Instantaneous Frequency Estimation Method Research Articles

Dynamic time warping based instantaneous frequency estimation for rotating machineries under non-stationary operating conditions

Abstract For tacho-less order tracking, it is crucial to achieve an accurate estimation of instantaneous phase and frequency under non-stationary conditions for fault diagnosis of mechanical systems. Therefore, an instantaneous frequency estimation method based on dynamic time warping for rotating machinery was proposed. Firstly, a mono-component signal is obtained by variational mode decomposition of multi-harmonic vibration signal generated in practical engineering. Then, a stationary reference signal is constructed by the mean frequency of the vibration signal, which is obtained by the signal spectrum. Aligning the vibration signal with the reference signal to obtain the rough phase estimation and warping path. Finally, a polynomial model is constructed to calculate the analytic expression of the smooth warping path to obtain the instantaneous frequency result. The simulation and experiment present that the dynamic time warping-based method can estimate the instantaneous frequency of the signal under non-stationary conditions. Compared with the classical Hilbert transform method, the proposed dynamic time-warping algorithm has stronger noise resistance.

Improved spectral amplitude modulation for tacholess estimation of rotation speed

Rotation speed is key information for rotating machinery, whether it’s a negative feedback control method based on speed or equipment status monitoring and fault diagnosis. In order to accurately and efficiently estimate the rotation speed under tacholess condition, a novel method named improved spectral amplitude modulation (ISAM) is proposed in this paper. In ISAM, the vibration signal is first filtered through a band-pass filter to eliminate interference from irrelevant frequency components. Subsequently, an appropriate weight, which modifies the amplitude spectrum of the time–frequency representation (TFR) to decompose different components based on their powers, is selected to enhance the instantaneous frequency (IF) of rotation speed. The modified signal is then reconstructed based on the edited amplitude and original phase. Finally, the IF can be extracted from the TFR of the modified signal with the multiple time–frequency curve extraction (MTFCE) technique. Additionally, entropy is utilized to guide the selection of weight. The effectiveness of the proposed method is evaluated through simulation analysis and case studies. The results show that the ISAM can accurately estimate rotation speed, with the mean absolute error (MAE) at 0.13. Compared to an existing IF estimation method, ISAM can reduce computation time by 95%.

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.

A fast method for robust estimation of gearbox instantaneous speed

Under non-stationary condition, estimating the instantaneous frequency (IF) of shaft from vibration signal is an ideal way for gearbox fault monitoring without tachometer. Among them, the method based on harmonic phase demodulation has always occupied a dominant position. In the low signal-to-noise ratio (SNR) condition, the noisy harmonic components may cause greater IF estimation error. In this work, an improved approach for estimating the IF of gearbox reference shaft is proposed based on the short time Fourier transform spectrogram. The theoretical derivation proves that the high-order meshing harmonic can reduce the IF estimation error of gearbox reference shaft. By constructing the spectrogram weight factor, an improved maximum tracking algorithm is proposed to redefine the IF estimation range. On this basis, the robust IF estimation of high-order meshing harmonic with low SNR is realized. Finally, it is equivalent to the estimated IF of gearbox reference shaft. The proposed strategy is compared with the two advanced IF estimation methods. The civil aircraft engine dataset and the wind turbine gearbox dataset provided in the Safran contest demonstrate that the proposed strategy works. In the case of heavy load and large speed change, the proposed method has extremely low consumption time while maintains good estimation accuracy.

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.

