I was reading a lot and wrote some stuff on the computer to make the DFT (discrete Fourier transform) in several ways.
My problem is that many errors are in literature. I gave up watching the scaling factors, because the flood of conventions is too big. The worst problem is that in some sources there is minus in front of the "i" and in others not.
Matlab does it with minus in Front and Mathematica does it without. Cambridge "Numerical Recipes” does it without minus.
It isn't the same thing! Sure the magnitude spectrum, the power spectrum and the absorption spectrum stay the same but the dispersion spectrum depends on if there is a minus or not.
(Dispersion Spectrum is Im(X) in function of frequency. You can do cosine Fourier transform as well to obtain it.)
With the knowledge how the Dispersion Spectrum should look like, there is no more to say that there has to be a minus in front of the i. So Mathematica and NR do it wrong.
I am chemist and there is something called IUPAC, which defines a lot of useful rules and conventions. Is there something similar in mathematics? A Group which make rules?
Can anybody tell me how complex DFT really is defined, and whether I am right that Mathematica is wrong.