Cohen's Class Research Articles

Local structure and effective dimensionality of time series data sets

The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points. The proposed tools include a total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator and an improvement of the Berezin-Lieb inequality using the projection functional of the data augmentation operator. The framework for our approach is provided by quantum harmonic analysis.

Seismic data filtering using deconvolutive short-time Fourier transform

Cohen class distributions are well-known time-frequency distributions that have been widely used to analyze different signals. However, they are impractical for signal filtering because they only provide amplitude spectra and suffer from cross terms. In this paper, deconvolutive short-time Fourier transform (DSTFT) is developed by estimating phase spectra and updating moduli to address residual cross terms. The DSTFT moduli is used as weights and apply a weighted least-squares technique to estimate a high-resolution and almost cross-term-free Wigner-Ville distribution. Through numerical tests, it has been found that choosing the optimal window length can minimize cross terms in short-time Fourier transform and DSTFT spectrograms. Furthermore, using thresholding or reweighting techniques can effectively eliminate weights associated with noise. The performance of our method is demonstrated using synthetic and two real seismic wavefield separation problems, including ground-roll removal in seismic shot records and polarization analysis in seismology. The results indicate the high performance of our method in estimating phase spectra and filtering seismic data.

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

Automatic Detection of K-Complexes Using the Cohen Class Recursiveness and Reallocation Method and Deep Neural Networks with EEG Signals.

Evoked and spontaneous K-complexes are thought to be involved in sleep protection, but their role as biomarkers is still under debate. K-complexes have two major functions: first, they suppress cortical arousal in response to stimuli that the sleeping brain evaluates to avoid signaling danger; and second, they help strengthen memory. K-complexes also play an important role in the analysis of sleep quality, in the detection of diseases associated with sleep disorders, and as biomarkers for the detection of Alzheimer’s and Parkinson’s diseases. Detecting K-complexes is relatively difficult, as reliable methods of identifying this complex cannot be found in the literature. In this paper, we propose a new method for the automatic detection of K-complexes combining the method of recursion and reallocation of the Cohen class and the deep neural networks, obtaining a recursive strategy aimed at increasing the percentage of classification and reducing the computation time required to detect K-complexes by applying the proposed methods.

A deep neural network classifier for P300 BCI speller based on Cohen’s class time-frequency distribution

This paper presents a new method of predicting the P300 component of an electroencephalography (EEG)signal to recognize the characters in a P300 brain-computer interface (BCI) speller accurately. This method consistsof a deep learning model and the nonlinear time-frequency features. It is believed that the combination of the deepmodel network and extracting the nonlinear features of the EEG led this research to a better prediction of the P300and, therefore, character recognition. Cohen's class distribution is used in order to extract the nonlinear features of theEEG. Evaluating all of the kernels, Butterworth found to be more informative and it produced better results. Basedon the differences observed between time-frequency responses of target and nontarget signals, specific subbands areselected to extract seven features. A deep-structured neural network, namely stacked sparse autoencoders, is appliedfor BCI character recognition. This deep network reduces the dimension of feature space by extracting unsupervisedfeatures. Then, the features are fed to a Softmax classifier. Afterward, the whole network passes a fine-tuning phase by asupervised backpropagation algorithm. For evaluating the work, Dataset II of BCI Competition III is utilized. Based onthe results, this approach would improve the accuracy in both P300 detection and character recognition. This researchresults in 82.7% and 93.5% accuracy for P300 classification and character recognition, respectively.

Instantaneous frequency extraction of highly nonstationary optical interferometric signal using Reassigned Smoothed Pseudo Wigner Ville Distribution

Instantaneous frequency (IF) of optical interferometric signal often carries information on various physical parameters such as velocity, strain and displacement. We propose smoothed Pseudo Wigner-Ville distribution method for extracting IF of highly non-stationary polynomial frequency modulated (FM) fringe signal. Cohen class signal representation is followed and time and frequency resolutions are independently optimized using separable kernel. Reassignment technique is further applied for sharpening of the spectrogram with these optimized kernels that improves accuracy in IF significantly. Quadratic FM along with strong nonlinear IF based fringe signal is considered for showing the efficacy of proposed method. Results are compared with that of existing Polynomial Wigner Ville Distribution method and Continuous Wavelet Transform (CWT) based approach and found to be more accurate in proposed technique. Finally fringe signal generated by varying the length of one of the legs of Michelson interferometer has been used to validate the proposed method experimentally.

