Complex number and fft

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?

Answer 1

[David Goodmanson]

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.

[Nmak]

Thank you for your response. Is there an easy workaround to avoid these issues? And do you know, why Matlab sometimes stores values as "0", sometimes as "0.000000000000000 + 0.000000000000000i", despite being in the same Matrix?

[David Goodmanson]

When you have a complex array, then two memory location are allocated for each element, whether that element has a a nonzero imaginary part or not. So:

A = 0:4
W = [0, 2+3i, 5, 0, 3+1i];
whos A W
Name      Size            Bytes  Class     Attributes

  A         1x5                40  double              
  W         1x5                80  double    complex

At eight bytes per real number for double precision, you can see two real numbers allocated for each element of W. Looking just at a single real element of W,

  W(1)
  ans = 0

there is no need to report out a zero imaginary part.