Compute fourrier transform in Julia

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:

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…and again be a bit careful, in the notation of said book (which I prefer) the inverse Fourier Transform has a factor of 1/N upfront, and (in theory) some prefer to also use the unitary DFT Discrete Fourier transform - Wikipedia or put the 1/N upfront the DFT instead of the inverse. But I think the one from the book should be the most common definition.