Signal Decomposition and Time-Frequency Representation Using Variable-Length Symmetric Filters

Milton José Porsani, Bjorn Ursin

Abstract


We present a time-frequency decomposition method to represent a time signal into a 2D (time X frequency) image, which describes how the frequency content varies along the time. This is done in two steps: firstly, by filtering the signal to obtain time-components; and secondly, by computing the average instantaneous frequency (AIF), which is used for moving the data components to the time-frequency plane. For the filtering process, we present an algorithm to generate a suite of symmetric filters that are computed recursively, starting with the high-frequency content of the signal, going down in frequency and leaving the lowest frequencies in the last filter component. This can be further decomposed by continuing the procedure. The symmetric impulse responses are zero-phase with positive frequency response, and they add up to a spike at the origin with a unit frequency response. The filtering procedure gives an exact decomposition of the signal and the traveltimes are preserved. Next, the analytic signal of each component is used for computing the AIF in sliding time windows, so that for each time sample, we have an associated AIF value. The 2D time-frequency plane is obtained by distributing and adding the data components along the frequency variable. Finally, by using the time X frequency distribution, a time-frequency filtering may be performed by stacking data of sub-domains with similar features. The new technique has been applied to two synthetic signals which have previously been analyzed by many authors using a variety of algorithms. The new signal decomposition algorithm and the AIF computation are simple and produce effective results on the synthetic data.

Keywords


time-series analysis; time-frequency representation; seismic noise

Full Text:

PDF

References


Andrade, M. C., M. J. Porsani, and B. Ursin, 2018, Complex autoregressive time-frequency analysis: estimation of time-varying periodic signal components: IEEE Signal Processing Magazine, 35, 142– 153, doi: 10.1109/MSP.2017.2783942.

Angelsen, B. A., 1981, Instantaneous frequency, mean frequency, and variance of mean frequency estimators for ultrasonic blood velocity Doppler signals: IEEE Transactions on Bio-medical Engineering, 28, 733–741, doi: 10.1109/TBME.1981.324853.

Auger, F., P. Flandrin, Y.-T. Lin, S. McLaughlin, S. Meignen, T. Oberlin, and H.-T. Wu, 2013, Time-frequency reassignment and synchrosqueezing: An overview: IEEE Signal Processing Magazine, 30, 32–41, doi: 10.1109/MSP.2013.2265316.

Burg, J. P., 1975, Maximum entropy spectral analysis.: PhD thesis, Stanford University. Castagna, J. P., S. Sun, and R. W. Siegfried, 2003, Instantaneous spectral analysis: Detection of low-frequency shadows associated with hydrocarbons: The Leading Edge, 22, 120–127, doi: 10.1190/1.1559038.

Chen, S. S., D. L. Donoho, and M. A. Saunders, 2001, Atomic decomposition by basis pursuit: SIAM Review, 43, 129–159, doi: 10.1137/S003614450037906X.

Cheng, J., and M. Sacchi, 2016, Fast and memory efficient singular spectrum analysis for seismic data reconstruction and denoising: SEG Technical Program Expanded Abstracts 2016, Society of Exploration Geophysicists, 4064–4068. doi: 10.1190/segam2016-13955076.1.

Cohen, L., 1989, Time-frequency distributions - a review: Proceedings of the IEEE, 77, 941–981, doi: 10.1109/5.30749.

Colominas, M. A., G. Schlotthauer, and M. E. Torres, 2014, Improved complete ensemble EMD: A suitable tool for biomedical signal processing: Biomedical Signal Processing and Control, 14, 19–29, doi: 10.1016/j.bspc.2014.06.009.

Dragomiretskiy, K., and D. Zosso, 2014, Variational mode decomposition: IEEE Transactions on Signal Processing, 62, 531–544, doi: 10.1109/TSP.2013.2288675.

Fomel, S., 2013, Seismic data decomposition into spectral components using regularized nonstationary autoregression: Geophysics, 78, O69–O76, doi: 10.1190/geo2013-0221.1.

