Digital Signal Processing Reference
In-Depth Information
Using
Figure 13.25
(
b), we obtain the plots shown in
Figure 13.27
.
2
-
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
0
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
0
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
0
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t (sec.)
FIGURE 13.27
Haar mother wavelet at different scales and translations.
A signal can be expanded by the father wavelets (scaling function and its translations) at level
j
.
More accuracy can be achieved by using the larger
j
. The expanded function is approximated by
N
c
j
ðkÞ
2
j=
2
fð
2
j
t kÞ
f ðtÞ
z
f
j
ðtÞ¼
(13.38)
k ¼
N
where the wavelet coefficients
c
j
ðkÞ
can be determined by an inner product:
c
j
ðkÞ¼
f ðtÞf
jk
ðtÞ
¼
Z
f ðtÞ
2
j=
2
fð
2
j
t kÞdt:
(13.39)
EXAMPLE 13.4
Approximate the following function using the Haar scaling function at level
j ¼ 1:
(
20 t < 0:5
10:5 t 1
f ðtÞ¼
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