Digital Signal Processing Reference
In-Depth Information
sequence. The result is that the noncyclic part of the convolution is about twice as long (nonsaturation is
as long at each end as the shorter of the overall lengths of the two sequences, minus one) and the cyclic
part has twice the amplitude.
2
0
−2
0
10
20
30
40
50
60
(a) Sample
2
0
−2
0
10
20
30
40
50
60
(b) Sample
2
0
−2
0
10
20
30
40
50
60
(c) Sample
Figure 3.23:
(a) A periodic sequence over eight samples; (b) An eight-sample sequence; (c) Linear
convolution, showing periodicity (in saturation) of eight samples.
3.16.2 CIRCULAR CONVOLUTION
The cyclic or periodic convolution, which we have arrived at via linear convolution of periodic sequences,
can also be computed (one period, that is) using a process called
Circular Convolution
, which we can
define as
N
−
1
[
]=
[
]
[
−
]
N
)
y
k
b
n
(x
k
n
(3.25)
n
=
0
where
b
[
n
]
and
x
[
n
]
are two
N
-point sequences and the expression
]
N
is evaluated modulo-
N
, which effectively extends the sequence
x
x
[
k
−
n
[
n
]
forward and backward so the
N
[
]
[
]
point convolution is a periodic convolution of
b
. The term circular convolution is used since
with the modulo index evaluation, a negative index effectively rotates to a positive one. For example, if
[
k
−
n
]
n
and
x
n
[
k
−
n
]
N
= 0, and so forth.
Using the MathScript function
mod(x, N)
, for example, a Command Line call such as
=0,
[
k
−
n
]
N
= 0;if
[
k
−
n
]
= -1,
[
k
−
n
]
N
=
N
-1;if
[
k
−
n
]
=
N
,
y = mod(-2,8)