Motion parameter estimation of AUV based on underwater acoustic Doppler frequency measured by single hydrophone

This paper describes the use of a single hydrophone to estimate the motion parameters of an autonomous underwater vehicle (AUV) from the underwater acoustic signal excited by its propulsion motor. First, the frequency range of the hydroacoustic signal radiated by the AUV motor is determined, and a detection and recognition model is designed. In the case of uniform linear motion of the AUV, the geometric relationship between the Doppler frequency shift curve of the sound source is derived and the motion model of the sound source and sound line propagation is established. An estimation algorithm for the motion parameters of multiple AUVs based on data from a single hydrophone is derived. Then, for Doppler underwater acoustic signals disturbed by independent identically distributed noise with an arbitrary probability distribution, a cumulative phase difference power amplification instantaneous frequency estimation method is proposed. This method is based on the sum of multiple logarithmic functions. Finally, the effectiveness and accuracy of the algorithm in estimating the motion parameters of multiple AUVs are verified through simulations and experiments.

Parameterized Instantaneous Frequency Estimation Method for Vibration Signal with Nonlinear Frequency Modulation

The vibration signal from the rotatory machinery condition monitoring under time-varying speed is usually amplitude-modulated (AM) and frequency-modulated (FM). It is important to efficiently and accurately estimate the instantaneous frequency (IF) of the vibration signal. In this paper, a novel parameterized IF estimation method is proposed. The method employs a high-order polynomial function to approximate the nonlinear IF and subsequently constructs overdetermined systems of linear equations by calculating the Fourier transform of the derivative of the signal. The IF can be estimated by using least squares estimation to solve the equations. The proposed method has high computational efficiency because it can obtain the estimation of IF over a period of time simultaneously; it differs from traditional time-frequency analysis methods that need to calculate the IF at each point in the time axis. It is demonstrated that the proposed method is not only particularly powerful for the nonlinear FM mono-component signal but also applicable to the multi-component signal constructed by multiple harmonics. The numerical simulation validates the effectiveness of the proposed method, and the experiment’s results show that the method is suitable for the IF estimation of the vibration signal from the varying-speed rotor system.

A robust method for instantaneous frequency estimation via inverse spectral decomposition

Instantaneous frequency is an important seismic attribute, which can indicate thin beds and lithofacies boundaries. However, instantaneous frequency attribute is susceptible to noise when it is obtained by the traditional Hilbert transform (HT) method. We propose a robust method for instantaneous frequency estimation. The method first obtains the time-frequency distribution of the seismic signal by inverse spectral decomposition (ISD) and then calculates the analytic signal through the window HT transform. Inverse spectral decomposition achieves high-resolution time-frequency distribution by adding sparse constraint to the corresponding inverse problem, and therefore the noise can be suppressed. The algorithm we choose to solve the mix ℓ2 − ℓ1 problem is the fast iterative shrinkage-thresholding algorithm (FISTA). Compared with the traditional iterative least squares (IRLS) algorithm, FISTA can achieve a better computational efficiency. We perform the method on a quadratic frequency modulation (QFM) signal, a synthetic data based on a wedge model and field data sets to demonstrate its performance, compared with the HT method and the time-frequency adaptive filtering method.

A novel instantaneous frequency estimation method for operational time-varying systems using short-time multivariate variational mode decomposition

Operational modal identification of time-varying systems plays a crucial role in assessing the health condition and controlling the dynamic properties of engineering structures. However, only the response is measurable, making it challenging. Based on the variational mode decomposition (VMD) theory, this paper presents a short-time multivariate or multi-channel VMD (STMVMD) method for instantaneous frequency (IF) identification of time-varying structures in the case of output-only measurements. The idea of short-time windows overcomes the shortcoming of many VMD-based methods that employ the narrowband assumption of intrinsic mode functions (IMFs) and cannot decompose non-stationary signals involving closely-spaced wideband IMFs. After obtaining the multivariate IMFs by STMVMD, an average scheme is employed to estimate IFs, reducing the noise sensitivity of Hilbert Transform. Moreover, by tracking the center frequencies of STMVMD at different moments, another more noise-robust IF estimation method is also presented. A series of numerical and experimental examples illustrate the advantages of the proposal.

