On Generating Tight Gabor Frames at Critical Density

Density of Gabor Frames

Gabor frames with rational density

We consider the frame property of the Gabor system G(g, α, β) = {eg(t − αm) : m,n ∈ Z} for the case of rational oversampling, i.e. α, β ∈ Q. A ’rational’ analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = n−1 n . Special attention is paid to the first Hermite function h1(t) = te −πt .

Relaxed quaternionic Gabor expansions at critical density

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal f in the Wiener space can be expanded into a unique `2 series on a lattice at critical density 1, provided one more poin...

FTN multicarrier transmission based on tight Gabor frames

A multicarrier signal can be synthesized thanks to a symbol sequence and a Gabor family (i.e., a regularly time–frequency shifted version of a generator pulse). In this article, we consider the case where the signaling density is increased such that inter-pulse interference is unavoidable. Over an additive white Gaussian noise channel, we show that the signal-to-interference-plus-noise ratio is...

Characterization and computation of canonical tight windows for Gabor frames

Let (gna,mb)n,m∈Z be a Gabor frame for L (R) for given window g. We show that the window h0 = S− 1 2 g that generates the canonically associated tight Gabor frame minimizes ‖g − h‖ among all windows h generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in ...