Shift covariant time-frequency distributions of discrete signals

Many commonly used time-frequency distributions are members of the Cohen (1989) class. This class is defined for continuous signals, and since time-frequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the class of all quadratic time-frequency distributions that are covariant to time shifts and frequency shifts. We extend this method to three types of discrete signals to derive what we call the discrete Cohen classes. The properties of the discrete Cohen classes differ from those of the original Cohen class. To illustrate these properties, we also provide explicit relationships between the classical Wigner distribution and the discrete Cohen classes     

Virtues and vices of quartic time-frequency distributions

We present results concerning three different types of quartic (fourth order) time-frequency distributions (TFDs). First, we present new results on the previously introduced local ambiguity function and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. Second, we introduce the class of quartic, shift-covariant, time-frequency distributions and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirp rate     

I.I.R. Volterra Filtering with Application to Bilinear Systems

A new nonlinear filtering technique by means of infinite impulse response (IIR) Volterra functionals is developed. It yields the projection onto the closed class of finite Volterra series with separable kernels of arbitrary degree k. Such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line. Moreover, if the original system model is itself bilinear, this computation is finitely recursive through higher moments of degree 2 k. Two simple illustrating examples are developed: (i) estimation of the covariance of the internal white noise driving a linear system and (ii) filtering of a non-Gaussian linear system (driven by a Poisson process). The robustness with respect to the observation noise distribution is finally examined     

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.     

Fast calculation of the Choi-Williams time-frequency distribution

The method presented allows faster calculation of any time-frequency distribution with a kernel that can be formulated in the time-lag plane. Specific examples are the Wigner and Choi-Williams distributions. The Choi-Williams distribution (CWD) uses an exponential kernel in the generalized class of bilinear time-frequency distributions to achieve a reduction in the cross-term components of the distribution. Matrix manipulations provide an intuitive approach and, when combined with parallel processing, improve the processing speed to allow real-time calculations of the CWD. The use of an outer product matrix with a weighting matrix is particularly useful when evaluating different weighting parameters. For any given signal, the outer product matrix needs to be calculated just once. The various weighting matrices can be stored and used with any signal when needed. Parallel processing architectures allow implementation of the algorithm with speeds that are appropriate for real-time, running window calculations     

Controlled multifunctional processing of causal signals on the basis of nonlinear (quadratic and cubic) time-frequency distributions

General laws of the controlled multifunctional processing based on nonlinear (quadratic and cubic) time-frequency distributions of finite duration arbitrarily shaped causal signals that do not coincide in time and that have different carrier frequencies are presented. These distributions are compared to the time-frequency representations used in the signal analysis based on the Wigner distribution, the uncertainty function, and other quadratic distributions. Examples of the real-time realization of controlled multifunctional processing of causal signals that is performed on the basis of quadratic and cubic distributions are given. These examples include direct and inverse Fourier transforms, convolution, spectral analysis with varied time and frequency scales, delay, compression, time-domain inversion, and other functions.     

A signal-dependent time-frequency representation: fast algorithmfor optimal kernel design

A time-frequency representation based on an optimal, signal-dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time-frequency distributions: that the best kernel and distribution depend on the signal to be analyzed. The optimization formulation for the signal-dependent kernel results in a linear program with a unique feature: a tree structure that summarizes a set of constraints on the kernel. The authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest. For a kernel with Q variables, the running time of the algorithm is O(Q log Q), which is several orders of magnitude less than any other known method for solving this class of linear program. This efficiency enables the computation of the signal-dependent, optimal-kernel time-frequency representation at a cost that is on the same order as a fixed-kernel distribution. An important property of the optimal kernel is that it takes on essentially only the values of 1 and 0     

On generalized-marginal time-frequency distributions

We introduce a family of time-frequency (TF) distributions with generalized marginals, i.e., beyond the time-domain and the frequency-domain marginals, in the sense that the projections of a TF distribution along one or more angles are equal to the magnitude squared of the fractional Fourier transforms of the signal. We present a necessary and sufficient condition for a TF distribution in Cohen's class to satisfy generalized marginals. We then modify the existing well-known TF distributions in Cohen's class, such as Choi-Williams (1989) and Page distributions, so that the modified ones have generalized marginals. Numerical examples are presented to show that the proposed TF distributions have the advantages of both Wigner-Ville and other quadratic TF distributions, which only have the conventional marginals. Moreover, they also indicate that the generalized-marginal TF distributions with proper marginals are more robust than the Wigner-Ville and the Choi-Williams distributions when signals contain additive noise     

