The W transform and its improved methods for time-frequency analysis of seismic data

The conventional linear time-frequency analysis method cannot achieve high resolution and energy focusing in the time and frequency dimensions at the same time,especially in the low frequency region.In order to improve the resolution of the linear time-frequency analysis method in the low-frequency region,we have proposed a W transform method,in which the instantaneous frequency is introduced as a parameter into the linear transformation,and the analysis time window is constructed which matches the instantaneous frequency of the seismic data.In this paper,the W transform method is compared with the Wigner-Ville distribution(WVD),a typical nonlinear time-frequency analysis method.The WVD method that shows the energy distribution in the time-frequency domain clearly indicates the gravitational center of time and the gravitational center of frequency of a wavelet,while the time-frequency spectrum of the W transform also has a clear gravitational center of energy focusing,because the instantaneous frequency corresponding to any time position is introduced as the transformation parameter.Therefore,the W transform can be benchmarked directly by the WVD method.We summarize the development of the W transform and three improved methods in recent years,and elaborate on the evolution of the standard W transform,the chirp-modulated W transform,the fractional-order W transform,and the linear canonical W transform.Through three application examples of W transform in fluvial sand body identification and reservoir prediction,it is verified that W transform can improve the resolution and energy focusing of time-frequency spectra.

Three-dimensional reconstruction of precession warhead based on multi-view micro-Doppler analysis

The warhead of a ballistic missile may precess due to lateral moments during release. The resulting micro-Doppler effect is determined by parameters such as the target's motion state and size. A three-dimensional reconstruction method for the precession warhead via the micro-Doppler analysis and inverse Radon transform(IRT) is proposed in this paper. The precession parameters are extracted by the micro-Doppler analysis from three radars, and the IRT is used to estimate the size of targe. The scatterers of the target can be reconstructed based on the above parameters. Simulation experimental results illustrate the effectiveness of the proposed method in this paper.

Time-Frequency Analysis of EEG Signals Evoked by Voluntary, Stimulated and Imaginary Motions

In order to investigate the characteristics of sensorimotor cortex during motor execution(ME), voluntary, stimulated and imaginary finger flexions were performed by ten volunteer subjects. Electroencephalogram(EEG) data were recorded according to the modified 10-20 International EEG System. The patterns were compared by the analysis of the motion-evoked EEG signals focusing on the contralateral(C3) and ipsilateral(C4) channels for hemispheric differences. The EEG energy distributions at alpha(8—13 Hz), beta(14—30 Hz) and gamma(30—50 Hz) bands were computed by wavelet transform(WT) and compared by the analysis of variance(ANOVA). The timefrequency(TF) analysis indicated that there existed a contralateral dominance of alpha post-movement event-related synchronization(ERS) pattern during the voluntary task, and that the energy of alpha band increased in the ipsilateral area during the stimulated(median nerve of wrist) task. Besides, the contralateral alpha and beta event-related desynchronization(ERD) patterns were observed in both stimulated and imaginary tasks. Another significant difference was found in the mean power values of gamma band(p<0.01)between the imaginary and other tasks. The results show that significant hemispheric differences such as alpha and beta band EEG energy distributions and TF changing phenomena(ERS/ERD) were found between C3 and C4 areas during all of the three patterns. The largest energy distribution was always at the alpha band for each task.

Effects of Gabor transform parameters on signa time-frequency resolution

In this paper, it is described that the time-frequency resolution of geophysical signals is affected by the time window function attenuation coefficient and sampling interval and how such effects are eliminated effectively. Improving the signal resolution is the key to signal time-frequency analysis processing and has wide use in geophysical data processing and extraction of attribute parameters. In this paper, authors research the effects of the attenuation coefficient choice of the Gabor transform window function and sampling interval on signal resolution. Unsuitable parameters not only decrease the signal resolution on the frequency spectrum but also miss the signals. It is essential to first give the optimum window and range of parameters through time-frequency analysis simulation using the Gabor transform. In the paper, the suggestions about the range and choice of the optimum sampling interval and processing methods of general seismic signals are given.

