Time-frequency Plane Research Articles

Time-Frequency Properties of the Short-Time Linear Canonical Transform and Its Application

The short-time linear canonical transform (STLCT) is a novel time-frequency analysis tool, which has attracted some attention recently. However, its applications in signal processing are limited because the time-frequency properties of the STLCT are still little known. Most existing studies focus on mathematical properties rather than time-frequency properties in signal processing. To handle this problem, first, we investigate some basic time-frequency properties such as 2-D resolution of the time-frequency plane, the STLCT domain support, computation of the STLCT, etc by generalizing the short-time Fourier transform to the STLCT. Then, based on these derived properties, we find that the Gaussian window is optimal window of the STLCT. Signal separation verified the results.

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.

Discrimination of Different AE and EMR Signals during Excavation of Coal Roadway Based on Wavelet Transform

During the process of coal road excavation, various interference signals, induced by environmental noise, drilling, and scraper loader, will affect the risk assessment of coal and gas outburst using acoustic emission (AE) and electromagnetic radiation (EMR) monitoring technology. To distinguish between different interference signals and danger signals, discrete wavelet transform (DWT) was used to decompose and reconstruct signals, and continuous wavelet transform (CWT) was used to obtain the time-frequency plane. The research results show that: (1) interference signals generally exhibit fluctuating changes within small ranges; in comparison, the intensity of AE and EMR signals caused by coal and rock fracture is found to continuously rise for a long period (longer than 2 h). (2) Different interference signals and danger signals differ significantly in their time-frequency plane. (3) Through decomposition and reconstruction of original signal, obvious precursor information can be found in the time-frequency plane of reconstructed signals.

Signal Useful Information Recovery by Overlapping Supports of Time-Frequency Representations

This paper presents a novel method for automatic extraction of useful information from time-frequency distributions of noisy signals. Signals' features are examined through the combined analysis of their time-frequency energy distributions and inverse complexity maps. The inverse complexity approach gives a new entropy-based insight into the signal structure. These two approaches result in mostly disjoint time-frequency supports, overlapping only in the proximity of the signal components. Locations of the signal components (i.e., useful information) are thus identified by low complexity and high amplitude occurring together in the time-frequency plane. Compared to existing methods using only the time-frequency energy distribution, useful information extraction by the proposed method exhibits a significantly reduced error rate.

Linear Canonical Stockwell Transform: Theory and Applications

The Stockwell transform (ST), which is a reversible method of time-frequency spectral representation, is an extension of the ideas of the wavelet transform (WT) and short-time Fourier transform (STFT). And yet, it represents the signal just in the time-frequency plane, which is unfavorable for nonstationary signals. In this paper, a new linear canonical Stockwell transform (LCST) is proposed based on the specific convolution structure in linear canonical transform (LCT) domain, which is a combination of the merits of ST and LCT to address this problem. It not only characterizes the signal in the time-linear canonical frequency plane, but more importantly, inherits the advantages of ST with a clear physical meaning. First, the theories about the continuous LCST are described at length, including its definition, basic properties and the time-LCT domain-frequency analysis. Next, the convolution theorem and cross-correlation theorem constructed in LCST domain are considered. Further, the discretization algorithm of the LCST is explored in order to realize it in the physical system. Finally, based on the proposed LCST, we study and discuss several applications of it, including time-frequency analysis and filtering of chirp signals. The rationality and validity of the work is verified out by simulations.

Analysis of Reassignment Operators Used in Synchrosqueezing Transforms: With an Application to Instantaneous Frequency Estimation

In this paper, our goal is first to investigate the behavior of reassignment operators used in synchrosqueezing transforms applied to multicomponent signals made of the superposition of amplitude and frequency modulated modes. Indeed, while these operators are associated with instantaneous frequency estimators very accurate on specific types of modes, the quality of the former worsens drastically when the modes depart from the ideal case they are designed for. We show in this paper that this particularly true when the modes interfere in the time-frequency plane or when some noise is present. Based on that analysis, we propose a novel instantaneous frequency estimator that only makes use of some specific points located on the ridges of synchrosqueezing transforms, and compare its performance with state-of-the-art techniques based on the same type of time-frequency representations.

Frequency Response Function Identification from Incomplete Data: A Wavelet-based Approach

Frequency Response Function (FRF) identification plays a crucial role in the design, the control, and the analysis of complex dynamical systems, including thermal and motion systems. Especially for applications that require long measurements, missing data samples, e.g., due to interruptions in the data transmission or sensor failure, often occur. The aim of this paper is to accurately identify nonparametric FRF models of periodically excited systems from noisy output measurements with missing samples. The presented method employs a wavelet-based transformation to address the identification problem in the time-frequency plane. A simulation example confirms that the developed techniques produce accurate estimates, even when many samples are missing.