Identification of a class of nonlinear systems under stationarynon-Gaussian excitation

This paper provides new solutions to the nonlinear system identification problem when the input to the system is a stationary non-Gaussian process. We propose the use of a model called the Hammerstein series, which leads to significant reductions in both the computational requirements and the mathematical tractability of the nonlinear system identification problem. We show that unlike the Volterra series, one can obtain closed-form expressions for the Hammerstein series kernels and the quadratic coherence function in the non-Gaussian case. Estimation of the kernels and quadratic coherence function is discussed. A comparison with a nonlinear system identification approach that uses the Volterra series is provided. An automotive engineering application illustrates the usefulness of the proposed method     

On the equivalence of the operator and kernel methods for jointdistributions of arbitrary variables

Generalizing the concept of time-frequency representations, Cohen (see Englewood Cliffs, NJ: Prentice-Hall, 1995) has proposed a method, based on operator correspondence rules, for generating joint distributions of arbitrary variables. As an alternative to considering all such rules, which is a practical impossibility in general, Cohen has proposed the kernel method in which different distributions are generated from a fixed rule via an arbitrary kernel. We derive a simple but rather stringent necessary condition, on the underlying operators, for the kernel method (with the kernel functionally independent of the variables) to generate all bilinear distributions. Of the specific pairs of variables that have been studied, essentially only time and frequency satisfy the condition; in particular, the important variables of time and scale do not. The results warrant further study for a systematic characterization of bilinear distributions in Cohen's method     

Resolution measure criteria for the objective assessment of the performance of quadratic time-frequency distributions

This paper presents the essential elements for developing objective methods of assessment of the performance of time-frequency signal analysis techniques. We define a measure for assessing the resolution performance of time-frequency distributions (TFDs) in separating closely spaced components in the time-frequency domain. The measure takes into account key attributes of TFDs, such as components mainlobes and sidelobes and cross-terms. The introduction of this measure allows to quantify the quality of TFDs instead of relying solely on visual inspection of their plots. The method of assessment of performance of TFDs also allows the improvement of methodologies for designing high-resolution quadratic TFDs for time-frequency analysis of multicomponent signals. Different TFDs, including the modified B distribution, are optimized using this methodology. Examples of a performance comparison of quadratic TFDs in resolving closely spaced components in the time-frequency domain, using the proposed resolution measure, are provided.     

二次时频表示中核函数的优化设计

二次时频分布是分析非平稳信号的有力工具，在具有许多优良特性的同时，存在严重的交叉干扰项。在Wigner-Ville分布及Cohen类时频分布具有固定核函数的基础上，研究了基于信号的核函数优化设计的两种方法，径向高斯核函数和最优相位核函数的设计方法。基于信号的核函数的时频表示可以有效地抑制或转移交叉分量，提高时频表示的可读估计，改善其主要性能。     

Positive Time-Frequency Distributions via Quadratic Programming

A new method for computing positive time-frequency distributions (TFDs) for nonstationary signals is presented. This work extends the earlier work of the author and his colleagues in computing positive TFDs [8,11]. This paper describes a general quadratic programming approach to the problem of computing these signal-dependent distributions. The method is based on an evolutionary spectrum formulation of positive TFDs. The minimization problem reduces to a linearly-constrained quadratic programming problem, for which standard solutions are widely available.     

