Hyperbolic Secants Yield Gabor Frames
✍ Scribed by A.J.E.M Janssen; Thomas Strohmer
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 105 KB
- Volume
- 12
- Category
- Article
- ISSN
- 1063-5203
No coin nor oath required. For personal study only.
✦ Synopsis
We show that (g 2 , a, b) is a Gabor frame when a > 0, b > 0, ab < 1, and g 2 (t) = ( 1 2 πγ ) 1/2 (cosh πγ t) -1 is a hyperbolic secant with scaling parameter γ > 0. This is accomplished by expressing the Zak transform of g 2 in terms of the Zak transform of the Gaussian g 1 (t) = (2γ ) 1/4 exp(-πγ t 2 ), 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 g 2 and g 1 are the same at critical density a = b = 1. Also, we display the "singular" dual function corresponding to the hyperbolic secant at critical density.