Digital Signal Processing Reference
In-Depth Information
x
p
[
k
]
h
p
[
k
]
55
3
3
3
2
2
2
1
1
1
k
k
−3
−2 −1
0
23
−3
−2 −1
0
23
−4
−4
periodic, or circular, convolution to convolve periodic sequences. We now show
how the periodic convolution can be calculated using the DTFS.
If
x
1
[
k
] and
x
2
[
k
] are two DT periodic sequences with the same fundamental
period
K
0
and the following DTFS pairs:
Fig. 11.15. Periodic sequences
x
p
[
k
] and
h
p
[
k
] used in Example
11.16.
DTFS
←→
DTFS
←→
D
x
1
n
D
x
n
,
x
1
[
k
]
and
x
2
[
k
]
then the periodic convolution property states that
DTFS
←−−→
K
0
D
x
n
D
x
n
.
x
1
[
k
]
⊗
x
2
[
k
]
(11.51)
We illustrate the application of the periodic convolution property by revisiting
Example 10.10.
Example 11.16
In Example 10.10, we calculated the periodic convolution
y
p
[
k
] of the two
periodic sequences
x
p
[
k
] and
h
p
[
k
], defined over one period (
K
0
=
4) as
x
p
[
k
]
=
k
,
0
≤
k
≤
3, and
h
p
[
k
]
=
5
,
0
≤
k
≤
1, in the time domain. Repeat Example
10.10 using the periodic convolution property.
Solution
The periodic sequences
x
p
[
k
] and
h
p
[
k
] are shown in Fig. 11.15. In part (i) of
Example 11.9, we calculated the DTFS coefficients of
x
p
[
k
] as follows:
=
3
=−
1
=−
1
=−
1
D
x
p
0
D
x
p
1
D
x
p
2
D
x
p
3
2
[1
+
j]
.
Similarly, in part (ii) of Example 11.9 we calculated the DTFS coefficients of
h
p
[
k
]:
2
,
2
[1
−
j]
,
2
,
and
=
5
=
5
=
5
D
h
p
0
D
h
p
1
D
h
p
2
D
h
p
3
4
[1
+
j]
.
Using the periodic convolution property, the DTFS coefficients of
y
p
[
k
] are
D
y
p
0
2
,
4
[1
−
j]
,
=
0
,
and
K
0
D
x
0
D
h
p
=
=
15;
0
D
y
p
1
K
0
D
x
1
D
h
p
=
=
j5;
1
D
y
p
2
K
0
D
x
p
2
D
h
p
2
=
=
0;
D
y
p
3
K
0
D
x
3
D
h
p
=
=−
j5
.
3
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