Connectivity in the set of Gabor frames

Connectivity in the Set of Gabor frames

In this paper we present a constructive proof that the set of Gabor frames is pathconnected in the L2(Rn)-norm. In particular, this result holds for the set of Gabor Parseval frames as well as for the set of Gabor orthonormal bases. In order to prove this result, we introduce a construction which shows exactly how to modify a Gabor frame or Parseval frame to obtain a new one with the same prope...

Density of Gabor Frames

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 ...

Nonstationary Gabor Frames

To overcome the limitation induced by the fixed timefrequency resolution over the whole time-frequency plane of Gabor frames, we propose a simple extension of the Gabor theory leading to the construction of frames with timefrequency resolution evolving over time or frequency. We describe the construction of such frames and give the explicit formulation of the canonical dual frame for some condi...

Gabor frames in finite dimensions

Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setti...