Accurate Instantaneous Frequency Research Articles

Multicomponent collaborative time-frequency state-space model for vibration signal decomposition under nonstationary conditions

The implementation of prognostics and health management strategies is essential for enhancing the safety and maintenance of rotating equipment in chemical plants. The examination of vibration signal behaviours under variable-speed conditions and the development of signal decomposition methods in such contexts are of substantial theoretical and practical relevance. This paper proposes a novel multicomponent collaborative time-frequency state-space method for the decomposition of vibration signals. The method employs a multi-component proportional model to accurately describe the synchronisation of high-frequency components with the rotational frequency component. Based on this framework, the instantaneous frequency change curves of multiple components are input into the Vold–Kalman filter algorithm for precise decomposition of multicomponent vibration signals under variable-speed conditions. Experimental results demonstrate the effectiveness of the proposed method in achieving accurate instantaneous frequency tracking and signal decomposition, showing clear advantages over traditional methods.

Synchronous odd symmetric transform for rolling bearing fault diagnosis

Time-frequency analysis (TFA) is an important tool to process non-stationary signals and detect bearing faults. For considering energy concentration, computational efficiency, and computational accuracy, a novel TFA technique, termed synchronous odd symmetric transform (SOST) is proposed in this paper. In the SOST, the odd symmetry extraction operator (OSEO) is defined according to the odd symmetric window based short-time Fourier transform (OSTFT). The main idea of OSEO is to obtain accurate instantaneous frequency (IF) ridges in a satisfactory computation time. Meanwhile, the STFT result is retained at the IF ridges, and the time–frequency representation (TFR) with high energy concentration is calculated. The performance of the developed technique is verified by a multi-component signal with strong frequency-modulated and amplitude-modulated (FM-AM) laws. Experimental analysis results show that the proposed SOST can display bearing fault-related time–frequency trajectories with high time–frequency resolution.

Local maximum synchrosqueezes form scaling-basis chirplet transform.

In recent years, time-frequency analysis (TFA) methods have received widespread attention and undergone rapid development. However, traditional TFA methods cannot achieve the desired effect when dealing with nonstationary signals. Therefore, this study proposes a new TFA method called the local maximum synchrosqueezing scaling-basis chirplet transform (LMSBCT), which is a further improvement of the scaling-basis chirplet transform (SBCT) with energy rearrangement in frequency and can be viewed as a good combination of SBCT and local maximum synchrosqueezing transform. A better concentration in terms of the time-frequency energy and a more accurate instantaneous frequency trajectory can be achieved using LMSBCT. The time-frequency distribution of strong frequency-modulated signals and multicomponent signals can be handled well, even for signals with close signal frequencies and low signal-to-noise ratios. Numerical simulations and real experiments were conducted to prove the superiority of the proposed method over traditional methods.

Self-matched extracting wavelet transform and signal reconstruction

Time-frequency (TF) analysis method provides a powerful tool to analyze non-stationary signals. However, for strongly time-varying signals, how to characterize the time-varying features of signals accurately, how to achieve a highly concentrated TF representation, and how to reconstruct signals with high accuracy are still challenging tasks. In this paper, in order to analyze strongly amplitude-modulated and frequency-modulated signals, we introduce an accurate instantaneous frequency (IF) estimator, further propose a high-resolution TF analysis method and three high-accuracy reconstruction methods. Firstly, we present signal parameter, chirp rate and IF estimation from the wavelet transform of Gaussian amplitude-modulated linear chirp model. Based on this, we introduce the self-matched extracting wavelet transform (SMEWT), which improves the energy concentration of TF representation, by only retaining TF information related to the IF of the signal. Furthermore, the signal parameter, chirp rate and IF estimation lead to accurate signal reconstruction formulas. In this regard, we provide a theoretical analysis and comparison of SMEWT reconstruction, and present two hybrid reconstruction methods of signal. It is well known that the reassignment method can improve the readability of the TF representation, but it cannot reconstruct the signal. It is worth mentioning that one of the proposed hybrid reconstruction methods is a combination of the reassignment method and deduced reconstruction formula, which not only captures the flavor and philosophy of the reassignment method, but also uses a hybrid way in signal reconstruction. Finally, simulated and real signals are employed to confirm the effectiveness of the proposed methods by comparing with some classical TF analysis methods.

