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Mj Martin Bastiaans - One of the best experts on this subject based on the ideXlab platform.
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Keynote 2: Time-Frequency Signal Representations by Martin J. Bastiaans
2014 Sixth International Conference on Computational Intelligence Communication Systems and Networks, 2014Co-Authors: Mj Martin BastiaansAbstract:This keynote paper presents an overview of two classes of time-frequency Signal representations. The first class, in which the Signal arises linearly, deals with the windowed Fourier transform and its sampled version (also known as the Gabor transform) and the inverse of the latter: Gabor's Signal Expansion. We will show how Gabor's Signal Expansion and the windowed Fourier transform are related and how they can benefit from each other. The second class, in which the Signal arises quadratically (or bilinearly, as it is often called), is based on the Wigner distribution. We will show some examples of the Wigner distribution and discuss some of its important properties. Being a bilinear Signal representation, the Wigner distribution shows artifacts in the case of multi-component Signals. To reduce these artifacts, a large class of bilinear Signal representations has been constructed, known as the shift-covariant Cohen class. We will consider this class and we will see how all its members can be considered as properly averaged versions of the Wigner distribution.
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Time frequency Signal representations
2009Co-Authors: Mj Martin Bastiaans, B. Zajc, A. TrostAbstract:This invited paper - of a tutorial and review character - presents an overview of two classes of time-frequency Signal representations. The first class, in which the Signal arises linearly, deals with the windowed Fourier transform and its sampled version (also known as the Gabor transform) and the inverse of the latter: Gabor's Signal Expansion. We will show how Gabor's Signal Expansion and the windowed Fourier transform are related and how they can benefit from each other. The second class, in which the Signal arises quadratically (or bilinearly, as it is often called), is based on the Wigner distribution. We will show some examples of the Wigner distribution and discuss some of its important properties. Being a bilinear Signal representation, the Wigner distribution shows artifacts in the case of multi-component Signals. To reduce these artifacts, a large class of bilinear Signal representations has been constructed, known as the shift-covariant Cohen class. We will consider this class and we will see how all its members can be considered as properly averaged versions of the Wigner distribution
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Gabor's Signal Expansion for a non-orthogonal sampling geometry
2003Co-Authors: Mj Martin Bastiaans, Van Aj Arno LeestAbstract:Gabor's Signal Expansion and the Gabor transform on a rectangular lattice have been introduced, along with the Fourier transform of the array of Expansion coefficients and the Zak transforms of the Signal and the window functions. Based on these Fourier and Zak transforms, the sum-of-products forms for the Gabor Expansion and the Gabor transform, which hold in the rationally oversampled case, have been derived. We have then studied Gabor's Signal Expansion and the Gabor transform based on a non-orthogonal sampling geometry. We have done this by considering the non-orthogonal lattice as a sub-lattice of an orthogonal lattice. This procedure allows us to use all the formulas that hold for the orthogonal sampling geometry. In particular we can use the sum-of-products forms that hold in the case of a rationally oversampled rectangular lattice.
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Gabor's Signal Expansion based on a non-orthogonal sampling geometry
Optical Processing and Computing: A Tribute to Adolf Lohmann, 2001Co-Authors: Mj Martin BastiaansAbstract:Gabor's Signal Expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice. The reason for doing so is that a non-orthogonal sampling geometry might be better adapted to the form of the window functions (in the time-frequency domain) than an orthogonal one: the set of shifted and modulated versions of the usual Gaussian synthesis window, for instance, corresponding to circular contour lines in the time-frequency domain, can be arranged more tightly in a hexagonal geometry than in a rectangular one. Oversampling in the Gabor scheme, which is required to have mathematically more attractive properties for the analysis window, then leads to better results in combination with less oversampling. The procedure presented in this paper is based on considering the non-orthogonal lattice as a sub-lattice of a denser orthogonal lattice that is oversampled by a rational factor. In doing so, Gabor's Signal Expansion on a non-orthogonal lattice can be related to the Expansion on an orthogonal lattice (restricting ourselves, of course, to only those sampling points that are part of the non-orthogonal sub-lattice), and all the techniques that have been derived for rectangular sampling - including an optical means of generating Gabor's Expansion coefficients via the Zak transform in the case of integer oversampling - can be used, albeit in a slightly modified form.
