Cohen's Class Research Articles

Windowed-Wigner Representations in the Cohen Class and Uncertainty Principles

For representations in the Cohen class, specific Cohen kernels depending only on one half of the variables are showed to produce two types of representations which can in a natural way be associated with time and frequency windows. This leads to the definition of representations with no interference for signals whose time-frequency content is confined in specific zones. We prove the main properties of these representations in the context of the Cohen class. We study then uncertainty principles at first in connection with support compactness and then in the framework of a general concept of duality among representations.

Measuring Corrosion with Laser-Ultrasound: Time-Frequency Analysis of Rayleigh Wave Propagation

Abstract Background: Laser - Ultrasound is a new non-contact technique used to detect the defects in the hot surfaces like hot billets etc. Determination of corrosion in the plates, using this non-contact technique seemed a promising research effort. Objectives: In this work, an inspection system has been presented that uses Laser-Ultrasound (LU) technique for Nondestructive testing (NDT) of metallic structures with specific interest in Oil & Gas sector. Methods: The developed system is the first one of its kind in the Middle- Eastern region. The nature of signals is quite unique as well and traditional signal processing runs into a lot of algorithmic complications with them. A new approach has been developed for this setup in order to efficiently enhance signal to noise ratio for the underlying signals so that any subsequent classification/intelligent-detection system can be based on the outcomes of this algorithm. Multiform Tiltable Exponential Distribution (MTED) kernel, which is a generalization of 2nd order Cohen's class functions in Time-Frequency Representation (TFR) space, has been used in this work to isolate the essential frequency components with temporal and frequency based masking filters. Results: While detecting defect points is quite similar to the conventional ultrasonic testing, the detection of corrosion is quite different. The reason being is the surface properties, and hence the surface vibrations, are quite different for a corroded surface as compared to a polished surface. In this respect, we have observed the propagation of the surface Rayleigh waves manifests a pattern that can be mapped to the corrosion concentration on the surface. Conclusion: Interesting observation has been made with coated corroded surfaces where the behavior has been found to be quite similar. Thus, the underlying technique can be applied without any need to remove the coatings from the sample under study.

A Time-Frequency-Based Approach to Phase and Phase Synchrony Estimation

Time-varying phase synchrony is an important bivariate measure that quantifies the dynamics between nonstationary signals and has been widely used in many applications including chaotic oscillators in physics and multichannel electroencephalography recordings in neuroscience. Current state-of-the-art in time-varying phase estimation uses either the Hilbert transform or the complex wavelet transform of the signals. Both of these methods have some major drawbacks such as the assumption that the signals are narrowband for the Hilbert transform and the nonuniform time-frequency resolution inherent to the wavelet analysis. In this paper, a new phase estimation method based on the Rihaczek distribution and Reduced Interference Rihaczek distribution belonging to Cohen's class is proposed. These distributions offer phase estimates with uniformly high time-frequency resolution which can be used for defining time and frequency dependent phase synchrony. Properties of the phase estimator and the corresponding phase synchrony measure are evaluated both analytically and through simulations showing the effectiveness of the new measures compared to existing methods.

Supports of Representations in the Cohen Class

In this paper we consider a version of the uncertainty principle concerning limitations on the supports of time-frequency representations in the Cohen class. In particular we obtain various classes of kernels with the property that the corresponding representations of non trivial signals cannot be compactly supported. As an application of our results we show that a linear partial differential operator applied to the Wigner distribution of a function f≠0 in the Schwartz class cannot produce a compactly supported function.

Preferred quantization rules: Born–Jordan versus Weyl. The pseudo-differential point of view

There has recently been evidence for replacing the usual Weyl quantization procedure by the older and much less known Born–Jordan rule. In this paper we discuss this quantization procedure in detail and relate it to recent results of Boggiato, De Donno, and Oliaro on the Cohen class. We begin with a discussion of some properties of Shubin’s τ-pseudo-differential calculus, which allows us to show that the Born–Jordan quantization of a symbol a is the average for \({\tau\in[0,1]}\) of the τ-operators with symbol a. We study the properties of the Born–Jordan operators, including their symplectic covariance, and give their Weyl symbol.

