Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transformation, performing standard processing techniques on the transformed signal, and then (in some cases) transforming the resulting output. The resulting unitarily equivalent systems focus on the critical signal characteristics in large classes of signals and, hence, prove useful for representing and processing signals that are not well matched by current techniques. As specific examples of this procedure, we generalize linear time-invariant systems, orthonormal basis and frame decompositions, and joint time-frequency and time-scale distributions, illustrating the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature. This work...

Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the “A ” content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen’s general construction for joint distributions of arbitrary variables.

Many commonly used time-frequency distributions are members of the Cohen 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. In this paper we extend this method to three types of discrete signals to derive what we will 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. I. Introduction I N signal analysis there are four types of signals commonly used. These four types are based on whether the signal is continuous or discrete, and whether the signal is aperiodic or periodic. The four signal types ...

Joint signal representations (JSRs) of arbitrary variables generalize time-frequency representations (TFRs) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSRs of arbitrary variables have been proposed by Cohen and Baraniuk. Cohen's method is a direct extension of his original formulation of TFRs, and Baraniuk's approach is based on a group theoretic formulation; both use the powerful concept of associating variables with operators. One of the main results of the paper is that despite their apparent differences, the two approaches to generalized JSRs are completely equivalent. Remarkably, the JSRs of the two methods are simply related via axis warping transformations, with the broad implication that JSRs with radically different covariance properties can be generated efficiently from JSRs of Cohen's method via simple pre- and post-processing. The development in this paper, illustrated with examples, also illuminates other related ...

We propose a straightforward characterization of all quadratic time-frequency representations covariant to an important class of unitary signal transforms (namely, those having two continuous-valued parameters and an underlying group structure). Thanks to a fundamental theorem from the theory of Lie groups, we can describe these representations simply in terms of unitary transformations of the well-known Cohen's and affine classes. This work was supported by the National Science Foundation, grant no. MIP--9457438, the Office of Naval Research, grant no. N00014--95--1--0849, and the Texas Advanced Technology Program, grant no. TX--ATP 003604-- 002. I. Introduction Quadratic time-frequency representations (TFRs) have found wide application in problems requiring time-varying spectral analysis [1, 2]. Since the distribution of signal energy jointly over time and frequency coordinates does not have a unique representation, there exist many different TFRs and many different ways to obtai...

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well-known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presente...

1 Generalizing the concept of time-frequency representations, Cohen has recently proposed a general 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. In this paper, 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. 1 Introduction Time-frequency representations...

Recently, Cohen has proposed a construction for joint distributions of arbitrary physical quantities, in direct generalization of joint time-frequency representations. Actually this method encompasses two approaches, one based on operator correspondences and one based on weighting kernels. The literature has emphasized the kernel method due to its ease of analysis; however, its simplicity comes at a price. In this paper, we use a simple example to demonstrate that the kernel method cannot generate all possible bilinear joint distributions. Our results suggest that the relationship between the operator method and the kernel method merits closer scrutiny. I. Introduction By representing signals in terms of several physical quantities simultaneously, joint distribution functions can reveal signal features that remain hidden from other methods of analysis. Distributions measuring joint timefrequency content, such as the Wigner distribution and the spectrogram from Cohen's class [1, 2] ha...

This letter introduces the spatial ambiguity functions (SAF's) and discusses their applications to direction finding and source separation problems. We emphasize two properties of SAF's that make them an attractive tool for array signal processing.