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Gabor's Signal Expansion and the Gabor transform on a non-separable time-frequency lattice
Journal of The Franklin Institute-engineering and Applied Mathematics, 2000Co-Authors: Arno J. Van Leest, Mj Martin BastiaansAbstract:Abstract Gabor's Signal Expansion and the Gabor transform are formulated on a general, non-separable time–frequency lattice instead of on the traditional rectangular lattice. The representation of the general lattice is based on the rectangular lattice via a shear operation, which corresponds to a description of the general lattice by means of a lattice generator matrix that has the Hermite normal form. The shear operation on the lattice is associated with simple operations on the Signal, on the synthesis and the analysis window, and on Gabor's Expansion coefficients; these operations consist of multiplications by quadratic phase terms. Following this procedure, the well-known bi-orthogonality condition for the window functions in the rectangular sampling geometry, can be directly translated to the general case. In the same way, a modified Zak transform can be defined for the non-separable case, with the help of which Gabor's Signal Expansion and the Gabor transform can be brought into product forms that are identical to the ones that are well known for the rectangular sampling geometry.
F. Hlawatsch - One of the best experts on this subject based on the ideXlab platform.
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Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes☆☆☆
Digital Signal Processing, 1998Co-Authors: Antonia Papandreou-suppappola, F. Hlawatsch, G. Faye Boudreaux-bartelsAbstract:Abstract We propose the generalized class of quadratic time-frequency representations (QTFRs) that satisfy the scale covariance property, which is important in multiresolution analysis, and the generalized time-shift covariance property, which is important in the analysis of Signals propagating through systems with specific dispersive characteristics. We discuss a formulation of the generalized class QTFRs in terms of two-dimensional kernel functions, a generalized Signal Expansion related to the generalized class time-frequency geometry, an important member of the generalized class, a set of desirable QTFR properties and their corresponding kernel constraints, and a “localized-kernel” generalized subclass that is characterized by one-dimensional kernels. Special cases of the generalized QTFR class include the affine class and the new hyperbolic class and power classes. All these QTFR classes satisfy the scale covariance property. In addition, the affine QTFRs are covariant to constant time shifts, the hyperbolic QTFRs are covariant to hyperbolic time shifts, and the power QTFRs are covariant to power time shifts. We present a detailed study of these classes that includes their definition and formulation, an associated generalized Signal Expansion, important class members, desirable QTFR properties and corresponding kernel constraints, and localized-kernel subclasses. Also, we investigate the subclasses formed by the intersection between the affine and hyperbolic classes, the affine and power classes, and the hyperbolic and power classes. These subclasses are important since their members satisfy additional desirable properties. We show that the hyperbolic class is obtained from Cohen's QTFR class using a “hyperbolic time-frequency warping” and that the power classes are obtained similarly by applying a “power time-frequency warping” to the affine class. The affine class is a special case of the power classes. Furthermore, we generalize the time-frequency warping so that when applied either to Cohen's class or to the affine class, it yields QTFRs that are always generalized time-shift covariant but not necessarily scale covariant.
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A UNIFIED FRAMEWORK FOR THE SCALE COVARIANT AFFINE, HYPERBOLIC, AND POWER CLASS USING GENERALIZED TIME SHIFTS* QUADRATIC TIME-FREQUENCY REPRESENTATIONS
1995Co-Authors: A. Papandreoul, F. Hlawatsch, G. F. Boudreaux-bartelslAbstract:We propose a framework that unifies and extends the affine, hyperbolic, a,nd power classes of quadratic time-frequency representa.tions (QTFRs). These QTFR classes satisfy the scale covariance property, important in multiresolution analysis, and a generalized time-shift covariance property, important in the analysis of Signals propagating through dispersive systems. We provide a general class formulation in terms of 2-D kernels, a generalized Signal Expansion, a list of desirable QTFR properties with kernel constraints, and a “central QTFR generalizing the Wigner distribution and the Altes-Marinovich Q-distribution. We also propose two generalizled time-shift covariant (not, in general, scale covariant) QTFR classes by applying a generalized warping to Cohen’s class and to the affine class.
