Digital Signal Processing Reference
In-Depth Information
f
V
1
fV
0
fV
1
t
t
t
2
1
1/2
Figure 2.8
Piecewise constant functions in
V
1
,
V
0
and
V
−1
.
L
2
(
R
);
[2
m
n,
2
m
(
n
+1)]
V
m
=
{
f
(
t
)
∈
f is constant in
∀
n
∈
.
(2.58)
Z
}
Figure 2.8 illustrates such a function.
We can easily see that
···
V
1
⊂
V
0
⊂
V
−1
···
and
f
(
t
)
∈
V
0
←→
f
(2
t
)
V
−1
, and that the inclusion property is fulfilled. The function
f
(2
t
) has the same shape as
f
(
t
) but is compressed to half the width.
The scaling function of the Haar wavelet
ϕ
(
t
)isgivenby
∈
ϕ
(
t
)=
1
,
0
≤
t
≤
1
(2.59)
0
,
else
and defines an orthonormal basis for
V
0
. Since for
n
=
m
,
ϕ
(
t
−
n
)and
ϕ
(
t
−
m
) do not overlap, we obtain
ϕ
(
t
−
n
)
ϕ
(
t
−
m
)
dt
=
δ
n−m
(2.60)
The Fourier transform of the scaling function yields
sin
ω/
2
ω/
2
Φ(
ω
)=
e
−j
2
.
(2.61)
Figure 2.9 shows that
ϕ
(
t
) can be written as the linear combination of
even and odd translations of
ϕ
(2
t
):
ϕ
(
t
)=
ϕ
(2
t
)+
ϕ
(2
t
−
1)
(2.62)
Since
V
−1
=
V
0
⊕
W
0
and
Q
0
f
=(
P
−1
f
−
P
0
f
)
∈
W
0
represent the
details from scale 0 to
−
1, it is easy to see that
ψ
(
t
−
n
)spans
W
0
.The
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