, 978–989 (1997)
Barbieri, R., Brown, E.: Analysis of heartbeat dynamics by point process adaptive filtering. IEEE Trans. Biomed. Eng. 53(1), 4–12 (2006)
Welch, P.: The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15, 70–73 (1967)
Thomson, D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70(9), 1055–1096 (1982)
Akselrod, S., Gordon, D., Ubel, F.A., Shannon, D.C., Barger, A.C., Cohen, R.J.: Power spectrum analysis of heart rate fluctuations: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222 (1981)
Saul, J.P., Arai, Y., Berger, R.D., Lilly, L.S., Colucci, W.S., Cohen, R.J.: Assessment of autonomic regulation in chronic congestive heart failure by heart rate spectral analysis. Am. J. Cardiol. 61(15), 1292–1299 (1988)
Task Force of the European Society of Cardiology the North American Society of Pacing: Heart rate variability: standards of measurement, physiological interpretation, and clinical use. Circulation 93(5), 1043–1065 (1996)
Montano, N., Ruscone, T.G., Porta, A., Lombardi, F., Pagani, M., Malliani, A.: Power spectrum analysis of heart rate variability to assess the changes in sympathovagal balance during graded orthostatic tilt. Circulation 90, 1826–1831 (1994)
Malliani, A., Pagani, M., Lombardi, F., Cerutti, S.: Cardiovascular neural regulation explored in the frequency domain. Circulation 84, 482–492 (1991)
Piccirillo, G., Ogawa, M., Song, J., Chong, V., Joung, B., Han, S., Magra, D., Chen, L., Lin, S.-F., Chen, P.-S.: Power spectral analysis of heart rate variability and autonomic nervous system activity measured directly in healthy dogs and dogs with tachycardia-induced heart failure. Heart Rhythm. 6(4), 546–552 (2009)
Orini, M., Bailon, R., Mainardi, L.T., Laguna, P., Flandrin, P.: Characterization of dynamic interactions between cardiovascular signals by time-frequency coherence. IEEE Trans. Biomed. Eng. 59, 663–673 (2012)
Grossman, P., Taylor, E.W.: Toward understanding respiratory sinus arrhythmia: Relations to cardiac vagal tone, evolution and biobehavioral functions. Biol. Psychol. 74, 263–285 (2007)
Eckberg, D.L.: Point: counterpoint: respiratory sinus arrhythmia is due to a central mechanism vs. respiratory sinus arrhythmia is due to the baroreflex mechanism. J. Appl. Physiol. 106, 1740–1742 (2009); discussion 1744
Julien, C.: The enigma of Mayer waves: facts and models. Cardiovasc. Res. 70, 12–21 (2006)
Eckberg, D.L.: Sympathovagal balance: a critical appraisal. Circulation 96(9), 3224–3232 (1997)
Malliani, A., Pagani, M., Montano, N., Mela, G.S.: Sympathovagal balance: a reappraisal. Circulation 98(23), 2640–2643 (1998)
Malliani, A., Julien, C., Billman, G.E., Cerutti, S., Piepoli, M.F., Bernardi, L., Sleight, P., Cohen, M.A., Tan, C.O., Laude, D., Elstad, M., Toska, K., Evans, J.M., Eckberg, D.L.: Cardiovascular variability is/is not an index of autonomic control of circulation. J. Appl. Physiol. 101(2), 684–688 (2006)
Flandrin, P.: Time-Frequency/Time-Scale Analysis. Academic Press, New York (1999)
Cohen, L.: Time-Frequency Analysis, vol. 778. Prentice Hall, Englewood Cliffs (1995)
Hlawatsch, F., Auger, F., (eds.): Time-Frequency Analysis, Concepts and Methods. Wiley, London (2008)
Cohen, L.: Time-frequency distributions-a review. Proc. IEEE 77, 941–981 (1989)
Hlawatsch, F., Boudreaux-Bartels, G.F.: Linear and quadratic time-frequency signal representations. IEEE Signal Process. Mag. 9, 21–67 (1992)
Auger, F., Flandrin, P., Gonçalvès, P., Olivier, L.: Time-Frequency Toolbox. Technical Report http://www.nongnu.org/tftb/
Flandrin, P.: Ambiguity Function. Time-Frequency Signal Analysis and Processing, pp. 160–167. Elsevier, Amsterdam (2003)
Auger, F., Chassande-Mottin, É.: Quadratic Time-Frequency Analysis I: Cohen’s Class, pp. 131–163. ISTE, London (2010)
Jeong, J., Williams, W.J.: Kernel design for reduced interference distributions. IEEE Trans. Signal Process. 40(2), 402–412 (1992)