Seismic Thin Interbeds Analysis Based on High-Order Synchrosqueezing Transform

Thin interbeds, which are made of several thin layers, are critical geological structures in seismic exploration. Due to the limited seismic resolution, thin interbeds cannot be easily analyzed with the standard time analysis or frequency analysis from a seismic dataset. Time-frequency (TF) analysis is a powerful technique for analyzing a nonstationary signal in the time and frequency domains simultaneously. Although it is tough to use the TF tool to estimate the thin layer’s thickness, TF analysis still has much potential for analyzing seismic thin interbeds from seismic data. The high-order synchrosqueezing transform (SST) has a significantly higher TF resolution than the linear TF transform and some commonly used SSTs. In this study, we propose to utilize the high-order SST to extract the TF features of thin interbeds’ seismic response. The TF analysis on the seismic response of cyclic thin interbeds shows that the thickness variation in thin interbeds causes a clear increasing (or decreasing) frequency curve in the TF plane. A seismic data example based on the Marmousi2 model demonstrates the proposed method can reflect the thickness variation with the depth. One real seismic data example has shown that the high-order SST can specifically extract the frequency component and be used to characterize the thickness variation, which is verified by two well’s velocity logs.

A Comprehensive Overview on Modified Versions of Stockwell Transform For Power Quality Monitoring

The increasing trends toward the accurate identification of power quality disturbances (PQD) via power quality (PQ) monitoring require an appropriate digital signal processing (DSP) technique and a robust classifier. To this end, Stockwell transform (ST), one of the most efficient feature extraction DSP tools, and its several variants play an utmost role in PQ assessment framework. Its time-varying spectral characteristics generally extract the local instantaneous frequency spectrum from the global temporary behavior of PQD signal. However, the Standard ST suffers from thepoor time-frequency resolution because of its frequency-dependent Gaussian window (GW). While the analysis of the statistically time-varying signals requires a suitable balance between time and frequency resolution. To this end, this paper provides a comprehensive literature review on several modified versions of Standard ST for the first time to reduce the computational complexity of the algorithm as well as maximize the energy concentration of the time-frequency plane. A comparative analysis of all the modified STs has been presented in tabular form to provide the key characteristics of each technique. Additionally, a case study has been presented to substantiate the highest accuracy of the proposed algorithm over the other ST variants. Apart from the PQD classification, miscellaneous applications of Standard ST and its modified variants have been indicated. This review paper may provide a valuable resource to the researchers for further improvement of the time-frequency resolution of ST not only in classifying PQD but also for its other wide applications.

The Frequency Spectrum and Time Frequency Analysis of Different Violins Classification as Tools for Selecting a Good-Sounding Violin

This work evaluates four violins from three distinct manufacturers, notably Eurostring, Stentor, and Suzuki, using a scientific approach. Eurostring1 and Eurostring2 were the names given to the two Eurostring units. The purpose of this study is to identify elements in various violins that could be used as tools for selecting a pleasantsounding violin by having them classified by a professional violinist. The signal’s time varying frequency was evaluated using a frequency spectrum and a time frequency plane, and the combination of frequency spectrum and time frequency domain is utilised. PicoScope oscilloscopes and Adobe Audition version 3 were used to record the acoustic spectra in terms of time and frequency. The time frequency plane is identified, and time frequency analysis (TFA) is produced by Adobe Audition spectrograms. The sound was processed in order to generate Fast Fourier Transform analysis: Fourier spectra (using PicoScope) and spectrograms (using Adobe Audition). Fourier spectra identify the intensity of the fundamental frequency and the harmonic spectra of the overtone frequencies. The highest frequencies that can be read are up to and including the 9th overtone. All violins have a constant harmonic overtone pattern with an uneven acoustic spectrum pattern. Eurostring1 showed inconsistent signal in the string G with 6th and 7th overtone missing, whereas Eurostring2 lack of the 6th overtone. Among the string D, only Eurostring1 display an exponential decay for the overtone. All the string A except for Suzuki showed nice and significant peak of fundamental and overtone. Stentor displays up to the 5th overtone. Among the string E, Suzuki showed inconsistent harmonic peak intensity. TFA revealed that the fundamental frequency of string E for Eurostring1 was lower than the first overtone. Only Eurostring1 has an uneven decay for the overtone frequency, whereas Eurostring2 exhibits a large exponential decay for the overtone frequency.

