Deformations of Gabor frames

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Abstract
The quantum mechanical Harmonic oscillator Hamiltonian H=(t2−∂2t)/2 generates a one-parameter unitary group W(θ)=eiθH in L2(R) which rotates the time–frequency plane. In particular, W(π/2) is the Fourier transform. When W(θ) 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 {ℱθ: 0≤θ<2π} of frames generated by the noncompactly supported windows gθ=W(θ)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 θ=π/2, ℱθ is the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family {ℱθ} therefore interpolates these two familiar examples.

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