Instantaneous Frequency Estimation Research Articles

Tacho-less spur gear condition monitoring at variable speed operation using the adaptive application of variational mode extraction

Effective condition monitoring of machine components contributes to a safer working environment for operators and assists in averting critical machinery shutdowns. In real-world industrial scenarios, fault detection must remain simplified, user-friendly and robust despite speed variations. The operational demands of machinery frequently result in speed fluctuations, leading to spectral smearing, thereby compromising the efficiency of the analysis methodology. Furthermore, the intricacies of parameter initialisation often increase the complexity of the analysis methods. This study introduces a fault diagnosis algorithm that requires minimal initialisation and operates independently of tacho pulses. The algorithm proposed in the study incorporates variational mode extraction with the maxima tracking algorithm for instantaneous frequency estimation. Hankel matrix-based selective spectral fusion is proposed to mitigate the impact of frequency tracking errors caused by transient noise. The results of spectral side-band-based fault severity analysis, conducted on an in-house spur gearbox test-bed with seeded tooth chips, underscore the superior performance of the proposed algorithm when compared to contemporary non-stationary analysis methods.

Radial vibration measurement of rotating shaft using constant density fringe pattern and line scan camera

The radial vibration signals of the rotor can provide abundant information about the health condition of the machine. In this paper, a simple vision-based measurement system is proposed to simultaneously measure two-dimensional displacements in radial directions for the rotating shaft, where the system consists of a constant density fringe pattern (CDFP), a line scan camera (LSC), and a lens. The CDFP should be installed around the surface of the rotating shaft to make the density of the fringe constant along the shaft axis, while the shaft axis is vertical to the optical axis of the LSC but not parallel to the line-array sensor of the LSC. Therefore, the density of the fringe imaged on the LSC is not constant because of the modulation of the circular surface of the shaft, and the distribution of the fringe density on the LSC is a U-shaped curve. Thus, the shaft centreline orbit can be tracked by the lowest point of the density distribution curve (DDC) of the fringe. Then, an efficient and accurate parameterized instantaneous frequency estimation method is employed to estimate the DDC of the fringe, because the variable density fringe can be regarded as an amplitude-modulated and frequency-modulated nonstationary signal whose instantaneous frequency function is equivalent to the DDC. Experimental results verify the feasibility and effectiveness of the proposed method by comparing it to the eddy current sensors.

Weight extracting transform for instantaneous frequency estimation and signal reconstruction

This paper introduces a novel time–frequency analysis method called the Weight Extracting Transform (WET). The primary objective of WET is to enhance the energy concentration in linear time–frequency representations for time-varying signals. By aggregating the time–frequency blur produced by the short-time Fourier transform onto the actual instantaneous frequency, WET improves the readability and accuracy of the time–frequency representation. Additionally, the algorithm is extended to a more general linear time–frequency transform known as the chirplet transform, which effectively handles fast-varying signals. The WET method excels at distinguishing components with closely spaced instantaneous frequencies and can reconstruct the original signal from its time–frequency representation. Experimental results using simulated and real-world signals demonstrate that WET achieves superior energy concentration and noise robustness compared to existing methods.

The Synchrosqueezed Method and Its Theory-Analysis-Based Novel Short-Time Fractional Fourier Transform for Chirp Signals

Time–frequency analysis is an important tool used for the processing and interpretation of non-stationary signals, such as seismic data and remote sensing data. In this paper, based on the novel short-time fractional Fourier transform (STFRFT), a new modified STFRFT is first proposed which can also generalize the properties of the modified short-time Fourier transform (STFT). Then, in the modified STFRFT domain, we derive the instantaneous frequency estimator for the chirp signal and present a new type of synchrosqueezing STFRFT (FRSST). The proposed FRSST presents many results similar to those of the synchrosqueezing STFT (FSST), and it extends the harmonic signal to a chirp signal that offers attractive new features. Furthermore, we provide a detailed analysis of the signal reconstruction, theories, and some properties of the proposed FRSST. Several experiments are conducted, and all of the results illustrate that the proposed FRSST is more effective than the FSST. Finally, based on the linear amplitude modulation and frequency modulation signal, we present a derivation for analyzing the limitations of the FRSST.