Efficient implementation of class of isotropic quadratic filters byusing Walsh-Hadamard transform

A method is proposed for the efficient implementation of a class of second-order Volterra filters where their quadratic kernels are recursively constructed from a set of isotropic subkernels (ISKs). The quadratic kernel of this class of filters can be factorised into a diagonal form by a Walsh-Hadamard transform (WHT) and be implemented using a transform and multiply structure. An algorithm for determining the quadratic ISK kernel for the nonlinear prediction of speech signals is presented     

基于时频重排-Hough变换的多分量Chirp信号的检测与参数估计

针对BTFD-Hough算法因交叉项形成伪尖峰引起的误检测;而平滑交叉项又造成参数估计精度降低和运行 时间增加的问题,本文提出了对BTFD进行时频重排再结合Hougb变换的多分量Chirp信号检测与参数估计的改进方法, 仿真实验结果表明,该方法不仅有效地检测多分量Chirp信号并估计其参数,还降低了H0ugIl变换时间,同时也改善了抗 噪声性能。     

Improving the readability of time-frequency and time-scalerepresentations by the reassignment method

In this paper, the use of the reassignment method, first applied by Kodera, Gendrin, and de Villedary (1976) to the spectrogram, is generalized to any bilinear time-frequency or time-scale distribution. This method creates a modified version of a representation by moving its values away from where they are computed, so as to produce a better localization of the signal components. We first propose a new formulation of this method, followed by a thorough theoretical study of its characteristics. Its practical use for a large variety of known time-frequency and time-scale distributions is then addressed. Finally, some experimental results are reported to demonstrate the performance of this method     

Hybrid linear/quadratic time-frequency attributes

We present an efficient method for robustly calculating time-frequency attributes of a signal, including instantaneous mean frequency, bandwidth, kurtosis, and other moments. Most current attribute estimation techniques involve a costly intermediate step of computing a (highly oversampled) two-dimensonal (2-D) quadratic time-frequency representation (TFR), which is then collapsed to the one-dimensonal (1-D) attribute. Using the principles of hybrid linear/quadratic time-frequency analysis (time-frequency distribution series), we propose computing attributes as nonlinear combinations of the (slightly oversampled) linear Gabor coefficients of the signal. The method is both computationally efficient and accurate; it performs as well as the best techniques based on adaptive TFRs. To illustrate, we calculate an attribute of a seismic cross section     

Quadratic weighted median filters for edge enhancement of noisy images.

Quadratic Volterra filters are effective in image sharpening applications. The linear combination of polynomial terms, however, yields poor performance in noisy environments. Weighted median (WM) filters, in contrast, are well known for their outlier suppression and detail preservation properties. The WM sample selection methodology is naturally extended to the quadratic sample case, yielding a filter structure referred to as quadratic weighted median (QWM) that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers arising in the higher order statistics of environment noise. Through statistical analysis of higher order samples, it is shown that, although the parent Gaussian distribution is light tailed, the higher order terms exhibit heavy-tailed distributions. The optimal combination of terms contributing to a quadratic system, i.e., cross and square, is approached from a maximum likelihood perspective which yields the WM processing of these terms. The proposed QWM filter structure is analyzed through determination of the output variance and breakdown probability. The studies show that the QWM exhibits lower variance and breakdown probability indicating the robustness of the proposed structure. The performance of the QWM filter is tested on constant regions, edges and real images, and compared to its weighted-sum dual, the quadratic Volterra filter. The simulation results show that the proposed method simultaneously suppresses the noise and enhances image details. Compared with the quadratic Volterra sharpener, the QWM filter exhibits superior qualitative and quantitative performance in noisy image sharpening.     

Evolutionary periodogram for nonstationary signals

Presents a novel estimator for the time-dependent spectrum of a nonstationary signal. By modeling the signal, at any given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion, the authors are able to compute a minimum mean-squared error estimate of this time-varying amplitude. Repeating the process over all frequencies, they obtain a power distribution as a function of time and frequency that is consistent with the Wold-Cramer evolutionary spectrum. Based on the model assumptions, the authors develop the evolutionary periodogram (EP) for nonstationary signals, an estimator analogous to the periodogram used in the stationary case. They also derive the time-frequency resolution of the new estimator. The approach is free of some of the drawbacks of the bilinear distributions and of the short-time Fourier transform spectral estimates. It is guaranteed to produce nonnegative spectra without the cross-term behavior of the bilinear distributions, and it does not require windowing of data in the time domain. Examples illustrating the new estimator are given