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Properties of the Discrete Fourier Transform 179
a. MATLAB Script for 5-point circular convolution:
>> x1 = [1,2,2]; x2 = [1,2,3,4]; y = circonvt(x1, x2, 5)
y =
9 4 9 14 14
Hence
x 1(n) 5○ x 2(n) ={9, 4, 9, 14, 14}
b. MATLAB Script for 6-point circular convolution:
>> x1 = [1,2,2]; x2 = [1,2,3,4]; y = circonvt(x1, x2, 6)
y =
1 4 9 14 14 8
Hence
x 1(n) 6○ x 2(n) ={1, 4, 9, 14, 14, 8}
c. A careful observation of 4-, 5-, and 6-point circular convolutions from
this and the previous example indicates some unique features. Clearly, an
N-point circular convolution is an N-point sequence. However, some samples
in these convolutions have the same values, while other values can be
obtained as a sum of samples in other convolutions. For example, the first
sample in the 5-point convolution is a sum of the first and the last samples
of the 6-point convolution. The linear convolution between x 1(n) and x 2(n)
is given by
x 1(n) ∗ x 2(n) ={1, 4, 9, 14, 14, 8}
which is equivalent to the 6-point circular convolution. These and other
issues are explored in the next section.
□
8. Multiplication: This is the dual of the circular convolution property.
It is given by
DFT [x 1 (n) · x 2 (n)] = 1 N X 1(k) N○ X 2 (k) (5.41)
in which the circular convolution is performed in the frequency domain.
The MATLAB functions developed for circular convolution can also be
used here since X 1 (k) and X 2 (k) are also N-point sequences.
9. Parseval’s relation: This relation computes the energy in the frequency
domain.
E x =
N−1
∑
n=0
|x(n)| 2 = 1 N
N−1
∑
k=0
|X(k)| 2 (5.42)
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