Appendix A
Gabor Expansion and Gabor Transform
LabVIEW Order Analysis Toolset User Manual
A-2
ni.com
The sampled STFT is also known as the Gabor transform and is represented
by the following equation.
(A-2)
where
∆
M
represents the time sampling interval and
N
represents the total
number of frequency bins.
The ratio between
N
and
∆
M
determines the Gabor sampling rate. For
numerical stability, the Gabor sampling rate must be greater than or equal
to one. Critical sampling occurs when
N
=
∆
M
. In critical sampling, the
number of Gabor coefficients
c
m,n
equals the number of original data
samples
s
[
k
]. Over sampling occurs when
N
/
∆
M
> 1. For over sampling,
the number of Gabor coefficients is more than the number of original data
samples. In over sampling, the Gabor transform in Equation A-2 contains
redundancy, from a mathematical point of view. However, the redundancy
in Equation A-2 provides freedom for the selection of better window
functions,
h
[
k
] and
γ
[k].
Notice that the positions of the window functions
h
[
k
] and
γ
[k] are
interchangeable. In other words, you can use either of the window functions
as the synthesis or analysis window function. Therefore,
h
[
k
] and
γ
[k] are
usually referred to as dual functions.
The method of the discrete Gabor expansion developed in this appendix
requires
in Equation A-2 to be a periodic sequence, as shown by the
following equation.
(A-3)
where
L
s
represents the length of the signal
s
[
k
] and
L
0
represents the period
of the sequence
L
0
is the smallest integer that is greater than or equal
to
L
s
.
L
0
must be evenly divided by the time sampling interval
∆
M
. For a
given window
h
[
k
] that always has unit energy, you can compute the
c
m n
,
s
˜
k
[ ]γ∗
k m
∆
M
–
[
]
e
j
2
π
nk
–
N
⁄
n
0
=
N
1
–
∑
=
s
˜
k
[ ]
s
˜
k iL
0
+
[
]
s k
[ ]
0
k L
s
<
≤
0
L
s
k L
0
<
≤
=
s
˜
k
[ ]
.
