Instantaneous frequency estimation of intersecting and close multi-component signals with varying amplitudes

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.

Instantaneous frequency estimation for wheelset bearings weak fault signals using second-order synchrosqueezing S-transform with optimally weighted sliding window

Chapter 10 - Instantaneous Frequency Estimation and Localization

A robust parameter estimation of FHSS signals using time–frequency analysis in a non-cooperative environment

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.

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

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.

Pipelined signal adaptive hardware implementation of an optimal time-frequency (TF) filter has been designed. It is based on the real-time results of TF analysis and on the TF analysis-based instantaneous frequency (IF) estimation. The implemented pipelining technique allows the filter to overlap in execution unconditional steps performimg in neighboring TF instants and, therefore, to significantly enhance time performance. The improvement corresponding to the one clock cycle by a TF point is achieved, which means that the improvement by a TF point can reach even 50% in some TF points. The design is tested on multicomponent signals.).

The synchrosqueezing transform is a time-frequency (TF) analysis tool to process non-stationary signals. Unfortunately, it does not produce accurate TF results when faced with LFM signals. In this paper, we propose synchrosqueezing-Hough transform (SS-HT) to address this problem. First, we introduce Hough transform to obtain more accurate instantaneous frequency (IF) estimation. Then, in order to improve the TF concentration, a new TF rearrangement operator is constructed. Furthermore, we demonstrated that SS-HT has the ability to reconstruct the signal. The experimental results prove that SSHT can not only obtain high TF concentration, but also improve the accuracy of IF estimation and parameters estimation of LFM signals.

This paper makes use of local polynomial Wigner-Ville distribution (LPWVD), originally designed for nonparametric instantaneous frequency (IF) estimation of transient signals, to propose a parametric IF estimation for polynomial phase signals (PPSs). Statistical performance such as asymptotic bias and variance of the LPWVD-based parametric IF estimator is derived in closed-form. Based on the analytical results, we extend the statistical efficiency of the Wigner-Ville distribution (WVD) for a second-order PPS only to that of the LP-WVD for an arbitrary order, when the IF is estimated at the middle of sample observations. Simulation results verify the analytical performance and comparisons with the polynomial Wigner-Ville distribution (PWVD) show that the LPWVD-based parametric IF estimator can provide better performance.

The paper presents a method for the estimation of instantaneous frequency (IF) of signals characterized by interfering IF trajectories. It would be a useful methodology for maintenance and troubleshooting of wireless networks sharing the same operating frequency interval or predictive maintenance of vibrating/rotating systems. Stemming from IF trajectories estimated through traditional or enhanced time-frequency representations, the method applies an agile digital signal processing approach in order to suitably merge different sections belonging to the same trajectory. Preliminary results show the efficacy of the method when signals characterized by multiple intersections in the time-frequency domain are experienced.

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.

XWD-algorithm for the instantaneous frequency estimation revisited: Statistical analysis

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

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.