Digital Signal Processing Reference
In-Depth Information
which are substituted in Eq. (10.18) to obtain
y
p
[0]
=
x
p
[0]
h
p
[0]
+
x
p
[1]
h
p
[
K
0
−
1]
+
x
p
[2]
h
p
[
K
0
−
2]
+
x
p
[
K
0
−
1]
h
p
[1]
,
y
p
[1]
=
x
p
[0]
h
p
[1]
+
x
p
[1]
h
p
[0]
+
x
p
[2]
h
p
[
K
0
−
1]
+
x
p
[
K
0
−
1]
h
p
[2]
,
y
p
[2]
=
x
p
[0]
h
p
[2]
+
x
p
[1]
h
p
[1]
+
x
p
[2]
h
p
[0]
+
x
p
[
K
0
−
1]
h
p
[3]
,
.
y
p
[
K
0
−
1]
=
x
p
[0]
h
p
[
K
0
−
1]
+
x
p
[1]
h
p
[
K
0
−
2]
+
x
p
[2]
h
p
[
K
0
−
3]
+
x
p
[
K
0
−
1]
h
p
[0]
.
(10.19)
These expressions require values from only one period (0
≤
k
≤
K
0
−
1) of
the input sequence
x
p
[
k
] and the impulse response
h
p
[
k
]. Therefore, we can
implement the periodic convolution from a single period of the convolving
functions. The main steps involved in such an implementation are listed in the
following algorithm.
Algorithm 10.3 Alternative procedure for computing the periodic convolution
(1) Sketch one period of the waveform for input
x
p
[
m
] by changing the inde-
pendent variable of
x
p
[
k
] from
k
to
m
within the range 0
≤
k
≤
K
0
−
1.
(2) Sketch one period of the waveform for the impulse response
h
p
[
m
]by
changing the independent variable from
k
to
m
within the range 0
≤
k
≤
K
0
−
1.
(3) Reflect
h
p
[
m
] such that
h
p
[
−
m
]
=
h
p
[
K
0
−
m
] as defined by the circular
reflection. Set
k
=
0.
(4) Using the circularly reflected function
h
p
[
−
m
], determine the waveform
for
h
p
[
k
−
m
]
=
h
p
[
−
(
m
−
k
)].
(5) Multiply the function
x
p
[
m
]by
h
p
[
k
−
m
] for 0
≤
m
≤
K
0
−
1 and plot
the product function
x
p
[
m
]
h
p
[
k
−
m
].
(6) Calculate the summation
K
0
−
1
m
=
0
x
p
[
m
]
h
p
[
k
−
m
] to determine
y
p
[
k
] for
the value of
k
selected in step (4).
(7) Increment
k
by one and repeat steps (4)-(6) till all values of
k
within the
range 0
≤
k
≤
K
0
−
1 are exhausted.
(8) Since
y
p
[
k
] is periodic with period
K
0
, the values of
y
p
[
k
] outside the range
0
≤
k
≤
K
0
−
1 are determined from the values obtained in steps (7).
We illustrate the alternative implementation by repeating Example 10.12 and
using the modified algorithm.
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