Diabetes, metformin and lactic acidosis

Metformin has long been thought to cause lactic acidosis (LA) but evidence from various sources has led researchers to question a direct causative relationship. We assessed the relationship of metformin prescription and other factors to the incidence of LA. All cases of LA at a single hospital were identified from laboratory lactate measurements. We compared patients classified as Cohen and Woods class A and B, patients with and without diabetes, and those taking metformin or not. LA was more common than in published analyses based on hospital coding of diagnoses. The incidence of LA was greater in diabetes than in the nondiabetic population but with no further increase in patients taking metformin. Lactate levels were no greater in patients on metformin than in patients with type 2 diabetes not on metformin even if patients with acute cardiorespiratory disturbance (Cohen and Woods class A) were excluded. Acidosis was greater in diabetes (hydrogen ion 94·9 ± 4·6 vs 83·2 ± 2·3 10(-9) m, P = 0·027) but factors besides lactate contributed. Acute cardiorespiratory illness, acute renal impairment and sepsis were the most common of the recognized precipitating factors. Age (P = 0·01), acute renal failure (P = 0·015) and sepsis (P = 0·005) were associated with mortality. Diabetes rather than metformin therapy is the major risk factor for the development of LA. Lactic acidosis occurs in association with acute illness particularly in diabetes. Current guidance for the prevention of lactic acidosis may overemphasize the role of metformin.

Power Quality Analysis Using Bilinear Time-Frequency Distributions

Bilinear time-frequency distributions (TFDs) are powerful techniques that offer good time and frequency resolution of time-frequency representation (TFR). It is very appropriate to analyze power quality signals which consist of nonstationary and multi-frequency components. However, the TFDs suffer from interference because of cross-terms. Many TFDs have been implemented, and there is no fixed window or kernel that can remove the cross-terms for all types of signals. In this paper, the bilinear TFDs are implemented to analyze power quality signals such as smooth-windowed Wigner-Ville distribution (SWWVD), Choi-Williams distribution (CWD), B-distribution (BD), and modified B-distribution (MBD). The power quality signals focused are swell, sag, interruption, harmonic, interharmonic, and transient based on IEEE Std, 1159-1995. A set of performance measures is defined and used to compare the TFRs. It shows that SWWVD presents the best performance and is selected for power quality signal analysis. Thus, an adaptive optimal kernel SWWVD is designed to determine the separable kernel automatically from the input signal.

A phase synchrony measure for quantifying dynamic functional integration in the brain

The temporal coordination of neural activity within structural networks of the brain has been posited as a basis for cognition. Changes in the frequency and similarity of oscillating electrical potentials emitted by neuronal populations may reflect the means by which networks of the brain carry out functions critical for adaptive behavior. A computation of the phase relationship between signals recorded from separable brain regions is a method for characterizing the temporal interactions of neuronal populations. Recently, different phase estimation methods for quantifying the time-varying and frequency-dependent nature of neural synchronization have been proposed. The most common method for measuring the synchronization of signals through phase computations uses complex wavelet transforms of neural signals to estimate their instantaneous phase difference and locking. In this article, we extend this idea by introducing a new time-varying phase synchrony measure based on Cohen's class of time-frequency distributions. This index offers improvements over existing synchrony measures by characterizing the similarity of signals from separable brain regions with uniformly high resolution across time and frequency. The proposed measure is applied to both synthesized signals and electroencephalography data to test its effectiveness in estimating phase changes and quantifying neural synchrony in the brain.

Weighted integrals of Wigner representations

We consider in this paper Wigner representations Wig τ depending on a parameter $${\tau\in[0,1]}$$ as defined in Boggiatto et al. (Trans Am Math Soc 362:4955–4981, 2010). Integrating these forms with respect to the parameter τ against a weight function Φ we obtain a new class of time–frequency representations Wig Φ. We give basic properties of Wig Φ as subclasses of the general Cohen class.

