Appendix A Gabor Expansion and GaborTransform
LabVIEW Order Analysis Toolset User Manual A-4 ni.com
Discrete Gabor-Expansion-Based Time-Varying Filter
Initially, discrete Gabor expansion seems to provide a feasible method
for converting an arbitrary signal from the time domain into the joint
time-frequency domain or vice versa. However, discrete Gabor expansion
is effective for converting an arbitrary signal from the time domain into the
joint time-frequency domain or vice versa only in the case of critical
sampling, ∆M = N. For over sampling, which is the case for most
applications, the Gabor coefficients are the subspace of two-dimensional
functions. In other words, for an arbitrary two-dimensional function, a
corresponding time waveform might not exist. For example, the following
equation represents a modified two-dimensional function.
where Φm, n denotes a binary mask function whose elements are either
0 or1. App lying the Gabor expansion to the modified two-dimensional
function results in the following equation.
The following inequality results from Gabor expansion.
The Gabor coefficients of the reconstructed time waveform are not
equal to the selected Gabor coefficients .
To overcome the problem of the reconstructed time waveform not equaling
the selected Gabor coefficients, use an iterative process. Complete the
following steps to perform the iterative process.
1. Determine a binary mask matrix for a set of two-dimensional Gabor
coefficients.
2. Apply the mask to the two-dimensional Gabor coefficients to preserve
desirable coefficients and remove unwanted coefficients.
3. Compute the Gabor expansion.
c
ˆ
mn,Φmn,cmn,
=
s
ˆk[] cmn,
n0=
N1–
∑
m
∑hk mT–[]ej2πnk N⁄
=
s
ˆ
m
∑k[]γkmT–[]ej2πnk–N⁄c
ˆmn,
≠
s
ˆ
k[]
c
ˆ
mn,