1993 DiscreteGaborTransform
- (Qian & Chen, 1993) ⇒ Shie Qian, and Dapang Chen. (1993). “Discrete Gabor Transform.” In: Signal Processing, IEEE Transactions on, 41(7).
Subject Headings: Gabor Transform.
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Abstract
A feasible algorithm for implementing the Gabor expansion, the coefficients of which are computed by the discrete Gabor transform (DGT), is presented. For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. By exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like DGT is obtained. As the discrete Fourier transform (DFT) is a discrete realization of the continuous-time Fourier transform, similarly, the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion
I. INTRODUCTION
Half century ago, Gabor [7] presented an approach to characterize a time function in time and frequency simultaneously, which later became known asthe Gabor expansion. For signal s (t),the Gabor expansion is defined as
- [math]\displaystyle{ s(t) = Crn.nhrn,n(t) }[/math]
- [math]\displaystyle{ h_{m,n}(t) = h(t - mT)e^{jn\Omega t} \ \ (1)\lt /math where T and \lt math\gt \Omega }[/math] represent time and frequency sampling intervals, respectively. The synthesis function [math]\displaystyle{ h(t) }[/math] is subject to a unit energy constraint. The existence of (1) has been found to be possible for arbitrary s (r) only for [math]\displaystyle{ T \Omega \lt = 2 \pi }[/math] [2], [9]. [math]\displaystyle{ T \Omega = 2 \pi }[/math] is called critical sampling and [math]\displaystyle{ T \Omega \lt 2 \pi }[/math] is oversampling.
Although the Gabor expansion has been recognized as very useful for signal processing, its applications were limited due to the difficulties associated with computing the Gabor coefficients [math]\displaystyle{ C_{m,n} }[/math]. According to the Balian-Low theorem, h,,,(t) do not form an orthogonal basis unless the corresponding elementary function h(t) is badly localized in either time or frequency [9]. Therefore, the selection of the Gabor coefficient Cm,n in general is not unique. There are two problems that have continued to draw much research - how to define the Gabor coefficients and to what extent the resulting coefficients represent the analyzed signal [1], [2], [7], [8].
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References
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| Author | volume | Date Value | title | type | journal | titleUrl | doi | note | year | |
|---|---|---|---|---|---|---|---|---|---|---|
| 1993 DiscreteGaborTransform | Shie Qian Dapang Chen | Discrete Gabor Transform | 1993 |