I am trying to find the analytic expression of the autocorrelation function for a signal $X(t)$ which is defined as follows.
- $X(t = 0) = 1$
- At random times jumps to $X(t) = 0$ happen
- The jump probability in each infinitesimal time-step is $\Gamma dt$
- After the jump the signal stays at $0$ for time $\tau$
- Then the signal jumps back to $1$
Do you know how or even if it is possible to calculate the autocorrelation function of such a signal?
The signal should describe the following scenario:
Imagine you have a house with a window. In front of the window there is a railway where trains can pass. Each train takes a fixed time $\tau$ to pass the window and trains arrive at random times. Assume that I am pointing a laser across the railway trough the window. If I can see the laser $X(t) = 1$ else the laser is hidden by the train and $X(t) = 0$. I want to calculate the autocorrelation function of such a signal.