Question 24.30: Use the convolution theorem and a computer package which cal......
Use the convolution theorem and a computer package which calculates d.f.t.s to find the circular convolution of the sequences f[n] = 1, 2, -1, 7 and g[n] = -1, 3, 2, -5.
You will need access to a computer package such as MATLAB to work through this example. The d.f.t. of f[n] can be calculated either directly, which is laborious, or using the MATLAB command fft().
Script File
>F=fft([1 2-17])
ans =
9.0000 2.0000+ 5.0000i -9.0000 2.0000- 5.00001
Hence F[k] = 9,2 + 5j, -9, 2 - 5j.
Similarly,
>G=fft([-1 3 2-5])
ans =
-1.0000 -3.0000- 8.00001 3.0000 -3.0000+ 8.00001
Hence G[k] = -1, -3 - 8j, 3, -3 + 8j. Then, the product of these d.f.t.s is calculated by multiplying corresponding terms together:
F[k] G[k]=-9,34-31 \mathrm{j},-27,34+31 \mathrm{j}
In MATLAB,
product = F.*G
product =
-9.0000 34.0000-31.00001 -27.0000 34.0000+31.00001
Finally, taking the inverse d.f.t., using the MATLAB command ifft(), gives
> ifft([-9 34-311 -27 34+311])
ans =
8 20 -26 -11
This is the circular convolution of f[n] and g[n], that is
f[n] \circledast g[n]=8,20,-26,-11
This example has illustrated how circular convolution can be achieved through the use of the d.f.t.