Instantaneous Frequency Estimation Method for the Vibration Signal of Rotating Machinery Based on STFTSC Algorithm

In this paper, an instantaneous frequency identification method known as STFTSC is developed by combining short-time Fourier transformation and the seam carving (SC) algorithm (widely used in image processing). In this method, the STFT is applied to analyze the time-frequency energy distribution of a vibration signal under variable-speed conditions. Subsequently, the energy gradient is calculated through the Sobel operator based on the time-frequency energy distribution. The energy gradient distribution contains multiple ridges, which are corresponding instantaneous frequencies of different orders. Finally, the targeted ridge extraction is transformed into an optimization problem, and the dynamic programming algorithm (DP) is used to search the targeted ridge with the minimum energy gradient to estimate the instantaneous frequency. The effectiveness of the proposed method is validated by a simulation experiment, moreover, a rotating machinery fault simulation test bench is employed to validate the method, which is then compared with polynomial chirplet transformation (PCT) and analyzed during the test process. The results show that the STFTSC instantaneous frequency estimation algorithm has a higher extraction accuracy and better application value in engineering problems than the PCT.

A generalized stochastic resonance based instantaneous frequency estimation method under low SNR

In this work, the instantaneous frequency estimation for the frequency modulated (FM) signal embedded in strong noise is considered. For engineering signals, current instantaneous frequency estimation methods are less satisfactory under low signal-to-noise ratio (SNR). To address this problem, a generalized stochastic resonance (GSR) based instantaneous frequency estimation method is proposed. First, the generalized stochastic resonance (GSR) is proposed, which overcomes the low frequency limitation of classical stochastic resonance and can enhance weak FM signals with time varying frequency. The GSR transforms the FM signal to a low-constant frequency signal by multiplying the parameterized demodulation operator with polynomial phase and then enhances it by the stochastic resonance. Second, based on the maximization of the GSR output signal, the instantaneous frequency can be estimated by global optimization of the polynomial coefficients in the parametrized demodulation operator. For the multi-component FM signal, it works by enhancing the objective component and suppressing the noise and other irrelevant components, in which the instantaneous frequency of each component is estimated in the order of component strengths and each component can be extracted from the signal. Numerical simulations and experimental signals validate that the proposed method outperforms the state-of-art methods in estimation accuracy for both mono-component and multi-component FM signals under lower SNR.

Estimating the Instantaneous Frequency of Linear and Nonlinear Frequency Modulated Radar Signals-A Comparative Study.

Automatic modulation recognition plays a vital role in electronic warfare. Modern electronic intelligence and electronic support measures systems are able to automatically distinguish the modulation type of an intercepted radar signal by means of real-time intra-pulse analysis. This extra information can facilitate deinterleaving process as well as be utilized in early warning systems or give better insight into the performance of hostile radars. Existing modulation recognition algorithms usually extract signal features from one of the rudimentary waveform characteristics, namely instantaneous frequency (IF). Currently, there are a small number of studies concerning IF estimation methods, specifically for radar signals, whereas estimator accuracy may adversely affect the performance of the whole classification process. In this paper, five popular methods of evaluating the IF–law of frequency modulated radar signals are compared. The considered algorithms incorporate the two most prevalent estimation techniques, i.e., phase finite differences and time-frequency representations. The novel approach based on the generalized quasi-maximum likelihood (QML) method is also proposed. The results of simulation experiments show that the proposed QML estimator is significantly more accurate than the other considered techniques. Furthermore, for the first time in the publicly available literature, multipath influence on IF estimates has been investigated.