Fourer, D., J. Harmouche, J. Schmitt, T. Oberlin, S. Meignen, F. Auger, and P. Flandrin, 2017, The ASTRES toolbox for mode extraction of non-stationary multicomponent signals: 2017 25th European Signal Processing Conference (EUSIPCO), 1130–1134. doi: 10.23919/EUSIPCO.2017.8081384.

Gabor, D., 1946, Theory of communication. Part 1: The analysis of information: Journal of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, 93, 429–441, doi: 10.1049/ji-3-2.1946.0074.

Golub, G. H., and C. F. V. Loan, 1996, Matrix computations, 3rd ed.: Johns Hopkins University Press.

Golyandina, N., and A. Zhigljavsky, 2020, Singular spectrum analysis for time series, 2nd ed.: Springer. 146 pp. doi: 10.1007/978-3-662-62436-4

Han, J., and M. van der Baan, 2013, Empirical mode decomposition for seismic time-frequency analysis: Geophysics, 78, O9–O19, doi: 10.1190/geo2012- 0199.1.

Harmouche, J., D. Fourer, F. Auger, P. Borgnat, and P. Flandrin, 2018, The sliding singular spectrum analysis: A data-driven nonstationary signal decomposition tool: IEEE Transactions on Signal Processing, 66, 251–263, doi: 10.1109/TSP.2017.2752720.

Harris, T. J., and H. Yuan, 2010, Filtering and frequency interpretations of singular spectrum analysis: Physica D: Nonlinear Phenomena, 239, 1958– 1967, doi: 10.1016/j.physd.2010.07.005.

Herrera, R. H., J. Han, and M. van der Baan, 2014, Applications of the synchrosqueezing transform in seismic time-frequency analysis: Geophysics, 79, V55–V64, doi: 10.1190/geo2013-0204.1.

Hu, H., S. Guo, R. Liu, and P. Wang, 2017, An adaptive aingular spectrum analysis method for extracting brain rhythms of electroencephalography: PeerJ, 5, e3474, doi: 10.7717/peerj.3474.

Huang, N. E., Z. Shen, and S. R. Long, 1999, A new view of nonlinear water waves: The Hilbert spectrum: Annual Review of Fluid Mechanics, 31, 417– 457, doi: 10.1146/annurev.fluid.31.1.417.

Huang, N. E., Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, 1998, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454, 903–995, doi: 10.1098/rspa.1998.0193.

Iatsenko, D., P. V. E. McClintock, and A. Stefanovska, 2015, Linear and synchrosqueezed time–frequency representations revisited: Overview, standards of use, resolution, reconstruction, concentration, and algorithms: Digital Signal Processing, 42, 1–26, doi: 10.1016/j.dsp.2015.03.004.

Lesage, P., 2008, Automatic estimation of optimal autoregressive filters for the analysis of volcanic seismic activity: Natural Hazards and Earth System Sciences, 8, 369–376, doi: 10.5194/nhess-8- 369-2008.

Lesage, P., F. Glangeaud, and J. Mars, 2002, Applications of autoregressive models and time–frequency analysis to the study of volcanic tremor and long-period events: Journal of Volcanology and Geothermal Research, 114, 391–417, doi: 10.1016/S0377- 0273(01)00298-0.

Liu, W., S. Cao, and Y. Chen, 2015, Seismic time–frequency analysis via empirical wavelet transform: IEEE Geoscience and Remote Sensing Letters, 13, 28–32, doi: 10.1109/LGRS.2015.2493198.

Liu, W., S. Cao, and Y. Chen, 2016, Applications of variational mode decomposition in seismic time-frequency analysis: Geophysics, 81, V365–V378, doi: 10.1190/geo2015-0489.1.

Mallat, S., 2008, A wavelet tour of signal processing: The sparse way, 3rd ed.: Academic Press.

Marple, L., 1980, A new autoregressive spectrum analysis algorithm: IEEE Transactions on Acoustics, Speech, and Signal Processing, 28, 441–454, doi: 10.1109/TASSP.1980.1163429.

Marple, S. L., 1987, Digital spectral analysis: With applications: Prentice-Hall.

