Fourier transforms and time shift There is probably something trivial behind this, but I am missing something. I need to create a stationary random time series data v(t) which is the the sum of another time series u(t) and u(t) with a time shift $\tau$, i.e

$v(t) = u(t) + u(t - \tau)$

The series u(t) is itself a stationary colored Gaussian time-series with some PSD $S_u(f)$. No problem! I wrote a small function (using Python) to make u(t) for a given PSD. I calculate the PSD of the the generated time series using Welch's method and it checks out. 



If we take the Fourier transform of u(t - \tau) it should just gain an extra phase compared to the Fourier transform of u(t)

$F[u(t - \tau)] = \int dt \, u(t -\tau) \, e^{2 \pi j f t } =  \int dt \, u(t) \, e^{2 \pi j f t }  e^{- 2 \pi j f \tau } = \tilde{u}(f) e^{- 2 \pi j f \tau }$

Since $v(t) = u(t) + u(t - \tau)$,  

$\tilde{v}(f) = \tilde{u} (f) \, (1 + e^{- 2 \pi j f \tau })  $

And the PSD of V should then be (using Euler identity)

$S_v (f) = 4 S_u (f) \cos^2 (\pi f \tau )$ 

In particular $S_v$ should vanish when $f = \frac{2n + 1}{2 \tau}$ with the smallest zero being at $\frac{1}{2 \tau}$. However when I test my this with the u(t) series I generated above the PSD looks different. There are zeros low frequencies which shouldn't exist. For example  here is the PSD of the v(t) series and the spectra I would expect for $\tau = 2$. (The first zero should be at 0.25 Hz)



Moreover the zeros in the data seems to change with the sampling frequency I am using. The v(t) series for the above plot has a sampling frequency of 0.25 Hz. Here is one with 0.5 Hz, keeping everything else the same. 



I am kind of baffled by this. Any help would be greatly appreciated. Sorry about the long post, and thanks in advance!!

EDIT: After using a higher sample rate as suggested by Hilmar below, and fixing the error I described in the comment everything looks good. Spectrum for the fixed case below.
Your math is correct, for a delay of 2 seconds, the first null in the PSD should be at 0.25 Hz.

I'm guessing the problem is in the code you use to generate the signals and/or the PSD. The sample rates seem awfully low. At a sample rate of 0.25 Hz, your 2 second delay is just half a sample, that means you would have to implement a fractional delay to generate the signal. T