Signal time–frequency representation and decomposition using partial fractions

Abstract

Summary The Z-transform of a complex time signal (or the analytic signal of a real signal) is equal to the Z-transform of a prediction error divided by the Z-transform of the prediction error operator. This inverse is decomposed into a sum of partial fractions, which are used to obtain impulse response operators formed by non-causal filters which complex-conjugate symmetric coefficients. The time-components are obtained by convolving the filters with the original signal, and the peak frequencies, corresponding to the poles of the prediction error operator, are used for mapping the time-components into frequency components. For non-stationary signals, this decomposition is done in sliding time windows, and the signal component values, in the middle of each window, are attributed to the peak value of its frequency response which corresponds to the pole of this partial fraction component. The result is an exact, but non-unique, time-frequency representation of the input signal. A sparse signal decomposition can be obtained by summing along the frequency axis in patches with similar characteristics in the time-frequency domain. The peak amplitude frequency of each new time component is obtained by computing a scalar prediction error operator in sliding time windows, resulting in a sparse time-frequency representation. In both cases, the result is a time-frequency matrix where an estimate of the frequency content of the input signal can be obtained by summation over the time variable. The performance of the new method is demonstrated with excellent results on a synthetic time signal, the LIGO gravitational wave signal, and on seismic field data.

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

“…Signal decomposition and time-frequency representation have numerous applications in geophysics (Chen et al, 2001;Lesage et al, 2002;Castagna et al, 2003;Lesage, 2008;Oropeza and Sacchi, 2011;Fomel, 2013;Han and van der Baan, 2013;Tary et al, 2013;Herrera et al, 2014;Mitrofanov and Priimenko, 2015;Liu et al, 2015Liu et al, , 2016Cheng and Sacchi, 2016;Ursin and Porsani, 2021), biomedical signal analysis (Angelsen, 1981;Colominas et al, 2014;Wu et al, 2016;Hu et al, 2017), and in other fields. Discussions of different methods are given by Marple (1987), Cohen (1989), Mallat (2008), Auger et al (2013), Tary et al (2014Tary et al ( , 2018, Iatsenko et al (2015), and Fourer et al (2017).…”