A new fractional wave packet transform

The fractional Fourier transform (FRFT) has been shown to be powerful tool for optics and signal processing. However, it fails in locating the fractional domain frequency contents which is required in some applications. In this paper, a new fractional wave packet transform (FrWPT) is proposed to address this problem, based on the wave packet transform (WPT) and FRFT. It preserves the properties of classical WPT and has better mathematical properties. First, we derived some fundamental results of the FrWPT, including its basic properties, inversion formula and admissibility condition and the reproducing kernel. Further, the relationship between the FrWPT and Wigner distribution is presented. Lastly, we show that the energetic form of the FrWPT the fractional wavepacketgram, i.e., the modulus square of the FrWPT is a member of the Cohen class time-frequency distribution where the kernel is a scale dependent ambiguity function.

An improved smooth-windowed Wigner-Ville distribution analysis for voltage variation signal

This paper outlines research conducted using bilinear time-frequency distribution (TFD), a smooth-windowed wigner-ville distribution (SWWVD) used to represent time-varying signals in time-frequency representation (TFR). Good time and frequency resolutions offer superiority in SWWVD to analyze voltage variation signals that consist of variations in magnitude. The separable kernel parameters are estimated from the signal in order to get an accurate TFR. The TFR for various kernel parameters is compared by a set of performance measures. The evaluation shows that different kernel settings are required for different signal parameters. Verification of the TFD that operated at optimal kernel parameters is then conducted. SWWVD exhibits a good performance of TFR which gives high peak-to-side lobe ratio (PSLR) and signal-to-cross-terms ratio (SCR) accompanied by low main-lobe width (MLW) and absolute percentage error (APE). This proved that the technique is appropriate for voltage variation signal analysis and it essential for development in an advanced embedded system.

Development of an Acoustic System for UAV Detection.

The purpose of this paper is to investigate the possibility of developing and using an intelligent, flexible, and reliable acoustic system, designed to discover, locate, and transmit the position of unmanned aerial vehicles (UAVs). Such an application is very useful for monitoring sensitive areas and land territories subject to privacy. The software functional components of the proposed detection and location algorithm were developed employing acoustic signal analysis and concurrent neural networks (CoNNs). An analysis of the detection and tracking performance for remotely piloted aircraft systems (RPASs), measured with a dedicated spiral microphone array with MEMS microphones, was also performed. The detection and tracking algorithms were implemented based on spectrograms decomposition and adaptive filters. In this research, spectrograms with Cohen class decomposition, log-Mel spectrograms, harmonic-percussive source separation and raw audio waveforms of the audio sample, collected from the spiral microphone array—as an input to the Concurrent Neural Networks were used, in order to determine and classify the number of detected drones in the perimeter of interest.

Generalized Born-Jordan distributions and applications

One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature is, however, at the origin of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them being members of the Cohen class. Among these, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called ghost frequencies are grandly damped, and the noise is, in general, reduced; it also seems to play a key role in quantum mechanics. The central insight relies on the kernel of such a distribution, which contains the sinus cardinalis sinc, the Fourier transform of the first B-spline B1. The idea is to replace the function B1 with the spline or order n, denoted by Bn, yielding the function (sinc)n when Fourier transformed, whose speed of decay at infinity increases with n. The related Cohen kernel is given by ${\Theta }^{n}(z_{1},z_{2})=\text {sinc}^{n}(z_{1}\cdot z_{2})$ , $n\in \mathbb {N}$ , and the corresponding time-frequency distribution is called generalized Born-Jordan distribution of ordern. We show that this new representation has a great potential to damp unwanted interference effects and this damping effect increases with n. Our proofs of these properties require an interdisciplinary approach, using tools from both microlocal and time-frequency analysis. As a by-product, a new quantization rule and a related pseudo-differential calculus are investigated.

