Statistical Synchrosqueezing Transform and Its Application to Seismic Thin Interbed Analysis

Generalized demodulation with tunable E-Factor for rolling bearing diagnosis under time-varying rotational speed

Synchrosqueezing transform (SST) is an effective time-frequency analysis (TFA) approach for the processing of nonstationary signals. The SST shows a satisfactory ability of the TF localization of the nonlinear signal with a slowly time-varying instantaneous frequency (IF). However, for the signal of which ridge curves in the TF domain are fast varying, or even almost parallel to the frequency axis, the SST will provide a blurred TF representation (TFR). To solve this issue, the transient-extracting transform (TET) was recently put forward. The TET can effectively characterize and extract transient features in the much concentrated TFR for the strongly frequency-modulated (FM) signal, especially the impulse-like signal. However, contrary to the SST, it is not suitable for weak FM modes. In this study, we propose a TFA method called the time-synchroextracting general chirplet transform (TEGCT). The TEGCT can achieve a highly concentrated TFR for strong FM signals as well as weak FM ones. Quantized indicators, the concentration measurement and the peak signal-to-noise ratio, are used to analyze the performances of the proposed method compared with those of other methods. The comparisons show that the TEGCT can provide a result with better TF localization. Then, the proposed method was applied to the spectrum analysis of the seismic data for oil reservoir characteristics. The horizontal slices of the offshore 3-D seismic data show that the TEGCT delineates more distinct and continuous subsurface channels in a fluvial-delta deposition system. All the results illustrate that our proposed method is a good potential tool for seismic processing and interpretation in the geoscience.

The use of linear time-frequency (TF) representa-tions as instantaneous frequency (IF) estimators arises in various fundamental disciplines, mainly thanks to their simplicity and immunity against interfering cross-terms. In this paper, we derive the optimal window width for IF estimation of noisy multicomponent signals based on a family of linear TF transforms that use Gaussian windowing functions and the Fourier oscillatory kernel. Closed-form formulas concerning the estimation bias and the variance are presented, which, thanks to their generality, describe the statistical performance of IF estimators based on transforms with fixed, time-adaptive, frequency-adaptive, or time-frequency adaptive windows. The optimal window is dependent on the unknown first derivative of the IF; therefore, we employ a low-complexity procedure to infer the derivative and optimize the width accordingly. Two adaptive and fully automated TF representations (TFRs) are developed; the first employs a time-adaptive window that minimizes the sum of the mean-squared errors (MSEs) of the IF estimates at each time instant, while in the second TFR, the window is adaptive over time and frequency, minimizing the estimation MSE at each location in the TF domain. Examples using synthetic and real-world signals demonstrate that the proposed algorithms may outperform many popular state-of-the-art techniques, including those that are signal-adaptive, in terms of IF estimation.

Under diverse conditions, the vibration signals of complex rotating machinery exhibit non-stationary behavior, multi-component characteristics, closely spaced frequencies, and non-proportionality, posing challenges to conventional time-frequency analysis (TFA) methods. These limitations hinder accurate instantaneous frequency (IF) estimation and time-frequency representation (TFR) construction, directly impacting machinery fault diagnosis. As such, we propose the Local Entropy Selection Scaling-Extracting Chirplet Transform (LESSECT), which optimizes entropy-based chirp rate (CR) selection to match non-proportional fundamental frequencies. By adaptively selecting multiple CRs at the same time center, LESSECT enhances TFR resolution and energy concentration, leading accurate IF identification. Experimental validation on bat echolocation, bearing fault, and planetary gearbox signals shows its superior performance in resolving non-proportional, closely spaced IFs. This significantly improves state estimation and enhances machinery diagnostics, contributing to greater system reliability.

