Multicomponent Signal Analysis Based on Polynomial Chirplet Transform

Slope synchronous chirplet transform and its application to tacho-less order tracking of rotating machineries

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.

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.

Self-paced learning for instantaneous frequencies estimation in heavy noise environments

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

Enhancement of adaptive mode decomposition via angular resampling for nonstationary signal analysis of rotating machinery: Principle and applications

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.

: Assessment of muscle fatigue involves the creation of indices based on the estimation of variables such as the instantaneous frequency (IF) and the instantaneous amplitude (IA), which can be estimated from a time-frequency (TF) distribution of a surface electromyographic signal (SEMG). Since during muscle fatigue there is a decrease of the IF and an increase of the IA, slopes from these parameters can be computed and used as indices to measure muscle fatigue. In this paper we present a comparison of four techniques used to build a TF distribution of SEMG signals, namely spectrogram, Wigner-Ville, Choi-Williams and smoothed pseudo Wigner-Ville. SEMG signals were recorded from thirty normal human subjects under specific physical routines to measure muscle fatigue, and TF distributions were computed for all the acquired signals. In order to confirm muscle fatigue, the computed indices were then correlated to the perceived discomfort levels reported by the subjects. Results show that although the spectrogram and smoothed pseudo Wigner-Ville have to overcome problems such as sizes of the time and frequency windows and cross terms, these two distributions provide equivalent slopes and valid indices of muscle fatigue for SEMG signals with low to medium nonstationarity.

This paper explores three groups of time–frequency distributions: the Cohen’s, affine, and reassigned classes of time–frequency representations (TFRs). This study provides detailed insight into the theory behind the selected TFRs belonging to these classes. Extensive numerical simulations were performed with examples that illustrate the behavior of the analyzed TFR classes in the joint time–frequency domain. The methods were applied both on synthetic and real-life non-stationary signals. The obtained results were assessed with respect to time–frequency concentration (measured by the Rényi entropy), instantaneous frequency (IF) estimation accuracy, cross-term presence in the TFRs, and the computational cost of the TFRs. This study gives valuable insight into the advantages and limitations of the analyzed TFRs and assists in selecting the proper distribution when analyzing given non-stationary signals in the time–frequency domain.

Time–frequency analysis and instantaneous frequency estimation using two-sided linear prediction

Multi-scale chirplet synchroextracting transform for accurate characterization of adjacent fault features in rotating machinery

Nonlinear squeezing time–frequency transform for weak signal detection

The Wigner distribution (WD) produces highly concentrated time-frequency (TF) representation of nonstationary signals. It may be used as an efficient signal analysis tool, including the cases of frequency modulated signals corrupted with the Gaussian noise. In some applications, a significant amount of impulse noise is present. Then, the WD fails to produce satisfactory results. The robust periodogram has been introduced for spectral estimation of this kind of noisy signals. It can produce good concentration for pure harmonic signals. However, it is not so efficient in the cases of signals with rapidly varying frequency. This is the motivation for introducing the robust WD. It is a reliable TF representation tool for wide class of nonstationary signals corrupted with impulse noise. This distribution produces good accuracy of the instantaneous frequency (IF) estimation. Using the Huber (1981) loss function, a generalization of the WD is presented. It includes both the standard and the robust WD as special cases. This distribution can be used for TF analysis of signals corrupted with a mixture of impulse and Gaussian noise. The presented theory is illustrated on examples, including applications on the IF estimation and time-varying filtering of signals corrupted with a mixture of the Gaussian and impulse noise. The case study analysis of the IF estimators' accuracy, based on the standard and the robust WD forms, is performed. In order to improve the IF estimation, a median filter is applied on the obtained IF estimate.

Multi-lifting synchrosqueezing transform for nonstationary signal analysis of rotating machinery

Although the original short-time Fourier-transform-based synchrosqueezing transform (FSST) and its second-order transform (FSST2) can effectively improve the readability of quasi-stationary signal and time-varying signal, respectively, the weak components of time–frequency representation are often submerged easily by noises and large-amplitude instantaneous frequencies (IFs). Moreover, aerospace engines always work in fierce vibration and non-steady states, and this easily causes the weak fault feature of rolling bearings obscured in the time–frequency domain. To solve this problem, we propose a time–frequency analysis algorithm called energy time-convexity second-order synchrosqueezing transform (ET-FSST2). This can sharpen the time-varying IFs like FSST2, and more importantly, it can extract the time-varying IFs with small amplitudes such as weak impulse-like components from multi-component vibration signals. The ET-FSST2 firstly calculates the energy convexity function in the time direction to extract the non-stationary IFs after employing FSST2. It then structures an optimization function by combining a hyperbolic tangent function with a chi-square distribution function as well as optimizing the targeted parameters, aiming to extract the weak components of the non-stationary IFs. Moreover, the effectiveness and robustness of the proposed method are validated by numerical simulation and rolling bearing fault tests. Finally, two case studies of weak fault diagnosis of Ni–Cu–Ag-based PVD-coated rolling bearings operating in cryogenic surroundings are given to illustrate the effectiveness of the proposed method for aerospace engine bearing fault diagnosis.