Aj. Van Leest et Mj. Bastiaans, Gabor's signal expansion and the Gabor transform on a non-separable time-frequency lattice, J FRANKL I, 337(4), 2000, pp. 291-301
Citations number
12
Categorie Soggetti
Engineering Management /General
Journal title
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
Gabor's signal expansion and the Gabor transform are formulated on a genera
l, nonseparable time-frequency lattice instead of on the traditional rectan
gular lattice. The representation of the general lattice is based on the re
ctangular 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 coefficients; these operations consist of multipl
ications by quadratic phase terms. Following this procedure, the well-known
bi-orthogonality condition for the window functions in the rectangular sam
pling geometry, can be directly translated to the general case. In the same
way, a modified Zak transform can be defined for the non-separable case? w
ith 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. (C) 2000 The Franklin Institut
e. Published by Elsevier Science Ltd. All rights reserved.