Deformations of Gabor Frames

The quantum mechanical harmonic oscillator Hamiltonian H = (t − ∂ t )/2 generates a one–parameter unitary group W (θ) = e in L(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 non–compactly supported windows g θ = 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. PACS numbers: 02.20.+b, 03.65.–w.