Digital Signal Processing Reference
In-Depth Information
H
0,0
(
t
)
1
t
0
0.5
1.0
H
1,0
(
t
)
1
H
1,1
(
t
)
1
t
t
0
0.5
1.0
0
0.5
1.0
H
2,0
(
t
)
1
H
2,1
(
t
)
1
H
2,2
(
t
)
1
H
2,3
(
t
)
1
t
t
t
t
0
0.5
1.0
0
0.5
1.0
0
0.5
1.0
0
0.5
1.0
Fig. P4.5. Haar functions for
m
The other Haar functions, at scale
m
and with translation
n
, are defined
using the mother Haar function as follows:
= 0, 1, and 2.
H
0
,
0
(2
m
t
−
n
)
,
=
0
,
1
,...,
(2
m
−
1)
.
H
m
,
n
(
t
)
=
n
The Haar functions for
m
=
0
,
1
,
2 are shown in Fig. P4.5.
Show that the Haar wavelet functions
H
m
,
n
(
t
)
,
m
=
0
,
1
,
2
,...,
n
=
0
,
1
,
2
,...
(2
m
−
1)
form a set of orthogonal functions over the interval
[0, 1] by proving the following:
1
−
m
2
m
=
p
,
n
=
q
=
H
m
,
n
(
t
)
H
p
,
q
(
t
)d
t
0
otherwise
.
0
4.6
Calculate the trigonometric CTFS coefficients for the periodic functions
shown in Figs. P4.6(a)-(e).
(a) Rectangular pulse train with period 2
π
:
3
for
0
≤
t
<π
x
1(
t
)
=
0
for
π
≤
t
<
2
π.
(b) Raised square wave with period 2
T
:
−
T
2
T
2
0
.
5
for
≤
t
<
x
2(
t
)
=
T
2
3
T
2
1
for
≤
t
<
.
(c) Half sawtooth wave with period
T
:
−
t
T
x
3(
t
)
=
1
for
0
≤
t
<
T
.
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