Enhanced Fourier Transform Using Wavelet Packet Decomposition

Many domains, including communication, signal processing, and image processing, use the Fourier Transform as a mathematical tool for signal analysis. Although it can analyze signals with steady and transitory properties, it has limits. The Wavelet Packet Decomposition (WPD) is a novel technique that we suggest in this study as a way to improve the Fourier Transform and get beyond these drawbacks. In this experiment, we specifically considered the utilization of Daubechies level 4 for the wavelet transformation. The choice of Daubechies level 4 was motivated by several reasons. Daubechies wavelets are known for their compact support, orthogonality, and good time-frequency localization. By choosing Daubechies level 4, we aimed to strike a balance between preserving important transient information and avoiding excessive noise or oversmoothing in the transformed signal. Then we compared the outcomes of our suggested approach to the conventional Fourier Transform using a non-stationary signal. The findings demonstrated that the suggested method offered a more accurate representation of non-stationary and transient signals in the frequency domain. Our method precisely showed a 12% reduction in MSE and a 3% rise in PSNR for the standard Fourier transform, as well as a 35% decrease in MSE and an 8% increase in PSNR for voice signals when compared to the traditional wavelet packet decomposition method.

Intelligibility evaluation of enhanced whisper in joint time-frequency domain

Some factors influencing the intelligibility of the enhanced whisper in the joint time-frequency domain are evaluated. Specifically, both the spectrum density and different regions of the enhanced spectrum are analyzed. Experimental results show that for a spectrum of some density, the joint time-frequency gain-modification based speech enhancement algorithm achieves significant improvement in intelligibility. Additionally, the spectrum region where the estimated spectrum is smaller than the clean spectrum, is the most important region contributing to intelligibility improvement for the enhanced whisper. The spectrum region where the estimated spectrum is larger than twice the size of the clean spectrum is detrimental to speech intelligibility perception within the whisper context.

DOPPLERLET BASED TIME-FREQUENCY REPRESENTATION VIA MATCHING PURSUITS

A new time-frequency representation called Dopplerlet transform, which uses the dilated, translated and modulated windowed Doppler signals as its basis functions, is proposed, and the Fourier transform, short-time Fourier transform (including Gabor transform), wavelet transform, and chirplet transform are formulated in one framework of Dopplerlet transform accordingly.It is proved that the matching pursuits based on Dopplerlet basis functions are convergent, and that the energy of residual signals yielded in the decomposition process decays exponentially. Simulation results show that the matching pursuits with Dopplerlet basis functions can characterize compactly a nonstationary signal.

IDENTIFICATION OF NONLINEAR DYNAMIC SYSTEMS:TIME-FREQUENCY FILTERING AND SKELETON CURVES

The nonlinear behavior varying with the instantaneous response was analyzed through the joint time-frequency analysis method for a class of S. D. O. F nonlinear system. A masking operator an definite regions is defined and two theorems are presented. Based on these, the nonlinear system is modeled with a special time-varying linear one, called the generalized skeleton linear system (GSLS). The frequency skeleton curve and the damping skeleton curve are defined to describe the main feature of the non-linearity as well. Moreover, an identification method is proposed through the skeleton curves and the time-frequency filtering technique.

Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

We present recent work of harmonic and signal analysis based on the complex Hardy space approach.

Wind Turbine Planetary Gearbox Fault Diagnosis via Proportion-Extracting Synchrosqueezing Chirplet Transform

Wind turbine planetary gearboxes usually work under time-varying conditions,leading to nonstationary vibration signals.These signals often consist of multiple time-varying components with close instantaneous frequencies.Therefore,high-quality time-frequency analysis(TFA)is needed to extract the time-frequency feature from such nonstationary signals for fault diagnosis.However,it is difficult to obtain high-quality timefrequency representations(TFRs)through conventional TFA methods due to low resolution and time-frequency blurs.To address this issue,we propose a new TFA method termed the proportion-extracting synchrosqueezing chirplet transform(PESCT).Firstly,the proportion-extracting chirplet transform is employed to generate highresolution underlying TFRs.Then,the energy concentration of the underlying TFRs is enhanced via the synchrosqueezing transform.Finally,wind turbine planetary gearbox fault can be diagnosed by analysis of the dominant time-varying components revealed by the concentrated TFRs with high resolution.The proposed PESCT is suitable for achieving high-quality TFRs for complicated nonstationary signals.Numerical and experimental analyses validate the effectiveness of the PESCT in characterizing the nonstationary signals from wind turbine planetary gearboxes.

