Digital Signal Processing Reference
In-Depth Information
step (1)
step (2)
step (3)
h
p
[−
m
]
5
5
h
p
[
m
]
x
p
[
m
]
3
2
1
m
m
m
0
1
23
0
1
23
0
1
23
step (4a)
step (4b)
x
p
[
m
]
h
p
[1 −
m
]
x
p
[
m
]
h
p
[0 −
m
]
h
p
[1 −
m
]
h
p
[−
m
]
m
m
m
m
0
1
23
0
1
23
0
1
23
0
1
23
step (4c)
step (4d)
x
p
[
m
]
h
p
[2 −
m
]
h
p
[3 −
m
]
x
p
[
m
]
h
p
[3 −
m
]
h
p
[2 −
m
]
m
m
m
m
0
1
23
0
1
23
0
1
23
0
1
23
steps (5)-(8)
y
p
[
k
]
25
25
25
15
15
15
15
15
15
5
5
5
Fig. 10.11. Periodic convolution
using circular shifting in Example
10.12.
k
−4
−3
−2 −1
0
0
2
2
56
Example 10.12
Using Algorithm 10.3, determine the periodic convolution of the periodic
sequences
5
k
=
0
,
1
x
p
[
k
]
=
k
(0
≤
k
≤
3)
and
h
p
[
k
]
=
0
k
=
2
,
3
,
with fundamental period
K
0
=
4.
Solution
Following steps (1) and (2), the applied input and the impulse response are
plotted as a function of
m
in Fig. 10.11, steps (1) and (2).
Following step (3), the circularly reflected impulse response
v
p
[
m
]
=
h
p
[
−
m
]
=
h
p
[
K
0
−
m
] for 0
≤
m
≤
3 is calculated as follows:
v
p
[0]
=
h
p
[0]
=
1;
v
p
[1]
=
h
p
[
−
1]
=
h
p
[3]
=
0;
v
p
[2]
=
h
p
[
−
2]
=
h
p
[2]
=
0;
and
v
p
[3]
=
h
p
[
−
3]
=
h
p
[1]
=
3
.
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