Appendix A
Gabor Expansion and Gabor Transform
©
National Instruments Corporation
A-5
LabVIEW Order Analysis Toolset User Manual
4.
Compute the new Gabor coefficients after you obtain the time
waveform.
5.
Repeat steps 1 through 4 until the time waveforms converge.
Without a loss of generality, rewrite the Gabor expansion from
Equation A-1 and the Gabor transform from Equation A-2 in matrix form,
as shown in the following equations.
where
H
denotes the analysis matrix and
G
denotes the synthesis matrix.
Notice that for over sampling, the following relationships are true.
Therefore, in the case of over sampling, the iterative process is described
by the following equations.
C
Gs
=
s
H
T
Gs
=
H
T
G
I
=
GH
T
I
≠
C
ˆ
1
Φ
C
=
s
1
H
T
C
ˆ
1
=
C
2
Gs
1
GH
T
Φ
C
=
=
C
ˆ
2
Φ
C
2
=
s
2
H
T
C
ˆ
2
=
C
3
Gs
2
GH
T
Φ
GH
T
Φ
C
GH
T
Φ
(
)
2
C
=
=
=
…
C
k
GH
T
Φ
(
)
k
1
–
C
=