Fractional S transform-Part 1:Theory

The S transform, which is a time-frequency representation known for its local spectral phase properties in signal processing, uniquely combines elements of wavelet transforms and the short-time Fourier transform （STFT）. The fractional Fourier transform is a tool for non-stationary signal analysis. In this paper, we define the concept of the fractional S transform （FRST） of a signal, based on the idea of the fractional Fourier transform （FRFT） and S transform （ST）, extend the S transform to the time-fractional frequency domain from the time- frequency domain to obtain the inverse transform, and study the FRST mathematical properties. The FRST, which has the advantages of FRFT and ST, can enhance the ST flexibility to process signals. Compared to the S transform, the FRST can effectively improve the signal time- frequency resolution capacity. Simulation results show that the proposed method is effective.

A technique to improve the empirical mode decomposition in the Hilbert-Huang transform

The Hilbert-based time-frequency analysis has promising capacity to reveal the time-variant behaviors of a sys- tem.To admit well-behaved Hilbert transforms,component decomposition of signals must be performed beforehand.This was first systematically implemented by the empirical mode decomposition(EMD)in the Hilbert-Huang transform,which can provide a time-frequency representation of the signals.The EMD,however,has limitations in distinguishing different components in narrowband signals commonly found in free-decay vibration signals.In this study,a technique for decompo- sing components in narrowband signals based on waves' beating phenomena is proposed to improve the EMD,in which the time scale structure of the signal is unveiled by the Hilbert transform as a result of wave beating,the order of component ex- traction is reversed from that in the EMD and the end effect is confined.The proposed technique is verified by performing the component decomposition of a simulated signal and a free decay signal actually measured in an instrumented bridge structure.In addition,the adaptability of the technique to time-variant dynamic systems is demonstrated with a simulated time-variant MDOF system.

Local Extrema of Periodic Function’s Wavelet Transform

The theory of detecling ridges in the modulus of the continuous wavelet transform is presented as well as reconstructing signal by using information on ridges,To periodic signal we suppose Morlet wavelet as basic wavelet, and research the local extreme point and extrema of the wavelet transform on periodic function for the collection of signal＇ s instantaneous amplitude and period.

High-frequency compensation for seismic data based on adaptive generalized S transform

The low-pass fi ltering eff ect of the Earth results in the absorption and attenuation of the high-frequency components of seismic signals by the stratum during propagation.Hence,seismic data have low resolution.Considering the limitations of traditional high-frequency compensation methods,this paper presents a new method based on adaptive generalized S transform.This method is based on the study of frequency spectrum attenuation law of seismic signals,and the Gauss window function of adaptive generalized S transform is used to fi t the attenuation trend of seismic signals to seek the optimal Gauss window function.The amplitude spectrum compensation function constructed using the optimal Gauss window function is used to modify the time-frequency spectrum of the adaptive generalized S transform of seismic signals and reconstruct seismic signals to compensate for high-frequency attenuation.Practical data processing results show that the method can compensate for the high-frequency components that are absorbed and attenuated by the stratum,thereby eff ectively improving the resolution and quality of seismic data.

Radar Signal Intra-Pulse Feature Extraction Based on Improved Wavelet Transform Algorithm

With the new system radar put into practical use, the characteristics of complex radar signals are changing and developing. The traditional analysis method of one-dimensional transformation domain is no longer applicable to the modern radar signal processing, and it is necessary to seek new methods in the two-dimensional transformation domain. The time-frequency analysis method is the most widely used method in the two-dimensional transformation domain. In this paper, two typical time-frequency analysis methods of short-time Fourier transform and Wigner-Ville distribution are studied by analyzing the time-frequency transform of typical radar reconnaissance linear frequency modulation signal, aiming at the problem of low accuracy and sen-sitivity to the signal noise of common methods, the improved wavelet transform algorithm was proposed.

