Gabor's signal expansion and the Gabor transform on a non-separable time–frequency lattice
✍ Scribed by Arno J. van Leest; Martin J. Bastiaans
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 123 KB
- Volume
- 337
- Category
- Article
- ISSN
- 0016-0032
No coin nor oath required. For personal study only.
✦ Synopsis
Gabor's signal expansion and the Gabor transform are formulated on a general, nonseparable time}frequency lattice instead of on the traditional rectangular lattice. The representation of the general lattice is based on the rectangular lattice via a shear operation, which corresponds to a description of the general lattice by means of a lattice generator matrix that has the Hermite normal form. The shear operation on the lattice is associated with simple operations on the signal, on the synthesis and the analysis window, and on Gabor's expansion coe$cients; these operations consist of multiplications by quadratic phase terms. Following this procedure, the well-known bi-orthogonality condition for the window functions in the rectangular sampling geometry, can be directly translated to the general case. In the same way, a modi"ed Zak transform can be de"ned for the non-separable case, with the help of which Gabor's signal expansion and the Gabor transform can be brought into product forms that are identical to the ones that are well known for the rectangular sampling geometry.