Adaptive short-time Fourier transform and synchrosqueezing transform for non-stationary signal separation

Abstract

The synchrosqueezing transform, a kind of reassignment method, aims to sharpen the timefrequency representation and to separate the components of a multicomponent non-stationary signal. In this paper, we consider the short-time Fourier transform (STFT) with a time-varying parameter, called the adaptive STFT. Based on the local approximation of linear frequency modulation mode, we analyze the well-separated condition of non-stationary multicomponent signals using the adaptive STFT with the Gaussian window function. We propose the STFT-based synchrosqueezing transform (FSST) with a time-varying parameter, named the adaptive FSST, to enhance the time-frequency concentration and resolution of a multicomponent signal, and to separate its components more accurately. In addition, we also propose the 2nd-order adaptive FSST to further improve the adaptive FSST for the non-stationary signals with fast-varying frequencies. Furthermore, we present a localized optimization algorithm based on our well-separated condition to estimate the time-varying parameter adaptively and automatically. Simulation results on synthetic signals and the bat echolocation signal are provided to demonstrate the effectiveness and robustness of the proposed method.where A k (t), φ k (t) > 0, has been a very active research area over the past few years. Note that the number of component K may change with time t, but it should be constant for long enough time intervals. The representation of x(t) in (1) with A k (t) and φ k (t) varying slowly or more slowly than φ k (t) is called an adaptive harmonic model (AHM) representation of x(t), where A k (t) are called the instantaneous amplitudes and φ k (t) the instantaneous frequencies (IFs). To decompose x(t) as an AHM representation (1) is important to extract information, such as the underlying dynamics, hidden in x(t). Time-frequency (TF) analysis is widely used in engineering fields such as communication, radar and sonar as a powerful tool for analyzing time-varying non-stationary signals [1]. Time-frequency analysis is especially useful for signals containing many oscillatory components with slowly timevarying amplitudes and instantaneous frequencies. The short-time Fourier transform (STFT), the continuous wavelet transform (CWT) and the Wigner-Ville distribution are the most typical TF analysis, see details in [1]-[6]. Other TF distributions of Cohen's class include the exponential distribution [7], a smoothed pseudo Wigner distribution [8] and the complex-lag distribution [9].In addition, the TF signal analysis and synthesis using the eigenvalue decomposition method has been studied [10,11]. In particular, an eigenvalue decomposition-based approach which enables the separation of non-stationary components with overlapped supports in the TF plane has been proposed in [12].Recently a number of new TF analysis methods such as the Hilbert spectrum analysis with empirical mode decomposition (EMD) [13], the reassignment method [14] and synchrosqueezed wavelet transform (SST) [15] have also been proposed ...

Analysis of adaptive short-time Fourier transform-based synchrosqueezing transform

“…is c, the IF of x(t). Hence, for a general x(t), at (t, η), [13] defines the real part of the quantity of ω adp,c x (t, η) in the above equation, denoted by ω adp x (t, η), as the phase transformation of the adaptive FSST:…”

“…Thus the authors of [13] define the real part of ω adp,2nd,c x as the phase transformation for the 2nd-order adaptive FSST. Namely, the phase transformation ω adp,2nd…”

“…Due to (30), there always exists C 1 (R) and bounded σ(t) such that (44) holds. For the sinusoidal function-based adaptive FSST, the authors in [13] suggest to choose σ(t) as…”

“…As an example, let us look at what are O k and α k (t) look like when g is the Gaussian function defined by (22). One can obtain for this g (see [13]),…”

Fault Diagnosis for Rolling Bearings Using Optimized Variational Mode Decomposition and Resonance Demodulation

“…The short-time Fourier transform is proposed to effectively identify the frequency and phase of sine wave in the local area of non-stationary signal, and its optimization processing [ 12 , 13 , 14 , 15 ]. Wavelet transform is a new transform analysis method, which inherits and develops the localization idea of short-time Fourier transform, and overcomes the shortcomings that the window size does not change with frequency [ 16 , 17 , 18 ].…”