Roll angular rate extraction based on modified spline-kernelled chirplet transform

The roll angular rate is much crucial for the guidance and control of the projectile.Yet the high-speed rotation of the projectile brings severe challenges to the direct measurement of the roll angular rate.Nevertheless,the radial magnetometer signal is modulated by the high-speed rotation,thus the roll angular rate can be achieved by extracting the instantaneous frequency of the radial magnetometer signal.The objective of this study is to find out a precise instantaneous frequency extraction method to obtain an accurate roll angular rate.To reach this goal,a modified spline-kernelled chirplet transform(MSCT)algorithm is proposed in this paper.Due to the nonlinear frequency modulation characteristics of the radial magnetometer signal,the existing time-frequency analysis methods in literature cannot obtain an excellent energy concentration in the time-frequency plane,thereby leading to a terrible instantaneous frequency extraction accuracy.However,the MSCT can overcome the problem of bad energy concentration by replacing the short-time Fourier transform operator with the Chirp Z-transform operator based on the original spline-kernelled chirplet transform.The introduction of Chirp Z-transform can improve the construction accuracy of the transform kernel.Since the construction accuracy of the transform kernel determines the concentration of time-frequency distribution,the MSCT can obtain a more precise instantaneous frequency.The performance of the MSCT was evaluated by a series of numerical simulations,high-speed turntable experiments,and real flight tests.The evaluation results show that the MSCT has an excellent ability to process the nonlinear frequency modulation signal,and can accurately extract the roll angular rate for the high spinning projectiles.

Extraction of MHD Signal Based on Wavelet Transform

Mirnov signals mixed with interferences are a kind of non-stationary signal. It cannot obtain satisfactory effects to extract MHD signals from mirnov signals by Fourier Transform. This paper suggests that the wavelet transform can be used to treat mirnov signals. Theoretical analysis and experimental result have indicated that using the time-frequency analysis characteristics of the wavelet transform to filter mirnov signals can remove effectively interferences and extract useful MHD signals.

Application of short-time Fourier transform to high-rise frame structural-health monitoring based on change of inherent frequency over time

The high-rise frame structure has become more and more widespread, like its damage from the complication of the environment. The traditional method of damage detection, which is only suitable for the stationary signal, does not apply to a high-rise frame structure because its damage signal is non-stationary. Thus, this paper presents an application of the short-time Fourier transform(STFT) to damage detection of high-rise frame structures. Compared with the fast Fourier transform, STFT is found to be able to express the frequency spectrum property of the time interval using the signal within this interval. Application of STFT to analyzing a Matlab model and the shaking table test with a twelve-story frame-structure model reveals that there is a positive correlation between the slope of the frequency versus time and the damage level. If the slope is equal to or greater than zero, the structure is not damaged. If the slope is smaller than zero, the structure is damaged, and the less the slope is, the more serious the damage is. The damage results from calculation based on the Matlab model are consistent with those from the shaking table test, demonstrating that STFT can be a reliable tool for the damage detection of high-rise frame structures.

SAR target detection based on the optimal fractional Gabor spectrum feature

In this paper,an algorithm based on a fractional time-frequency spectrum feature is proposed to improve the accuracy of synthetic aperture radar(SAR)target detection.By extending the fractional Gabor transform(FrGT)into two dimensions,the fractional time-frequency spectrum feature of an image can be obtained.In the achievement process,we search for the optimal order and design the optimal window function to accomplish the two-dimensional optimal FrGT.Finally,the energy attenuation gradient(EAG)feature of the optimal time-frequency spectrum is extracted for high-frequency detection.The simulation results show the proposed algorithm has a good performance in SAR target detection and lays the foundation for recognition.

Output-only identification of the modal and physical properties of structures using free vibration response

The viability of a complete structural characterization of civil structures is explored and discussed. In particular, the identification of modal （i.e. natural frequencies, damping ratios and modal shapes） and physical properties （i.e. mass and stiffness） using only the structure＇s free decay response is studied. To accomplish this, modal analysis from flee vibration response only （MAFVRO） and mass modification （MM） methodologies are engaged along with Wavelet based techniques for optimal signal processing and modal reconstruction. The methodologies are evaluated using simulated and experimental data. The simulated data are extracted from a simple elastic model of a 5 story shear building and from a more realistic nonlinear model of a RC frame structure. The experimental data are gathered from shake table test of a 2-story scaled shear building. Guidelines for the reconstruction procedure from the data are proposed as the quality of the identified properties is shown to be governed by adequate selection of the frequency bands and optimal modal shape reconstruction. Moreover, in cases where the structure has undergone damage, the proposed identification scheme can also be applied for preliminary assessment of structural health.