Application of second order multi-synchrosqueezing transform for seismic data analysis

Time-frequency methods, which allow us to observe the seismic signal in time and frequency domains simultaneously, can efficiently discern the seismic response features of geological bodies in different frequency bands. In this paper, a newly developed time-frequency method named second order multi-synchrosqueezing transform (SMSST) is introduced to analyze seismic data. SMSST combines the second-order instantaneous frequency estimation and the iterative idea of multi-synchrosqueezing transform to approximate the true instantaneous frequency trajectory. It solves the problems of non-reassignment points and the sensitivity to noise at the same time. Synthetic example is employed to validate the superiority of SMSST in time-frequency analysis. Results obtained from four different methods show that SMSST offer the best performance in terms of energy-concentration and noise robustness. Applications on field data further demonstrate that SMSST-based method is suitable and effective to highlight geological information and stratigraphic features compared to the previous techniques. The capabilities of SMSST for accurate reservoir localization with high resolution, making it a promising and potential tool in seismic interpretation.

Two-dimensional synchrosqueezing transform and its application to tight channel sandstone reservoir characterization

Spectral analyses can effectively highlight the morphological characteristics of tight-channel sandstone reservoirs, which are crucial development targets in the field of unconventional oil and gas. A majority of these methods focus on trace-by-trace analyses of seismic data without lateral constraints. However, they do not depict the morphological characteristics of the channels in detail. Therefore, this study proposes a novel time-frequency (TF) analysis (TFA) method: two-dimensional synchrosqueezing transform (TDSST). The TDSST introduces a space-time window to capture the lateral variation of seismic signals. Then, from a two-dimensional (2D) purely harmonic signal, the instantaneous frequency (IF) and instantaneous wavenumber (IK) estimation formulas were strictly derived in the two-dimensional short-time Fourier transform domain. Furthermore, we define a new two-dimensional synchrosqueezing operator (TDSTO) that can squeeze the TF coefficients into the estimated IF and IK trajectories. Finally, the TDSST can effectively reflect the vertical and horizontal changes, and according to the relationship between the wave number and frequency, the wave number parameter can be determined by maximizing the energy of the TDSST. TDSST can provide a highly energy-concentrated TF representation (TFR) while allowing the reconstruction of signals. Compared to other TFA methods, TDSST can provide a more laterally continuous spectral decomposition that clearly describes tight-channel sandstone reservoirs, even in a strongly noisy environment. The reliability of TDSST is validated through the utilization of both synthetic model and seismic field data.

Instantaneous Frequency and Amplitude Estimation in Multicomponent Signals Using an EM-Based Algorithm

Instantaneous Frequency and Amplitude Estimation in Multicomponent Signals Using an EM-Based Algorithm

Error Analysis of Instantaneous Frequency Estimation of Second-Order SST and Its Application on Analyzing Gear Meshing Stiffness From Motor Current

Error Analysis of Instantaneous Frequency Estimation of Second-Order SST and Its Application on Analyzing Gear Meshing Stiffness From Motor Current

Multisynchrosqueezing short-time fractional Fourier transform and its application in rolling bearing instantaneous frequency estimation

Multisynchrosqueezing transform (MSST) enhances the time-frequency energy concentration by using iterative reassignment operations in time-frequency analysis (TFA). However, its effectiveness is limited for signals with rapidly changing instantaneous frequency. To address this issue, this paper presents a novel time-frequency representation (TFR) method called multisynchrosqueezing short-time fractional Fourier transform, which offers improved TF concentration for strongly frequency-modulated signals. Firstly, a high-resolution TFR of the signal is obtained by locally optimized short-time fractional Fourier transform (STFrFT). Secondly, iterative synchrosqueezing operations are introduced to further enhance the STFrFT energy concentration, with a termination strategy relying on Rényi entropy proposed to ascertain the optimal number of iterations. Finally, the ideal TFA with high energy concentration is achieved. The proposed method was validated using multi-scene simulated signals and variable-speed bearing signals. The results show that the proposed method exhibits superior time-frequency energy concentration and instantaneous frequency estimation accuracy. The estimation error of the method is consistently at least 40% lower than that of the compared short-time Fourier transform-based methods, as assessed through the evaluation criteria of maximum relative error, mean square error and symmetric mean absolute percentage error.

