EE4206/EE5806 Digital Image Processing : Assignment Q3

EE4206/EE5806 Digital Image Processing : Assignment Q3

3. An FFT computer program is available for computing the DFT of 1D real data.

(a) Show how such a program could be used to compute the DFT of 1D complex data.

ans : by using Euler's equation : e^-jx = cos(x) - jsin(x), the real part is cos(x), the imaginary part is -jsin(x).

First, we square the real part , ie., cos^2(x).

Second, use 1 to minus it. 1 - (cos^2(x) ).

Since cos^2(x) + sin^2(x) =1, therefore, 1 - cos^2(x) = sin^2(x). We then take the square root and negate it, this is the imaginary part of the DFT.

j = sqrt( sin^2(x) );
j = sin(x);
j = -sin(x);


(b) Show how such a program could be used to compute the DFT of a 2D digital image whose pixels are real numbers. (20 marks)

ans : The 2-D DFT of f(x,y) can be obtained by computing the 1-D transform of EACH row of f(x,y) and then computing the 1-D transform along EACH column of the result.

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