Digital Signal Processing Reference
In-Depth Information
equally spaced at angles of around the outer circle in a counterclockwise direc-
tion. The
N
values of are then written, equally spaced at angles of in a
clockwise direction on the inner circle. We calculate by multiplying the corre-
sponding values on each radial line and then adding the products. We find succeed-
ing values of in the same way after rotating the inner circle counterclockwise
through the angle and finding the sum of products of the corresponding val-
ues. The circular convolution process is demonstrated in the next example.
2p/N
h[n]
2p/N
y[0]
y[n]
2pn/N
Circular convolution of two discrete sequences
EXAMPLE 12.16
We wish to evaluate the circular convolution of the sequences
x
1
[n] = [1, 2, 3, 4]; x
2
[n] = [0, 1, 2, 3];
y[n] = x
1
[n]
*
x
2
[n].
We will do this by two methods—first, by using the circular convolution process as described
previously and, second, by using the DFT and IDFT.
The first value in the convolution sequence,
y[0],
is calculated from Figure 12.23(a):
y[0] = (1) (0) + (2) (3) + (3) (2) + (1) (4) = 16.
1
1
0
1
2
3
1
4
2
0
2
4
2
3
3
(a)
3
(b)
1
1
2
3
2
1
3
4
2
2
0
4
0
1
3
(c)
3
(d)
Figure 12.23
Circular convolution for
Example 12.16.