Sea Clutter Suppression Using Smoothed Pseudo-Wigner–Ville Distribution–Singular Value Decomposition during Sea Spikes

The detection of small targets within the background of sea clutter is a significant challenge faced in radar signal processing. Small target echoes are weak in energy, and can be submerged by sea clutter and sea spikes, which are caused by overturning waves and breaking waves. This severely affects the radar target detection performance. This paper proposes a smoothed pseudo-Wigner–Ville distribution–singular value decomposition (SPWVD-SVD) method for sea clutter suppression. This method determines the instantaneous frequency range of the target by contrasting the time–frequency characteristics of the sea spike and the target. Subsequently, it employs a singular value difference spectrum to reduce the rank of the Hankel matrix, thereby reducing the computational burden of the instantaneous frequency estimation step in the experiment. Based on the instantaneous frequency range of the target in the time–frequency domain, the singular values of the target signal are retained, while the singular values of clutter are set to zero. This process accomplishes the reconstruction of radar echo signals and effectively achieves the suppression of sea clutter. The suppression effect is verified using simulation data alongside ten sets of Intelligent Pixel processing X-band (IPIX) radar data against the background of sea spikes. By contrasting the clutter amplitudes before and after suppression, the SPWVD-SVD algorithm demonstrated an average clutter suppression of 15.06 dB, which proves the effectiveness of the proposed algorithm in suppressing sea clutter.

Instantaneous Frequency and Chirp Rate Estimation for Noisy Quadratic FM Signals by CNN

Deep learning and machine learning are widely employed in various domains. In this paper, Artificial Neural Network (ANN) and Convolution Neural Network (CNN) are used to estimate the Instantons Frequency (IF), Linear Chirp Rate (LCR), and Quadratic Chirp Rate (QCR) for Quadratic Frequency Modulated (QFM) signals under Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (ASαS) noise. SαS distributions are impulsive noise disturbances except for a few circumstances, lack a closed-form Probability Density Function (PDF), and an infinite second-order statistic. Geometric SNR (GSNR) is used to determine the impulsiveness of mixture noise for Gaussian and SαS noise. ANN is a machine learning classifier with few layers that reduce FE, LCRE, and QCRE complexity and achieve high accuracy. CNN is a deep learning classifier that is built with multiple layers of FE, LCRE, and QCRE. CNN is more accurate than ANN when dealing with large amounts of data and determining optimal features. The results reveal that SαS noise is substantially more damaging to FE, LCRE, and QCRE than Gaussian noise, even when the magnitude is modest, and it is less damaging when alpha is greater than one. After training DCNN for FE, LCRE, and QCRE estimation of QFM signals. The 2D-CNN model accuracy achieved 98.7603 and 1D-CNN is 75.8678 for ten epochs. ANN model accuracy achieved 37.5 for 1000 epochs. The accuracy of TFD (spectrogram & pspectrum) for frequency estimation of QFM signals was 38.4254 by spectrogram and 38.6746 by pspectrum.

Adaptive scale chirplet transform and its application to bearing fault analysis

In response to the problems of biased estimation of instantaneous frequency (If) and poor noise immunity in current time–frequency (Tf) analysis methods, the adaptive scale chirplet transform (ASCT) is proposed in this paper. The core idea of the proposed algorithm is to use a frequency-dependent quadratic polynomial kernel function to approximate the IF of the signal and to use the time-varying window length to overcome the frequency resolution problem due to the change in signal modulation. This method can dynamically select suitable parameters and overcome the disadvantage of unfocused energy of TF distribution. The experimental results show that the ASCT algorithm has high TF aggregation and can suppress noise interference well. In practical signal processing, the advantage of the ASCT algorithm is that it can accurately depict the characteristic frequency of the signal and detect the fault in the bearing signal. Both simulation and experimental results prove the strong realistic relevance of this algorithm.

전파 거리에 따른 신호의 감쇠와 분산이 고려된 순시 주파수 추정 알고리즘 및 장거리 전파용 반사파 계측법 시뮬레이터 개발

Time-Frequency domain Reflectometry(TFDR) is used for the purpose of fault detection and evaluating the integrity in cables. TFDR can diagnose live wire in nondestructive. but, the incident signal cannot aviod attenuation and dispersion according to increasing propagation distance, propagation velocity changes with propagation distance. attenuation and dispersion in long distance lead to decresing falut detection performance of cable. therefore, Developing instantaneous frequency Estimation algorithm of signal and signal restoration algorithm in order to determine tendency of propagation velocity for propagation distance, we demonstrate the performance of algorithm by implementing TFDR simulator possible long length.

