Chair: Patrick Flandrin, ENS Lyon (FRANCE)
Tzu-Hsien Sang, University of Michigan (USA)
William J. Williams, University of Michigan (USA)
Renyi uncertainty measure has been proposed to be a measurement of complexity of signals. We further suggest that it can be used to evaluate the performance of different time-frequency distributions (TFDs). We provided two schemes of normalization in calculating Renyi uncertainty measure. For the first one, TFDs are normalized by their energy, while for the second one, normalized with their volume. The behavior of Renyi uncertainty measure under several situations is studied. A signal-dependent algorithm is developed to achieve TFDs with a minimal uncertainty measure.
Antonio H. Costa, University of Massachusetts - Dartmouth
G. Faye Boudreaux-Bartels, University of Rhode Island (USA)
In this paper, we formulate and provide design equations for novel time-frequency representations (TFRs) with multiform, tiltable (MT) Chebyshev kernels whose passband support regions are capable of attaining a wide diversity of iso-contour shapes in the ambiguity function (AF) plane, e.g., parallel strips, crosses, tilted and untilted ellipses, diamonds, hyperbolas, rectangles, etc. Simple constraints on the parameters of the new kernels can be used to guarantee many desirable properties of TFRs.
Jeffrey C. O'Neill, University of Michigan (USA)
William J. Williams, University of Michigan (USA)
In time-frequency analysis there are four bilinear domains commonly used: time-frequency, ambiguity, temporal auto-correlation, and spectral auto-correlation. This paper introduces four new domains that are quadralinear in the signal. These four domains are each a function of three of the four variables used in the bilinear domains. Properties of these quantities are developed and an application is given for the design of adaptive time-varying kernels.
Patrick J. Loughlin, University of Pittsburgh
James W. Pitton, AT&T Bell Laboratories
Blake Hannaford, University of Pittsburgh (USA)
We present a general approach to approximating positive time-frequency distributions through nonlinear combinations of spectrograms. Closed-form solutions for the combinations are obtained via optimization of entropy functionals subject to an energy constraint. We apply two such combinations to generating approximate TFDs for whale sounds and speech. Through these applications, it can be seen that these methods give results superior to that achieved with individual spectrograms, and remarkably close to the positive TFDs obtained via computationally-intensive multivariate nonlinear optimization.
Berkant Tacer, University of Pittsburgh (USA)
Patrick J. Loughlin, University of Pittsburgh (USA)
It is generally stated that the conditional mean frequency of a time-frequency distribution (TFD) should equal the instantaneous frequency of the signal. The commonly accepted definition of instantaneous frequency as the derivative of the phase of the analytic signal sometimes leads to curious results. Although it is commonly held that positivity of the TFD and satisfaction of the so-called "instantaneous frequency constraint" are generally incompatible, we show that one can always find a complex signal, the real part of which is the given signal, for which the derivative of the phase is consistent with the marginals and positivity. Furthermore, for the cases considered, the derivative of the phase of this signal, which by design equals the conditional mean frequency of a positive TFD, is a reasonable, readily interpretable choice for instantaneous frequency.
A. Papandreou, University of Rhode Island (USA)
F. Hlawatsch, Technische Universitat Wien (Austria)
G. F. Boudreaux-Bartels, University of Rhode Island (USA)
We propose a framework that unifies and extends the affine, hyperbolic, and power classes of quadratic time-frequency representations (QTFRs). These QTFR classes satisfy the scale covariance property, important in multiresolution analysis, and a generalized time-shift covariance property, important in the analysis of signals propagating through dispersive systems. We provide a general class formulation in terms of 2-D kernels, a generalized signal expansion, a list of desirable QTFR properties with kernel constraints, and a ``central QTFR'' generalizing the Wigner distribution and the Altes-Marinovich Q-distribution. We also propose two generalized time-shift covariant (not, in general, scale covariant) QTFR classes by applying a generalized warping to Cohen's class and to the affine class.
Richard G. Baraniuk, Rice University (USA)
Recently, Cohen has proposed a method for constructing joint distributions of arbitrary physical quantities, in direct generalization of joint time-frequency representations. In this paper, we investigate the covariance properties of this procedure and caution that in its present form it cannot generate all possible distributions. Using group theory, we extend Cohen's construction to a more general form that can be customized to satisfy specific marginal and covariance requirements.
Franz Hlawatsch, Technische Universitat Wien (AUSTRIA)
Helmut Bolcskei, Technische Universitat Wien (AUSTRIA)
Important classes of quadratic time-frequency representations (QTFRs), such as Cohen's class and the affine, hyperbolic, and power classes, are special cases within a general theory based on the concept of time-frequency displacement operators (DOs). The present paper considers the important separable case where a DO can be decomposed into two ``partial DOs'' (PDOs). We define marginal properties associated to the PDOs and derive necessary and sufficient constraints on the QTFR kernels. We also show that, in the case of ``conjugate'' PDOs, our theory coincides with the characteristic function method of Cohen and Baraniuk. In this case, QTFR classes that are both displacement-covariant and satisfy marginal properties can be constructed in a systematic way.
M.S. Richman, Cornell University
T.W. Parks, Cornell University
R.G. Shenoy, Schlumberger-Doll Research (USA)
A discrete-time, discrete-frequency Wigner distribution is derived using a group-theoretic approach. It is based upon a study of the Heisenberg group generated by the integers mod $N$, which represents the group of discrete-time and discrete-frequency shifts. The resulting Wigner distribution satisfies several desired properties. An example demonstrates that it is a full-band time-frequency representation, and, as such, does not require special sampling techniques to suppress aliasing. It also exhibits some interesting and unexpected interference properties. The new distribution is compared with other discrete-time, discrete-frequency Wigner distributions proposed in the literature.
Douglas L. Jones, University of Illinois (USA)
Akbar M. Sayeed, University of Illinois (USA)
Time-frequency based methods, particularly quadratic (Cohen's-class) representations, are often considered for detection in applications ranging from sonar to machine monitoring. We propose a method of obtaining near-optimal quadratic detectors directly from training data using Fisher's optimal linear discriminant to design a quadratic detector. This detector is optimal in terms of Fisher's scatter criterion as applied to the quadratic outer product of the data vector, and in early simulations appears to closely approximate the true optimal quadratic detector. By relating this quadratic detector to an equivalent operation on the Wigner distribution of a signal, we derive near-optimal time-frequency detectors. A simple example demonstrates the excellent performance of the method.