Cohen's Class Research Articles

Adaptive diffusion using Hesse matrix for detecting nonstationary signals

Adaptive diffusion, using a gradient‐square tensor, was proposed to improve the readability of time frequency representations (TFRs) of the Cohen class and the affine class in 2005. It works with a signal‐dependent kernel and can adapt to a wide range of nonstationary signals. Here we substitute the Hesse matrix for the gradient‐square tensor after making an analysis of local diffusion behavior in an adaptive diffusion process, in close conjunction with the local signature of the auto terms and the interference terms. The eigenvectors of the Hesse matrix not only can give the local average gradient direction and thus the local diffusion direction, but also can provide an explicit mathematical explanation for local diffusion behavior. The Hesse method is capable of providing TFRs of better readability than the gradient square tensor. The validity of the new method is verified by computational simulations.

Adaptive Signal Analysis Based on Radial Parabola Kernel

Because of the deficiency of fixed kernel in bilinear time-frequency distribution (TFD), i.e. for each mapping, the resulting time-frequency representation is satisfactory only for a limited class of signals, a new adaptive kernel function named the radial parabola kernel (RPK), is proposed. The RPK can adopt the optimizing method to filter cross-terms adaptively according to the signal distribution, obtain good time-frequency resolution, and offer improved TFD for a large class of signals. Compared with traditional fixed -kernel functions, such as Wigner-Ville distribution, Choi-Willams distribution and Cone-kernel distribution, the superiority of the RPK function is obvious. At last, the RPK function is applied to the analysis of vibration signals of bearing, and the result proves the RPK function an effective method in analyzing signals.

Wigner-Ville distribution and its application in seismic attenuation estimation

The attenuation of seismic signals is often characterized in the frequency domain using statistical measures of the power spectrum. However, the conventional Fourier transform-based power spectrum estimation methods suffer from time-frequency resolution problems. Wigner-Ville distribution, which is a member of Cohen class time-frequency distributions, possesses many appealing properties, such as time-frequency marginal distribution, time-frequency localization, etc. Therefore, Wigner-Ville distribution offers a new way for estimating the attenuation of seismic signals. This paper initially gives a brief introduction to Wigner-Ville distribution and the smoothed Wigner-Ville distribution that is effective in reducing the cross-term effect, and then presents a method for seismic attenuation estimation based on the instantaneous energy spectrum of the Wigner-Ville distribution. A real data example from central Tarim Basin in western China is presented to illustrate the effectiveness of the proposed method. The results show that the Wigner-Ville distribution-based seismic attenuation estimation method can effectively detect the difference between reef, shoal and lagoon facies by their attenuation properties, indicating that the estimated seismic attenuation can be used for reef and shoal carbonate reservoir characterization.

Combining Adaptive Diffusion with Difference for Detecting Nonstationary Signals

Time-frequency analysis aims to construct a density function of time and frequency to reveal the frequency components in signals to be analyzed and the evolution of the frequency of signals with time. The Wigner distribution (WD) is one of the most fundamental and widely used methods for analyzing nonstationary signals in the fields of radar, communication, etc. However, the application of the WD is greatly limited by the existence of interference terms. The adaptive diffusion method proposed to remove the interference terms of the WD by Julien Gosme, et al. is to be invalid in the presence of interference terms generated by signals, whose distributions are interwoven together in the time-frequency plane of the WD. We combine the diffusion technique with difference method for removing these interference terms to improve the resolution and readability of the time-frequency representation of the Cohen class for detecting nonstationary signals.

Temporal coherence and time-frequency distributions in spectroscopic optical coherence tomography

Traditional analysis of spectroscopic optical coherence tomography (SOCT) signals is limited by an uncertainty relationship between time (depth) and frequency (wavelength). The use of a bilinear time-frequency distribution for analysis, such as those that compose Cohen's class of functions, may provide a way to avoid this limitation. Here we present the relationship between traditional SOCT analysis and the relevant Cohen class functions: the Wigner and Choi-Williams distributions. While cross terms that arise in these bilinear time-frequency distributions have been viewed as an artifact, here we identify these terms with temporal coherence, which contains significant information about the signal through phase relationships. The utility of time-frequency distributions is illustrated through analysis of calculated signals.