Multi-sensor random sample consensus for instantaneous frequency estimation of multi-component signals

An instantaneous frequency estimation algorithm is developed for multi-component signals acquired through multiple sensors. The proposed method estimates the mixing matrix for the strongest source in the given multi-sensor recording. The information of the mixing matrix is used to extract the corresponding source through spatial filtering. The spatially filtered signal is then analyzed using adaptive directional time-frequency distribution. The instantaneous frequencies of the extracted sources are then estimated through the Random sample consensus (RANSAC) algorithm that involves the detection of peaks, fitting of the instantaneous frequency curve using Fourier series approximation, and evaluating the instantaneous frequency estimate using an objective function. Both the mixing matrix and instantaneous frequency information is used to remove the corresponding source from the mixture signal by employing time-frequency and spatial filtering. The process mentioned above is then repeated till the instantaneous frequencies of all the sources have been estimated. Once the instantaneous frequencies of all the sources have been estimated, each source's instantaneous frequency is re-estimated by removing all the other components using time-frequency and spatial filtering. Till convergence, the process is iterated. Through numerical results, it is shown that the proposed method performs better than the relevant methods in the state of the art.

Synchronous fault feature extraction for rolling bearings in a generalized demodulation framework

Generalized demodulation (GD) has the potential to process non-stationary vibration signals since it can demodulate a signal with a curved time–frequency (TF) ridge into a signal, with a TF ridge parallel to a time axis with an improved time-frequency representation (TFR) energy concentration level. However, current GD methods require iteration operations and cannot simultaneously deal with vibrations from multiple components of rolling bearings. This paper proposes a method based on the GD framework, which can simultaneously demodulate multiple components of interest using the Hadamard product between matrices. A synchronous extractor is also constructed to post-process TFRs of generalized demodulated signals to further improve the TF aggregation. Unlike the conventional synchronous extraction transform, the synchronous extractor in this paper can be directly applied to TF ridges parallel to the time axis without the estimation of instantaneous frequencies (IFs). Then, the post-processed TF ridges are backward demodulated to restore the actual IF. The proposed synchronous fault feature extraction method in the GD framework also allows for the signal reconstruction. Both simulated and experimental signals are applied to validate the effectiveness of the proposed method for rolling bearing fault diagnosis.

Deep learning-based adaptive mode decomposition and instantaneous frequency estimation for vibration signal

Under time-varying environments and loads, vibration monitoring data of structures exhibit nonstationarity with multiple frequency components, and may have obvious nonlinearity under some extreme loads. The time–frequency analysis method is ideal for analyzing these nonlinear and nonstationary signals. However, the vibration signals of large-scale structures typically contain a variety of frequency components, making it challenging for effective mode decomposition without the interference of false mono-components by existing adaptive mode decomposition methods. False mono-components will severely decrease the accuracy of instantaneous frequency. To address the mode mixing and false mono-components in mode decomposition, a data-driven approach based on deep learning for signal adaptive mode decomposition and instantaneous frequency estimation is proposed in this paper. By transforming the time–frequency analysis problem into an optimization task of a deep neural network, built-in redundant adaptive filters with sparse regularization are designed, then the accurate intrinsic mode functions and time-varying frequencies can be estimated. The proposed method is verified by the numerical nonstationary signal and vibration signal of an experimental long cable model. The results demonstrate the superior capability of the proposed method in decomposing a signal into mono-components adaptively without false mono-components and achieving high accuracy of corresponding instantaneous frequencies. From the examples, an accuracy of 98.52% is achieved for a noise-free signal with spectral overlap. The proposed method is more robust to noise, and the instantaneous frequency identification error is 4.90%, which is much lower than EEMD and VMD under a signal-to-noise ratio of 5 dB.

