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Example of the bat echolocation signal. Top-left: waveform; Top-right: conventional STFT with σ = 8×10 −5 ; Bottom-left: conventional 2nd-order FSST with σ = 8×10 −5 ; Bottom-right: 2nd-order adaptive FSST with time-varying parameter σ est (t) obtained by our proposed Algorithm 1.
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The continuous wavelet transform (CWT)-based synchrosqueezing transform (SST) is a special type of the reassignment method which not only enhances the energy concentration of CWT in the time-frequency plane, but also separates the components of multicomponent signals. The “bump wavelet” and Morlet's wavelet are commonly used continuous wavelets for...
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... a given real-world signal, how to select an appropriate constant σ such that the resulting conventional SST or 2nd-order SST has a sharp representation is probably not very simple. Here we choose σ = 8 × 10 −5 , which is close to the mean of σ est (t) obtained by Algorithm 1. Fig.6 shows the TF representations of the echolocation signal: ...
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... Zhang [11] applied this method to the resonance band feature extraction of sound signals in planetary gearboxes. Li et al. [12,13] proposed the Adaptive SST (ASST) and the Adaptive SST based on STFT (AFSST), then extending them to the second order. It has been verified that this method not only improves the time-frequency concentrate on and resolution of multi-component signals but also separates their components. ...
... According to Equation (6), the length of the STFT window ( , ) G t ω is determined by the length of the signal s. The size of the signal data to be processed depends on the computer's computational memory, thus limiting SST to handling only short signals [11][12][13][14]. However, in practical engineering applications, vibration signals from variable-speed rolling bearings are mostly long signals. ...
... The analytical signal of the vibration signal can be represented by taking the original vibration signal as the real part and its Hilbert transform as the imaginary part. Therefore, the analytical signal of the filtered vibration signal is as follows: (12) where s is the filtered signal, ( ) H s is the Hilbert transform of s, and, thus, the envelope of the signal is y . From the above, it can be deduced that the frequency after envelope transformation only contains positive frequency components. ...
Synchrosqueezed transform (SST) is a time–frequency analysis method that can improve energy aggregation and reconstruct signals, which has been applied in the fields of medical treatment, fault diagnosis, and seismic wave processing. However, when dealing with time-varying signals, SST suffers from poor time–frequency resolution and is unable to deal with long signals. In order to accurately extract the characteristic frequency of variable speed rolling bearing faults, this paper proposes a synchrosqueezed transform method based on fast kurtogram and demodulation and piecewise aggregate approximation (PAA). The method firstly filters and demodulates the original signal using fast kurtogram and Hilbert transform to reduce the influence of background noise and improve the time–frequency resolution. Then, it compresses the signal by using piecewise aggregate approximation, so that the SST can deal with long signals and, thus, extract the fault characteristic frequency. The experimental data verification results indicate that the method can effectively identify the fault characteristic frequency of variable-speed rolling bearings.
... The resulting 2D scalogram images derived from the CWT prove beneficial in scenarios where capturing non-stationary features is essential for accurate interpretation and decision-making, such as in various vibration signal analysis areas, including denoising, structural and ground motion analysis, fault diagnosis, and damage detection (Gurley and Kareem, 1999;Peng and Chu, 2004). We conducted a comparative analysis of CWT methods using the following mother wavelet functions and each function is illustrated in Figure 1: Bump Wavelet: Bump wavelet is a bandlimited function defined in the frequency domain with parameters μ and σ and window w [n] (Li et al., 2020). It has a wider variance in time and a narrower frequency variance. ...
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... Recently, a multisynchrosqueezing generalized S-transform has been introduced for tight sandstone gas reservoir identification, which concentrates the energy of TFR in a stepwise manner through an iterative reassignment procedure [4]. In addition, there are a series of time-frequency analysis tools with high energy concentration [17,22,23,57], which provide more options for non-stationary signal processing. Unfortunately, the abilities of this methods to analyze signal are also limited in TF plane. ...
