Apply the Discrete Fourier Transform as a Matrix Multiplication in MATLAB

Question

0

I have a vector x of length N x 1, I need to perform the iDCT operation for it using MATALB. I performed that using the below method:

x = [1+i;1-i;-1+i;1+i];    %Assume N = 4
y = sqrt(4)*idct(x); 

Then I performed it using the other way such that:

x = [1+i;1-i;-1+i;1+i];    %Assume N = 4
D= dctmtx(4)/sqrt(4); 
y = D'*x;

I don't find the value of y equals in the two ways ?? However for the case of of FFT operations, that gives the same results!

Is there a special case for DCT operation?

Answer 1

Pay attention that by default MATLAB use DCT Type II hence the inverse is basically DCT Type III:

vX = [1 + 1i; 1 - 1i; -1 + 1i; 1 + 1i];    %Assume N = 4
vY = dct(vX); 
mD = dctmtx(length(vX));
vYY = mD * vX;

vYY ./ vY
max(abs(vY - vYY))

vY = idct(vX);
vYY = mD.' * vX;

vYY ./ vY
max(abs(vY - vYY))

The result:

ans =

   1.0000 + 0.0000i
   1.0000 + 0.0000i
   1.0000 - 0.0000i
   1.0000 + 0.0000i


ans =

   4.9651e-16


ans =

   1.0000 - 0.0000i
   1.0000 - 0.0000i
   1.0000 + 0.0000i
   1.0000 + 0.0000i


ans =

   4.5776e-16

So everything works as expected.

Pay attention that in your code using the transform you multiply by factor sqrt(4) and for the matrix form you divide by this factor.

So:

vY = sqrt(4) * idct(vX);
vYY = (mD' / sqrt(4)) * vX;

vY ./ vYY

ans =

   4.0000 + 0.0000i
   4.0000 + 0.0000i
   4.0000 + 0.0000i
   4.0000 - 0.0000i

So just don't divide and use the same factor and all is good.