Gabor expansion on orthogonal bases

Einziger, P.D.; Raz, S.; Farkash, S.

doi:10.1049/el:19890058

Cited by 9 publications

(3 citation statements)

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“…We therefore present and analyze a scheme with kernels that are not necessarily exponential. The case of a continuous-time single-window scheme with critical sampling was introduced in [23].…”

Section: F Representation Functions With Nonexponential Kemelsmentioning

confidence: 99%

“…To show that equality is obtained in (59), we find a function f ( i ) for which an equality is satisfied. Let i , , , be such that in (23), A = X3(imtn) for some j , i.e., A3(imin) is the minimum eigenvalue of S(imln). Choose f ( i ) such that f(imln) is the corresponding normalized eigenvector of the minimum eigenvalue and zero for other values of i.…”

Section: Appendix B Proof Of (19) and (20)mentioning

confidence: 99%

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Discrete multiwindow Gabor-type transforms

Zibulski

Zeevi

1997

IEEE Trans. Signal Process.

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The discrete (finite) Gabor scheme is generalized by incorporating multiwindows. Two approaches are presented for the analysis of the multiwindow scheme: the signal domain approach and the Zak transform domain approach. Issues related to undersampling, critical sampling, and oversampling are considered. The analysis is based on the concept of frames and on generalized (Moore-Penrose) inverses. The approach based on representing the frame operator as a matrix-valued function is far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame. DFT-based algorithms, including complexity analysis, are presented for the calculation of the expansion coefficients and for the reconstruction of the signal in both signal and transform domains. The scheme is further generalized and incorporates kernels other than the complex exponential. Representations other than those based on the dual frame and nonrectangular sampling of the combined space are considered as well. An example that illustrates the advantages of the multiwindow scheme over the single-window scheme is presented.

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Section: F Representation Functions With Nonexponential Kemelsmentioning

confidence: 99%

Section: Appendix B Proof Of (19) and (20)mentioning

confidence: 99%

Discrete multiwindow Gabor-type transforms

Zibulski

Zeevi

1997

IEEE Trans. Signal Process.

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“…Clearly, the sequence { be j2pmbx } is a particular example of such a sequence, since it constitutes an orthonormal basis. For a single window and a critical sampling case, a scheme involving orthogonal bases was presented in [34]. The interesting (but not surprising) property of such a generalization is that the frame property is not changed, as indicated by the following theorem.…”

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confidence: 99%

Analysis of Multiwindow Gabor-Type Schemes by Frame Methods

Zibulski

Zeevi

1997

Applied and Computational Harmonic Analysis

112

131

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The Gabor scheme is generalized to incorporate several window functions as well as kernels other than the exponential. The properties of the sequence of representation functions are characterized by an approach based on the concept of frames. Utilizing the piecewise Zak transform (PZT), the frame operator associated with the multiwindow Gabor-type frame is examined for a rational oversampling rate by representing the frame operator as a finite-order matrixvalued function in the PZT domain. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculations of the frame bounds and the dual frame, as well as the issue of tight frames, are considered. It is shown that the properties of the sequence of representation functions are essentially not changed by replacing the widely used exponential kernel with other kernels. Some examples and the issue of a different sampling rate for each window are also considered. The so-called Balian-Low theorem is generalized to consideration of a scheme of multiwindows which makes it possible to overcome in a way the constraint imposed by the original theorem in the case of a single window. ᭧ 1997 Academic Press

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Synthesis of signals from wigner distributions: representation on biorthogonal bases

Raz

1990

Signal Processing

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Gabor expansion on orthogonal bases

Cited by 9 publications

References 1 publication

Discrete multiwindow Gabor-type transforms

Discrete multiwindow Gabor-type transforms

Analysis of Multiwindow Gabor-Type Schemes by Frame Methods

Synthesis of signals from wigner distributions: representation on biorthogonal bases

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