Second-order Synchrosqueezing Modified S Transform for wind turbine fault diagnosis

Under the working condition of variable speed, vibration signals of wind turbine gearboxes are characterized by instability coupled with noise interference, which has made it difficult to identify early fault characteristics through common Time-Frequency (TF) Analysis (TFA) methods. Stockwell Transform (ST) has such advantages as undergoing inverse transformation without loss and being free from cross-term interference when processing signals. Therefore, this paper has put forward Modified Stockwell Transform (MST), which enables the window width to change proportionally with the varying frequency. Moreover, the change caused by parameter adjustment is much more purposeful. It ensures that TF analysis windows are more flexible and changeable at different positions in the TF plane. Furthermore, to obtain much more accurate Instantaneous Frequency (IF) estimation of multi-component vibration signals, this paper uses the second-order partial derivative of phase function to modify it. Eventually, in order to achieve a concentrated Time-Frequency Representations (TFR) of wind turbines vibration signals, this paper proposes Second-order Synchrosqueezing Modified S Transform, namely SSMST2, through making use of the theoretical framework of Synchrosqueezing Transform (SST) and second-order IF estimations to extend MST. By reassigning the TF transform coefficients of MST, the proposed SSMST2 can better show the time-varying features. To verify this method is feasible, this paper will apply it to fault diagnosis of bearing test rig and to vibration signal processing of rolling bearing in 2 MW wind turbine. The result demonstrates that SSMST2 has yielded great outcome in TFR performance of variable speed vibration signals.

High-Order Synchroextracting Chirplet Transform for Accurate Instantaneous Frequency Estimation and Its Application in Fault Diagnosis of Rotary Machinery

Time-frequency analysis (TFA) technology is an effective tool for extracting time-varying characteristics of a signal, but for strong AM-FM signals, time-frequency representation (TFR) with high time-frequency (TF) resolution is often difficult to generate. The measured vibration signals of mechanical equipment often contain a large number of non-stationary features and are polluted by noise, which hinders the use of TFA technologies. In this paper, we propose a TFA method termed high-order synchroextracting chirplet transform (HSECT) to effectively process strong AM-FM signals. On the basis of Chirplet transform (CT), we first determine the best linear demodulation operator at each moment through the local maximum amplitude of the generated TFR. Subsequently, the amplitude and phase of the signal are subjected to high-order Taylor expansion, and a new frequency estimation operator termed high-order synchroextracting chirplet operator (HSECO) is obtained by the high-order approximation of the time-varying laws of the signal. HSECO can accurately extract the instantaneous frequency of strong AM-FM signal and the construction of HSECT is completed by Dirichlet function at last. HSECT can generate TFR with high energy concentration from which the time-varying law of strong AM-FM signal can be extracted accurately. Meanwhile, the signal reconstruction operation can be completed concisely. In addition, the proposed method is applied to the fault diagnosis of the rotor rub impact and variable speed equipment and compared with other TFA methods. Simulation and experiments can verify the effectiveness and competitiveness of the proposed method.

Adaptive multiple second-order synchrosqueezing wavelet transform and its application in wind turbine gearbox fault diagnosis

Wind turbines usually operate in harsh environments and in working conditions of variable speed, which easily causes their key components such as gearboxes to fail. The gearbox vibration signal of a wind turbine has nonstationary characteristics, and the existing time-frequency (TF) analysis (TFA) methods have some problems such as insufficient concentration of TF energy. In order to obtain a more apparent and more congregated time-frequency representation (TFR), this paper proposes a new TFA method, namely adaptive multiple second-order synchrosqueezing wavelet transform (AMWSST2). Firstly, a short-time window is innovatively introduced on the foundation of classical continuous wavelet transform, and the window width is adaptively optimized by using the center frequency and scale factor. After that, a smoothing process is carried out between different segments to eliminate the discontinuity and thus adaptive wavelet transform is generated. Then, on the basis of the theoretical framework of synchrosqueezing transform and accurate instantaneous frequency estimation by the utilization of second-order local demodulation operator, adaptive second-order synchrosqueezing wavelet transform (AWSST2) is formed. Considering that the quality of actual TFA is greatly disturbed by noise components, through performing multiple synchrosqueezing operations, the congregation of TFR energy is further improved, and finally, the AMWSST2 algorithm studied in this paper is proposed. Since synchrosqueezing operations are performed only in the frequency direction, this method AMWSST2 allows the signal to be perfectly reconstructed. For the verification of its effectiveness, this paper applies it to the processing of the vibration signal of the gearbox of a 750 kW wind turbine.

