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I want to calculate the autocorrelation function $R_{xx}$ of the signal $X\in\{-1,1\}$ that jumps with probability p between the two values:

\begin{equation*}X_{k+1} = \begin{cases} X_k &1-p \\ -X_k & p \end{cases}\end{equation*}

In addition the time step $\Delta t_{k+1} = t_{k+1} -t_k$ varies according to

\begin{equation*}\Delta t_{k} = \begin{cases} \Delta t &\mbox{if }\ X_k = 1 \\ \alpha \Delta t &\mbox{if }\ X_k = -1 \end{cases}\end{equation*}

where $\alpha \in \mathbf{N}$. Do you know how to calculate the autocorrelation function of such a process?

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  • $\begingroup$ Your definition of the process needs clarification. The process is a continuous-time process whose value at any time $t$ depends on its past history and which bit is being worked on right now. But you are defining it as a discrete-time process with $k$ denoting an integer in which case the time step $\Delta t$ is irrelevant. Do you want the autocorrelation of the discrete-time process or the continuous-time process? $\endgroup$ – Dilip Sarwate Jun 10 '14 at 13:54
  • $\begingroup$ I changed alpha to be an integer number, I guess that was the problem. This means that the signal stays longer in the -1 state. In the limiting case $\alpha \rightarrow \infty$ at least the average should change to $-1$ as the signal would almost always display a $-1$. Does this clarify what you meant? $\endgroup$ – physicsGuy Jun 10 '14 at 14:16
  • $\begingroup$ You could also reformulate my definition of the process so that if a jump from $1$ to $-1$ occurs or the signal stays in $-1$ the next $\alpha$ steps will also be $-1$. After those $\alpha$ steps a jump can again happen with probability $p$. $\endgroup$ – physicsGuy Jun 10 '14 at 14:19
  • $\begingroup$ A random process is a collection of random variables $\{X_t \colon t \in \mathbb T\}$ where the index set $\mathbb T$ is usually the set of integers (or the set of nonnegative integers) or the real line (or the positive real line), etc. You need to tell us what $\mathbb T$ is. You specify $X_k$ where $k$ is clearly meant to be an integer, but you talk of switching times which may not be integer-valued, and so the process may change value from $+1$ to $-1$ at $\Delta t$ or it might stay the same. In one case, the next possible transition comes at $\Delta t + \alpha\Delta t$ while .... $\endgroup$ – Dilip Sarwate Jun 10 '14 at 14:29
  • $\begingroup$ ... in the other case, the next possible transition comes at $2\Delta t$. So, it seems that the random variables $X_t$ of the process need to be defined not just at integer values of $t$ as you have done but also for other time instants, and indeed the value at any time $t$ will depend on the past history of the process. So, pick your favorite values of $\alpha$ and $\Delta t$ and tell us what the value of the process is at time $t=3.1416$. I am voting to (temporarily) close this question because it is unanswerable as it stands. $\endgroup$ – Dilip Sarwate Jun 10 '14 at 14:33

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