Estimation of Time-Varying Coherence and Its Application in Understanding Brain Functional Connectivity

Time-varying coherence is a powerful tool for revealing functional dynamics between different regions in the brain. In this paper, we address ways of estimating evolutionary spectrum and coherence using the general Cohen's class distributions. We show that the intimate connection between the Cohen's class-based spectra and the evolutionary spectra defined on the locally stationary time series can be linked by the kernel functions of the Cohen's class distributions. The time-varying spectra and coherence are further generalized with the Stockwell transform, a multiscale time-frequency representation. The Stockwell measures can be studied in the framework of the Cohen's class distributions with a generalized frequency-dependent kernel function. A magnetoencephalography study using the Stockwell coherence reveals an interesting temporal interaction between contralateral and ipsilateral motor cortices under the multisource interference task.

Approximating the Time-Frequency Representation of Biosignals with Chirplets

A new member of the Cohen's class time-frequency distribution is proposed. The kernel function is determined adaptively based on the signal of interest. The kernel preserves the chirp-like components while removing interference terms generated due to the quadratic characteristic of Wigner-Ville distribution. This approach is based on the chirplet as an underlying model of biomedical signals. We illustrate the method using a number of common biological signals including echo-location and evoked potential signals. Finally, the results are compared with other techniques including chirplet decomposition via matching pursuit and the Choi-Williams distribution function.

Time-frequency representations of Wigner type and pseudo-differential operators

We introduce a τ \tau -dependent Wigner representation, Wig τ \operatorname {Wig}_\tau , τ ∈ [ 0 , 1 ] \tau \in [0,1] , which permits us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other. The scheme includes various types of time-frequency representations, among the others the classical Wigner and Rihaczek representations and the most common classes of pseudo-differential operators. We show further that the integral over τ \tau of Wig τ \operatorname {Wig}_\tau yields a new representation Q Q possessing features in signal analysis which considerably improve those of the Wigner representation, especially for what concerns the so-called “ghost frequencies”. The relations of all these representations with respect to the generalized spectrogram and the Cohen class are then studied. Furthermore, a characterization of the L p L^p -boundedness of both τ \tau -pseudo-differential operators and τ \tau -Wigner representations are obtained.

Radar Micro-Doppler Signature Analysis with HHT

Radar micro-Doppler signature analysis of vibrating targets with the joint time-frequency methods can provide useful information for target detection, classification, and recognition. We employ a well-known method for nonlinear and nonstationary signals, Hilbert-Huang transform (HHT) as a new approach in micro-Doppler analysis. Simulation and experimental results show that it can achieve better performance than the traditional Cohen's class time-frequency methods.

Recovery of structural degradation indicators of human bones using elastic waves.

Non-ionizing techniques of bone characterization have mostly been based on ultrasonic wave propagation. In this study, bone is characterized via its linear response to vibratory solicitations. The analysis of the response of bones to transient mechanical excitation is made both theoretically and experimentally. An orthotropic three-dimensional finite element interaction model (FEIM) using computerized tomography scan image geometries of excised dry human tibiae bones is developed. The model parameters (nine elastic constants) are recovered by solving an inverse eigenvalue problem using resonance frequencies from vibration spectroscopy. Transient waves are excited and signals acquired along the diaphysis of the tibiae, experimentally by employing piezoelectric transducers and numerically using FEIM and a time integration scheme. The elastic waves propagating as guided modes in the bone are investigated using reassigned bilinear time-frequency distribution. To avoid ambiguity in determining the elastic wave phase velocities due to the dispersive nature of propagation in bone, the longitudinal L(0,1) and flexural F(1,1) phase velocities, measured at their asymptotes, i.e., the bar and Rayleigh velocities respectively, are proposed as indicators for assessing the strength of long bones. Their sensitivity to changes in bone state like the effect of cortical thickness variation due to aging and osteoporosis is examined.

