Gabor's signal expansion and the Gabor transform on a non-separable time-frequency lattice

A.J. Leest, van, M.J. Bastiaans

Research output: Contribution to journalArticleAcademicpeer-review

7 Citations (Scopus)

Abstract

Gabors signal expansion and the Gabor transform are formulated on a general, non-separable 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 coefficients; these operations consist of multiplications by quadratic phase terms. Following this procedure, the well-known biorthogonality condition for the window functions in the rectangular sampling 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, 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.
Original languageEnglish
Pages (from-to)291-301
Number of pages11
JournalJournal of the Franklin Institute
Volume337
Issue number4
DOIs
Publication statusPublished - 2000

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Gabor Transform
Nonseparable
Sampling
Geometry
Zak Transform
Biorthogonality
Product Form
Hermite
Normal Form
Multiplication
Generator
Synthesis
Coefficient
Term

Cite this

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abstract = "Gabors signal expansion and the Gabor transform are formulated on a general, non-separable 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 coefficients; these operations consist of multiplications by quadratic phase terms. Following this procedure, the well-known biorthogonality condition for the window functions in the rectangular sampling 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, 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.",
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Gabor's signal expansion and the Gabor transform on a non-separable time-frequency lattice. / Leest, van, A.J.; Bastiaans, M.J.

In: Journal of the Franklin Institute, Vol. 337, No. 4, 2000, p. 291-301.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Bastiaans, M.J.

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AB - Gabors signal expansion and the Gabor transform are formulated on a general, non-separable 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 coefficients; these operations consist of multiplications by quadratic phase terms. Following this procedure, the well-known biorthogonality condition for the window functions in the rectangular sampling 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, 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.

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