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PaleyWiener theorems and uncertainty principles for the windowned linear canonical transform
 Mathematical Methods in the Applied Sciences
"... In a recent paper the authors have introduced the windowed linear canonical transform and shown its good properties together with some applications such as Poisson summation formulas, sampling interpolation and series expansion. In this paper we prove the PaleyWiener theorems and the uncertainty pr ..."
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In a recent paper the authors have introduced the windowed linear canonical transform and shown its good properties together with some applications such as Poisson summation formulas, sampling interpolation and series expansion. In this paper we prove the PaleyWiener theorems and the uncertainty principles for the (inverse) windowed linear canonical transform. They are new in literature and has some consequences that are now under investigation. Copyright c
Fractional Fourier transform as a signal processing tool: An overview of recent developments
, 2011
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Anechoic Blind Source Separation Using Wigner Marginals
"... Blind source separation problems emerge in many applications, where signals can be modeled as superpositions of multiple sources. Many popular applications of blind source separation are based on linear instantaneous mixture models. If specific invariance properties are known about the sources, for ..."
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Blind source separation problems emerge in many applications, where signals can be modeled as superpositions of multiple sources. Many popular applications of blind source separation are based on linear instantaneous mixture models. If specific invariance properties are known about the sources, for example, translation or rotation invariance, the simple linear model can be extended by inclusion of the corresponding transformations. When the sources are invariant against translations (spatial displacements or time shifts) the resulting model is called an anechoic mixing model. We present a new algorithmic framework for the solution of anechoic problems in arbitrary dimensions. This framework is derived from stochastic timefrequency analysis in general, and the marginal properties of the WignerVille spectrum in particular. The method reduces the general anechoic problem to a set of anechoic problems with nonnegativity constraints and a phase retrieval problem. The first type of subproblem can be solved by existing algorithms, for example by an appropriate modification of nonnegative matrix factorization (NMF). The second subproblem is solved by established phase retrieval methods. We discuss and compare implementations of this new algorithmic framework for several example problems with synthetic and realworld data, including music streams, natural 2D images, human motion trajectories and twodimensional shapes.
A nonparametric approach
, 2006
"... Decompounding random sums: A nonparametric approach by ..."
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AN ALGORITHM FOR FRESNEL DIFFRACTION COMPUTING BASED ON FRACTIONAL FOURIER TRANSFORM
"... Abstract: The fractional Fourier transform (FrFT) is used for the solution of the diffraction integral in optics. A scanning approach is proposed for finding the optimal FrFT order. In this way, the process of diffraction computing is speeded up. The basic algorithm and the intermediate results at e ..."
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Abstract: The fractional Fourier transform (FrFT) is used for the solution of the diffraction integral in optics. A scanning approach is proposed for finding the optimal FrFT order. In this way, the process of diffraction computing is speeded up. The basic algorithm and the intermediate results at each stage are demonstrated. Key words: Fresnel diffraction, fractional Fouriertransform ACM Classification Keywords: G.1.2 Fast Fourier transforms (FFT)
Explicit Hermitetype eigenvectors of the discrete Fourier transform
, 2015
"... The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis s ..."
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The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a solution to these problems. First, we construct an explicit basis of (nonorthogonal) eigenvectors of the discrete Fourier transform, thus extending the results of [9]. Applying the GramSchmidt orthogonalization procedure we obtain an orthogonal eigenbasis of the discrete Fourier transform. We prove that the first eight eigenvectors converge to the corresponding Hermite functions, and we conjecture that this convergence result remains true for all eigenvectors.
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"... Analysis and separation of timefrequency components in signals with chaotic behavior ..."
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Analysis and separation of timefrequency components in signals with chaotic behavior