Mitrofanov, G., and V. Priimenko, 2015, Prony filtering of seismic data: Acta Geophysica, 63, 652–678, doi: 10.1515/acgeo-2015-0012.

Morf, M., B. Dickinson, T. Kailath, and A. Vieira, 1977, Efficient solution of covariance equations for linear prediction: IEEE Transactions on Acoustics, Speech, and Signal Processing, 25, 429–433, doi: 10.1109/TASSP.1977.1162989.

Oropeza, V., and M. Sacchi, 2011, Simultaneous seismic data denoising and reconstruction via multi-channel singular spectrum analysis: Geophysics, 76, V25–V32, doi: 10.1190/1.3552706.

Porsani, M. J., B. Ursin, and M. G. Silva, 2019, Signal decomposition and time-frequency representation using iterative singular spectrum analysis: Geophysical Journal International, 217, 748–765, doi: 10.1093/gji/ggz046.

Robinson, E. A., and S. Treitel, 2000, Geophysical signal analysis: Society of Exploration Geophysicists. 481 pp. doi: https://doi.org/10.1190/1.9781560802327

Rodrigues, P. C., P. G. S. E. Tuy, and R. Mahmoudvand, 2018, Randomized singular spectrum analysis for long time series: Journal of Statistical Computation and Simulation, 88, 1921–1935, doi: 10.1080/00949655.2018.1462810.

Taner, M. T., F. Koehler, and R. E. Sheriff, 1979, Complex seismic trace analysis: Geophysics, 44, 1041–1063, doi: 10.1190/1.1440994.

Tary, J. B., R. Herrera, and M. Baan, 2013, Time-varying autoregressive model for spectral analysis of microseismic experiments and long-period volcanic events: Geophysical Journal International, 196, 600–611, doi: 10.1093/gji/ggt400.

Tary, J. B., R. H. Herrera, J. Han, and M. van der Baan, 2014, Spectral estimation —What is new? What is next?: Reviews of Geophysics, 52, 723–749, doi: 10.1002/2014RG000461.

Tary, J. B., R. H. Herrera, and M. van der Baan, 2018, Analysis of time-varying signals using continuous wavelet and synchrosqueezed transforms: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376, 20170254, doi: 10.1098/rsta.2017.0254.

Torres, M. E., M. A. Colominas, G. Schlotthauer, and P. Flandrin, 2011, A complete ensemble empirical mode decomposition with adaptive noise: 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4144–4147. doi: 10.1109/ICASSP.2011.5947265.

Ursin, B., and M. J. Porsani, 2021, Signal time–frequency representation and decomposition using partial fractions: Geophysical Journal International, 226, 617–626, doi: 10.1093/gji/ggab115.

Vesnaver, A., 2017, Instantaneous frequency and phase without unwrapping: Geophysics, 82, F1–F7, doi: 10.1190/geo2016-0185.1.

Wu, H.-T., G. F. Lewis, M. I. Davila, I. Daubechies, and S. W. Porges, 2016, Optimizing estimates of instantaneous heart rate from pulse wave signals with the synchrosqueezing transform: Methods of Information in Medicine, 55, 463–472, doi: 10.3414/ME16-01-0026.

-----------------------------------------------------------




DOI: http://dx.doi.org/10.22564/brjg.v40i1.2138

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.







>> Brazilian Journal of Geophysics - BrJG (online version): ISSN 2764-8044
a partir do v.37n.4 (2019) até o presente

Revista Brasileira de Geofísica - RBGf (online version): ISSN 1809-4511
v.15n.1 (1997) até v.37n.3 (2019)

Revista Brasileira de Geofísica - RBGf (printed version): ISSN 0102-261X
v.1n.1 (1982) até v.33n.1 (2015)

 

Brazilian Journal of Geophysics - BrJG
Sociedade Brasileira de Geofísica - SBGf
Av. Rio Branco 156 sala 2509
Rio de Janeiro, RJ, Brazil
Phone/Fax: +55 21 2533-0064
E-mail: editor@sbgf.org.br

 

Creative Commons