Digital Signal Processing Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
0.2
0.4
5
4
3
2
1
0
1
2
3
4
5
t
Figure 2.4
Mexican-hat wavelet.
C
ψ
<
∞
, then the wavelet is called admissible. Then for the gain we get
Ψ(0) =
∞
−∞
ψ
(
t
)
dt
= 0
(2.28)
We immediately see that
ψ
(
t
) corresponds to the impulse response of a
bandpass filter and has a decay rate of
|
1−ε
. It is important to note that
based on the admissibility condition, it can be shown that the CWT is
complete if
W
f
(
a, b
)isknownforall
a, b
.
The
Mexican-hat wavelet
|
t
ψ
(
t
)=(
2
t
2
)
e
−
t
2
√
3
π
−
4
)(1
−
(2.29)
is visualized in figure 2.4. It has a distinctive symmetric shape, and it
has an average value of zero and dies out rapidly as
|
t
|→∞
.Thereis
no scaling function associated with the Mexican hat wavelet.
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