Least-squares Wavelet and Cross-wavelet Analyses of VLBI Baseline Length and Temperature Time Series: Fortaleza–Hartebeesthoek–Westford–Wettzell

Extensive least-squares wavelet and cross-wavelet analyses are performed on the Very Long Baseline Interferometry (VLBI) baseline length and temperature time series for a network of four VLBI antennas located in different continents. These analyses do not rely on any pre-processing of the measurements including interpolations or gap-filling, and they can provide accurate instantaneous frequency information along with phase differences in the time–frequency domain. Out of four antennas mounted in Fortaleza, Hartebeesthoek, Westford, and Wettzell, the most and the least possibly impacted VLBI time series by the annual temperature variation are Westford–Wettzell and Fortaleza–Hartebeesthoek, respectively. Furthermore, the annual components of the temperature time series in Westford and Wettzell lead the ones in the corresponding VLBI time series by nearly one month, where one of the baseline sides is either in Westford or Wettzell.

Optimization of the Brillouin instantaneous frequency measurement using convolutional neural networks.

The Brillouin instantaneous frequency measurement (B-IFM) is used to measure instantaneous frequencies of an arbitrary signal with high frequency and broad bandwidth. However, the instantaneous frequencies measured using the B-IFM system always suffer from errors, due to system defects. To address this, we adopt a convolutional neural network (CNN) that establishes a function mapping between the measured and nominal instantaneous frequencies to obtain a more accurate instantaneous frequency, thus improving the frequency resolution, system sensitivity, and dynamic range of the B-IFM. Using the proposed CNN-optimized B-IFM system, the average maximum and root mean square errors between the optimized and nominal instantaneous frequencies are less than 26.3 and 15.5MHz, which is reduced from up to 105.8 and 57.0MHz. The system sensitivity is increased from 12.1 to 7.8dBm for the 100MHz frequency error, and the dynamic range is larger.

An iterative approach for spectrogram reassignment of frequency modulated multicomponent signals

In this paper, an evolution law for the modulus of the Short Time Fourier Transform, referred as spectrogram, of a frequency modulated multicomponent signal is proposed. Based on this result, an iterative reallocating method for the enhancement of spectrogram resolution is proposed. Compared to the standard reassignment method, the presented procedure allows us to obtain a time–frequency representation which better localizes the individual modes of a multicomponent signal even in the non separable case. The proposed method is computationally advantageous and robust to modes interference, then it could be employed for accurate instantaneous frequency estimation, which is a fundamental goal in applications dealing with non stationary signals such as radar, surveillance and audio signals.

Multiple squeezes from adaptive chirplet transform

Time-frequency (TF) analysis method is an effective tool to analyze non-stationary signals. However, how to generate a clear TF representation for strongly time-varying signals is still a challenging task. In this paper, we propose a new TF analysis method, named multi-synchrosqueezing chirplet transform (MSSCT), to study the non-stationary features of strongly time-varying signals. By relaxing the hypothesis made on the amplitude and frequency modulations, the proposed method produces a more accurate instantaneous frequency (IF) estimator to correct the deviation caused by TF ridge. MSSCT is an upgraded version of the recently developed multi-synchrosqueezing transform (MSST) method, and can be viewed as a good combination of adaptive chirplet transform and synchrosqueezing technique. Moreover, we prove that MSSCT retains the signal reconstruction ability. Simulated and real-life signals are employed to validate the advantages of MSSCT by comparing with some advanced TF methods. The experimental results demonstrate that MSSCT can provide a better time-varying description, obtaining a more precise IF estimate, being stable for the selection of window length, and having a stronger noise robustness.

Time-Frequency demodulation analysis via Vold-Kalman filter for wind turbine planetary gearbox fault diagnosis under nonstationary speeds

Abstract Wind turbine planetary gearbox fault diagnosis under nonstationary speeds is a challenging topic, because of the high complexity and strong time variability of vibration signals. In order to resolve time-varying gear fault features, a quality time-frequency analysis method is in demand. Time-frequency representations based on Hilbert transform and analytic signal approach have fine time-frequency resolution, and are free from both outer (cross-term) and inner (auto-term) interferences, thus providing an effective approach to nonstationary signal analysis. However, they rely on accurate instantaneous frequency estimation, and thereby are subject to mono-component constraint. To address this issue, Vold-Kalman filter is exploited to construct time-frequency representation, by virtue of its capability to decompose the multi-component vibration signal of rotating machinery into constituent mono-component harmonic waves. Even so, intricate time-varying sidebands inherent with raw planetary gearbox vibration signals in joint time-frequency domain are still a hurdle, because they do not link to gear fault frequency directly. To solve this problem, the proposed time-frequency analysis method is further extended to generate time-varying amplitude and frequency demodulated spectra, inspired by the fact that gear fault frequency is manifested straight by the amplitude and frequency modulating frequencies. The proposed method is illustrated by numerical simulation, and is further validated using lab experimental signals of a wind turbine planetary gearbox. Both the localized and distributed faults on gears are successfully diagnosed under nonstationary speeds.

