Digital Signal Processing Reference
In-Depth Information
The inner circle is rotated counterclockwise through
2p/4 = p/2
radians to form Figure 12.23(b)
for calculation of the second term:
y[1] = (1) (1) + (2) (4) + (3) (3) + (0) (2) = 18.
The inner circle is rotated counterclockwise an additional
p/2
radians to form Figure 12.23(c)
for calculation of the third term:
y[2] = (2) (1) + (3) (4) + (0) (3) + (1) (2) = 16.
Repeating the process for the fourth term, we have
y[3] = (3) (1) + (0) (4) + (1) (3) + (2) (2) = 10.
Note that the sequence
y[n] = [16, 18, 16, 10]
is not the same as would result from linear convolution of the two sequences. Linear convo-
lution would give
x
1
[n]
*
x
2
[n] = [0, 1, 4, 10, 16, 17, 12].
It is seen that circular convolution yields a four-element sequence, whereas linear convolu-
tion yields seven elements.
The DFT and IDFT can be used to compute the circular convolution of the two se-
quences. From (12.47),
Y[k] = X
1
[k]X
2
[k],
where
X
1
[k] =
5
x
1
[n]
6
= [10, -2 + j2, -2, -2 - j2],
as found in Example 12.9, and
X
2
[k] =
5
x
2
[n]
6
= [6, -2 + j2, -2, -2 - j2].
The two discrete-frequency sequences are multiplied term by term to find
Y[k] = [(10) (6), (-2 + j2) (-2 + j2), (-2) (-2), (-2 - j2) (-2 - j2)]
= [60, -j8, 4, j8].
Then,
y[n] =
t
5
Y[k]
6
= [16, 18, 16, 10],
as was found from circular convolution of the discrete-time sequence.
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