Variance analysis of noisy LFM signal in linear canonical Cohen's class

Exact expression for the discrete-time linear canonical Cohen's class's average variance of signals in additive stationary white circular Gaussian noise processes is derived. It is shown that the average variance depends on signal form, kernel type and linear canonical transform (LCT) free parameters. For a kind of special analytic signal form, linear frequency-modulated signals, the effect of kernels and LCT free parameters on the average variance is observed. By focusing on two kinds of minimum variance kernels, the frequency marginal constrained kernel and the time marginal, frequency marginal and time support constrained kernel, a correlation analysis between the average variance and LCT free parameters is further conducted.

Radar Signal Intra-Pulse Modulation Recognition Based on Convolutional Neural Network and Deep Q-Learning Network

Intra-pulse modulation recognition of radar signals is an important part of modern electronic intelligence reconnaissance and electronic support systems. With the increasing density of radar signals, the analysis and processing of multi-component radar signals have become an urgent problem in the current radar reconnaissance system. In this paper, an intra-pulse modulation recognition approach for single-component and dual-component radar signals is proposed. First, in order to adapt to the time-frequency energy distribution characteristics of various radar signals, we propose to extract the time-frequency images (TFIs) of received signals by Cohen class time-frequency distribution (CTFD) with multiple kernel functions. Besides, the image processing methods are used to suppress noise and adjust the size and amplitude of the TFIs. Second, we design and pre-train a TFI feature extraction network for radar signals based on a convolutional neural network (CNN). Finally, to improve the probability of successful recognition (PSR) of the recognition system in the pulse overlapping environment, a multi-label classification network based on a deep Q-learning network (DQN) is explored. Besides, two sub-networks take TFIs based on special kernel functions as input and re-judge the recognition results of some specific signals to further enhance the recognition effect of the recognition system. The proposed approach can identify 8 kinds of random overlapping radar signals. The simulation results show that the overall PSR of dual-component radar signals and single-component radar signals can reach 94.83% and 94.43%, respectively, when the signal-to-noise ratio (SNR) is −6 dB.

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Recently we introduced mixed-state localization operators associated with a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution that is based on the Cohen class distribution associated with the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies’ localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen’s class since they rely on the reproducing kernel property of the short-time Fourier transform. Our approach is based on quantum harmonic analysis on phase space, which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators, the accumulated Cohen’s class distributions.

Linear perturbations of the Wigner distribution and the Cohen class

The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

We study mixed-state localization operators from the perspective of Werner’s operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner’s convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen’s class.

Velocity synchronous bilinear distribution for planetary gearbox fault diagnosis under non-stationary conditions

This paper proposes a time-frequency method, named velocity synchronous bilinear distribution (VSBD), for planetary gearbox fault diagnosis under non-stationary conditions. This method is based on the Cohen's class bilinear distribution (CCBD). The current CCBDs are able to provide a time-frequency representation (TFR) with good time-frequency resolution for multi-component signal with constant frequencies. However, their poor time-frequency resolution in dealing with time-varying frequencies has limited their applications. As such, the VSBD is proposed to solve this problem. This method firstly uses the generalized demodulation to demodulate the signal under analysis into several signals based on the shaft rotational speed to meet the constant-frequency requirement of the bilinear distribution. The CCBD is then applied to the demodulated signals to obtain the TFRs of the demodulated signals and finally the TFR of the original signal is restored from the demodulated TFRs. This method is free from smear effect and has improved time-frequency resolution in comparison with other existing time-frequency methods. This method only considers planetary gearbox systems running under time-varying speed and fixed external load conditions. Therefore, the industrial applications of the proposed method may be restricted to such conditions.

