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Properties of the Discrete Fourier Transform 177
Hence
x 1(n) 4○ x 2(n) ={15, 12, 9, 14}
• Frequency-domain approach: In this approach we first compute 4-point DFTs
of x 1(n) and x 2(n), multiply them sample by sample, and then take the
inverse DFT of the result to obtain the circular convolution.
DFT of x 1(n)
DFT of x 2(n)
Now
x 1(n) ={1, 2, 2, 0} =⇒ X 1(k) ={5, −1 − j2, 1, −1+j2}
x 2(n) ={1, 2, 3, 4} =⇒ X 2(k) ={10, −2+j2, −2, −2 − j2}
Finally after IDFT,
X 1(k) · X 2(k) ={50, 6+j2, −2, 6 − j2}
which is the same as before.
x 1(n) 4○ x 2(n) ={15, 12, 9, 14}
□
Similar to the circular shift implementation, we can implement the
circular convolution in a number of different ways. The simplest approach
would be to implement (5.39) literally by using the cirshftt function
and requiring two nested for...end loops. Obviously, this is not efficient.
Another approach is to generate a sequence x ((n − m)) N
for each n in
[0,N − 1] as rows of a matrix and then implement (5.39) as a matrixvector
multiplication similar to our dft function. This would require
one for...end loop. The following circonvt function incorporates these
steps.
function y = circonvt(x1,x2,N)
% N-point circular convolution between x1 and x2: (time-domain)
% -------------------------------------------------------------
% [y] = circonvt(x1,x2,N)
% y = output sequence containing the circular convolution
% x1 = input sequence of length N1 = the length of x1’)
end
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