3

Consider a white noise process $x(t)$ defined by the Dirac Delta function $\delta$, were

$$cov(x(t),x(t_1))=\delta(t-t_1)$$

What is the variance of this noise process ? We have that $$cov(x(t),x(t))=\delta(t-t)=\delta(t-t)=+\infty$$but I know the dirac delta is a heuristic function and I was wondering if the variance is defined differently, since I saw in a paper that they claim the variance here is 1.

raK1
  • 275
  • You are aware that the object you call white noise has no rigorous mathematical definition? – Did May 09 '17 at 20:24
  • @Did No i was not. So how can we calculate or define variance in the process above ? or how do people typically do it ? – raK1 May 09 '17 at 20:26
  • 2
    To a physicist, the power spectral density of a white noise is nonzero and constant hence its total power (its integral over all frequencies) is infinite. This is the variance of your $x(t)$. – Did May 09 '17 at 20:35
  • Worth checking: https://math.stackexchange.com/q/1154940/532409 https://math.stackexchange.com/q/3463795/532409 https://math.stackexchange.com/q/3277965/532409 – Quillo Feb 25 '23 at 16:22

0 Answers0