MATLAB: Complex number and fft

complex numberfft

So I noticed during processing some images in matlab, that the angle phase images after fft+ifft are not the same as the original anymore. When looking at the matrix, I saw that Matlab stores values sometimes as "0", sometimes as "0.000000000000000 + 0.000000000000000i". I was wondering, what is the difference? Isn´t the whole matrix supposed to contain complex numbers?
I also tried a simpler example myself
V = [0, 2];
W = [0, 2+3i, 5, 0, 3+1i];
fft_V = fft(V);
fft_W = fft(W);
vv = ifft(fft_V);
fs_V = fftshift(fft(V));
vvv = ifft(ifftshift(fs_V));
ww = ifft(fft_W);
fs_W = fftshift(fft_W);
www = ifft(ifftshift(fs_W));
w2 = ifft(fft(ww));
While for V everything is normal, 2 things about W make me wonder:
"www" is the same as W, but why are the signs in front of the "0"s inverted? (from + to -)
If one checks "w2 == ww", Matlab returns: "[0 1 1 1 1]", which seems weird. All 5 values are the same, but somehow, the first entry of the vector returns false?

Best Answer

Hi Nmak,
you are just getting into standard numerical precision issues. The fft and ifft involove complex variable calculations. Getting things to agree in double precision after a bunch of such calculations doesn't always work exactly. For example
ww(2:end)-w2(2:end)
ans =
   0     0     0     0
These elements are truly equal.
w(1)-w2(1)
ans =
   0 + 8.8818e-17i
not quite equal. Hence the results of the ww==w2 test.
Sometimes what you see is the result of formatting,
>> W(1)
ans =
     0
>> format long
>> W
W =
  Columns 1 through 2
  0.000000000000000 + 0.000000000000000i  2.000000000000000 + 3.000000000000000i
  Columns 3 through 4
  5.000000000000000 + 0.000000000000000i  0.000000000000000 + 0.000000000000000i
  Column 5
  3.000000000000000 + 1.000000000000000i
W(1) is still 0 of course, but since the some of the other elements of W are complex, W(1) is listed in a 0 + 0i format.
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