Time-Frequency Localization and the Gabor Transform

Details Time-Frequency Localization and the Gabor Transform Time-Frequency Localization and the Gabor Transform

1993-02-18

arxiv.org/abs/funct-an/9302004v1

Mathematics

Functional Analysis

14 pages, LaTeX

Scientific paper

The time-frequency content of a signal can be measured by the Gabor transform or windowed Fourier transform. This is a function defined on phase space that is computed by taking the Fourier transform of the product of the signal against a translate of a fixed window. The problem of finding signals with Gabor transform that are maximally concentrated within a given region of phase space is discussed. It is well known that such problems give rise to an eigenvalue problem for an associated self-adjoint, positive {\em concentration} operator that has its spectrum contained in the unit interval. In this paper, the asymptotic behavior of these eigenvalues as the concentration region gets large is studied. The smoothness of the eigenfunctions is also examined.

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