О МНОГОМЕРНОМ ПРЕДСТАВЛЕНИИ СИГНАЛОВ АВТОМАТИЧЕСКИХ ДОЗИРУЮЩИХ УСТРОЙСТВ

The article describes the development of a special approach based on using multidimensional wavelet distributions principle to monitor and control the feed dozing processes in the mix preparation unit. As a key component, this approach uses the multidimensional time-frequency Wigner-Ville distribution, which is the part of Cohen's class distributions. The research focuses on signals characterizing mass transfer processes in the form of material flow measuring signals in relevant points of the unit. Wigner-Ville distribution has been shown in time terms as Fourier transform of products of multiplied parts of the signal under consideration for past and future time moments; corresponding distribution for the frequency spectrum is shown as Fourier transform of the products of signal parts for high-frequency and low-frequency fragments of the signal spectrum. It has been noted that when using a complex model of a dozing signal, discrete values (samples) of the latter are considered as its real values. The description of the signal parameters (amplitude, phase, frequency) has been carried out with the help of Hilbert transform. In Cohen's class distributions which represent one-dimensional non-stationary flow signals, the concept of ‘instantaneous frequency’ has been introduced. A graphical explanation for the transformation of a process flow signal from a one-dimensional time domain to a time-frequency 2 D/ 3 D -space is presented. The technology of developing a multidimensional image in the form of Wigner distribution for one-dimensional signals of continuous spiral or screw-type feeders has been examined in detail. There have been considered the features to support Wigner distribution, which allow to guess the presence or absence of time-frequency distribution elements in the interval of signal recording. There has been demonstrated how Wigner distribution can be obtained for a continuous-intermittent feeding signal. It has been concluded that for a certain types of the signal for zero fragments of the latter, non-zero time-frequency elements (i.e. virtual, anomalous ones) appear on the distribution. In addition to Wigner distribution, two other distributions - of Rihachek and Page - are considered. They display the same signal and also contain virtual elements, but in different domains of the time-frequency space. A generalized multidimensional compound signal distribution with a so-called distribution kernel available in it is presented, which includes a correction parameter that allows controlling the intensity of the virtual signal energy.

Development of the method of frequency­temporal representation of fluctuations of gaseous medium parameters at fire

The method of operative frequency-temporal representation of fluctuations of gaseous media parameters at an early stage of fire at premises was developed. The basic assumptions about the peculiarities of dynamics of hazardous factors of gaseous medium at early ignition at premises were stated. The authors created theoretical framework for development of the method, based on the fact that violation of equilibrium state of gaseous medium is translated by the medium to the zone of sensors’ localization and responds to emergence of an ignition in premises. The fire source in this case is considered a moving source of disturbances and parameters of the medium carry information about temporal and frequency shifts of disturbances. It was shown that these shifts of disturbances are characterized by the correspondent uncertainty function, which is an invariant with respect to the double Fourier transformation, determined by squared modulus of frequency-temporal energy density of the parameter. The proposed method is a further development of frequency-temporal representations of the Cohen class in case of fluctuations of gaseous medium parameters at early ignitions in premises. The main features of the method are its relative simplicity and the use of data in real time. The verification of the developed method was performed based on the experimental data of the main parameters of the gaseous medium at an early ignition of alcohol, paper, wood, and textiles in the simulation chamber.

Cohen class of time-frequency representations and operators: Boundedness and uncertainty principles

This paper presents a proof of an uncertainty principle of Donoho–Stark type involving ε-concentration of localization operators. More general operators associated with time-frequency representations in the Cohen class are then considered. For these operators, which include all usual quantizations, we prove a boundedness result in the Lp functional setting and a form of uncertainty principle analogous to that for localization operators.

Radar Signal Intra-Pulse Modulation Recognition Based on Convolutional Neural Network

In this paper, to solve the problem of the low recognition rate of the existing approaches at low signal-to-noise ratio (SNR), an intra-pulse modulation recognition approach for radar signal is proposed. The approach identifies the modulation of radar signals using the techniques of time-frequency analysis, image processing, and convolutional neural network (CNN). Through Cohen class time-frequency distribution (CTFD), the time-frequency images (TFIs) of received signals are extracted. In order to obtain the high-quality TFIs of received signals, we introduce a new kernel function for the CTFD, which has stronger anti-noise ability than Choi–Williams time-frequency distribution. A series of image processing techniques, including 2-D Wiener filtering, bilinear interpolation, and Otsu method, are applied to remove the background noise of the TFI and obtain a fixed-size binary image that contains only morphological features of the TFI. We design a CNN classifier to identify the processed TFIs. The proposed approach can identify up to 12 kinds of modulation signals, including frequency modulation, phase modulation, and composite modulation. Simulation results show that, for 12 kinds of modulation signals, the proposed approach achieves an overall probability of successful recognition of 96.1% when SNR is −6 dB.