A Novel Approach for Seismic Time-Frequency Analysis Based on High-Order Synchrosqueezing Transform

Time-frequency analysis always plays a central role in the field of seismic processing due to the advantage in characterizing nonstationary signals. In this letter, we present a novel technique for seismic time-frequency analysis based on the high-order synchrosqueezing transform, which obtains more accurate instantaneous frequencies by using the higher order approximations for both amplitude and phase in order to achieve a highly energy-concentrated time-frequency representation. A synthetic example is employed to demonstrate the validity of the proposed method in sharpening time-frequency representation. Application on field data example further proves its potential in enhancing time-frequency resolution and delineating stratigraphic characteristics with higher precision and renders that this technique is promising for seismic data analysis.

Time–Frequency Analysis of Pressure Pulsation Signal in the Chamber of Self-Resonating Jet Nozzle

In this paper, time–frequency (TF) characteristics of pressure pulsation in the self-resonating jet nozzle were thoroughly checked and analyzed by two TF analysis methods. The frequency spectrum including frequency-domain structure and approximate frequency-scope was obtained via the classical fast Fourier transform (FFT) method. Due to the typical nonstationary and time-varying nature, the FFT method is inefficient in analyzing such signals. TF analysis method was introduced to obtain the accurate instantaneous frequency (IF) of the self-resonating jet signal, which makes the quantitative analysis of the resonance frequency possible. In addition, a comparative study on the IF results were carried out by coupling the Morlet wavelet transforms, synchrosqueezed wavelet transforms (SWTs) and the modulus maximum methods together, and the coherence of frequency-domain structure was demonstrated. The result showed that SWT is good at acquiring the frequency distribution of the self-resonating jet with high readability and accuracy, and the TF analysis method is suitable for researching jet nozzle’s TF characteristics with high efficiency.

A modified tacho-less order tracking method for the surveillance and diagnosis of machine under sharp speed variation

Tacho-less order tracking (TLOT) is an effective technique to address the surveillance and diagnosis of mechanical fault on speed variation conditions without tachometers. Nevertheless, fault diagnosis with sharp speed fluctuations is still a challenging task, since the accurate instantaneous frequency (IF) cannot be obtained from the acquired condition signals. In this work, a modified TLOT method based on nonlinear compensating demodulation transform (NCDT) is proposed for fault diagnostics in extreme condition monitoring of machinery. During NCDT, the demodulation functions based on opposite cubic polynomial are investigated and performed to improve IF estimation. Then, a regression filter (RF) is employed to extract the interested harmonic component according to the estimated IF and instantaneous bandwidth for signal resampling. Finally, order spectrum analysis is used to analyze the resampled signal for mechanical fault diagnosis. Simulations, experiments and industrial cases are given to verify the effectiveness and efficiency of the proposed method. The results are as effective as those obtained by tacho-based method under sharp speed fluctuation.

Joint High-Order Synchrosqueezing Transform and Multi-Taper Empirical Wavelet Transform for Fault Diagnosis of Wind Turbine Planetary Gearbox under Nonstationary Conditions.

Wind turbines usually operate under nonstationary conditions, such as wide-range speed fluctuation and time-varying load. Its critical component, the planetary gearbox, is prone to malfunction or failure, which leads to downtime and repair costs. Therefore, fault diagnosis and condition monitoring for the planetary gearbox in wind turbines is a vital research topic. Meanwhile, the signals measured by the vibration sensors mounted in the gearbox exhibit time-varying and nonstationary features. In this study, a novel time-frequency method based on high-order synchrosqueezing transform (SST) and multi-taper empirical wavelet transform (MTEWT) is proposed for the wind turbine planetary gearbox under nonstationary conditions. The high-order SST uses accurate instantaneous frequency approximations to obtain a sharper time-frequency representation (TFR). As the acquired signal consists of many components, like the meshing and rotating components of the gear and bearing, the fault component may be masked by other unrelated components. The MTEWT is used to separate the fault feature from the masking components. A variety of experimental signals of the wind turbine planetary gearbox under nonstationary conditions have been analyzed to demonstrate the effectiveness and robustness of the proposed method. Results show that the proposed method is effective in diagnosing both gear and bearing faults.

