National Instruments Order Analysis Toolset - Manual (Page 54)

Appendix A Gabor Expansion and Gabor Transform
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
Φ
m, n
denotes a binary mask function whose elements are either
0 or 1. Applying 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.
cˆ
m n ,
Φ
m n ,
c
m n ,
=
sˆ k [ ] c
m n ,
n 0 =
N 1 – ∑
m
∑
h k mT –
[ ]e
j2 πnk N ⁄
=
sˆ
m
∑
k [ ]γ k mT
–
[ ]e j2 πnk – N ⁄
cˆ
m n ,
≠
sˆ k [ ]
cˆ
m n ,