© National Instruments Corporation A-1 LabVIEW Order Analysis Toolset User Manual
A

Gabor Expansion and

Gabor Transform

This appendix presents an overview of the Gabor expansion and the Gabor
transform methods used in the LabVIEW Order Analysis Toolset. This
appendix also describes application issues associated with using the
discrete Gabor-expansion-based time-varying filter.

Gabor Expansion and Gabor Transform Basics

The Gabor expansion characterizes a signal jointly in the time and
frequency domains. Although Dennis Gabor introduced the Gabor
expansion more than half a century ago, its implementation was an open
research topic until Bastiaans discovered the relationship between the
Gabor expansion and the short-time Fourier transform (STFT) in the
early 1980s.
Over the years, many different implementation schemes for the discrete
Gabor expansion were proposed. The LabVIEW Order Analysis Toolset
uses an extension of the method originally developed by Wexler and Raz
to implement the discrete Gabor expansion. In the method used by the
LabVIEW Order Analysis Toolset, lengths of the analysis and synthesis
window functions are the same, while perfect reconstruction is guaranteed.
For a discrete sample s[k], the corresponding Gabor expansion is defined
by the following equation.
(A-1)
where the Gabor coefficients cm,n are computed by a sampled STFT.
sk[] cmn,
n0=
N1–

∑

m
∑hk m∆M–[]ej2πnk N⁄
=