Baraniuk, R.G., Jones, D.L.: A signal-dependent time-frequency representation: optimal kernel design. IEEE Trans. Signal Process. 41, 1589–1602 (1993)
Baraniuk, R.G., Jones, D.L.: A signal-dependent time-frequency representation: fast algorithm for optimal kernel design. IEEE Trans. Signal Process. 42(1), 134–146 (1994)
Cunningham, G.S., Williams, W.J.: Kernel decomposition of time-frequency distributions. IEEE Trans. Signal Process. 42(6), 1425–1442 (1994)
Costa, A., Boudreau-Bartels, G.: Design of time-frequency representations using a multiform, tiltable exponential kernel. IEEE Trans. Signal Process. 43, 2283–2301 (1995)
Arce, G.R., Hasan, S.R.: Elimination of interference terms of the discrete wigner distribution using nonlinear filtering. IEEE Trans. Signal Process. 48(8), 2321–2331 (2000)
Aviyente, S., Williams, W.J.: Multitaper marginal time-frequency distributions. Signal Process. 86(2), 279–295 (2006)
Wahlberg, P., Hansson, M.: Kernels and multiple windows for estimation of the Wigner-Ville spectrum of gaussian locally stationary processes. IEEE Trans. Signal Process. 55(1), 73–84 (2007)
Hlawatsch, F.: Interference terms in the Wigner distribution. In: Cappellini, V., Constantinides, A. (eds.) Digital Signal Processing, vol. 84, pp. 363–267. North-Holland, Amsterdam (1984)
Hlawatsch, F., Flandrin, P.: The interference structure of the Wigner distribution and related time-frequency signal representations. The Wigner Distribution - Theory and Applications in Signal Processing, pp. 59–113. Elsevier, Amsterdam (1997)
Flandrin, P.: Some features of time-frequency representations of multicomponent signals. In: Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP’84, vol. 9, pp. 266–269 (1984)
Orini, M., Bailón, R., Mainardi, L., Mincholé, A., Laguna, P.: Continuous quantification of spectral coherence using quadratic time-frequency distributions: error analysis and application. In: International Conference on Computers in Cardiology, pp. 681–684 (2009)
Martin, W., Flandrin, P.: Wigner-ville spectral analysis of nonstationary processes. IEEE Trans. Acoust. Speech Signal Process. 33(6), 1461–1470 (1985), ID: 1
Janssen, A., Claasen, T.: On positivity of time-frequency distributions. IEEE Trans. Acoust. Speech Signal Process. 33, 1029–1032 (1985)
Matz, G., Hlawatsch, F.: Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations. IEEE Trans. Inf. Theory 52(3), 1067–1086 (2006). ID: 1
Mainardi, L.T.: On the quantification of heart rate variability spectral parameters using time-frequency and time-varying methods. Philos. Transact. A Math. Phys. Eng. Sci. 367(1887), 255–275 (2009)
Pola, S., Macerata, A., Emdin, M., Marchesi, C.: Estimation of the power spectral density in nonstationary cardiovascular time series: assessing the role of the time-frequency representations (TFR). IEEE Trans. Biomed. Eng. 43, 46–59 (1996)
Orini, M., Bailon, R., Laguna, P., Mainardi, L., Barbieri, R.: A multivariate time-frequency method to characterize the influence of respiration over heart period and arterial pressure. EURASIP J. Adv. Signal Process. 2012(1), 214 (2012)
Carter, G.C.: Coherence and time delay estimation. Proc. IEEE 75, 236–255 (1987)
Di Rienzo, M., Parati, G., Radaelli, A., Castiglioni, P.: Baroreflex contribution to blood pressure and heart rate oscillations: time scales, time-variant characteristics and nonlinearities. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367, 1301–1318 (2009)
Orini, M., Mainardi, L.T., Gil, E., Laguna, P., Bailon, R.: Dynamic assessment of spontaneous baroreflex sensitivity by means of time-frequency analysis using either rr or pulse interval variability. Conf. Proc. IEEE Eng. Med. Biol. Soc. 1, 1630–1633 (2010)
Keissar, K., Maestri, R., Pinna, G.D., Rovere, M.T.L., Gilad, O.: Non-invasive baroreflex sensitivity assessment using wavelet transfer function-based time-frequency analysis. Physiol. Meas. 31, 1021–1036 (2010)
