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High-speed railways are “lifeline projects” that shoulder the heavy responsibility of transporting relief supplies and medical forces for the first time after earthquakes. To ensure the train’s safety after earthquakes, it is of great urgency to ascertain a post-earthquake speed threshold. To that end, a target model for seismic irregularity emerges as a key parameter. In this paper, a null phase assumption-based technique for the mutual conversion between evolutionary power spectral density and non-stationary signal was proposed. Taking a high-speed railway track-bridge system as the research object, the target model of seismic irregularities was constructed based on the proposed technique. The rationality of the target model of seismic irregularities was verified, and the construction parameter settings were discussed. Moreover, a simplified frequency-domain fitting method for the target model of seismic irregularities was proposed based on the spectral decomposition theory. According to the research findings, the null phase assumption-based technique is capable of performing interconversion between seismic irregularity and its evolutionary power spectral density with satisfactory accuracy. It is recommended to set the minimum number of seismic irregularities, spatial sampling interval, hop size, and length of window function as 50, 0.25m, 1, and 40 to 100, respectively.

We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.

Modal identification of bridges based on the responses of moving vehicles, termed as the indirect method, has drawn much attention in recent years. This indirect method has been proven to be capable of identifying bridge frequencies and mode shapes in the feasibility research. However, in the presence of additional factors such as vehicle speed and road surface roughness, it is difficult to identify high-order frequencies and high-resolution mode shapes of the bridge. The authors previously proposed a tractor-double-trailers model and the time–domain subtraction method to obtain the relative displacement of two trailers for bridge modal identification. In this study, the authors develop a methodology to improve the identification performance by combining the EMD technique and the time–frequency analysis method. The EMD technique is applied to separate the multi-source response signals of the trailers to obtain the signal component that contains only the bridge vibration. Then, different time–frequency analysis methods such as the short-time Fourier transform (STFT), Wigner–Ville distribution (WVD), and continuous wavelet transform (CWT) are adopted to construct the high-resolution bridge mode shapes. Numerical studies with one-dimensional and three-dimensional bridge models are conducted to verify the proposed method and investigate the effects of different factors. Finally, a laboratory test is implemented on a scaled bridge to verify the performance of the proposed method. The results show that the proposed method is able to separate the vehicle frequency and bridge frequencies, which facilitates the identification of bridge frequencies from the dynamic response of moving vehicles. Meanwhile, the CWT method performs the best in the identification of modal shapes among the three time–frequency analysis methods.

This paper describes a signal processing technique aimed at complementing cardiac auscultation in the detection of heart valve disease. The method provides a means for keeping objective records by analyzing the characteristics of the cardiac murmurs. The Short-time Fourier Transform (STFT) is used here to provide a graphic representation of the time-frequency information of the cardiac murmurs from eight different pathology cases. The graphic representation obtained shows the variation in frequency and intensity during the murmur. Some interesting observations show characteristic rising and falling tones, suggesting degrees of the pathology severity.

A soft-computing method attenuating noise from heart sound (HS) signal for wearable e-healthcare device is proposed. The HS signal is decomposed by third-level wavelet packet transform (WPT). An automatic HS cycle detection algorithm is applied to find HS cycles in the (3, 0) leaf signal of WPT tree. Based on the quasi-cyclic feature of HS, short-time Fourier transform is implemented for cycles of each WPT tree leaf signal to decompose each cycle into time-frequency fragments which are called particles. Furthermore, the novel cuboid method is proposed to identify constituents of HS and noise from such generated particles. The particles representing HS are then retained and merged into noise-quasi-free WPT tree leaf signals. Eventually the inverse WPT (IWPT) is employed to build the noise-quasi-free HS signal. The method is assessed using mean square error (MSE) and compared with wavelet multi-threshold method (WMTM) and Tang's method. The experimental results show that the proposed method not only filters HS signal effectively but also well retains its pathological information.

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling–Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from time-frequency analysis.

We introduce and study continuity properties of the Gabor product ${♮}_g$ and relate it to the well-known product formula for the short-time Fourier transform (STFT). We derive a phase space representation of the cubic nonlinear Schrödinger equation in terms of the Gabor product, and discuss how the Gabor product can be used in the study of nonlinear dynamics of mixed quantum states.

