Previous work has shown that various flavours of Independent Component Analysis, when applied to natural images, all result in broadly similar localised, oriented band-pass feature detectors, which have been likened to wavelets or edge detectors.

We present a theory of linear and bilinear/quadratic timefrequency (TF) representations that satisfy a covariance property with respect to \TF displacement operators." These operators cause TF displacements such as (possibly dispersive) TF shifts, dilations/compressions, etc. Our covariance theory establishes a uni ed framework for important classes of linear TF representations (e.g., short-time Fourier transform and continuous wavelet transform) as well as bilinear TF representations (e.g., Cohen's class and ane class). It yields a theoretical basis for TF analysis and allows the systematic construction of covariant TF representations.

Amongst the myriad of time-frequency distributions, the Wigner distribution stands alone in satisfying many desirable mathematical properties. Attempts to extend definitions of the Wigner distribution to discrete signals have not been completely successful. In this letter, we propose an alternative definition for the Wigner distribution that extends to discrete signals in a straightforward manner. Under this definition, we show that the Wigner distribution does not exist for certain classes of discrete signals. Corresponding Author Jeffrey C. O'Neill Laboratoire de Physique Ecole Normale Sup'erieure de Lyon 46, all'ee d'Italie 69364 Lyon Cedex 07 FRANCE EDICS Number: SPL.SP.2.3 Time-Frequency Signal Analysis y This research has been supported in part by the Office of Naval Research, ONR contract nos. N00014-90-J-1654 and N00014-97-1-0072. I Introduction In signal processing, one is often interested in four different types of signals characterized by being either continuous or d...

We formulate the problem of approximating a signal with a sum of chirped Gaussians, the so-called chirplets, under the framework of maximum likelihood estimation. For a signal model of one chirplet in noise, we formulate the maximum likelihood estimator (MLE) and compute the Cram'er-Rao lower bound. An approximate MLE is developed, based on time-frequency methods, and is applied sequentially to obtain a decomposition of multiple chirplets. The decomposition is refined after each iteration with the expectation-maximization algorithm. A version of the algorithm, which is O(N) for each chirplet of the decomposition, is applied to a data set of whale whistles. I. Introduction Chirplets are a class of signals that consists of Gaussians that are translated in time and frequency, scaled, and chirped. They are defined as s t;!;c;d = s(n; t; !; c; d) = ( p 2d) \Gamma 1 2 exp n \Gamma \Gamma n\Gammat 2d \Delta 2 + j c 2 (n \Gamma t) 2 + j!(n \Gamma t) o : where t, !, and c...

In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the auto-correlation function of the signal, and the auto-correlation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, lag, and doppler. In time-frequency analysis, these functions of one variable are extended to quadratic functions of two variables. In this paper, we investigate a method for creating quartic functions of three of these variables and also a quartic function of all four variables. These quartic functions provide a meaningful representation of the signal that goes beyond the well known quadratic functions. The quartic functions are applied to the design of signal-adaptive kernels for the Cohen class and shown to provide improvements over previous methods. Corresponding Author Jeffrey C. O'Neill Laboratoire de Physique Ecole Normale Sup'erieure 46 All'ee d'Italie 69364 Lyon Cedex 07 FRANCE Tel: (+33) 4 ...

The Cohen class of time-frequency distributions for continuous signals has recently been to extended to discrete signals using both an axiomatic approach and an operator theory approach. In this paper, we investigate the formulation of several classical timefrequency distributions (Wigner, Rihaczek, Margenau-Hill, Page, Levin, Born-Jordan, spectrogram) in the discrete Cohen classes. The main result of this paper concludes that there does not exist a formulation of the Wigner distribution in all of the discrete Cohen classes. 1. INTRODUCTION There are four types of signals often used in signal processing, and to analyze these signals, there are four types of Fourier transforms. In Table 1 we list the four types of signals along with their properties and the appropriate Fourier transform. Since the Fourier transform is linear, the discrete Fourier transforms are samples of the continuous Fourier transform under the appropriate sampling conditions. The Cohen class of time-frequency distr...

Future wireless communications systems are expected to provide ever higher data rates. Still, devices have to be produced at reasonable cost in order to be affordable to customers. The widely known impairments -- "dirt effects" -- in analog RF tend to aggravate as we go for the large transmission bandwidths and high carrier frequencies that usually come with an increased data throughput.

In recent years, there has been an enormous effort put in the definition and analysis of fractional or fractal operators. Fractional calculus is for example a flourishing field of active research. In this paper we restrict ourselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. The book by H.M. Ozaktas, Z. Zalevsky, and M.A. Kutay, The fractional Fourier transform, John Wiley, 2001 gives a state of the art of 2001. Because this field is still in full expansion, we want to summarize in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored aspects.