Planetary Gearbox Fault diagnosis via Joint Amplitude and Frequency Demodulation Analysis Based on Variational Mode Decomposition

Planetary gearbox vibration signals have strong modulation features due to the amplitude modulation and frequency modulation (AM-FM) effect of gear faults, as well as the amplitude modulation (AM) effect of time-varying vibration transfer paths, on gear meshing vibrations. This results in an involute sidebands structure in Fourier spectrum, possibly misleading fault diagnosis. The modulating frequency of both amplitude modulation (AM) and frequency modulation (FM) parts is closely related to the gear fault characteristic frequency. This inspires the idea of joint amplitude and frequency demodulation analysis, thus addressing the complex sidebands issue inherent in Fourier spectrum. Demodulation analysis requires mono-component signals for accurate estimation of instantaneous frequency, and proper selection of an AM-FM component sensitive to gear fault. To this end, we firstly decompose the complex signal into intrinsic mode functions (IMFs) via variational mode decomposition (VMD), by exploiting its capability in decomposing complex modulated signal into constituent AM-FM components. For effective application of VMD in complex planetary gearbox signal analysis, we propose a method to determine a key parameter in VMD, i.e. the number of IMFs to be separated. For accurate instantaneous frequency estimation, we decompose IMFs via empirical AM-FM decomposition, to remove the influence of AM on instantaneous frequency estimation. Then, we select the sensitive IMF that contains the main gear fault information for further demodulation analysis. In order to properly select the sensitive IMF, we propose a criterion based on the gear vibration characteristics and the VMD properties. Finally, we obtain the amplitude and frequency demodulated spectra by applying Fourier transform to the amplitude envelope and instantaneous frequency of the selected sensitive IMF. According to the characteristics exhibited in the demodulated spectra, we can detect planetary gearbox fault. The proposed method is illustrated via a numerical simulated planetary gearbox vibration signal, and is further validated using lab experimental vibration signals of a planetary gearbox. Faults on all the three types of gear (sun, planet and ring) are successfully identified.

A two-level method for sparse time-frequency representation of multiscale data

Based on the recently developed data-driven time-frequency analysis (Hou and Shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function (IMF) and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some extent. We also present a method to reduce the end effects.

Zoom Synchrosqueezing Transform for Instantaneous Speed Estimation of High Speed Spindle

Instantaneous speed (IS) is of great significance of fault diagnosis and condition monitoring of the high speed spindle. In this paper, we propose a novel zoom synchrosqueezing transform (ZST) for IS estimation of the high speed spindle. Due to the limitation of the Heisenberg uncertainty principle, the conventional time-frequency analysis (TFA) methods cannot provide both good time and frequency resolution at the whole frequency region. Moreover, in most cases, the interested frequency component of a signal only locates in a narrow frequency region, so there is no need to analyze the signal in the whole frequency region. Different from conventional TFA methods, the proposed method arms to analyze the signal in a specific frequency region with both excellent time and frequency resolution so as to obtain accurate instantaneous frequency (IF) estimation results. The proposed ZST is an improvement of the synchrosqueezing wavelet transform (SWT) and consists of two steps, i.e., the frequency-shift operation and the partial zoom synchrosqueezing operation. The frequency-shift operation is to shift the interested frequency component from the lower frequency region to the higher frequency to obtain better time resolution. The partial zoom synchrosqueezing operation is conducted to analyze the shifted signal with excellent frequency resolution in a considered frequency region. Compared with SWT, the proposed method can provide satisfactory energy concentrated time-frequency representation (TFR) and accurate IF estimation results. Additionally, an application of the proposed ZST to the IS fluctuation estimation of a motorized spindle was conducted, and the result showed that the IS estimated by the proposed ZST can be used to detect the quality of the finished workpiece surface.

Sparse time–frequency representation for signals with fast varying instantaneous frequency

Time–frequency (TF) distributions have been used for providing high-resolution representation in a large number of signal processing applications. However, high resolution and accurate instantaneous frequency (IF) estimation usually depends on the employed distribution and complexity of signal phase function. To ensure an efficient IF tracking for various types of signals, a class of complex-time distributions (CTD) has been developed. These distributions facilitate analysis in cases when standard distributions cannot provide satisfactory results (e.g. for highly non-stationary signal phase). In that sense, an ambiguity-based form of the fourth-order CTD is considered, in a new compressive sensing (CS) context. CS is an intensively growing approach in signal processing that allows efficient analysis and reconstruction of randomly under-sampled signals. In this study, randomly chosen ambiguity domain coefficients serve as CS measurements. By exploiting sparsity in the TF plane, it is possible to obtain highly concentrated IF using just small number of randomly chosen coefficients from the ambiguity domain. Moreover, in noisy signal case, this CS approach can be efficiently combined with the L-statistics producing robust TF representations. Noisy coefficients are first removed using the L-statistics and then reconstructed by using the CS algorithms. The theoretical considerations are illustrated using experimental results.