Synchronous Spline-kernelled Chirplet Squeezing Transform and Its Application for Seismic Data Analysis

Time-frequency (TF) analysis (TFA) to nonstationary signals can reveal the nonlinearly changing instantaneous frequencies (IFs) of signals; it is, therefore, widely used for rotating machinery fault diagnosis under time-varying speed conditions. However, the traditional TFA methods may only reveal the outline of IFs due to the extra TF diffusion caused by limited TF resolution, hindering the fault diagnosis of rotating machinery. To enhance the readability of the time-frequency representation (TFR) of the signals with IF trajectories linearly time-varying, linear chirplet transform has been proved effective. To effectively tackle the signal with nonlinearly changing IFs, a string of chirp-rates is preferred, where the final TFR is obtained by superposition of each corresponding sub-TFR at each TF point. However, the extra cross-term interferences resulted from a string of chirp-rates cannot be neglected on TFRs. Aiming at alleviating the cross-terms from the TFR, matching and reassignment based TF enhancement strategy is proposed, where only the appropriate chirp-rate and its corresponding TFR slice is retained at each time instant. The appropriate chirp-rate is adaptively selected by the index—spectral kurtosis. To further increase the readability of the resulting TFR, a reassignment technique synchrosqueezing transform is integrated with the proposed matching strategy. By iteratively employing reassignments, TFR with the enhanced energy and sharp IF ridges can be generated. The effectiveness of the proposed method is validated by both simulated and experimental analyses. It is shown that the proposed method is effective in processing time-varying signals and can provide more accurate IF estimation, which paves the way for rotating machinery fault diagnosis under nonstationary speed conditions.

Cohen's class of time-frequency representations (TFRs) is reformulated into a discrete-time, discrete-frequency, computer-implementable form. It is shown how, in this form, many of the properties of the continuous-time, continuous-frequency formulation are either lost or altered. Intuitions applicable in the continuous-time case do not necessarily carry over to the discrete-time case examined. The properties of the discrete variable formulation examined are the presence and form of cross-terms, instantaneous frequency estimation, and relationships between Cohen's class of TFRs. A parameterized class of distributions which is a blending between the short-time Fourier transform (STFT) and the Wigner-Ville distribution. The two main conclusions are that all TFRs of Cohen's class implementable in the given form (which includes all commonly used TFRs) possess cross-terms and that instantaneous frequency estimation using periodic moments of these TFRs is purposeless, since simpler methods obtain the same result. >

A time-frequency distribution (TFD) signal processor, developed at the Applied Physics Laboratory, is currently under evaluation using simulated signals and actual laser vibration sensor (LVS) data that we collected on various ship targets. Preliminary results for one instantaneous frequency (IF) estimator implementation, the smoothed cross Wigner-Ville Distribution (XWVD), indicate 8 to 10 dB demodulation (CNR) advantage compared to a digital FM limiter-discriminator. A second approach, using the unsmoothed XWVD TFD, demonstrated a 3-5 dB advantage. Regarding spectral estimation, we are investigating performance of our reduced interference distribution (RID) implementation through comparison with the short-time Fourier transform (STFT). From the LVS data processed, indications are that a significant increase in spectral and temporal resolution exists using our RID approach. Our processor also provided improved detectability over the STFT for transient signals and short-lived sinusoids. Significant correlation between accepted acoustic lines and LVS-derived vibration lines are indicated. Details are presented that describe our signal simulation, the LVS measurements, and signal processing implementations along with assumptions based on measured speckle-induced amplitude modulation.

Traditional signal postprocessing methods suffer from the repeated assignment problem (RAP), which can result in inaccurate instantaneous frequency (IF) estimation and signal recovery. In this article, to solve this problem, a novel time-frequency (TF) analysis (TFA) method called the synchro-reassigning transform (SRT) is proposed. This method aims to obtain the ideal TF representation (TFR) by utilizing derivatives of the constructed amplitude function and a three-step selection rule to adaptively extract the TF coefficients on the IF trajectories in the TF plane, and reassigning these TF coefficients into a new TFR. In this way, SRT can eliminate the RAP, thus obtaining an approximately ideal TFR that helps to realize more accurate IF estimation and signal reconstruction. Furthermore, SRT shows satisfactory performance on signals with high nonlinear IFs or relatively close IFs, even under strong noise conditions. Two simulated signal and three real-life signals were used to demonstrate the performance of SRT.