Time-Width Versus Frequency Band Mapping of Energy Distributions

Most of the signal processing operations involve some kind of a transformation or approximation of the signal. The transform coefficients or the approximation parameters reveal many hidden characteristics of a signal that can be appropriately processed to extract useful information. In recent years, adaptive time-frequency (TF) transformations have significantly contributed to this area. The TF transformation can be classified into two main categories based on 1) signal decomposition approaches and 2) bilinear TF distributions (also known as Cohen's class). TF distributions are nonparametric in nature and mainly used for visualization purposes. On the other hand, decomposition approaches are parametric in nature and highly suitable for objective feature extraction. This paper focuses on a particular TF decomposition approach (adaptive TF transformation) based on matching pursuit-type algorithm. Using this decomposition approach, a novel time-width versus frequency band (TWFB) energy mapping is proposed that possesses both parameterization benefits and meaningful visual patterns with favorable properties for pattern recognition. This organized mapping of the TF decomposition parameters allows the application of pruning algorithms such as local discriminant bases (LDB) to identify application specific TF subspaces. The identification of these subspaces enables efficient processing of information and reduces the computational effort considerably. The visual patterns of the TWFB mappings exhibit high potential of becoming a powerful pattern analysis tool. The paper covers in detail the formulation of the TWFB mapping and some of its properties. Experiments performed with speech and synthetic signals produced desirable results demonstrating the benefits of TWFB for pattern recognition related applications

Uncertainty principle, positivity and [formula omitted]-boundedness for generalized spectrograms

In this paper we are concerned with the properties of positivity, uncertainty principle and continuity in L p spaces of a generalized spectrogram. In particular we study the connections of a generalized spectrogram, as a subclass of the Cohen class, with the Rihaczek and the Wigner representations. We also consider the behavior of the generalized spectrogram with respect to the positivity and the L p boundedness of the corresponding localization operators.

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen's class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman's sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter

Estimation of modal parameters using bilinear joint time–frequency distributions

In this paper, a new method is proposed for modal parameter estimation using time–frequency representations. Smoothed Pseudo Wigner–Ville distribution which is a member of the Cohen's class distributions is used to decouple vibration modes completely in order to study each mode separately. This distribution reduces cross-terms which are troublesome in Wigner–Ville distribution and retains the resolution as well. The method was applied to highly damped systems, and results were superior to those obtained via other conventional methods.

Quadratic Cohen representations in spectral analysis of serration process in Al–Mg alloys

Important from mechanical point of view the Portevin–Le Chatelier serration phenomenon is being characterized by a complicated spectral profile. As a typical example of nonstationary processes it demands a special treatment allowing to follow the evolution of energy of stress fluctuations as a function of strain. The authors suggest the utilization of a compact system of quadratic transformations, known as Cohen class, as a technique enabling the reliable analysis of serration processes. In this elaboration a comparison of the application of various Cohen decompositions to analyze frequency bands has been introduced.

Feature Extraction From Underwater Signals Using Time-Frequency Warping Operators

Processing marine-mammal signals for passive oceanic acoustic tomography or species classification and monitoring are problems that have recently attracted attention in scientific literature. For these purposes, it is necessary to use a method which could be able to extract the useful information about the processed data, knowing that the underwater environment is highly nonstationary. In this context, time-frequency (TF) or time-scale methods constitute a potential approach. Practically, it has been observed that the majority of TF structures of the marine-mammal signals are highly nonlinear. This fact affects dramatically the performances achieved by the Cohen's class methods, these methods being efficient in the presence of linear TF structures. Fortunately, thanks to the warping operator principle, it is possible to generate other class of time-frequency representations (TFRs). The new TFRs may analyze nonlinear chirp signals better than Cohen's class does. In spite of its mathematical elegance, this principle is limited in real applications by two major elements. First, as we will see, its implementation leads to a considerable growth of the signal length. Consequently, from operational point of view, this principle is limited to short synthetic signals. Second, the design of a single warping operator can be inappropriate if the analyzed signal is multicomponent. Furthermore, the choice of "adapted" warping operator becomes a problem when the signal components have different TF behaviors. In this paper, we propose a processing method of marine-mammal signals, well adapted to a real passive underwater context. The method tries to overcome the two aforementioned limitations. Also, the first step consists in data size reducing by the detection of the TF regions of interests (ROIs). Furthermore, in each ROI, a technique which combines some typical warping operators is used. The result is an analytical characterization of the instantaneous frequency laws (IFLs) of signal components. The simulations on real underwater data show the performances of this method in comparison with classical ones

The T-class of time–frequency distributions: Time-only kernels with amplitude estimation

This paper presents a class of time–frequency distributions (TFDs) characterized by time-lag kernels which are functions of time only. If the parameters of the time-only kernels are properly chosen, their corresponding TFDs, the T-distributions, are more efficient than their two-dimensional counterparts in terms of cross-terms suppression while keeping a high-energy concentration (resolution) around the IF law of non-stationary signals. The proposed class is a subclass of Cohen's Class of quadratic TFDs. We have shown that separable time-lag kernels should be lag-independent (or time-only) for best resolution. In addition, non-parametric amplitude estimation is possible directly from the T-distributions in case of FM signals, a property that is not verified by other TFDs. Two examples of the T-distributions are given and their performance is compared to other TFDs with numerical examples using synthetic and real-life signals.

