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Virtues and Vices of Quartic TimeFrequency Distributions
 in IEEE Trans. on Signal Processing
, 2000
"... We present results concerning three different types of quartic (fourth order) timefrequency 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. S ..."
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We present results concerning three different types of quartic (fourth order) timefrequency 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, timefrequency distributions, and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirprate. I. Introduction T HE notion of a timefrequency 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...
Sparse Representations with Chirplets via Maximum Likelihood Estimation
"... We formulate the problem of approximating a signal with a sum of chirped Gaussians, the socalled 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'erRao lower bound. ..."
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We formulate the problem of approximating a signal with a sum of chirped Gaussians, the socalled 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'erRao lower bound. An approximate MLE is developed, based on timefrequency methods, and is applied sequentially to obtain a decomposition of multiple chirplets. The decomposition is refined after each iteration with the expectationmaximization 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...
CramérRao Lower Bounds for Atomic Decomposition
 In Proc. of the IEEE Int. Conf. on Acoust., Speech, and Signal Processing
, 1999
"... 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 CramerRao lower bounds for estimating the seven chirp parameters, and the results of a simulation showing that our suboptimal, b ..."
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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 CramerRao lower bounds for estimating the seven chirp parameters, and the results of a simulation showing that our suboptimal, but computationally tractable, estimators perform well in comparison to the bound at low signaltonoise ratios. We also show that methods based on signal dictionaries will require much higher computations to perform well in low signaltonoise 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...
Bayesian Estimation of Chirplet . . .
"... 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 me ..."
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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 CramerRao bounds. The obtained results have been excellent.
Approximation of Real, BandLimited Signals
"... In this paper we present algorithms for approximating real bandlimited 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 positivevalued maxima and negativevalued minima of a signed a ..."
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In this paper we present algorithms for approximating real bandlimited 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 positivevalued maxima and negativevalued 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 multidimensional searches to obtain suitable choices of chirp parameters. 1
CHIRPLET APPROXIMATION OF BANDLIMITED, REAL SIGNALS MADE EASY ∗
"... Abstract. In this paper we present algorithms for approximating real bandlimited 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 positiv ..."
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Abstract. In this paper we present algorithms for approximating real bandlimited 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 positivevalued maxima and negativevalued 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. bandlimited signals, chirplet decomposition, Paley–Wiener class
Chirplet Decomposition
"... Many signals that exist in nature are inherently nonstationary or transient like Bat and Whale echoes, evokedpotentials, RADAR clutters. There exist many methods to analyze such signals including wavelettransforms, shorttime Fourier transform, adaptive timefrequency distributions, Matching Purs ..."
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Many signals that exist in nature are inherently nonstationary or transient like Bat and Whale echoes, evokedpotentials, RADAR clutters. There exist many methods to analyze such signals including wavelettransforms, shorttime Fourier transform, adaptive timefrequency distributions, Matching Pursuits