Hyperbolic secants yield Gabor frames

2003/01/13 by A. J. E. M. Janssen, Thomas Strohmer · 1 citation
Mathematics · #math.FA

paper · pdf

published as Appl. Comp. Harm. Anal., 12(2): 259--267, 2002

arxiv created 2003/01/13 · arxiv updated 2009/11/30

Abstract

We show that (g₂,a,b) is a Gabor frame when a>0, b>0, ab<1 and g₂(t)=(1/2πγ)^1/2 (\cosh πγt)^-1 is a hyperbolic secant with scaling parameter γ>0. This is accomplished by expressing the Zak transform of g₂ in terms of the Zak transform of the Gaussian g₁(t)=(2γ)^1/4 exp (-πγt²), together with an appropriate use of the Ron-Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to g₂ and g₁ are the same at critical density a=b=1. Also, we display the ``singular'' dual function corresponding to the hyperbolic secant at critical density.

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