Digital Signal Processing Reference
In-Depth Information
Fig. 10.12. Periodic convolution
using circular shifting in Example
10.13.
step (1)
step (3)
step (4)
x
p
[
k
]
h
p
[
k
]
h
[
k
]
x
[
k
]
3
3
3
3
2
2
2
2
−1
1
−2
2
−1
1
−2
2
k
k
k
k
0
2345 6
−1
0
1
345
0
−1
0
1
−1
−1
−1
−1
−1
−1
−1
−1
step (5)
y
p
[
k
]
5
5
5
5
5
5
1
1
1
1
1
1
−8
−6
−2
02
345
6
8
10
14
k
−9
−7
−5 −4 −3
−1
1
7
9
11 12 13
−2
−2
−2
−5
−5
−5
−5
−5
−5
(5) Calculate the periodic convolution
y
p
[
k
]
=
x
p
[
k
]
⊗
h
p
[
k
]. The result of
the linear convolution is obtained by selecting the range
k
ℓ
1
+
k
ℓ
2
≤
k
≤
k
u
1
+
k
u
2
of
y
p
[
k
].
Example 10.13 illustrates the aforementioned procedure.
Example 10.13
Compute the linear convolution of the following DT sequences:
2
k
=
0
2
k
=
0
3
k
=
1
x
[
k
]
=
−
1
k
=
1
and
h
[
k
]
=
−
1
k
=
2
0
otherwise
0
otherwise
,
using the periodic convolution method outlined in Algorithm 10.4.
Solution
The DT sequences
x
[
k
] and
h
[
k
] are plotted in Fig. 10.12, step (1). We observe
that the length
K
x
of
x
[
k
] is 3, while the length
K
h
of
h
[
k
]is5.
Based on step (2), the value of
K
0
≥
3
+
5
−
1 or 7. We select
K
0
=
8.
Following step (3), we form
x
p
[
k
] by padding
x
[
k
] with
K
0
−
K
x
or five
zeros. The resulting sequence
x
p
[
k
] is shown in Fig. 10.12, step (3).
Following step (4), we form
h
p
[
k
] by padding
h
[
k
] with
K
0
−
K
h
, or three
zeros. The resulting sequence
h
p
[
k
] is shown in Fig. 10.12, step (4).
Following step (5), the periodic convolution of the DT sequences
x
p
[
k
] and
h
p
[
k
] is performed using the sliding tape method. The final result is shown
in Table 10.3, where only one period (
K
0
=
8) of each sequence within the
duration
k
=
[
−
3, 4] is considered.
The sliding tape approach illustrated in Table 10.3 is slightly different from
that of Table 10.2. The reflection and shifting operations in Table 10.3 are
based on circular reflection and circular shifting since periodic sequences are
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