Digital Signal Processing Reference
In-Depth Information
For
k
=
0, the DT sequence
h
p
[
k
−
m
]
=
h
p
[
−
m
]. The value of the output
response at
k
=
0isgivenby
y
p
[0]
=
K
0
−
1
x
p
[
m
]
h
p
[
−
m
]
=
0(5)
+
1(0)
+
2(0)
+
3(5)
=
15
.
m
=
0
For
k
=
1, the DT sequence
h
p
[
k
−
m
]
=
h
p
[1
−
m
]. The new sequence
h
p
[1
−
m
]
=
h
p
[
−
(
m
−
1)] is obtained by circularly shifting
h
p
[
−
m
] towards
the right by one sample, with the last sample at
m
=
3 taking the place of the
first sample at
m
=
0. The sequence
h
p
[1
−
m
] is plotted in Fig. 10.11, step
(4b). Multiplying by
h
p
[
m
], the value of the output response at
k
=
1isgiven
by
y
p
[1]
=
K
0
−
1
x
p
[
m
]
h
p
[1
−
m
]
=
0(5)
+
1(5)
+
2(0)
+
3(0)
=
5
.
m
=
0
For
k
=
2, the DT sequence
h
p
[
k
−
m
]
=
h
p
[2
−
m
]. The new sequence
h
p
[2
−
m
] is obtained by circularly shifting
h
p
[1
−
m
] towards the right by
one sample, with the last sample at
m
=
3 taking the place of the first sample at
m
=
0. The sequence
h
p
[2
−
m
] is plotted in Fig. 10.11, step (4c). Multiplying
by
h
p
[
m
], the value of the output response at
k
=
2isgivenby
y
p
[2]
=
K
0
−
1
x
p
[
m
]
h
p
[2
−
m
]
=
0(0)
+
1(5)
+
2(5)
+
3(0)
=
15
.
m
=
0
For
k
=
3, the DT sequence
h
p
[
k
−
m
]
=
h
p
[3
−
m
]. The new sequence
h
p
[3
−
m
] is obtained by circularly shifting
h
p
[2
−
m
] towards the right by
one sample, with the last sample at
m
=
3 taking the place of the first sample at
m
=
0. The sequence
h
p
[3
−
m
] is plotted in Fig. 10.11, step (4d). Multiplying
by
h
p
[
m
], the value of the output response at
k
=
3isgivenby
y
p
[3]
=
K
0
−
1
x
p
[
m
]
h
p
[3
−
m
]
=
0(0)
+
1(0)
+
2(5)
+
3(5)
=
25
.
m
=
0
The final output
y
p
[
k
], obtained from steps (5)-(8) of Algorithm 10.3, is
plotted in Fig. 10.11, Steps (5)-(8). Observe that the result is identical to that in
Fig. 10.9, which was obtained using the full periodic convolution.
10.6.1 Linear convolution through periodic convolution
In this chapter, we have introduced two types of DT convolution. The linear
convolution, defined in Eq. (10.14), is used to convolve aperiodic sequences,
while the periodic convolution, defined in Eq. (10.15), is used for convolving
periodic sequences. Definition 10.3 states a condition under which the results
of the periodic and linear convolution are the same.
Definition 10.3
Assume that x
[
k
]
and h
[
k
]
are two aperiodic DT sequences of
finite length such that the following are true.
Search WWH ::
Custom Search