Digital Signal Processing Reference
In-Depth Information
3.16
LINEAR, PERIODIC, AND CIRCULAR
CONVOLUTION AND THE DF T
The mathematical formula for convolution, which we now further denote, for purposes of this discussion,
as
Linear Convolution
, is, for two sequences
b
[
n
]
and
x
[
n
]
and Lag Index
k
, defined as
N
−
1
y
[
k
]=
b
[
n
]
x
[
k
−
n
]
(3.24)
n
=
0
Figure 3.22 shows two sequences of length eight, and their linear convolution, which has a length
equal to 15 (i.e., 2
N
-1).
2
0
−2
0
5
10
15
(a) Sample
2
0
−2
0
5
10
15
(b) Sample
2
0
−2
0
5
10
15
(c) Sample
Figure 3.22:
(a) Sequence 1; (b) Sequence 2; (c) Linear Convolution of Sequences 1 and 2.
3.16.1 CYCLIC/PERIODIC CONVOLUTION
Consider the case in which two sequences are both periodic over
N
samples. In this case, the linear
convolution (when in steady-state or saturation) produces a convolution sequence that is also periodic
over
N
samples.
Figure 3.23 shows a simple scheme wherein one sequence is just a repetition of an eight sample
subsequence, and the other is a single period of an eight sample sequence. The linear convolution of
these two sequences results in a periodic, or cyclic, convolution sequence when the two sequences are
in saturation (steady-state response). If you look closely, you'll see that the two end (transient-response)
segments of the overall convolution would actually, by themselves (removing the cyclic part of the con-
volution and bringing the two end segments together) constitute the linear convolution of the two eight
sample sequences. Figure 3.24 shows more or less the same thing only with two periods of the second
