Iterative algorithms to approximate canonical Gabor windows: Computational aspects

A.J.E.M Janssen, Peter Lempel Søndergaard

    Research output: Book/ReportReportResearch

    Abstract

    In this paper we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g,a,b). The iterations start with the window g while the iteration steps comprise the window g, the k^th iterand \gamma_{k}, the frame operators S and S_{k} corresponding to (g,a,b) and (\gamma_{k},a,b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged convergence order m of the method. We consider two strategies for scaling the terms in the iteration step: norm scaling, where in each step the windows are normalized, and initial scaling where we only scale in the very beginning. Norm scaling leads to fast, but conditionally convergent methods, while initial scaling leads to unconditionally convergent methods, but with possibly suboptimal convergence constants. The iterations, initially formulated for time-continuous Gabor systems, are considered and tested in a discrete setting in which one passes to the appropriately sampled-and-periodized windows and frame operators. Furthermore, they are compared with respect to accuracy and efficiency with other methods to approximate canonical windows associated with Gabor frames.
    Original languageEnglish
    Place of PublicationLyngby
    PublisherInstitut for Matematik, DTU
    Publication statusPublished - 2006
    SeriesMAT Preprints
    Number2006-2

    Cite this

    Janssen, A. J. E. M., & Søndergaard, P. L. (2006). Iterative algorithms to approximate canonical Gabor windows: Computational aspects. Lyngby: Institut for Matematik, DTU. MAT Preprints, No. 2006-2
    Janssen, A.J.E.M ; Søndergaard, Peter Lempel. / Iterative algorithms to approximate canonical Gabor windows: Computational aspects. Lyngby : Institut for Matematik, DTU, 2006. (MAT Preprints; No. 2006-2).
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    abstract = "In this paper we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g,a,b). The iterations start with the window g while the iteration steps comprise the window g, the k^th iterand \gamma_{k}, the frame operators S and S_{k} corresponding to (g,a,b) and (\gamma_{k},a,b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged convergence order m of the method. We consider two strategies for scaling the terms in the iteration step: norm scaling, where in each step the windows are normalized, and initial scaling where we only scale in the very beginning. Norm scaling leads to fast, but conditionally convergent methods, while initial scaling leads to unconditionally convergent methods, but with possibly suboptimal convergence constants. The iterations, initially formulated for time-continuous Gabor systems, are considered and tested in a discrete setting in which one passes to the appropriately sampled-and-periodized windows and frame operators. Furthermore, they are compared with respect to accuracy and efficiency with other methods to approximate canonical windows associated with Gabor frames.",
    author = "A.J.E.M Janssen and S{\o}ndergaard, {Peter Lempel}",
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    language = "English",
    publisher = "Institut for Matematik, DTU",

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    Janssen, AJEM & Søndergaard, PL 2006, Iterative algorithms to approximate canonical Gabor windows: Computational aspects. MAT Preprints, no. 2006-2, Institut for Matematik, DTU, Lyngby.

    Iterative algorithms to approximate canonical Gabor windows: Computational aspects. / Janssen, A.J.E.M; Søndergaard, Peter Lempel.

    Lyngby : Institut for Matematik, DTU, 2006. (MAT Preprints; No. 2006-2).

    Research output: Book/ReportReportResearch

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    Janssen AJEM, Søndergaard PL. Iterative algorithms to approximate canonical Gabor windows: Computational aspects. Lyngby: Institut for Matematik, DTU, 2006. (MAT Preprints; No. 2006-2).