Is there a difference in the output values of the functions fft vs fft2 vs fftn

When applying an FFT to a 3D matrix will the information in the output be different if one were to apply fftn to the entire matrix or if one were to use an fft2 on each slice of the 3D matrix and then stack up the output?

If one were to use the second approach would it be valide to directly compare the magnitudes in one slice with those from another, or would the relative magnitudes between frequency peaks only be valide within any given slice unless a normalization is applied?

To take it one step further if one were to apply a 1D fft on each dimension of a slice how could the two outputs be combined to produce the output provided by an fft2.

The documentation for fftn states "The N-D transform is equivalent to computing the 1-D transform along each dimension of X."

But I am unclear as to how the output of 1D transforms could be combined to give the output of the fft2 or fftn transforms, and if the multidimensional transforms offer any advantages.

Answer 1

[Walter Roberson]

For a 3d matrix, the code for fft2 is literally

fft(fft(x,[],2),[],1)

For a 2d matrix, fft2 calls fftn

The only remaining question is then whether calling that above fft code on a 2d array will give you the same result as calling fftn. The answer is that mathematically they are the same, but that fftn calls into a highly optimized multi-threaded library that might do operations in a different order and so might lead to slightly different roundoff. Just like 0.1+0.2-0.3 leads to different roundoff than 0.1-0.3+0.2

[Anthony Hosein]

Hello Mr. Roberson,

Thank you for your reply.

Would you also be able to tell me if the magnitudes obtained from an fft2 being applied to different slices of the 3D matrix be directly comparable, or would they first have to be normalized.

Thank you.

Anthony

[Walter Roberson]

They are directly comparable under the assumption that the means are the zero (and possibly other circumstances that I would have to think about more.)

However, there are cases where it can make sense to normalize the power first.

[Anthony Hosein]

Thank you.

Anthony