UploadFile_6417

Properties of the Discrete Fourier Transform 175
10
8
Original Sequence
x(n)
6
4
2
0
0 5 10 15
n
Circularly Shifted Sequence, N=15
10
x((n-6) mod 15)
8
6
4
2
0
0 5 10 15
n
FIGURE 5.18 Circularly shifted sequence in Example 5.12
that contains a circular shift is called the circular convolution and is
given by
x 1 (n) N○ x 2 (n) =
N−1
∑
m=0
x 1 (m)x 2 ((n − m)) N
, 0 ≤ n ≤ N − 1 (5.39)
Note that the circular convolution is also an N-point sequence. It has
a structure similar to that of a linear convolution. The differences
are in the summation limits and in the N-point circular shift. Hence
it depends on N and is also called an N-point circular convolution.
Therefore the use of the notation N○ is appropriate. The DFT property
for the circular convolution is
[
]
DFT x 1 (n) N○ x 2 (n) = X 1 (k) · X 2 (k) (5.40)
An alternate interpretation of this property is that when we multiply
two N-point DFTs in the frequency domain, we get the circular
convolution (and not the usual linear convolution) in the time domain.
□ EXAMPLE 5.13 Let x 1(n) = {1, 2, 2} and x 2(n) = {1, 2, 3, 4}. Compute the 4-point circular
convolution x 1(n) 4○ x 2(n).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.