A time-varying instantaneous frequency fault features extraction method of rolling bearing under variable speed

Chapter 10 - Instantaneous Frequency Estimation and Localization

Summary Seismic instantaneous frequency (IF) is a useful attribute for characterizing depositional features from seismic data. However, commonly used IF estimations methods are sensitive to noise and also suffer from meaningless values. In this paper, we propose an IF regularization method based on time-frequency analysis. The Stockwell transform (ST), which has an advantage in providing multi-resolution time-frequency analysis while retaining the absolute phase of each frequency component, is used for IF regularization. We firstly decompose the estimated IF into the ST domain. By considering that most geologic changes are expressed only in certain local spectral ranges, we utilize an instantaneous amplitude parameterized low pass filter to identify the spectral ranges of IF from the multi-resolution results. After inverse ST is taken, noise and meaningless values are removed, and the regularized IF becomes more useful for describing geological features. The synthetic and real data examples demonstrate the effectiveness of our method.

On the use of short-time fourier transform and synchrosqueezing-based demodulation for the retrieval of the modes of multicomponent signals

Accurate extraction of time-varying instantaneous frequency (IF) is crucial for fault diagnosis in rotating machinery operating under time-varying speed conditions, as real-world signals often exhibit nonlinearity and complexity. Variational nonlinear chirp mode decomposition (VNCMD) demonstrates significant capability and adaptability in extracting time-varying IFs by converting broadband signals into narrowband signals through a demodulation technique. However, its accuracy is limited by the fact that it only enforces smoothness on the demodulated signals, and it does not consider the inherent proportional relationships among IFs in rotating machine vibration signals. These limitations hinder VNCMD from accurately extracting IFs and result in poor noise robustness, a strong dependence on the initial IFs, and difficulties in handling nonlinear chirp signals (NCSs) with closely spaced IFs. To overcome these limitations, this article proposes hybrid constraints embedding VNCMD (HCVNCMD), which places further constraints on demodulated signals and IFs. Specifically, a sparse constraint is first applied to the demodulated signals. This improves their estimation accuracy and reduces errors in the subsequent IF estimates. Second, a low-rank constraint is applied to the IFs, effectively capturing their proportional relationships and optimizing inaccurate initial IFs to achieve the true IFs. The hybrid constraints embedded in HCVNCMD significantly enhance noise robustness and improve IF estimation accuracy, enabling effective processing of NCSs with closely spaced IFs. Both simulation and experimental cases demonstrate that the HCVNCMD outperforms existing time–frequency analysis (TFAs) methods in accurately estimating the IFs of NCSs. Furthermore, it successfully identifies fault features in gear and bearing under time-varying speed conditions, confirming its effectiveness and practicability.

The main factors responsible for the nonstationarity of seismic signals are the nonstationarity of the geologic structural sequences and the complex pore structure. Time-frequency analysis can identify various frequency components of seismic data and reveal their time-variant features. Choosing a proper time-frequency decomposition algorithm is the key to analyze these nonstationarity signals and reveal the geologic information contained in the seismic data. According to the Heisenberg uncertainty principle, we cannot obtain the finest time location and the best frequency resolution at the same time, which results in the trade-off between the time resolution and the frequency resolution. For instance, the most commonly used approach is the short-time Fourier transform, in which the predefined window length limits the flexibility to adjust the temporal and spectral resolution at the same time. The continuous wavelet transform (CWT) produces an “adjustable” resolution of time-frequency map using dilation and translation of a basic wavelet. However, the CWT has limitations in dealing with fast varying instantaneous frequencies. The synchrosqueezing transform (SST) can improve the quality and readability of the time-frequency representation. We have developed a high-resolution and effective time-frequency analysis method to characterize geologic bodies contained in the seismic data. We named this method the SST, and the basic wavelet is the three-parameter wavelet (SST-TPW). The TPW is superior in time-frequency resolution than those of the Morlet and Ricker wavelets. Experiments on synthetic and field data determined its validity and effectiveness, which can be used in assisting in oil/gas reservoir identification.