DFT-Based Channel Estimation with Channel Response Mirroring for MIMO OFDM Systems

In this paper, DFT-Based channel estimation with channel response mirroring is proposed and analyzed. In General, pilot symbols for channel estimation in MIMO(Multi-Input Multi-Output) OFDM(Orthogonal Frequency-Division Multiplexing) Systems have a diamond shape in the time-frequency plane. An interpolation technique to estimate the channel response of sub-carriers between reference symbols is needed. Various interpolation techniques such as linear interpolation, low-pass filtering interpolation, cubic interpolation and DFT interpolation are employed to estimate the non-pilot sub-carriers. In this paper, we investigate the conventional DFT-based channel estimation for noise reduction and channel response interpolation. The conventional method has performance degradation by distortion called “edge effect” or “border effect”. In order to mitigate the distortion, we propose an improved DFT-based channel estimation with channel response mirroring. This technique can efficiently mitigate the distortion caused by the DFT of channel response discontinuity. Simulation results show that the proposed method has better performance than the conventional DFT-based channel estimation in terms of MSE.

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

Trans-dimensional inversion for sediment attenuation using modal dispersion data collected by an underwater glider

This paper inverts modal-dispersion data collected by an underwater glider to estimate a layered seabed geoacoustic model including sediment attenuation as well as sound speed and density. The data considered here were collected during the 2017 Seabed Characterization Experiment (SBCEX17) conducted on the New England Mud Patch. The distance between the (combustive) sound source and the hydrophone-equipped (Teledyne Webb Research Slocum) glider was approximately 8 km. The frequency range considered was from 20 to 392 Hz, and 6 modes were resolved from the data using warping. Previous work has demonstrated that the positions of dispersive acoustic modes in the time-frequency plane provide information to estimate seabed sound-speed and density profiles. Including the relative energy distribution in the time-frequency plane (normalized modal amplitudes and/or amplitude ratios between modes) provides additional information to also estimate sediment attenuation, which is carried out here using a trans-dimensional Bayesian inversion. In addition, uncertainties of the seabed sound speed and density with/without modal energy information are compared. [Work supported by ONR.]

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle (UP), which offers information about a signal and its Fourier transform in the time-frequency plane, is particularly powerful in mathematics, physics and signal processing community. Under the polar coordinate form of quaternion-valued signals, the UP of the two-sided quaternion linear canonical transform (QLCT) is strengthened in terms of covariance. The condition giving rise to the equal relation of the derived result is obtained as well. The novel UP with covariance can be regarded as one in a tighter form related to the QLCT. It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.

ECG-based machine-learning algorithms for heartbeat classification

Electrocardiogram (ECG) signals represent the electrical activity of the human hearts and consist of several waveforms (P, QRS, and T). The duration and shape of each waveform and the distances between different peaks are used to diagnose heart diseases. In this work, to better analyze ECG signals, a new algorithm that exploits two-event related moving-averages (TERMA) and fractional-Fourier-transform (FrFT) algorithms is proposed. The TERMA algorithm specifies certain areas of interest to locate desired peak, while the FrFT rotates ECG signals in the time-frequency plane to manifest the locations of various peaks. The proposed algorithm’s performance outperforms state-of-the-art algorithms. Moreover, to automatically classify heart disease, estimated peaks, durations between different peaks, and other ECG signal features were used to train a machine-learning model. Most of the available studies uses the MIT-BIH database (only 48 patients). However, in this work, the recently reported Shaoxing People’s Hospital (SPH) database, which consists of more than 10,000 patients, was used to train the proposed machine-learning model, which is more realistic for classification. The cross-database training and testing with promising results is the uniqueness of our proposed machine-learning model.

On a new Wigner-Ville distribution associated with linear canonical transform

Linear canonical transform as a general integration transform has been considered into Wigner-Ville distribution (WVD) to show more powerful ability for non-stationary signal processing. In this paper, a new WVD associated with linear canonical transform (WVDL) and integration form of WVDL (IWVDL) are presented. First, the definition of WVDL is derived based on new autocorrelation function and some properties are investigated in details. It removes the coupling between time and time delay and lays the foundation for signal analysis and processing. Then, based on the characteristics of WVDL over time-frequency plane, a new parameter estimation method, IWVDL, is proposed for linear modulation frequency (LFM) signal. Two phase parameters of LFM signal are estimated simultaneously and the cross term can be suppressed well by integration operator. Finally, compared with classical WVD, the simulation experiments are carried out to verify its better estimation and suppression of cross term ability. Error analysis and computational cost are discussed to show superior performance compared with other WVD in linear canonical transform domain. The further application in radar imaging field will be studied in the future work.