A basis for efficient representation of the S-transform

The S-transform is a time-frequency representation known for its local spectral phase properties. A key feature of the S-transform is that it uniquely combines a frequency dependent resolution of the time-frequency space and absolutely referenced local phase information. This allows one to define the meaning of phase in a local spectrum setting, and results in many desirable characteristics. One drawback to the S-transform is the redundant representation of the time-frequency space and the consumption of computing resources this requires (a characteristic it shares with the continuous wavelet transform, the short time Fourier transform, and Cohen's class of generalized time-frequency distributions). The cost of this redundancy is amplified in multidimensional applications such as image analysis. A more efficient representation is introduced here as a orthogonal set of basis functions that localizes the spectrum and retains the advantageous phase properties of the S-transform. These basis functions are defined to have phase characteristics that are directly related to the phase of the Fourier transform spectrum, and are both compact in frequency and localized in time. Distinct from a wavelet approach, this approach allows one to directly collapse the orthogonal local spectral representation over time to the complex-valued Fourier transform spectrum. Because it maintains the phase properties of the S-transform, one can perform localized cross spectral analysis to measure phase shifts between each of multiple components of two time series as a function of both time and frequency. In addition, one can define a generalized instantaneous frequency (IF) applicable to broadband nonstationary signals. This is the first time a channel IF has been integrated in an orthogonal local spectral representation. A direct comparison between these basis functions and complex wavelets is performed, highlighting the advantages of this approach. The relationship between this basis set and the fully redundant S-transform is demonstrated highlighting the ability to arbitrarily sample the time-frequency space. The introduction of this basis set leads to efficient analysis routines that may find use in a wide range of fields.

Fractional time-varying energy distributions of ECG signals

We introduce here fractional Cohen class of time-frequency distributions (FCCTFDs) containing fractional modulations which is kernel of fractional Fourier transform (FFT). The fractional modulation depends on angular parameter α and can be interpreted as a rotation by an angle α in time-frequency plane. This distribution promotes to track time-variant energy of a biological signals and represents it in time-frequency domain. It uses the fractional ambiguity function (FAF) of signal multiplied by a suitable kernel which is designed for the biological signals generally having multi-non-stationary components. This result improves and generalizes some of the previous time-frequency distributions derived in the literature.

Adaptive diffusion as a versatile tool for time-frequency and time-scale representations processing: a review

Inspired by the work on image processing by Perona and Malik, diffusion-based models were first investigated by Goncalve/spl grave/s and Payot to improve the readability of Cohen class time-frequency representations. They rely on signal-dependent partial differential equations that yield adaptive smoothed representations with sharpened time-frequency components. Here, we demonstrate the versatility and utility of this family of methods, and we propose new adaptive diffusion processes to locally control both the orientation and the strength of smoothing. Our approach is an improvement on previous works as it provides a unified framework not only for the Cohen class but for the affine class as well. The latter is of particular interest because, except for some special techniques such as the reassignment method, no signal-dependent smoothing technique exists to process bilinear time-scale distributions, nor even a transposition of the adaptive optimal-kernel method proposed by Baraniuk and Jones.

Multitaper marginal time–frequency distributions

Time–frequency distributions (TFDs) belonging to Cohen's class yield a frequency marginal that is equivalent to the periodogram of the signal. It is well-known that the periodogram is not a good spectral estimator since it is not a consistent estimate, i.e. its variance does not decrease with the sample size. Thomson addressed this issue by introducing a multitaper spectrum estimator with high resolution and statistical stability [D.J. Thomson, Spectrum estimation and harmonic analysis, Proc. IEEE 70 (9) (1982) 1055–1096]. In recent years, various approaches have been developed to extend such multitaper spectral estimators to the area of nonstationary signal analysis [F. Çakrak, P. Loughlin, Multiple window nonlinear time-varying spectral analysis, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, 1998, pp. 2409–2412; F. Çakrak, P. Loughlin, Multiple window nonlinear time-varying spectral analysis, IEEE Trans. Signal Process. 49 (2) (2001) 448–453; J.W. Pitton, Time–frequency spectrum estimation: an adaptive multitaper method, in: Proceedings of IEEE International Symposium on Time–frequency and Time-scale analysis, 1998, pp. 665–668; J.W. Pitton, Nonstationary spectrum estimation and time–frequency concentration, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, 1998, pp. 2425–2428; G. Frazer, B. Boashash, Multiple window spectrogram and time–frequency distributions, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, 1994, pp. 293–296; M. Bayram, R.G. Baraniuk, Multiple window time–frequency analysis, in: Proceedings of the IEEE International Symposium on Time–frequency and Time-scale analysis, 1996, pp. 173–176.]. In this paper, a new method that approaches the problem from the perspective of frequency marginals is introduced. A class of time–frequency distributions, multitaper marginal TFD (MTM-TFD), is constructed for analyzing time-varying signals in noise with statistically stable frequency marginals. A kernel design method yielding any desired frequency marginal, such as provided by Thomson's spectrum estimator, is derived for a given signal. The improvement in the performance of this new class of time–frequency distributions compared to the conventional time–frequency distributions is illustrated through examples.