Instantaneous Frequency Estimation Based on Modified Kalman Filter for Cone-Shaped Target

The instantaneous frequency (IF) is a vital parameter for the analysis of non-stationary multicomponent signals, and plays an important role in space cone-shaped target recognition. For a cone-shaped target, IF estimation is not a trivial issue due to the proximity of the energy of the IF components, the intersections among different IF components, and the existence of noise. Compared with the general parameterized time-frequency (GPTF), the traditional Kalman filter can perform better when the energy of different signal components is close. Nevertheless, the traditional Kalman filter usually makes association mistakes at the intersections of IF components and is sensitive to the noise. In this paper, a novel IF estimation method based on modified Kalman filter (MKF) is proposed, in which the MKF is used to associate the intersecting IF trajectories obtained by the synchroextracting transform (SET). The core of MKF is the introduction of trajectory correction strategy in which a trajectory survival rate is defined to judge the occurrence of association mistakes. When the trajectory survival rate is below the predetermined threshold, it means that an association mistakes occurs, and then the new trajectories generated by the random sample consensus algorithm are used to correct the wrong associations timely. The trajectory correction strategy can effectively obviate the association mistakes caused by the intersections of IF components and the noise. The windowing technique is also used in the trajectory correction strategy to improve computational speed. The experimental results based on the electromagnetic computation data show that the proposed method is more robust and precise than the traditional Kalman filter. Moreover, the proposed method has great performance advantages compared with other methods (i.e., the multiridge detection, the ant colony optimization, and the GPTF methods) especially in the case of low signal noise ratio (SNR).

Analysis of adaptive short-time Fourier transform-based synchrosqueezing transform

Recently, the study of modeling a non-stationary signal as a superposition of amplitude and frequency-modulated Fourier-like oscillatory modes has been a very active research area. The synchrosqueezing transform (SST) is a powerful method for instantaneous frequency estimation and component separation of non-stationary multicomponent signals. The short-time Fourier transform-based SST (FSST) reassigns the frequency variable to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. Very recently the FSST with a time-varying parameter, called the adaptive FSST, was introduced. The simulation experiments show that the adaptive FSST is very promising in instantaneous frequency estimation of the component of a multicomponent signal, and in accurate component recovery. However, the theoretical analysis of the adaptive FSST has not been carried out. In this paper, we study the theoretical analysis of the adaptive FSST and obtain the error bounds for the instantaneous frequency estimation and component recovery with the adaptive FSST and the second-order adaptive FSST.

Research on a Signal Separation Method Based on Vold-Kalman Filter of Improved Adaptive Instantaneous Frequency Estimation

The fault vibration signal of rotating machinery system under strong background noise has the characteristics of non-stationary, non-Gaussian and complex components. In view of these characteristics, an improved method of signal separation based on Vold-Kalman filter (VKF) of adaptive instantaneous frequency estimation is proposed. First, a method for adaptive multiridge extraction of peaks detection based on synchro-squeezing wavelet transform (SWT) is proposed as the high-precision adaptive instantaneous frequency (IF) estimation method. The high precision IF estimation is used as the instantaneous frequency parameter of VKF, so that the complex multi-component non-stationary signal can be separated directly in the time domain and transformed into a signal combination composed of multiple stationary single-component signals and signal residues. Secondly, an improved method is proposed combining the adaptive IF estimation method with order tracking analysis and diagonal slice of bispectrum. In the improved method, the corresponding IF estimation of each component signal is taken as the reference frequency of its order tracking and the order spectrum analysis of each component signal is carried out respectively. Meanwhile, the signal residual is analyzed by diagonal slice of bispectrum, so as to suppress Gaussian noise and effectively separate and extract fault features in the vibration signal. Finally, the method is verified on simulation data and experimental data under different conditions. The results show that the improved method has higher extraction accuracy than other traditional methods. It has the superiority and the great potential for practical applications.

