Chair: Steve Elgar, Washington State University (USA)
Naoki Saito, Schlumberger-Doll Research
Ronald R. Coifman, Yale University (USA)
We describe extensions to the ``best-basis'' method which select orthonormal bases suitable for signal classification and regression problems from a large collection of orthonormal bases. For classification problems, we select the basis which maximizes relative entropy of time-frequency energy distributions among classes. For regression problems, we select the basis which tries to minimize the regression error. Once these bases are selected, a small number of most significant coordinates are fed into a traditional classifier or regression method such as Linear Discriminant Analysis (LDA) or Classification and Regression Tree (CART). The performance of these statistical methods is enhanced since the proposed methods reduce the dimensionality of the problems without losing important information for the problem at hand. Here, the basis functions which are well-localized in the time-frequency plane are used as feature extractors. We also compare their performance with the traditional methods using a synthetic example.
Akbar M. Sayeed, University of Illinois (USA)
Douglas L. Jones, University of Illinois (USA)
Joint time-frequency representations have proven very useful for analyzing signals in terms of time-frequency content. Recently, in an attempt to tailor joint representations to a richer class of signals, two approaches (Cohen's and Baraniuk's) to obtaining joint representations of arbitrary variables have been proposed. Baraniuk's generalization appears broader than Cohen's, since the latter can be recovered from the former as a special case. One of the main results of this paper is that, despite being apparently quite different, the two approaches to generalized joint representations are exactly equivalent. We explicitly characterize the mapping which relates the representations of the two methods, and also determine the corresponding relationship between the operators of the two methods. A practical implication of the results is that one can avoid the group transforms in Baraniuk's approach, which may not be computationally efficient, by replacing them with Fourier transforms in Cohen's method.
Aydin Akan, University of Pittsburgh (USA)
L. F. Chaparro, University of Pittsburgh (USA)
In this paper, we present a connection between the discrete Gabor expansion and the evolutionary spectral theory. Including a scale parameter in the Gabor expansion we obtain a new representation for deterministic signals that is analogous to the Wold-Cramer decomposition for non-stationary processes. The energy distribution resulting from the expansion is easily calculated from the Gabor coefficients. By choosing gaussian windows and appropriate scales, the expansion can represent narrow-band and wide-band signals, as well as their combination. As an application, we consider the masking of signals in the time-frequency space and provide an approximate implementation using the new Gabor expansion. Examples illustrating the time-frequency analysis and the masking are given.
Sokbom Han, Arizona State University (USA)
Douglas Cochran, Arizona State University (USA)
Motivated by their utility in coding of digital communication signals, numerous examples of orthonormal wavelet sets that are generated from a bandlimited mother wavelet have been constructed. In most of these examples, orthogonality of the replicates at different scales is achieved by ensuring that the frequency domain supports at every scale are disjoint. Given the mother wavelet for such a set, this paper describes how it can be extended to yield a new bandlimited mother wavelet which generates an orthonormal set and whose dilated replicates overlap in the frequency domain. The utility of these wavelets to improve robustness of wavelet-coded communication signals with respect to demodulation by an unintended receiver is also discussed. The possibility of developing a new class of symbols by unitary warping of the frequency axis is also considered, but is shown not to be viable.
J.-C. Pesquet, CNRS/UPS (FRANCE)
In this paper we investigate a M-band wavelet decomposition of second order random processes. In particular, we propose an extension of results which are known for the dyadic wavelet transform. The statistical properties of the M-band wavelet coefficients are listed and recursive relations are derived and used to compute their multiscale characteristics. Special attention is also paid to the multiscale analysis of linear parametric models.
Heinrich Kirchauer, Vienna University of Technology
Franz Hlawatsch, Vienna University of Technology
Werner Kozek, University of Vienna (AUSTRIA)
The nonstationary Wiener filter (WF) is the optimum linear system for estimating a nonstationary signal contaminated by nonstationary noise. We propose a time-frequency (TF) formulation of nonstationary WFs for the important case of underspread, nonstationary processes. This TF formulation extends the spectral representation of stationary WFs to the nonstationary case, and it allows an approximate TF design of nonstationary WFs. For underspread processes, the performance obtained with the approximate TF design is shown to be close to that of the exact WF.
C. S. Detka, University of Pittsburgh (USA)
A. El-Jaroudi, University of Pittsburgh (USA)
Evolutionary spectral theory provides a means for defining a decomposition of signal energy jointly over time and frequency for processes which exhibit an oscillatory behavior. The oscillatory model of behavior effectively assumes that a process is composed of sinusoidal components with slowly-varying time-dependent amplitudes. In this paper we expand evolutionary spectral theory by allowing the frequency of the sinusoidal components to vary with time. An estimator of this generalized spectrum is described and examples are presented that illustrate the relative merits of this new approach.
Eugene J. Zalubas, University of Michigan (USA)
William J. Williams, University of Michigan (USA)
The scale transform introduced by Cohen is a special case of the Mellin transform. The scale transform has mathematical properties desirable for comparison of signals for which scale variation occurs. In addition to the scale invariance property of the Mellin transform many properties specific to the scale transform have been presented. A procedure is presented in this paper for complete implementation of the scale transformation for discrete signals. This complements discrete Mellin transforms and delineates steps whose implementation are specific to the scale transform.
Hamid Krim, Massachusetts Institute of Technology
S. Mallat, Courant Institute
D. Donoho, Stanford University
A.S. Willsky, Massachusetts Institute of Technology (USA)
We propose a Best Basis Algorithm for Signal Enhancement (BBASE) in white gaussian noise. We base our search of best basis on a criterion of minimal reconstruction error of the underlying signal. We subsequently compare our simple error criterion to the Stein unbiased risk estimator, and provide a subtantiating example to demonstrate its performance.
A.A. (Louis) Beex, Virginia Tech (USA)
Min Xie, Virginia Tech (USA)
We aim to recover a multi-frequency component nonstationary signal from its broadband noise-corrupted measurements using a time-varying optimal Wiener filter. A new method for realizing the Wiener filter is proposed, based on our multiresolution parametric spectral estimator (MPSE). Conventional estimators for contaminated AR processes are all fixed resolution based methods, which are mostly suitable for stationary situations. In nonstationary applications, the estimator must not only locate the signal components in frequency but also in time. MPSE offers better time resolution than conventional fixed resolution parametric estimators. The MPSE frequency band splitting reduces necessary model orders and improves SNR. The Wiener filter is given in terms of the MPSE parameters. Experiments show that the performance of the MPSE Wiener filter lies much closer to the ideally possible performance than for a Wiener filter based on fixed resolution AR modeling.