# 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. • plot your data on a linear frequency scale. the nulls are periodic in frequency
– user28715
Jun 22 '19 at 12:10
• At a sample rate of 0.25Hz and a delay of 2 seconds, there shouldn't be any null at all Jun 22 '19 at 12:11

• You are totally right!! The low sampling frequency should have had required a fractional delay. But I had further messed up the time delay part. In the code the time delay should be by something like $f_s \tau$ samples of the time series, where as I was doing $\frac{\tau}{f_s}$ samples. While this gave me an integer delay for the time series, it was of course the wrong time shift. I added a plot when I do the right time shift with a higher sampling frequency and things look okay. Thank you very much! Jun 22 '19 at 13:21