Rolling bearing fault diagnosis via STFT and improved instantaneous frequency estimation method

Bearing vibration signals exhibit instantaneous modulation features under variable rotating frequency, making it difficult to identify the characteristic frequency components. As such, a novel rolling bearing fault diagnosis method via short time Fourier transform (STFT) and improved instantaneous frequency estimation algorithm is proposed. The novelty is that it can convert all the trajectories of instantaneous components, e.g., fault characteristic frequency (FCF) and modulation rotating frequency, to linear path in the time-frequency domain. First, a band pass filter is applied to separate the optimal frequency band which is determined by fast kurtogram. Second, the envelope time-frequency representation (TFR) is obtained by jointly adopting Hilbert transform and STFT to the filtered signal. Next, the instantaneous fault characteristic frequency (IFCF) is extracted from the filtered TFR based on improved instantaneous frequency estimation algorithm. Then, the frequency representation is calculated by dealing with the TFR of the envelope. Finally, the proportion of FCF to rotational frequency on the frequency representation is computed and then compared with the theoretical characteristic coefficient. The method is evaluated via the experimental data. The result demonstrates that bearing fault pattern can be identified via the proposed method.

Direction of arrival estimation of sources with intersecting signature in time–frequency domain using a combination of IF estimation and MUSIC algorithm

Time–frequency (TF) approaches are frequently employed for source localization at low signal to noise ratio. However, TF approaches fail to achieve the desired performance for sparsely sampled signals or signals corrupted by heavy noise in an under-determined scenario when sources are not TF separable. In this study, we propose a new TF method for direction of arrival (DOA) estimation of sources with closely spaced and overlapping TF signature. The proposed method uses a combination of a high-resolution time–frequency distribution and instantaneous frequency estimation method for extraction of sources with intersecting and closely spaced time–frequency signatures. Once sources are extracted, their DOAs are estimated using a well known multiple signal classification (MUSIC) algorithm. Experimental results demonstrate that the proposed source localization method achieves better performance as compared to the conventional time–frequency MUSIC algorithm.

Widely Linear Adaptive Instantaneous Frequency Estimation in Vector Hydrophones

To solve the problem that the time-frequency resolution of Short-Time Fourier Transform (STFT) is constrained by the window length and the moving step of the short time window, and to utilize the merits of a widely linear method, a novel instantaneous frequency estimation method in vector hydrophone was proposed. In this paper, a complex variable was constructed. It is composed of sound pressure and particle velocity as its real part and imaginary part, respectively. The constructed variable was approved to be second order noncircular (improper). For the modelling of noncircular signals, the standard linear estimation is not adequate and the pseudo-covariance matrix should also be taken into consideration. As a result, a widely linear adaptive instantaneous frequency estimation algorithm and its three solutions based on the augmented complex least mean square (ACLMS) method are presented to estimate the instantaneous frequency in vector hydrophones. The results of simulations and laboratory experiments prove that this approach based on a widely linear model performs better compared to STFT and strict linear filter methods.

An Improved Time‐Frequency Analysis Method for Instantaneous Frequency Estimation of Rolling Bearing

Instantaneous frequency estimation of rolling bearing is a key step in order tracking without tachometers, and time‐frequency analysis method is an effective solution. In this paper, a new method applying the variational mode decomposition (VMD) in association with the synchroextracting transform (SET), named VMD‐SET, is proposed as an improved time‐frequency analysis method for instantaneous frequency estimation of rolling bearing. The SET is a new time‐frequency analysis method which belongs to a postprocessing procedure of the short‐time Fourier transform (STFT) and has excellent performance in energy concentration. Considering nonstationary broadband fault vibration signals of rolling bearing under variable speed conditions, the time‐frequency characteristics cannot be obtained accurately by SET alone. Thus, VMD‐SET method is proposed. Firstly, the signal is decomposed into several intrinsic mode functions (IMFs) with different center frequency by VMD. Then, effective IMFs are selected by mutual information and kurtosis criteria and are reconstructed. Next, the SET method is applied to the reconstructed signal to generate the time‐frequency representation with high resolution. Finally, instantaneous frequency trajectory can be accurately extracted by peak search from the time‐frequency representation. The proposed method is free from time‐varying sidebands and is robust to noise interference. It is proved by numerical simulated signal analysis and is further validated by lab experimental rolling bearing vibration signal analysis. The results show this method can estimate the instantaneous frequency with high precision without noise interference.