I'm trying to find out how to "properly" fourier-transform a velocity-auto-correlation function (VAC), for calculation of phonon density of states (DOS), from molecular dynamics simulation. The problem I'm running into is that my calculated DOS: a) doesn't fall of to zero as the frequency goes to zero (i.e. as ω → 0) b) doesn't have a sharp fall-off at some cut-off frequency. I've tried to read up and found two ways to proceed: 1) Pad the signal, i.e. add suitable number of zeros to the end of the velocity data. This didn't influence a) or b) but only decreased the amplitude of the transformed VAC. 2) DFFT assumes periodic signal. So I to replicated and "reflected" my velocity array in two ways to obtain a periodic signal: 2.1) First doing it "oddly" so that the end-point and the starting point are continuous and have continuous first derivative. --> seem to give nice ω → 0 behavior but influences high frequency in unsatisfactory way. 2.2) Secondly doing it "evenly" so that the end-point and the starting point are continuous but dosen't have continuous first derivative. --> better result for high frequencies but no improvement for ω → 0. I think the origin to my problem is due to that a VAC have VAC(t→0)=1, VAC(t→∞)=0, and matching these two conditions with the assumption of periodicity in DFFT gives raise to a lot of overtones to try to capture the steplike jump. Calculating VAC and FFTs thereof is very well established but I haven't been able to find any similar problem description. Any suggestion for resources to read? Experience from similar problems? I'm grateful for all input!