Digital Signal Processing Reference
In-Depth Information
step (1)
x
p
[
m
]
step (2)
h
p
[
m
]
step (3)
h
p
[−
m
]
5
5
3
3
3
2
2
2
1
1
1
m
m
m
−4
−3
−2
−1
0
1234567
−4
−3
−2
−1
0
12345
67
−4
−3
−2
−1
0
1234567
step (4a)
x
p
[−
m
]
h
p
[0 −
m
]
step (4b)
x
p
[−
m
]
h
p
[1 −
m
]
step (4c)
x
p
[−
m
]
h
p
[2−
m
]
m
m
m
−4
−3
−2
−1
0
1234567
−4
−3
−2
−1
0
1234567
−4
−3
−2
−1
0
1234567
y
p
[
k
]
step (4d)
step (7)
step (8)
y
p
[
k
]
25
25
25
25
25
25
25
x
p
[−
m
]
h
p
[3 −
m
]
15
15
15
15
15
15
15
15
15
15
15
15
15
15
5
5
5
5
5
5
5
5
k
k
m
k
0
0
−4
−3
−2
−1
0
1234567
2
2
−4
−3
−2
−1
0
1234567
By comparing the aforementioned procedure for computing the periodic con-
volution with the procedure specified for evaluating the linear convolution in
Section 10.5, we observe that steps (4), (6), and (7) are different in the two
algorithms. In the linear convolution, the summation
∞
Fig. 10.9. Periodic convolution
of the periodic sequences
x
[
k
]
and
h
[
k
] in Example 10.10.
x
[
m
]
h
[
k
−
m
]
m
=−∞
is computed within the limits
m
=
[
−∞
,
∞
] for different values of
k
in the
range
−∞ ≤
k
≤∞
. In the periodic convolution, however, the summation is
computed over one complete period, say
m
=
[
m
0
,
m
0
+
K
0
−
1] for a reduced
range (0
≤
k
≤
K
0
−
1).
Example 10.10
Determine the periodic convolution between the following periodic sequences:
5
k
=
0
,
1
x
p
[
k
]
=
k
,
for
0
≤
k
≤
3
and
h
p
[
k
]
=
0
k
=
2
,
3
,
with the fundamental period
K
0
=
4.
Solution
Following steps (1)-(3), the periodic sequences
x
p
[
m
],
h
p
[
m
], and its reflected
version
h
p
[
−
m
] are plotted in Fig. 10.9, steps (1)-(3). Since the fundamental
period
K
0
=
4, we compute the result of the periodic convolution as follows:
3
y
p
[
k
]
=
x
p
[
k
]
⊗
h
p
[
k
]
=
x
p
[
m
]
h
p
[
k
−
m
]
(10.16)
m
=
0
for 0
≤
k
≤
3. The DT periodic sequences
h
p
[
k
−
m
] and
x
p
[
m
] for
k
=
0, 1,
2, and 3 are plotted, respectively, in Fig. 10.9, steps 4(a)-(d). The convolution
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