Also I might leave these formula I found in the recommended book (Numerical Fourier Analysis | SpringerLink ) here, maybe they are usefull to someone encountering the same problem.

The discrete fourrier transform is given by:

Whereas if I want to actually want to approximate the fourrier transform from a dataset, it is given by:

where \omega_{nN}^{jk} = \exp(i \dots)

so we could see which shifts and normalisation one would need to perform.

here is the formula for the DFT used by fftw: