Cohen's Class Research Articles

Alias-free generalized discrete-time time-frequency distributions

A definition of generalized discrete-time time-frequency distribution that utilizes all of the outer product terms from a data sequence, so that one can avoid aliasing, is introduced. The new approach provides (1) proper implementation of the discrete-time spectrogram, (2) correct evaluation of the instantaneous frequency of the underlying continuous-time signal, and (3) correct frequency marginal. The formulation provides a unified framework for implementing members of Cohen's class, which was formulated in the continuous-time domain. Some requirements for the discrete-time kernel in the new approach are discussed in association with desirable distribution properties. Some experimental results are provided to illustrate the features of the proposed method. >

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. >

Computing time-frequency distributions (signal analysis)

Recently, numerous strategies have been proposed for computing discrete time-frequency distributions such as the Wigner distribution. The author describes an efficient and straightforward strategy for computing time-frequency distributions that are members of Cohen's class. The strategy is based on the work by A.H. Nuttall (Naval Underwater Syst. Center, Tech. Rep., 8533, April 1989) which has resolved the questions concerning aliasing and required sampling rate for the Wigner distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the convergence of the estimated posterior density weighted algorithm

Sufficient conditions are established for the convergence of the algorithm when applied to the problem of radar imaging as suggested by D.L. Snyder and H.J. Whitehouse (1985), and the problem of enhancement of nonstationary spectrum estimators based on L. Cohen's (1966) class of time-frequency distributions, as proposed by L.B. White (1989). In the latter case, a regularization step must be included to ensure good behavior. The sufficient conditions may always be met in practice.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transition kernels for bilinear time-frequency distributions

The author proposes a sequence of time-frequency signal representations which offer a continuous transition from time smoothed pseudo-Wigner-Ville distributions (PWVD) to a specified class of positive representations. This allows the selection of a representation which offers the desired tradeoff between the inherent localization properties of the PWVD and the lack of interference terms present in the class of positive representations, a cause of some criticism of the method. The sequence converges uniformly to the specified positive distribution. >

Cross spectral analysis of nonstationary processes

Consideration is given to the generalization of stationary cross spectral analysis methods to a class of nonstationary processes, specifically, the class of semistationary finite energy processes possessing sample functions that are of finite energy almost surely. A new quantity called the time-frequency coherence (TFC) is defined, and it is demonstrated that its properties are analogous to those possessed by the stationary coherence function. The problem of estimating the TFC by using elements of L. Cohen's (1966) class of joint time-frequency representations is investigated. It is shown that the only admissible estimators are those based on the class of time-frequency smoothed periodograms. Thus, the familiar procedure of segmentation and (smoothed) short-time Fourier analysis cannot be improved upon (within the framework considered) by the use of the higher-resolution nonparametric time-frequency methods. Procedures for selection of the appropriate estimators and a possible application are suggested. >

Time-frequency distributions for a wide-sense stationary random signal

It is shown that a wide class of time-frequency distributions gives the power spectrum for the case of a wide-sense stationary random signal. A simple criterion on the Cohen (1966) kernel for determining when a bilinear time-frequency distribution gives the power spectrum of the signal is derived. >

Improved time-frequency representation of multicomponent signals using exponential kernels

The authors introduce a time-frequency distribution of L. Cohen's (1966) class and examines its properties. This distribution is called exponential distribution (ED) after its exponential kernel function. First, the authors interpret the ED from the spectral-density-estimation point of view. They then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain, and they analyze the ED for two specific types of multicomponent signals: sinusoidal signals and chirp signals. Next, they define the ED for discrete-time signals and the running windowed exponential distribution (RWED), which is computationally efficient. Finally, the authors present numerical examples of the RWED using the synthetically generated signals. It is found that the ED is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time-frequency distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On positivity of time-frequency distributions

This correspondence addresses the problem of how to regard the fundamental impossibility with time-frequency energy distributions of Cohen's class always to be nonnegative and, at the same time, to have correct marginal distributions. It is shown that the Wigner distribution is the only member of a large class of bilinear time-frequency distributions that becomes nonnegative after smoothing in the time-frequency plane by means of Gaussian weight functions with BT product equal to unity.