... Rényi entropy is regarded as an effective tool to measure the energy concentration of TFR [23]. A smaller value implies a higher concentration and resolution. ...
... In many cases, it is necessary to separate the signal to a finite number of mono-components [5,25]. Recently, some signal separation methods based on the SST framework have been investigated [22,23]. In this paper, we propose a novel technique to separate multicomponent chirp signal using SSFrST. ...
The synchrosqueezing transform (SST) is an advanced post-processing method to sharpen the time–frequency representation (TFR). However, it still processes the signal in frequency domain. Therefore, it cannot effectively analyze signals whose energy is not well concentrated in frequency domain. The fractional S-transform (FrST) inherits the merits of the short-time fractional Fourier transform and the continuous wavelet transform, processing signals in fractional frequency domain. In this paper, a novel non-stationary signal processing tool, synchrosqueezing fractional S-transform (SSFrST) has been proposed, which combines the advantages of SST and FrST. First, we introduce a novel FrST and derive its fundamental properties. Second, based on the novel FrST, we propose SSFrST and discuss its reconstruction formula and implementation. It can not only improve time–frequency resolution, but also process signals in the time-fractional–frequency plane. Finally, we provide several applications to validate the effectiveness of our methods, including chirp signal parameters estimation, signal separation, filtering and noise separation.
... More over, time-frequency (TF) analysis tools have been used extensively on non-stationary signals for the tasks mentioned. These tools include the synchrosqueezing transform (SST) [11,12], and its variants like the demodulation SST [13,14] and the adaptive SST [15,16]. Recently, the chirplet transform has been proposed for component retrieval of signals with components overlapping in the time-frequency domain [17,18]. ...
... As mentioned before, this regression problem is nonlinear on the model coefficients given Eq. (16). The optimal solution is obtained by using an iterative non-linear curve fitting procedure based on the Levenberg-Marquardt algorithm [38]. ...
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... Then for each tile, the Rènyi entropy is computed for each window and the one with the best resolution is selected for that tile discarding any other. On a different approach, the authors in [14] [15] propose adapting the wavelet to increase the energy concentration. For this case, a Matlab implementation is also provided in [16]. ...
... : Compute the energy E N (k) according to (12) [or (14)] 5: Normalize E N (k) as in (13) marked are distributed among the remaining bands through an integer distribution method as before, but analyzing only the energy across the surviving bands. As in the proportional method, these operations are repeated over the new analysis bands until there are no more frequencies to reassign, or when the maximum number of iterations is reached. ...
... One option is to use a TFR with better performance to replace STFT or CWT. For example, adaptive CWT [8], adaptive STFT [9], and adaptive S-transform [10] are employed instead of the original TFR. Another alternative is to carry out the second-order or highorder Taylor expansion on the signal to generate a more accurate estimate of the instantaneous frequency (IF), such as the second-order SST [11] and high-order SST [12] applications to gravitational-wave signal. ...
In this letter, based on the variational model, we propose a novel time-frequency post-processing technique to approximate the ideal time-frequency representation. Our method has the advantage of modularity, enabling "plug and play", independent of the performance of specific time-frequency analysis tool. Therefore, it can be easily generalized to the fractional Fourier domain and the linear canonical domain. Additionally, high-resolution is its merit, which depends on the specific instantaneous frequency estimation method. We reveal the relationship between instantaneous frequency estimation and reassignment method. The effectiveness of the proposed method is verified on both synthetic signals and real world signal.
... The synchrosqueezing transform [9,10], proposed by I. Daubechies in 1996, is based on a continuous wavelet transform, which rearranges the energy in the scale direction of the signal into the energy in the frequency and greatly improves the aggregation of TF energy. A modified second-order SST is proposed in [21], which can be used to describe the formation more accurately. The synchrosqueezing transform based on a STFT was proposed by T. Oberlin in 2014 [23]. ...