Gil, E., Orini, M., Bailon, R., Vergara, J.M., Mainardi, L., Laguna, P.: Photoplethysmography pulse rate variability as a surrogate measurement of heart rate variability during non-stationary conditions. Physiol. Meas. 31(9), 1271–1290 (2010)
Matz, G., Hlawatsch, F.: Time-frequency coherence analysis of nonstationary random processes. In: Proceedings of the Tenth IEEE Workshop Statistical Signal and Array Processing, pp. 554–558 (2000)
White, L.B., Boashash, B.: Cross spectral analysis of nonstationary processes. IEEE Trans. Inf. Theory 36, 830–835 (1990)
Faes, L., Pinna, G.D., Porta, A., Maestri, R., Nollo, G.: Surrogate data analysis for assessing the significance of the coherence function. IEEE Trans. Biomed. Eng. 51, 1156–1166 (2004)
Schreiber, T., Schmitz, A.: Surrogate time series. Phys. D 142(3–4), 346–382 (2000)
Lachaux, J.P., Rodriguez, E., Martinerie, J., Varela, F.J.: Measuring phase synchrony in brain signals. Hum. Brain Mapp. 8(4), 194–208 (1999)
Lachaux, J., Rodriguez, E., Quyen, M.L.V., Lutzand, A., Martinerie, J., Varela, F.: Studying single-trials of phase-synchronous activity in the brain. Int. J. Bifurcation Chaos 10, 2429–2439 (2000)
Quyen, M.L.V., Foucher, J., Lachaux, J., Rodriguez, E., Lutz, A., Martinerie, J., Varela, F.J.: Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. J. Neurosci. Methods 111, 83–98 (2001)
Rudrauf, D., Douiri, A., Kovach, C., Lachaux, J.-P., Cosmelli, D., Chavez, M., Adam, C., Renault, B., Martinerie, J., Quyen, M.L.V.: Frequency flows and the time-frequency dynamics of multivariate phase synchronization in brain signals. Neuroimage 31, 209–227 (2006)
Aviyente, S., Bernat, E.M., Evans, W.S., Sponheim, S.R.: A phase synchrony measure for quantifying dynamic functional integration in the brain. Hum. Brain Mapp. 32(1), 80–93 (2011)
Aviyente, S., Evans, W.S., Bernat, E.M., Sponheim, S.: A time-varying phase coherence measure for quantifying functional integration in the brain. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing ICASSP 2007, vol. 4, pp. IV–1169–IV–1172, 15–20 (2007)
Shin, Y., Gobert, D., Sung, S.-H., Powers, E.J., Park, J.B.: Application of cross time-frequency analysis to postural sway behavior: the effects of aging and visual systems. IEEE Trans. Biomed. Eng. 52, 859–868 (2005); Time-frequency phase cross time-frequency analysis
Lachaux, J.-P., Lutz, A., Rudrauf, D., Cosmelli, D., Quyen, M.L.V., Martinerie, J., Varela, F.: Estimating the time-course of coherence between single-trial brain signals: an introduction to wavelet coherence. Neurophysiol. Clin. 32, 157–174 (2002)
Orini, M., Laguna, P., Mainardi, L., Bailón, R.: Characterization of the dynamic interactions between cardiovascular signals by cross time-frequency analysis: phase differences, time delay and phase locking. In: International Conference on Numerical Method in Engineering (2011)
Orini, M., Bailon, R., Mainardi, L.T., Laguna, P.: Time-frequency phase differences and phase locking to characterize dynamic interactions between cardiovascular signals. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2011, 4689–4692 (2011)
Bendat, J.S., Piersol, A.G.: Multiple-input/output relationships. In: Random Data, pp. 201–247. Wiley, New York (2012)
Orini, M., Taggart, P., Lambiase, P.D.: A multivariate time-frequency approach for tracking QT variability changes unrelated to heart rate variability. In: 2016 IEEE 38th Annual International Conference of the Engineering in Medicine and Biology Society (EMBC), pp. 924–927. IEEE, Orlando. doi:10.1109/EMBC.2016.7590852
Orini, M., Bailón, R., Enk, R., Koelsch, S., Mainardi, L., Laguna, P.: A method for continuously assessing the autonomic response to music-induced emotions through HRV analysis. Med. Biol. Eng. Comput. 48, 423–433 (2010)
Bailon, R., Garatachea, N., De La Iglesia, I., Casajus, J.A., Laguna, P.: Influence of running stride frequency in heart rate variability analysis during treadmill exercise testing. IEEE Trans. Biomed. Eng. 60, 1796–1805 (2013). doi:10.1109/TBME.2013.2242328
