Appendix A
Gabor Expansion and Gabor Transform
LabVIEW Order Analysis Toolset User Manual
A-6
ni.com
C
k
and
converge.
with
. For
with
, the support of
C
k
in the time-frequency domain is inside the
masked area if and only if the following equation is true.
(A-6)
for 0
≤
k
<
N
and 0
≤
m
<
. Two trivial cases for Equation A-6 are
critical sampling and when
γ
[
k
] =
h
[
k
].
In critical sampling,
N
=
∆
M
. Notice that in the case of critical sampling,
the analysis and synthesis windows cannot both be localized in the joint
time-frequency domain.
In the case of
γ
[
k
] =
h
[
k
], the Gabor coefficients
C
2
, after the first iteration,
are the closest in terms of the LMSE to the masked Gabor coefficients
Φ
C
.
The masked Gabor coefficients are the desirable Gabor coefficients. The
case of
γ
[
k
] =
h
[
k
]
usually implies considerable over sampling, which
results in a huge amount of redundancy. The amount of redundancy causes
slow computation speed and huge memory consumption, making
γ
[
k
] =
h
[
k
]
an impractical Gabor transform method.
Usually, the LabVIEW Order Analysis Toolset uses the orthogonal-like
representation introduced at the end of the
section. For commonly used window functions, such as
the Gaussian and Hanning windows, the difference between the analysis
and synthesis windows is negligible when the over sampling rate is four.
s
k
C
k
Φ
C
k
=
k
∞
→
C
k
Φ
C
k
=
k
∞
→
γ∗
i
0
=
L
N
----
1
–
∑
iN k
+
[
]
h iN k m
∆
M
+
+
[
]
=
h
∗
i
0
=
L
N
----
1
–
∑
iN k
+
[
]γ
iN k m
∆
M
+
+
[
]
L
∆
M
---------