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Time-frequency projection filters and time-frequency Signal Expansions
IEEE Transactions on Signal Processing, 1994Co-Authors: F. Hlawatsch, Werner KozekAbstract:We consider the problems of designing a linear, time-varying filter with a specified "time-frequency (TF) pass region" and of constructing an orthonormal basis for the parsimonious Expansion of Signals located in a given TF support region. These problems of TF filtering and TF Signal Expansion are reduced to the problem of designing a "TF subspace", i.e., a linear Signal space comprising all Signals located in a given TF legion. Specifically, the TF filter is taken to be the orthogonal projection operator on the TF subspace. We present an optimum design of TF subspaces that is based on the Wigner distribution of a linear Signal space and is an extension of the well-known Signal synthesis problem. The optimum TF subspace is shown to be an "eigenspace" of the TF region, and some properties of eigenspaces are discussed. The performance of TF projection filters and TF Signal Expansions is studied both analytically and via computer simulation. >
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ICASSP - A unified framework for the scale covariant affine, hyperbolic, and power class quadratic time-frequency representations using generalized time shifts
1995 International Conference on Acoustics Speech and Signal Processing, 1Co-Authors: A. Papandreou, F. Hlawatsch, G.f. Boudreauz-bartelsAbstract:We propose a framework that unifies and extends the affine, hyperbolic, and power classes of quadratic time-frequency representations (QTFRs). These QTFR classes satisfy the scale covariance property, important in multiresolution analysis, and a generalized time-shift covariance property, important in the analysis of Signals propagating through dispersive systems. We provide a general class formulation in terms of 2-D kernels, a generalized Signal Expansion, a list of desirable QTFR properties with kernel constraints, and a "central QTFR" generalizing the Wigner distribution and the Altes-Marinovich Q-distribution. We also propose two generalized time-shift covariant (not, in general, scale covariant) QTFR classes by applying a generalized warping to Cohen's (1966) class and to the affine class.
G.f. Boudreauz-bartels - One of the best experts on this subject based on the ideXlab platform.
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ICASSP - A unified framework for the scale covariant affine, hyperbolic, and power class quadratic time-frequency representations using generalized time shifts
1995 International Conference on Acoustics Speech and Signal Processing, 1Co-Authors: A. Papandreou, F. Hlawatsch, G.f. Boudreauz-bartelsAbstract:We propose a framework that unifies and extends the affine, hyperbolic, and power classes of quadratic time-frequency representations (QTFRs). These QTFR classes satisfy the scale covariance property, important in multiresolution analysis, and a generalized time-shift covariance property, important in the analysis of Signals propagating through dispersive systems. We provide a general class formulation in terms of 2-D kernels, a generalized Signal Expansion, a list of desirable QTFR properties with kernel constraints, and a "central QTFR" generalizing the Wigner distribution and the Altes-Marinovich Q-distribution. We also propose two generalized time-shift covariant (not, in general, scale covariant) QTFR classes by applying a generalized warping to Cohen's (1966) class and to the affine class.
Bruno Torrésani - One of the best experts on this subject based on the ideXlab platform.
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Random models for sparse Signals Expansion on unions of bases with application to audio Signals
IEEE Transactions on Signal Processing, 2008Co-Authors: Matthieu Kowalski, Bruno TorrésaniAbstract:A new approach for Signal Expansion with respect to hybrid dictionaries, based upon probabilistic modeling is proposed and studied, with emphasis on audio Signal processing applications. The Signal is modeled as a sparse linear combination of waveforms, taken from the union of two orthonormal bases, with random coefficients. The behavior of the analysis coefficients, namely inner products of the Signal with all basis functions, is studied in details, which shows that these coefficients may generally be classified in two categories: significant coefficients versus unsignificant coefficients. Conditions ensuring the feasibility of such a classification are given. When the classification is possible, it leads to efficient estimation algorithms, that may in turn be used for de-noising or coding purpose. The proposed approach is illustrated by numerical experiments on audio Signals, using MDCT bases.
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Random Models for Sparse Signals Expansion on Unions of Bases With Application to Audio Signals
IEEE Transactions on Signal Processing, 2008Co-Authors: Matthieu Kowalski, Bruno TorrésaniAbstract:A new approach for Signal Expansion with respect to hybrid dictionaries, based upon probabilistic modeling is proposed and studied. The Signal is modeled as a sparse linear combination of waveforms, taken from the union of two orthonormal bases, with random coefficients. The behavior of the analysis coefficients, namely inner products of the Signal with all basis functions, is studied in details, which shows that these coefficients may generally be classified in two categories: significant coefficients versus insignificant coefficients. Conditions ensuring the feasibility of such a classification are given. When the classification is possible, it leads to efficient estimation algorithms, that may in turn be used for denoising or coding purposes. The proposed approach is illustrated by numerical experiments on audio Signals, using MDCT bases. However, it is general enough to be applied without much modifications in different contexts, for example in image processing.