In radar signal processing, detecting the moving targets in a cluttered background remains a challenging task due to the moving out and entry of targets, which is highly unpredictable. In addition, detection of targets and estimation of the parameters have become a major constraint due to the lack of required information. However, the appropriate location of the targets cannot be detected using the existing techniques. To overcome such issues, this paper presents a developed Deep Convolutional Neural Network-enabled Neuro-Fuzzy System (Deep CNN-enabled Neuro-Fuzzy system) for detecting the moving targets using the radar signals. Initially, the received signal is presented to the Short-Time Fourier Transform (STFT), matched filter, radar signatures-enabled Deep Recurrent Neural Network (Deep RNN), and introduced deep CNN to locate the targets. The target location output results are integrated using the newly introduced neuro-fuzzy system to detect the moving targets effectively. The proposed deep CNN-based neuro-fuzzy system obtained effective moving target detection results by varying the number of targets, iterations, and the pulse repetition level for the metrics, like detection time, missed target rate, and MSE with the minimal values of 1.221s, 0.022, and 1,952.15.

This paper is concerned with the spectral decomposition and the adaptive analysis of data coming from car crash simulations. The mathematical ingredient of the proposed signal processing technique is the flexible Gabor-wavelet transform or the α-transform that reliably detects both high and low frequency components of such complicated short-time signals. We go from the functional treatment of this wavelet-type transform to its numerical implementation and we show how it can be used as an improved tool for spectral investigations compared to the short-time Fourier transform or the classical wavelet transform.

This paper establishes a relationship between the phase and log-magnitude of the Stockwell transform (S-transform). The proposed relationship is derived by defining the S-Transform in terms of wavelet functions. The proposed work is an extension of the study [Holighaus N., Koliander G., Průša Z., Abreu L. D., Characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm, IEEE Trans. Signal Process. 67(15):3894–3908, 2019] carried out to establish a relationship between the phase and magnitude of the continuous wavelet transform. Our methodology exploits the relationship between partial derivatives of the real and imaginary parts of the wavelet and S-transform for a couple of window functions (Gaussian and bi-Gaussian). Apart from the continuous case, these relationships are explicitly shown for the discrete version of the S-transform.

Aiming at the problem of poor fault diagnosis of rolling bearing, combined with the non-stationary characteristics of vibration signals, a bearing fault diagnosis method based on STFT-SPWVD and an improved convolutional neural network is proposed. Firstly, short-time Fourier transform and smoothed pseudo Wigner-Ville distribution are performed on the vibration signals of rolling bearings, and then through the analysis of the two methods, it is proposed to use the STFT-SPWVD method to obtain high time-frequency aggregation performance and no crossover. The time-frequency analysis results in distinct and distinct features. Finally, it is detected using an improved convolutional neural network. From the experimental results on the bearing dataset of Case Western Reserve University, it can be seen that the accuracy rate of the proposed method reaches 98.14%, which can better distinguish different faults.

Localization operators have been object of study in quantum mechanics, in PDE and signal analysis recently. In engineering, a natural language is given by time-frequency analysis. Arguing from this point of view, we shall present the theory of these operators developed so far. Namely, regularity properties, composition formulae and their multilinear extension shall be highlighted. Time-frequency analysis will provide tools, techniques and function spaces. In particular, we shall use modulation spaces, which allow “optimal” results in terms of regularity properties for localization operators acting on L²(ℝ^d).

The deformation failure and mechanical vibration is closely related to power transformer windings. The problem of a transformer is poor accuracy in the diagnosis of mechanical deformation and difficult to accurately judge the winding fault type. In this paper, as the research object of the S11-M-500/35 type distribution transformer, to obtain the vibration information through the obstructing of transformer winding insulation, low voltage winding compression and loose winding deformation failure. Two signal processing methods, wavelet packet energy spectrum entropy and the short-time Fourier transform are adopted respectively to extract the feature of different deformation information. Using the method of fuzzy c-means clustering analysis to compare the classification of the two methods of feature extraction to determine the one with more effectiveness. The results demonstrate that the wavelet packet energy spectrum entropy feature extraction method can achieve the best classification results for the winding failure deformation type. Its membership degree is above 0.94, providing the basis for the later mechanical failure diagnosis of winding type.