Digital Signal Processing Reference
In-Depth Information
/
√
a
)
with
¯
ψ
∗
(
ψ
a
(
t
)
=
(1
−
t
/
a
), and where
W
(
a
,
b
) is the wavelet coefficient of the function
f
(
t
)
ψ
∗
(
t
) is its complex conjugate
ψ
(
t
) is the analyzing wavelet and
∈ R
+
\{
a
0
}
is the scale parameter
b
∈ R
is the position parameter
In the Fourier domain, we have
W
(
a
=
√
a f
(
)
ˆ
ψ
∗
(
a
,ν
)
ν
ν
)
.
(2.2)
When the scale
a
varies, the filter
ˆ
ψ
∗
(
a
ν
) is only reduced or dilated while keeping
the same pattern.
2.2.2 Properties
The CWT is characterized by the following three properties:
1. CWT is a linear transformation; for any scalar
ρ
1
and
ρ
2
,
if
f
(
t
)
=
ρ
1
f
1
(
t
)
+
ρ
2
f
2
(
t
) then
W
f
(
a
,
b
)
=
ρ
1
W
f
1
(
a
,
b
)
+
ρ
2
W
f
2
(
a
,
b
)
.
2. CWT is covariant under translation:
if
f
0
(
t
)
=
f
(
t
−
t
0
) then
W
f
0
(
a
,
b
)
=
W
f
(
a
,
b
−
t
0
)
.
3. CWT is covariant under dilation:
1
√
s
W
f
(
sa
if
f
s
(
t
)
=
f
(
st
) then
W
f
s
(
a
,
b
)
=
,
sb
)
.
The last property makes the wavelet transform very suitable for analyzing hierar-
chical structures. It is like a mathematical microscope with properties that do not
depend on the magnification.
2.2.3 The Inverse Transform
Consider now a function
W
(
a
b
), which is the wavelet transform of a given function
f
(
t
). It has been shown (Grossmann and Morlet 1984) that
f
(
t
) can be recovered
using the inverse formula
,
t
da
+∞
+∞
1
C
χ
1
√
a
W
(
a
−
b
db
a
2
.
f
(
t
)
=
,
b
)
χ
,
(2.3)
a
−∞
0
where
+∞
0
ˆ
ˆ
ψ
∗
(
ψ
∗
(
ν
)ˆ
χ
(
ν
)
ν
)ˆ
χ
(
ν
)
C
χ
=
d
ν
=
d
ν.
(2.4)
ν
ν
−∞
0
Reconstruction is only possible if
C
χ
is finite (admissibility condition), which
ˆ
implies that
0; that is, the mean of the wavelet function is 0. The wavelet
is said to have a zero moment property and appears to have a bandpass profile.
Generally, the function
ψ
(0)
=
χ
(
t
)
=
ψ
(
t
), but other choices can enhance certain features