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Random models for audio Signals Expansion on hybrid MDCT dictionaries
2007Co-Authors: Matthieu Kowalski, Bruno TorrésaniAbstract:A new approach for Signal Expansion with respect to hybrid dictionaries, based upon probabilistic modeling is proposed and studied, with emphasis on audio Signal processing applications. The Signal is modeled as a sparse linear combination of waveforms, taken from the union of two orthonormal bases, with random coefficients. The behavior of the analysis coefficients, namely inner products of the Signal with all basis functions, is studied in details, which shows that these coefficients may generally be classified in two categories: significant coefficients versus unsignificant coefficients. Conditions ensuring the feasibility of such a classification are given. When the classification is possible, it leads to efficient estimation algorithms, that may in turn be used for de-noising or coding purpose. The proposed approach is illustrated by numerical experiments on audio Signals, using MDCT bases.
A.j. Van Leest - One of the best experts on this subject based on the ideXlab platform.
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ICASSP - Gabor's Signal Expansion on a quincunx lattice and the modified Zak transform
1999 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999Co-Authors: A.j. Van Leest, Mj Martin BastiaansAbstract:Gabor's Expansion of a Signal on a quincunx lattice with oversampling by a rational factor is presented for continuous-time Signals. It is shown how a modified Zak transform instead of the ordinary Zak transform can be helpful in determining Gabor's Signal Expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.
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ISSPA - On the non-separable discrete Gabor Signal Expansion and the Zak transform
ISSPA '99. Proceedings of the Fifth International Symposium on Signal Processing and its Applications (IEEE Cat. No.99EX359), 1Co-Authors: A.j. Van Leest, Mj Martin BastiaansAbstract:The discrete Gabor Signal Expansion on a lattice that is obtained by linear combinations of two independent vectors, and its relation with the discrete Zak transform are presented. It is shown how the Zak transform can be helpful in determining Gabor's Signal Expansion coefficients and how it can be used in finding the dual window that corresponds to a given window for this (generally non-separable) lattice.
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ISSPA - On the non-separable Gabor Signal Expansion and filter banks
ISSPA '99. Proceedings of the Fifth International Symposium on Signal Processing and its Applications (IEEE Cat. No.99EX359), 1Co-Authors: A.j. Van LeestAbstract:Gabor's discrete Signal Expansion and the discrete Gabor transform are formulated on a lattice that can be obtained by linear combinations of two independent vectors, and are implemented by using modulated filter banks. As an example, the filter bank in the quincunx case is given.
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ISSPA - Gabor's Signal Expansion and the Gabor transform based on a non-orthogonal sampling geometry
Proceedings of the Sixth International Symposium on Signal Processing and its Applications (Cat.No.01EX467), 1Co-Authors: Mj Martin Bastiaans, A.j. Van LeestAbstract:Gabor's (1946) Signal Expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice. The reason for doing so is that a non-orthogonal sampling geometry might be better adapted to the form of the window functions (in the time-frequency domain) than an orthogonal one. Oversampling in the Gabor scheme, which is required to have mathematically more attractive properties for the analysis window, then leads to better results in combination with less oversampling. The new procedure presented in this paper is based on considering the non-orthogonal lattice as a sub-lattice of a denser orthogonal lattice that is oversampled by a rational factor. In doing so, Gabor's Signal Expansion on a non-orthogonal lattice can be related to the Expansion on an orthogonal lattice (restricting ourselves, of course, to only those sampling points that are part of the non-orthogonal sub-lattice), and all the techniques that have been derived for rectangular sampling can be used, albeit in a slightly modified form.
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ICASSP - Gabor's discrete Signal Expansion and the discrete Gabor transform on a non-separable lattice
2000 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings (Cat. No.00CH37100), 1Co-Authors: A.j. Van Leest, Mj Martin BastiaansAbstract:Gabor's (1946) discrete Signal Expansion and the discrete Gabor transform are formulated on a general, non-separable time-frequency lattice instead of on the traditional rectangular lattice. The representation of the general lattice is based on the rectangular lattice via a shear operation, which corresponds to a description of the general lattice by means of a lattice generator matrix that has the Hermite normal form. The shear operation on the lattice is associated with simple operations on the Signal, on the synthesis and the analysis window, and on Gabor's Expansion coefficients; these operations consist of multiplications by quadratic phase terms. This procedure makes it possible to reuse algorithms, which are designed for a rectangular lattice only, to calculate the analysis window, Gabor's Expansion coefficients and Gabor's Expansion on a general non-separable lattice.
