1
$\begingroup$

I have a stochastic process completely uncorrelated. Why the autocorrelation function has a delta of dirac in the origin? Which is the reason of that?

$R_{XX}({\tau})=A{\delta(\tau)}$

where $R_{XX}$ is the autocorrelation function of $X$ process and $A$ is a constant dependent by process spectrum

$\endgroup$
3
$\begingroup$

It has a delta at the origin because you are multiplying your time series with itself (i.e. there is no time delay). Any non-zero time series correlated with itself will give a positive value.

| improve this answer | |
$\endgroup$
1
$\begingroup$

As you noted, the wide-sense stationary stochastic process $X(t)$ is uncorrelated with itself for all nonzero time lags $\tau$. For $\tau=0$, it's impossible for a process to be uncorrelated with itself (unless $X(t)$ is the trivial process $X(t)=0$). Look at the definition for $R_{XX}(\tau)$:

$$ R_{XX}(\tau) = E(X(t)X^*(t+\tau) $$

$$ R_{XX}(0) = E(|X(t)|^2) $$

By definition, $|X(t)|^2$ must be $\ge 0$, so its expectation must also satisfy that condition.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.