Digital Signal Processing Reference
In-Depth Information
100
100
0
0
-100
-100
0
0.5
1
0
0.5
1
100
100
0
0
-100
-100
0
0.5
1
0
0.5
1
100
100
0
0
-100
-100
0
0.5
1
0
0.5
1
Time (s ec . )
Time (s ec . )
FIGURE 13.28
Signal expanded by Haar father wavelets.
Finally, we have
f
1
ðtÞ¼
1
:
2
2
1
=
2
2
1
=
2
fð
2
tÞþ
0
:
4
2
1
=
2
2
1
=
2
fð
2
t
1
Þ¼
2
:
4
fð
2
tÞþ
0
:
8
fð
2
t
1
Þ
z
f ðtÞ
It is clear that there is a numerical error. The error can be reduced when a smaller time interval
Dt
is
adopted.
Figure 13.28
demonstrates the approximation of a sinusoidal delaying function using the scaling
functions (Haar father wavelets) at different scales, i.e.,
j ¼
0
;
1
;
2
;
4
;
5.
Now, let us examine the function approximation at resolution
j ¼
1:
N
p
2
p
2
p
2
f
1
ðtÞ
z
c
1
ðkÞ
fð
2
t kÞ¼c
1
ð
0
Þ
fð
2
tÞþc
1
ð
1
Þ
fð
2
t
1
Þ
k ¼
N
We also look at another possibility at a coarser scale with both the scaling functions (father wavelets)
and mother wavelets, that is,
j ¼
0:
k¼
N
c
0
ðkÞf
0
k
ðtÞþ
N
N
f
1
ðtÞ
z
k¼
N
d
0
ðkÞj
0
k
ðtÞ
¼ c
0
ð
0
Þf
00
ðtÞþd
0
ð
0
Þj
00
ðtÞ¼c
0
ð
0
ÞfðtÞþd
0
ð
0
ÞjðtÞ
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