A Second-Order Synchrosqueezing Transform with a Simple Form of Phase Transformation

Abstract

To model a non-stationary signal as a superposition of amplitude and frequency-modulated Fourier-like oscillatory modes is important to extract information, such as the underlying dynamics, hidden in the signal. Recently, the synchrosqueezed wavelet transform (SST) and its variants have been developed to estimate instantaneous frequencies and separate the components of non-stationary multicomponent signals. The short-time Fourier transform-based SST (FSST for short) reassigns the frequency variable to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. However FSST works well only with multicomponent signals having slowly changing frequencies. To deal with multicomponent signals having fast-changing frequencies, the secondorder FSST (FSST2 for short) was proposed. The key point for FSST2 is to construct a phase transformation of a signal which is the instantaneous frequency when the signal is a linear chirp. In this paper we consider a phase transformation for FSST2 which has a simple expression than that used in the literature. In the study the theoretical analysis of FSST2 with this phase transformation, we observe that the proof for the error bounds for the instantaneous frequency estimation and component recovery is simpler than that with the conventional phase transformation. We also provide some experimental results which show that this FSST2 performs well in non-stationary multicomponent signal separation.

Instantaneous Frequency-Embedded Synchrosqueezing Transform for Signal Separation

“…x (t, η) in ( 13) as the phase transformation for the 2ndorder FSST. Very recently a simple phase transformation for the 2nd-order FSST was proposed in [18].…”

“…The corresponding phase transformation in [43] is different from our ω I x (t, η) defined in (21). By (18) in Theorem 1, we know the input signal x(t) can be recovered from its IFE-FSST as shown in the following: For x(t) ∈ L 2 (R),…”

“…Many works on SST have been carried out since the publication of the seminal article [1]. For example, the short-time Fourier transform (STFT)-based SST (FSST) [13][14][15], the 2nd-order SST [16][17][18], the higher-order FSST [19,20], a hybrid EMD-WSST [21], the WSST with vanishing moment wavelets [22], the multitapered SST [23], the synchrosqueezed wave packet transform [24] and the synchrosqueezed curvelet transform [25] were proposed. Furthermore, the adaptive SST with a window function having a changing parameter was proposed in [26][27][28][29][30][31].…”