On conditional spectral moments of Gaussian and damped sinusoidal atoms in adaptive signal decomposition

It is well known that the convergence (with different speeds) of the matching pursuit (MP) signal decomposition algorithm for any dense dictionary is guaranteed. In this paper, we have analysed (in theory and through simulations) the performance and properties of both Gaussian and damped sinusoidal atoms for the MP signal decomposition. We have examined the decomposed signal in ambiguity space (to look for auto-terms concentrated around the origin), and investigated the requirement to have a positive time–frequency representation. We are thus able to propose what kind of dictionary might be more suitable for MP signal decomposition. We have also derived general formulae for the first and second conditional spectral moments, which are useful generalizations of the concept of instantaneous frequency and instantaneous bandwidth, respectively. While the second conditional moment is not positive for many bilinear time–frequency distributions, thus making useless its interpretation as instantaneous bandwidth, we have proved that for MP decomposition based on Gaussian or damped sinusoidal atoms, it is always guaranteed positive.

Decomposing ERP time–frequency energy using PCA

Objective Time–frequency transforms (TFTs) offer rich representations of event-related potential (ERP) activity, and thus add complexity. Data reduction techniques for TFTs have been slow to develop beyond time analysis of detail functions from wavelet transforms. Cohen's class of TFTs based on the reduced interference distribution (RID) offer some benefits over wavelet TFTs, but do not offer the simplicity of detail functions from wavelet decomposition. The objective of the current approach is a data reduction method to extract succinct and meaningful events from both RID and wavelet TFTs. Methods A general energy-based principal components analysis (PCA) approach to reducing TFTs is detailed. TFT surfaces are first restructured into vectors, recasting the data as a two-dimensional matrix amenable to PCA. PCA decomposition is performed on the two-dimensional matrix, and surfaces are then reconstructed. The PCA decomposition method is conducted with RID and Morlet wavelet TFTs, as well as with PCA for time and frequency domains separately. Results Three simulated datasets were decomposed. These included Gabor logons and chirped signals. All simulated events were appropriately extracted from the TFTs using both wavelet and RID TFTs. Varying levels of noise were then added to the simulated data, as well as a simulated condition difference. The PCA-TFT method, particularly when used with RID TFTs, appropriately extracted the components and detected condition differences for signals where time or frequency domain analysis alone failed. Response-locked ERP data from a reaction time experiment was also decomposed. Meaningful components representing distinct neurophysiological activity were extracted from the ERP TFT data, including the error-related negativity (ERN). Conclusions Effective TFT data reduction was achieved. Activity that overlapped in time, frequency, and topography were effectively separated and extracted. Methodological issues involved in the application of PCA to TFTs are detailed, and directions for further development are discussed. Significance The reported decomposition method represents a natural but significant extension of PCA into the TFT domain from the time and frequency domains alone. Evaluation of many aspects of this extension could now be conducted, using the PCA-TFT decomposition as a basis.

The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals

The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame/spl acute/r-Loe/spl grave/ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohen's class of bilinear TFDs.

Source localization of non-stationary acoustic data using time-frequency analysis

An improvement in temporal locality of the generalized cross-correlation (GCC) for angle of arrival (AOA) estimation can be achieved by employing 2-D cross-correlation of infrasonic sensor data transformed to its time-frequency (TF) representation. Intermediate to the AOA evaluation is the time delay between pairs of sensors. The signal class of interest includes far field sources which are partially coherent across the array, nonstationary, and wideband. In addition, signals can occur as multiple short bursts, for which TF representations may be more appropriate for time delay estimation. The GCC tends to smooth out such temporal energy bursts. Simulation and experimental results will demonstrate the improvement in using a TF-based GCC, using the Cohen class, over the classic GCC method. Comparative demonstration of the methods will be performed on data captured on an infrasonic sensor array located at NASA Langley Research Center (LaRC). The infrasonic data sources include Delta IV and Space Shuttle launches from Kennedy Space Center which belong to the stated signal class. Of interest is to apply this method to the AOA estimation of atmospheric turbulence. [Work supported by NASA LaRC Creativity and Innovation project: Infrasonic Detection of Clear Air Turbulence and Severe Storms.]