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.

  • $\begingroup$ Is $\tau$ fixed and deterministic? $\endgroup$
    – Matt L.
    Jun 11 '14 at 9:26
  • 2
    $\begingroup$ To little informations about your signal to give specific answer, but you might find this useful: Random Rectangular Pulse Train ACF $\endgroup$
    – jojek
    Jun 11 '14 at 9:35
  • 1
    $\begingroup$ @physicsGuy: Most importantly please answer Matt's question - it's crucial for understanding. $\endgroup$
    – jojek
    Jun 11 '14 at 9:54
  • 2
    $\begingroup$ Apart from all the other matters discussed in the comments, a very important issue is "What is meant by At random times jumps to...?" There is no such thing as a random number, or a positive random number, unless the distribution is specified: the common response that "random means that all values are equally likely to be chosen" does not work. For example, in the link pointed out by @jojek, the random time is uniformly distributed on $[-T/2,T/2]$. For the problem posed here, "random means $U(0,T)$ gives vastly different results than "random means $\text{Exponential}(\lambda)$". $\endgroup$ Jun 11 '14 at 11:42
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    $\begingroup$ It might be fruitful to consider your signal as a convolution of a boxcar (rectangular) filter with a signal of Dirac delta pulses. The time between delta pulses can be no less than $\tau$ $\endgroup$ Jun 11 '14 at 13:07

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