Say we have stochastic differential equation
$\frac{dx}{dt} = n(t)$
where $n(t)$ is a noise process.
$n(t)$ has a correlation function $R(t – t') = <n(t)n(t')>$
If the noise process is white noise then we have
$<n(t)n(t')> = \delta(t – t')$
where $\delta$ is the dirac delta function.
I don't understand what this means? I would imagine it is supposed to imply that there is no correlation between the value of $n$ at $t$ and $t'$, simply because its called 'white noise'. But I don't see how the dirac delta function gives us that?
Best Answer
Correlation is an index to show how much shape of two functions are similar.
Ideal noise is similar to nothing. Otherwise it is not noise.
$<n(t),n(t')> = \delta(t - t')$
Means that if you shift a white noise it has zero correlation to itself. And if you don't shift it, it is identical to itself.