# Fourier transform, range of indices

1. Jun 22, 2011

### Derivator

hi,

could someone explain the following statement, please?

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?

2. Jun 24, 2011

### 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.

3. Jun 24, 2011

### Derivator

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

4. Jun 24, 2011

### MisterX

Oops, those should be multiplications, not divisions.

5. Jun 25, 2011

### Derivator

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