Novel multivariate methods to track frequency shifts of neural oscillations in EEG/MEG recordings

Instantaneous and peak frequency changes in neural oscillations have been linked to many perceptual, motor, and cognitive processes. Yet, the majority of such studies have been performed in sensor space and only occasionally in source space. Furthermore, both terms have been used interchangeably in the literature, although they do not reflect the same aspect of neural oscillations. In this paper, we discuss the relation between instantaneous frequency, peak frequency, and local frequency, the latter also known as spectral centroid. Furthermore, we propose and validate three different methods to extract source signals from multichannel data whose (instantaneous, local, or peak) frequency estimate is maximally correlated to an experimental variable of interest. Results show that the local frequency might be a better estimate of frequency variability than instantaneous frequency under conditions with low signal-to-noise ratio. Additionally, the source separation methods based on local and peak frequency estimates, called LFD and PFD respectively, provide more stable estimates than the decomposition based on instantaneous frequency. In particular, LFD and PFD are able to recover the sources of interest in simulations performed with a realistic head model, providing higher correlations with an experimental variable than multiple linear regression. Finally, we also tested all decomposition methods on real EEG data from a steady-state visual evoked potential paradigm and show that the recovered sources are located in areas similar to those previously reported in other studies, thus providing further validation of the proposed methods.

Method for Automatic Estimation of Instantaneous Frequency and Group Delay in Time–Frequency Distributions with Application in EEG Seizure Signals Analysis

Instantaneous frequency (IF) is commonly used in the analysis of electroencephalogram (EEG) signals to detect oscillatory-type seizures. However, IF cannot be used to analyze seizures that appear as spikes. In this paper, we present a novel method for the automatic estimation of IF and group delay (GD) in order to detect seizures with both spike and oscillatory characteristics. Unlike previous methods that use IF alone, the proposed method utilizes information obtained from localized Rényi entropies (LREs) to generate a binary map that automatically identifies regions requiring a different estimation strategy. The method combines IF estimation algorithms for multicomponent signals with time and frequency support information to improve signal ridge estimation in the time–frequency distribution (TFD). Our experimental results indicate the superiority of the proposed combined IF and GD estimation approach over the IF estimation alone, without requiring any prior knowledge about the input signal. The LRE-based mean squared error and mean absolute error metrics showed improvements of up to 95.70% and 86.79%, respectively, for synthetic signals and up to 46.45% and 36.61% for real-life EEG seizure signals.

Mechanical varying non-stationary signal separation method based on instantaneous frequency estimation

Mechanical equipment often works on variable speed condition, its corresponding vibration signal presents multi-component, modulation coupling with fast time-varying instantaneous frequency (IF), how to effectively compute IF and realize fasting varying non-stationary signal decoupling separation plays an important role in mechanical system fault diagnosis. In this paper, a sparse representation method called multi-scale chirp sparse representation (MSCSR) is introduced to identify, extract, and trend IF for achieving a highly concentrated time-frequency energy. Simulation demonstrates that the proposed method performs better than traditional IF estimation method. Furthermore, an adaptive time-varying filter is constructed using the extracted instantaneous frequency to decouple non-stationary fast signal. Ultimately, by rapid instantaneous frequency fluctuation experiment, the effectiveness of proposed method for fast strong time-varying signal is validated, it can effectively extract rapid oscillation instantaneous frequency, and the error is less than 10%.

A fault diagnosis method for variable speed gearbox bearing based on SET improved multi-source ridge line

Due to the noise interference and frequency ambiguity of gearbox vibration signal under variable speed, the traditional instantaneous frequency estimation method is prone to frequency mutation and low ridge line fusion accuracy. This paper proposes an improved method of extracting multi-source time–frequency ridge lines based on time–frequency analysis by exploring the relationship between the vibration signal of the gearbox without a tachometer and the fluctuation law of the bearing rotation frequency. In the proposed method, the Variable Mode Decomposition (VMD) filter and Synchroextracting Transform (SET) are first used to analyze the time frequency of the gearbox vibration signal. Secondly, the local peak search algorithm obtains multiple time–frequency ridge lines from low-frequency and high-frequency resonance frequency bands. Thirdly, the synchronization process is carried out based on the maximum energy spectrum in the time spectrum. The multi-source ridge line extraction algorithm is improved with reduced frequency mutation and optimized ridge line fusion accuracy. Finally, the optimized speed curve is used to track the vibration signal order of the gearbox. The bearing fault characteristic order is obtained from the angular order spectrum, and the effectiveness of the proposed method is verified by the simulation and the experiment.