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...

We present results concerning three different types of quartic (fourth order) time-frequency distributions. First, we present new results on the recently introduced local ambiguity function, and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. Second, we introduce the class of quartic, shiftcovariant, time-frequency distributions, and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirp-rate. I. Introduction T HE notion of a time-frequency distribution (TFD) [1], [2], [3] is inherently a concept that is not well defined [4]. A frequency is something that is measured over a period of time (e.g. how many times does the heart beat in a minute), and we would like to specify this frequency description at an instant of time (e.g. how fast is the heart beating right now). Nevertheless, TFD's have proven to be useful in many applications [5]. TFD's have been defin...

In a previous paper [1] we presented a method for atomic decomposition with chirped, Gabor functions based on maximum likelihood estimation. In this paper we present the Cramer-Rao lower bounds for estimating the seven chirp parameters, and the results of a simulation showing that our sub-optimal, but computationally tractable, estimators perform well in comparison to the bound at low signal-to-noise ratios. We also show that methods based on signal dictionaries will require much higher computations to perform well in low signal-to-noise ratios. 1. INTRODUCTION Given a signal, x(n), our goal is to find a sparse decomposition of the signal as a weighted sum of chirped, Gabor functions x = x(n) = M X i=1 A i e jOE i s(n; t i ; ! i ; c i ; d i ); where s t;!;c;d = s(n; t; !; c; d) = i p 2ßd j \Gamma 1 2 exp n \Gamma \Gamma n\Gammat 2d \Delta 2 + j c 2 (n \Gamma t) 2 + j!(n \Gamma t) o : The parameters t, !, c, and d represent, respectively, the location in time, t...

We address the problem of parameter estimation of chirplets, which are chirp signals with Gaussian shaped envelopes. The procedure we propose is an extension of our previous work on estimation of chirp signals [5], and it is based on MCMC sampling. For fast convergence of the MCMC sampling based method, a critical step is the initialization of the method. Since the chirplets have nite durations and may or may not overlap in time, we propose initialization procedures for each of these cases. Wehave tested the method by extensivesimulations and compared it with Cramer-Rao bounds. The obtained results have been excellent.

In this paper we present algorithms for approximating real band-limited signals by multiple Gaussian Chirps. These algorithms are hierarchial and, at each stage, the number of terms in a given approximation depends only on the number of positive-valued maxima and negative-valued minima of a signed amplitude function characterizing part of the signal. Like the algorithms used in [1] and unlike previous methods, our chirp approximations require neither a complete dictionary of chirps nor complicated multi-dimensional searches to obtain suitable choices of chirp parameters. 1

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Abstract. In this paper we present algorithms for approximating real band-limited signals by multiple Gaussian chirps. These algorithms do not rely on matching pursuit ideas. They are hierarchical, and, at each stage, the number of terms in a given approximation depends only on the number of positive-valued maxima and negative-valued minima of a signed amplitude function characterizing part of the signal. Like the algorithms used in [J. M. Greenberg, Z. Wang, and J. Li, IEEE Trans. Signal Process., 55 (2007), pp. 734–741] and unlike previous methods, our chirplet approximations require neither a complete dictionary of chirps nor complicated multidimensional searches to obtain suitable choices of chirp parameters. Key words. band-limited signals, chirplet decomposition, Paley–Wiener class

Many signals that exist in nature are inherently non-stationary or transient like Bat and Whale echoes, evoked-potentials, RADAR clutters. There exist many methods to analyze such signals including wavelet-transforms, short-time Fourier transform, adaptive time-frequency distributions, Matching Pursuits