Signal decomposition and analysis via extraction of frequencies

Abstract

Time–frequency analysis is central to signal processing, with standard adaptation to higher dimensions for imaging applications, and beyond. However, although the theory, methods, and algorithms for stationary signals are well developed, mathematical analysis of non-stationary signals is almost nonexistent. For a real-valued signal defined on the time-domain RR, a classical approach to compute its instantaneous frequency (IF) is to consider the amplitude–frequency modulated (AM–FM) formulation of its complex (or analytic) signal extension, via the Hilbert transform. In a popular paper by Huang et al., the so-called empirical mode decomposition (EMD) scheme is introduced to separate such a signal as a sum of finitely many intrinsic mode functions (IMFs), with a slowly oscillating signal as the remainder, so that more than one IFs of the given signal can be computed by extending each IMF to an AM–FM signal component. Based on the continuous wavelet transform (CWT), the notion of synchrosqueezing transform (SST), introduced by Daubechies and Maes in 1996, and further developed by Daubechies, Lu, and Wu (DLW) in a 2011 paper, provides another approach to extract more than one IFs of the signal on RR. Furthermore, by introducing a list of fairly restrictive conditions on the adaptive harmonic model (AHM), the DLW paper also derives a theory for estimating the signal components according to this model, by using the IFs with estimates from the SST. The objective of our present paper is to introduce another mathematical theory, along with rigorous methods and computational schemes, to achieve a more ambitious goal than the SST approach, first to extract the polynomial-like trend from the source signal, then to compute the exact number of signal components according to a less restrictive AHM model, then to obtain better estimates of the IFs and instantaneous amplitudes (IAs) of the signal components, and finally to separate the signal components from the (blind) source signal. Furthermore, our computational scheme can be realized in near-real-time, and our mathematical theory has direct extension to the multivariate setting.

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