Phase-space representation of partially coherent imaging systems using the Cohen class distribution

We develop a phase-space representation (termed phase-space imager) suitable for analysis and design of imaging systems that use large illumination apertures (partially coherent illumination). The representation, developed from the transmission cross-coefficient model, falls in the general Cohen class of distributions and elegantly captures the bilinear nature of image formation in partially coherent systems. It uses only the requisite number of dimensions and leads to an efficient algorithm for calculating images. Computed partially coherent images of a double slit and a two-dimensional sinusoidal spoke are presented.

Optimisation of features using evolutionary algorithm for EEG signal classification

Stochastic optimisation plays a significant role in analysis of complex problems. EEG data is very noisy and has different types of artefacts. In this paper, we have evaluated the various time-frequency analysis of different signals as the features. Since the EEG signals are non-stationary in nature, time-frequency transformations have been suggested to extract the common features for a particular mental task performed by different subjects. The major contribution of this paper is the optimisation of different time-frequency kernels belonging to Cohen's class. A comparative assessment of the classification performance with the conventional Gaussian kernels in time as well as frequency domain has been also performed. It has been found that the Wigner-Ville type time-frequency kernel exhibit the best performance with an accuracy of 94%, followed by STFT. Comparative simulation results demonstrate a significant improvement in the classification accuracy in case of these optimised kernels.

Spectral decomposition of seismic data with reassigned smoothed pseudo Wigner–Ville distribution

Seismic signals are nonstationary mainly due to absorption and attenuation of seismic energy in strata. Referring to spectral decomposition of seismic data, the conventional method using short-time Fourier transform (STFT) limits temporal and spectral resolution by a predefined window length. Continuous-wavelet transform (CWT) uses dilation and translation of a wavelet to produce a time-scale map. However, the wavelets utilized should be orthogonal in order to obtain a satisfactory resolution. The less applied, Wigner–Ville distribution (WVD) being superior in energy distribution concentration, is confronted with cross-terms interference (CTI) when signals are multi-component. In order to reduce the impact of CTI, Cohen class uses kernel function as low-pass filter. Nevertheless it also weakens energy concentration of auto-terms. In this paper, we employ smoothed pseudo Wigner–Ville distribution (SPWVD) with Gauss kernel function to reduce CTI in time and frequency domain, then reassign values of SPWVD (called reassigned SPWVD) according to the center of gravity of the considering energy region so that distribution concentration is maintained simultaneously. We conduct the method above on a multi-component synthetic seismic record and compare with STFT and CWT spectra. Two field examples reveal that RSPWVD potentially can be applied to detect low-frequency shadows caused by hydrocarbons and to delineate the space distribution of abnormal geological body more precisely.

Computationally Efficient Scale Covariant Time-Frequency Distributions

Scale is a physical attribute of a signal which occurs in many natural settings. Time-frequency distributions (TFDs) belonging to Cohen's class are invariant to time and frequency shifts, but are not necessarily covariant to the time scalings of the signal. Conditions on the time-frequency kernel for yielding a scale covariant distribution have been previously derived (Cohen, 1995). In this paper, a new class of computationally efficient scale covariant distributions is introduced. These distributions are constructed using the eigendecomposition of time-frequency kernels (Burrus et al., 1997). The performance of this new class of distributions is illustrated with examples and is compared to conventional scale covariant distributions.

Kaiser window based kernel for time-frequency distributions

A novel sinhc kernel, which generates a Kaiser window based time-frequency distribution for non-stationary signal analysis is introduced. The Kaiser distribution belongs to the Cohen class of time-frequency distributions, and satisfies desirable distribution properties. By controlling the shape of the Kaiser window, auto-term resolutions and cross-term magnitudes can be successfully traded.

Unifying distribution functions: some lesser known distributions

We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.