Digital Signal Processing Reference
In-Depth Information
10.6 Periodic conv
olution
Linear convolution is used to convolve aperiodic sequences. If the convolv-
ing sequences are periodic, the result of linear convolution is unbounded. In
such cases, a second type of convolution, referred to as
periodic
or
circular
convolution
, is generally used.
Consider two periodic sequences
x
p
[
k
] and
h
p
[
k
], with identical fundamental
period
K
0
. The subscript p denotes periodicity. The relationship for the periodic
convolution between two periodic sequences is defined as follows:
y
p
[
k
]
=
x
p
[
k
]
⊗
h
p
[
k
]
=
x
p
[
m
]
h
p
[
k
−
m
]
,
(10.15)
m
=
K
0
where the summation on the right-hand side of Eq. (10.15) is defined over
one complete period
K
0
. In calculating the summation, we can, therefore, start
from any arbitrary position (say
m
=
m
0
) as long as one complete period of
the sequences is covered by the summation. For the lower limit
m
=
m
0
, the
upper limit is given by
m
=
m
0
+
K
0
−
1. In the text, the periodic convolu-
tion is denoted by the operator
⊗
, whereas the linear convolution is denoted
by
∗
.
The steps involved in calculating the periodic convolution are given in the
following algorithm.
Algorithm 10.2 Graphical procedure for computing the periodic convolution
(1) Sketch the waveform for input
x
p
[
m
] by changing the independent vari-
able of
x
p
[
k
] from
k
to
m
and keep the waveform for
x
p
[
m
] fixed during
steps (2)-(7).
(2) Sketch the waveform for the impulse response
h
p
[
m
] by changing the inde-
pendent variable from
k
to
m
.
(3) Reflect
h
p
[
m
] about the vertical axis to obtain the time-inverted impulse
response
h
p
[
−
m
]. Set the time index
k
=
0.
(4) Shift the function
h
p
[
−
m
] by a selected value of
k
. The resulting sequence
represents
h
p
[
k
−
m
].
(5) Multiply input sequence
x
p
[
m
]by
h
p
[
k
−
m
] and plot the product function
x
p
[
m
]
h
p
[
k
−
m
].
(6) Calculate the summation
m
=
K
0
x
p
[
m
]
h
p
[
k
−
m
] for
m
=
[
m
0
,
m
0
+
K
0
−
1] 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
in the specified
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 (6) and
(7).
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