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Deformations of Gabor Frames | | Gerald Kaiser
; | | Date: |
20 Aug 2001 | | Journal: | J. Math. Phys. 35, 1172-1376, 1994 | | Subject: | Mathematical Physics; Group Theory MSC-class: 41-XX, 44-XX, 81-XX | math-ph math.GR math.MP | | Abstract: | The quantum mechanical harmonic oscillator Hamiltonian generates a one-parameter unitary group W( heta) in L^2(R) which rotates the time-frequency plane. In particular, W(pi/2) is the Fourier transform. When W( heta) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family of frames F_ heta generated by the non-compactly supported windows g_ heta=W(theta)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When heta=pi/2, we obtain the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family F_ heta therefore interpolates these two familiar examples. | | Source: | arXiv, math-ph/0108012 | | Services: | Forum | Review | PDF | Favorites | |
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