Digital Signal Processing Reference
In-Depth Information
Equation (10.18) is referred to as
periodic
or
circular reflection
. Before pro-
ceeding with the alternative algorithm for periodic convolution, we explain
circular reflection in more detail.
h
p
[
k
]
k
0
1
23
Example 10.11
For the periodic sequence
(a)
h
p
[
k
]
5
k
=
0
,
1
h
p
[
k
]
=
0
k
=
2
,
3
,
with fundamental period
K
0
=
4, determine the circularly reflected sequence
h
p
[
−
k
] and the circular shifted sequence
h
p
[
k
−
1].
k
0
1
23
Solution
Let
v
p
[
k
] denote the circular reflected sequence
h
p
[
−
k
]. Using
v
p
[
k
]
=
h
p
[
−
k
]
=
h
p
[
K
0
−
k
], the values of the circularly reflected signals are given
by
(b)
h
p
[−
k
]
k
=
0
v
p
[0]
=
h
p
[
K
0
]
=
h
p
[0]
=
5;
k
01
23
k
=
1
v
p
[1]
=
h
p
[
K
0
−
1]
=
h
p
[3]
=
0;
(c)
k
=
2
v
p
[2]
=
h
p
[
K
0
−
2]
=
h
p
[2]
=
0;
h
p
[
k
]
k
=
3
v
p
[3]
=
h
p
[
K
0
−
3]
=
h
p
[1]
=
5
.
The original sequence
h
p
[
k
] is plotted in Fig. 10.10(a), and the circularly
reflected sequence
h
p
[
−
k
] is plotted in Fig. 10.10(c). Note that the circu-
larly reflected signal
h
p
[
−
k
] can be obtained directly from
h
p
[
k
] by keep-
ing the value of
h
p
[0] fixed and then reflecting the remaining values of
h
p
[
k
] for 1
≤
k
≤
K
0
−
1 about
k
=
K
0
/
2
.
This procedure is illustrated in
Fig. 10.10(b).
Substituting 0
≤
k
≤
K
0
−
1, the values for the circularly shifted signal
w
p
[
k
]
=
h
p
[
k
−
1] are obtained as follows:
k
0
1
23
(d)
h
p
[
k
− 1]
k
=
0
w
p
[0]
=
h
p
[
−
1]
=
h
p
[
K
0
−
1]
=
0;
k
k
=
1
w
p
[1]
=
h
p
[0]
=
5;
01
23
(e)
k
=
2
v
p
[2]
=
h
p
[1]
=
5;
Fig. 10.10. Circular reflection
and shifting for a periodic
sequence. (a) Original periodic
sequence
h
p
[
k
]. (b) Procedure
to determine circularly reflected
sequence
h
p
[−
k
] from
h
p
[
k
].
(c) Circularly reflected sequence
h
p
[−
k
]. (d) Procedure to
determine circularly shifted
sequence
h
p
[
k
− 1] from
h
p
[
k
].
(e) Circularly shifted sequence
h
p
[
k
− 1].
k
=
3
v
p
[3]
=
h
p
[2]
=
0
.
The circularly shifted sequence
h
p
[
k
−
1] is plotted in Fig. 10.10(e). The circu-
larly shifted signal
h
p
[
k
−
1] can also be obtained directly from
h
p
[
k
] by shift-
ing
h
p
[
k
] towards the left by one time unit and moving the overflow value of
h
p
[
K
0
−
1] back into the sequence. This procedure is illustrated in Fig. 10.10(d).
To derive the alternative algorithm for periodic convolution, we substitute
different values of
k
within the range 1
≤
k
≤
K
0
−
1 in Eq. (10.18). The
resulting equations are given by
h
p
[
−
1]
=
h
p
[
K
0
−
1];
h
p
[
−
2]
=
h
p
[
K
0
−
2];
...
;
h
p
[
−
(
K
0
−
1)]
=
h
p
[1]
,
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