Analysis of adaptive synchrosqueezing transform with a time-varying parameter
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Analysis of adaptive synchrosqueezing transform with a time-varying parameter Jian Lu1 · Qingtang Jiang2
· Lin Li3
Received: 11 January 2020 / Accepted: 6 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The synchrosqueezing transform (SST) was developed recently to separate the components of non-stationary multicomponent signals. The continuous wavelet transform-based SST (WSST) reassigns the scale variable of the continuous wavelet transform of a signal to the frequency variable and sharpens the time-frequency representation. The WSST with a time-varying parameter, called the adaptive WSST, was introduced very recently in the paper “Adaptive synchrosqueezing transform with a time-varying parameter for non-stationary signal separation.” The well-separated conditions of non-stationary multicomponent signals with the adaptive WSST and a method to select the time-varying parameter were proposed in that paper. In addition, simulation experiments in that paper show that the adaptive WSST is very promising in estimating the instantaneous frequency of a multicomponent signal, and in accurate component recovery. However, the theoremretical analysis of the adaptive WSST has not been studied. In this paper, we carry out such analysis and obtain error bounds for component recovery with the adaptive WSST and the 2nd-order adaptive WSST. These results provide a mathematical guarantee to non-stationary multicomponent signal separation with the adaptive WSST. Keywords Adaptive continuous wavelet transform · Adaptive synchrosqueezing transform · Instantaneous frequency estimation · Non-stationary multicomponent signal separation Mathematics Subject Classification (2010) 42C40 · 42C15 · 42A38
Communicated by: Gitta Kutyniok This work was supported in part by the National Natural Science Foundation of China under grants 61373087, 11871348, 61872429 and Simons Foundation under grant 353185. Qingtang Jiang
[email protected]
Extended author information available on the last page of the article.
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Adv Comput Math
(2020) 46:72
1 Introduction Most real signals such as EEG and bearing signals are non-stationary multicomponent signals given by: x(t) = A0 (t) +
K
xk (t),
xk (t) = Ak (t)ei2πφk (t) ,
(1)
k=1
with Ak (t), φk (t) > 0, where A0 (t) is the trend, and Ak (t), 1 ≤ k ≤ K, are called the instantaneous amplitudes and φk (t) the instantaneous frequencies. Modeling a nonstationary signal x(t) as in (1) is important to extract information hidden in x(t). The empirical mode decomposition (EMD) algorithm along with the Hilbert spectrum analysis (HSA) is a popular method to decompose and analyze non-stationary signals [19]. EMD decomposes a non-stationary signal as a superposition of intrinsic mode functions (IMFs) and then the instantaneous frequency of each IMF is calculated by HSA which results in a representation of the signal as in (1). The properties of EMD have been studied and variants of EMD have been proposed to improve the performance in many articles;
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