Gabor frames generated by Random-Periodic time-frequency shifts
Sarthak Raj, S. Sivananthan
TL;DR
The paper addresses the existence of Gabor frames under random-periodic time-frequency shifts by introducing $\Lambda=\mathbb{Z}+\{x_1,\dots,x_m\}$ and showing that, for well-behaved windows $g$ with sufficient decay, the system $\mathcal{G}(g,\Lambda\times\mathbb{Z})$ forms a frame with high probability when the number of random shifts $m$ is large. Central to the approach are the Zak transform and Hoeffding's inequality, which reduce the frame condition to a finite, probabilistically controlled bound on $L^2(Q)$ and yield explicit probabilistic frame constants. The main result provides explicit, high-probability lower and upper frame bounds in terms of $m$, $q$, and $\beta-q/2$, along with deterministic assumptions on $g$ (periodic summability, square summability away from zero, and Lipschitz-type Zak bounds). The work broadens deterministic Gabor frame theory by delivering a practical randomized construction applicable to Hermite functions, totally positive functions, and B-splines, with explicit constants and probabilistic guarantees that are suitable for applications requiring robust, grid-like time-frequency representations.
Abstract
In this article, we consider a variation of the existence of Gabor frames in a probabilistic setting, in which we consider time-frequency shifts taken over random-periodic sets. We demonstrate that the method of selecting random-periodic time-frequency shifts is successful with high probability for specific categories of well-behaved functions, notably including Hermite functions, totally positive functions, and B-spline functions. In particular, we show that if $x_1, x_2, \ldots ,x_m$ are independent and uniformly distributed in $[0,1),$ with $m$ sufficiently large, then the set of time-frequency shifts $Λ\times \ZZ, $ where $Λ=\ZZ + \{x_1, x_2, \ldots, x_m\},$ forms Gabor frame with high probability.