Experimental Investigations of Offshore Sand Production Monitoring Based on the Analysis of Vibration in Response to Weak Shocks

Sand production is a problem that is often encountered in unconventional oil and gas exploitation and that is difficult to effectively solve. Accurate online monitoring of sand production is one of the keys to ensuring the safety and long-term production of oil wells as well as efficient production throughout the life cycle of production wells. This paper proposes a method for monitoring sand production in offshore oil wells that is based on the vibration response characteristics of sand-carrying fluid flow impinging on the pipe wall. This method uses acceleration sensors to obtain the weak vibration response characteristics of sand particles impinging on the pipe wall on a two-dimensional time-frequency plane. The time-frequency parameters are further optimized, and the ability to identify weakly excited vibration signals of sand particles in the fluid stream is enhanced. The difference between the impact response of the sand particles and the impact response of the fluid flow to the pipe wall is identified, and corresponding indoor verification experiments are carried out. Under different sand contents, particle sizes, and flow rates (sand content 0-2‰, sand particle size 96-212 μm, and flow velocity 1-3 m/s), the impact response frequency of sand particles to the pipe wall exhibits good consistency. The characteristic frequency band of sand impacting the pipe wall is 30-50 kHz. A statistical method is used to establish the response law of the noise signal of the fluid. Based on this knowledge, a real-time calculation model of sand production in offshore oil wells is constructed, and the effectiveness of this model is verified. Finally, a field test is carried out with a self-developed sand production signal dynamic time-frequency response software system on 4 wells of an oil production platform in the Bohai Sea. This system can effectively distinguish sand-producing wells from non-sand-producing wells. The dynamic time-frequency response, field test results, and actual laboratory results are consistent, verifying the effectiveness of the method proposed in this paper and further providing a theory for improving the effectiveness of the sand production monitoring method under complex multiphase flow conditions. This study also provides technical guidance for the industrial application of sand production monitoring devices in offshore oil wells.

Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces

We obtain estimates for the Lp-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of R2, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions T×W of the time-frequency plane using the Fourier transform to intertwine time and frequency, we localize the representation of the signals in terms of the short-time Fourier transform in sets Δ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for f∈L2(R) from its STFT values on arbitrary discs.

Multiscale holospectrum convolutional neural network-based fault diagnosis of rolling bearings with variable operating conditions

In recent years, the convolutional neural network (CNN) has been widely used in the field of intelligent fault diagnosis. However, each convolutional layer of CNN cannot take the overall and local information into account, and the feature extraction ability of CNN with fewer layers is weak. These circumstances lead to poor performance of CNN in practical fault diagnosis with variable operating conditions. To solve these problems, this paper proposes a multiscale holospectrum CNN (MH-CNN) based on the methods of two-dimensional multiscale feature fusion and decision-level feature fusion. First, the continuous wavelet transform is used to map the time-domain signal to the time-frequency plane to fully reflect the complex information contained in the signal. Then the two-dimensional multiscale feature fusion is introduced to extract features at different scales, which can take both overall and local information into account. Finally the decision-level feature fusion is introduced to fuse the features from signal in X, Y directions in the decision-level of CNN, which serves to enhance the features. By combining these methods, the proposed MH-CNN can extract more distinguishable features with a shallow structure, which can ensure the classification capability while avoiding the overfitting problem caused by overly complex networks. The effectiveness of the MH-CNN is verified using complicated data sets consisting of 16 rolling bearings with four different health conditions, two speeds and three loads. Results show that the proposed MH-CNN achieves a correct rate of 99.8% for rolling bearing fault diagnosis under variable operating conditions, which is much higher than other comparative methods.

Enhancement of time-frequency post-processing readability for nonstationary signal analysis of rotating machinery: Principle and validation

Rotating machinery signals are often composed of multiple components and are nonstationary under practical time-varying conditions. In most cases, the constituent frequency components are close to each other on time–frequency plane. As such, better time–frequency readability is necessary to discover the underlying physical nature of such complex nonstationary signals. Nevertheless, conventional time–frequency analysis methods suffer from limited time–frequency resolution and/or cross-term interferences, thus cannot achieve desirable time–frequency readability. In recent decades, some time–frequency post-processing methods (such as time–frequency reassignment, synchrosqueezing transform, multi-tapering, concentration of frequency and time, and higher-order synchrosqueezing transform) have been proposed to improve the time–frequency readability from two aspects: reducing cross-term interferences and enhancing time–frequency energy concentration. However, they still surfer from time-frequency blurs for the signal whose time–frequency ridges are close to each other on time–frequency plane. To address this issue, we propose a framework to improve the post-processing methods. Firstly, the Vold-Kalman filter is employed to extract constituent frequency components, by exploiting its capability to separate mono-components. Then, the time–frequency representation (TFR) of each mono-component is obtained via time–frequency post-processing method separately. Finally, the TFR of raw signal is constructed by superposing the TFR of all mono-components. This framework achieves a TFR of high time–frequency resolution, free from cross-term interferences, and therefore better time–frequency readability. As such, it extends time–frequency post-processing methods to analyze complex nonstationary signals. The proposed framework is illustrated through numerical simulation, and validated using typical rotating machine vibration data (including the experimental signals of a planetary gearbox and an induction motor, ground test data of a civil aircraft engine and in-situ measurement of a hydraulic turbine rotor). The analysis results show the good capability of proposed framework to reveal the frequency contents and their time-varying features of complex nonstationary signals.