Orini, M., Hanson, B., Taggart, P., Lambiase, P.: Detection of transient, regional cardiac repolarization alternans by time-frequency analysis of synthetic electrograms. In: 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 3773–3776 (2013)
Zhao, H., Lu, S., Zou, R., Ju, K., Chon, K.H.: Estimation of time-varying coherence function using time-varying transfer functions. Ann. Biomed. Eng. 33, 1582–1594 (2005)
Zhao, H., Cupples, W.A., Ju, K.H., Chon, K.H.: Time-varying causal coherence function and its application to renal blood pressure and blood flow data. IEEE Trans. Biomed. Eng. 54, 2142–2150 (2007)
Chen, Z., Purdon, P., Harrell, G., Pierce, E., Walsh, J., Brown, E., Barbieri, R.: Dynamic assessment of baroreflex control of heart rate during induction of propofol anesthesia using a point process method. Ann. Biomed. Eng. 39, 260–276 (2011). doi:10.1007/s10439-010-0179-z
Orini, M., Bailon, R., Laguna, P., Mainardi, L.T.: Modeling and estimation of time-varying heart rate variability during stress test by parametric and non parametric analysis. In: Proceedings of Computers in Cardiology, pp. 29–32, Sept. 30 2007–Oct. 3 2007
Grinsted, A., Moore, J.C., Jevrejeva, S.: Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Process. Geophys. 11, 561–566 (2004)
Gallet, C., Chapuis, B., Barrès, C., Julien, C.: Time-frequency analysis of the baroreflex control of renal sympathetic nerve activity in the rat. J. Neurosci. Methods 198(2), 336–343 (2011)
Keissar, K., Davrath, L.R., Akselrod, S.: Coherence analysis between respiration and heart rate variability using continuous wavelet transform. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367(1892), 1393–1406 (2009)
Zhan, Y., Halliday, D., Jiang, P., Liu, X., Feng, J.: Detecting time-dependent coherence between non-stationary electrophysiological signals–a combined statistical and time-frequency approach. J. Neurosci. Methods 156, 322–332 (2006)
Bigot, J., Longcamp, M., Dal Maso, F., Amarantini, D.: A new statistical test based on the wavelet cross-spectrum to detect time-frequency dependence between non-stationary signals: application to the analysis of cortico-muscular interactions. Neuroimage 55, 1504–1518 (2011)
Brittain, J.S., Halliday, D.M., Conway, B.A., Nielsen, J.B.: Single-trial multiwavelet coherence in application to neurophysiological time series. IEEE Trans. Biomed. Eng. 54, 854–862 (2007)
Xiao, J., Flandrin, P.: Multitaper time-frequency reassignment for nonstationary spectrum estimation and chirp enhancement. IEEE Trans. Signal Process. 55, 2851–2860 (2007)
Thomson, D.J.: Jackknifing multitaper spectrum estimates. IEEE Signal Process. Mag. 24, 20–30 (2007)
Xu, Y., Haykin, S., Racine, R.J.: Multiple window time-frequency distribution and coherence of EEG using Slepian sequences and hermite functions. IEEE Trans. Biomed. Eng. 46(7), 861–866 (1999)
Lovett, E.G., Ropella, K.M.: Time-frequency coherence analysis of atrial fibrillation termination during procainamide administration. Ann. Biomed. Eng. 25(6), 975–984 (1997) multitaper, spectrogram,coherence.
Faes, L., Nollo, G.: Multivariate frequency domain analysis of causal interactions in physiological time series. Biomedical Engineering, Trends in Electronics, Communications and Software. InTech, Rijeka (2011)
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The Matlab code to conduct the analysis and create the figures shown in this chapter is are available at http://www.micheleorini.com/matlab-code/.
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Orini, M., Laguna, P., Mainardi, L.T., Bailón, R. (2017). Time-Frequency Analysis of Cardiovascular Signals and Their Dynamic Interactions. In: Barbieri, R., Scilingo, E., Valenza, G. (eds) Complexity and Nonlinearity in Cardiovascular Signals. Springer, Cham. https://doi.org/10.1007/978-3-319-58709-7_9
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