The instantaneous frequency (IF) provides important characteristic information for analyzing signals. To improve the accuracy of IF estimation, time–frequency (TF) analysis with high resolution and high aggregation is necessary. A new IF estimation method is proposed in this paper, which is referred to as ridge path regrouping based on an adaptive short-time Fourier high-order synchrosqueezing transform (AFSST). In the TF analysis of the non-stationary signal, we optimize the window function of the short-time Fourier transform to be adaptive to signal characteristics, and then perform the high-order synchrosqueezing transform. Based on the high-resolution time–frequency distribution (TFD), the frequency ridge of the signal is extracted, and the ridge of the modal aliasing at the intersection is discarded and reorganized according to the fundamentals. Experiments on multicomponent nonstationary signals were performed, and the results show that AFSST has satisfactory TF aggregation and TF resolution. IF estimation experiment based on AFSST, the results indicate that the proposed method can estimate the IF accurately even under a low signal-to-noise ratio (SNR).
... Moreover, a study of SST applied to multivariate data was done in [34]. An adaptive SST with a time-varying parameter was introduced in [35,36], which obtains the well-separated condition for multicomponent signals using the linear frequency modulation to approximate a non-stationary signal at every time instant. The theoretical analysis of adaptive SST was studied in [37,38]. ...
High-concentration time-frequency (TF) representation provides a valuable tool for characterizing multi-component non-stationary signals. In our previous work, we proposed using an instantaneous frequency (IF) equation to sharpen the TF distribution, and experiments verified its effectiveness. In this paper, we systematically discuss why the IF equation-based TF analysis methods work and how to use the IF equation to improve TF sharpness. By the analysis of the properties of the IF equation, we prove that a good IF equation can unify the well-known IF and group delay estimators and provides an effective way to characterize the mixture of time-varying and frequency-varying signals. By discussing the post-processing techniques based on the IF equation, we can prove that many popular TF post-processing methods, such as the synchroextracting transform, the multi-synchrosqueezing transform, and the time extracting transform, fall into the IF equation-based category. We also propose a novel approach to combine different IF equations to minimize energy spreading based on local sparsity. Numerical simulations and practical experiments are presented to illustrate the performance of the proposed IF equation-based TF analysis method.
... While SWT can reconstruct the interesting signals but suffers from blurred TFRs for signals with fast varying IF. Furthermore, new extensions of SWT have been developed [7,[27][28][29][30][31][32], among these improved methods, recently synchroextracting transform (SET) [32] has attracted great attention. Differing from the squeezing manner of SWT, SET only utilizes a small number of TF coefficients, but achieves a more concentrated TFR. ...
... In the last five years, however, a new technology has appeared in the field of vibration measurement of structures that allows us to continuously measure with high frequency the response of a structure in all three axes. The article [37][38][39][40][41][42][43][44][45][46] describe data capture methods that allow measurements of up to 20,000 Hz and more, from which we can obtain highly accurate and reliable results. The problem with these methods is their range and their difficult installation on demanding structures. ...
This paper describes the non-contact vibration monitoring of prestressed reinforced concrete railway sleepers. The monitoring was carried out using physical measuring equipment consisting of a seismograph and geophones, a robotic total station (RTS) and scanning laser Doppler vibrometry (SLDV) equipment. Measurements of the dynamic response of sleepers to the dynamic loading induced by the running of train sets provides data that give an insight into the actual state of the sleeper structure at the moment when it is most stressed. The main objective of the study was to identify the most appropriate form of monitoring to monitor events related to the occurrence of cracks in the rail sill, which can also be caused by the impact of vibration. As is well-known, monitoring the dynamic response of railway sleepers is very important for the sustainable management and maintenance of railway lines. The vibration analysis was carried out in the field on the Ormož–Ptuj section of the railway line, in Velika Nedelja and in the laboratory of the UM FGPA, where the vibration was simulated using a hydraulic kit with a static and pulsed force of 100 kN. Several cycles of measurement were carried out and the results were compared with the results of the field monitoring. The vibration parameters measured for the concrete sleepers are necessary to develop a realistic dynamic model of the railway line, which will be able to predict its response to impact loads and the possible occurrence of damage, as has recently been observed for concrete sleepers in several European Union countries.
