Fourier transform, range of indices

[Derivator]

hi,

could someone explain the following statement, please?

In the mathematical literature sums in Fourier transformation formulas typically run from
−N to N or N −1. In all numerical FFTs indices run from 0 to N −1. For all the real
data this just implies a shift whereas for data in Fourier space it means that the negative
frequencies are in the second half of the data set as shown below for the case of N=4:

x_0,x_1,x_2,x_3,x_4,x_{-3},x_{-2},x_{-1}​

Why is the real data only shifted, but the Fourier space data is 'wrapped around'?

The only difference should be: exp(k*x*2*i*Pi/N) in reals space vs. exp(-k*x*2*i*Pi/N) in Fourier space. Both have a periodicity of N. So why is there any difference?

 

[MisterX]

For the "real data" (I'm not sure I like this term), going from the described "mathematical literature" notation indices to the FFT indices, -N would become 0, -N + 1 would become 1, etc; this is a simple shift to make it start at 0.

In the Fourier domain for FFTs, the element with index 0 will be the 0 frequency element, then the positive frequency elements are next, and then the negative frequency elements will follow those, beginning with the most negative frequency. If you really want a rationale, having it this way makes some things easier then they might be otherwise. For example locating the 0 frequency element is easy as it is just the element with index 0. Also, converting the index of a positive frequency element to its corresponding frequency is easier this way. The frequency is just index/fstep instead of something such as (index - index0)/fstep.

 

[Derivator]

ah, so this index choice is just definition and follows not from any mathematical principle?

 

[MisterX]

Oops, those should be multiplications, not divisions.

 

[Derivator]

So this is just definition and is not due to any mathematical reason?