Autocorrelazione dell'uscita di un filtro

Nel capitolo è stato affermato che, quando un processo attraversa un filtro, il processo di uscita è caratterizzato da $\mathcal {R}$_y$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$ = $\mathcal {R}$_x$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*$\mathcal {R}$_H$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$. Mostriamo che è vero.

$\mathcal {R}$_y$\left(\vphantom{ t,t+\tau }\right.$t, t + $\tau$ $\left.\vphantom{ t,t+\tau }\right)$ = E$\left\{\vphantom{ y\left( t\right) y\left( t+\tau \right) }\right.$y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$y$\left(\vphantom{ t+\tau }\right.$t + $\tau$ $\left.\vphantom{ t+\tau }\right)$ $\left.\vphantom{ y\left( t\right) y\left( t+\tau \right) }\right\}$ = E$\left\{\vphantom{ \int h\left( \alpha \right) x\left( t-\alpha \right) d\alpha \int h\left( \beta \right) x\left( t+\tau -\beta \right) d\beta }\right.$$\int$h$\left(\vphantom{ \alpha }\right.$$\alpha$ $\left.\vphantom{ \alpha }\right)$x$\left(\vphantom{ t-\alpha }\right.$t - $\alpha$ $\left.\vphantom{ t-\alpha }\right)$d$\alpha$$\int$h$\left(\vphantom{ \beta }\right.$$\beta$ $\left.\vphantom{ \beta }\right)$x$\left(\vphantom{ t+\tau -\beta }\right.$t + $\tau$ - $\beta$ $\left.\vphantom{ t+\tau -\beta }\right)$d$\beta$ $\left.\vphantom{ \int h\left( \alpha \right) x\left( t-\alpha \right) d\alpha \int h\left( \beta \right) x\left( t+\tau -\beta \right) d\beta }\right\}$ =

= $\int$$\int$h$\left(\vphantom{ \alpha }\right.$$\alpha$ $\left.\vphantom{ \alpha }\right)$h$\left(\vphantom{ \beta }\right.$$\beta$ $\left.\vphantom{ \beta }\right)$E$\left\{\vphantom{ x\left( t-\alpha \right) x\left( t+\tau -\beta \right) }\right.$x$\left(\vphantom{ t-\alpha }\right.$t - $\alpha$ $\left.\vphantom{ t-\alpha }\right)$x$\left(\vphantom{ t+\tau -\beta }\right.$t + $\tau$ - $\beta$ $\left.\vphantom{ t+\tau -\beta }\right)$ $\left.\vphantom{ x\left( t-\alpha \right) x\left( t+\tau -\beta \right) }\right\}$d$\alpha$d$\beta$ = (se x$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ è stazionario)

= $\int$h$\left(\vphantom{ \alpha }\right.$$\alpha$ $\left.\vphantom{ \alpha }\right)$$\int$h$\left(\vphantom{ \beta }\right.$$\beta$ $\left.\vphantom{ \beta }\right)$$\mathcal {R}$_x$\left(\vphantom{ \tau +\alpha -\beta }\right.$$\tau$ + $\alpha$ - $\beta$ $\left.\vphantom{ \tau +\alpha -\beta }\right)$d$\beta$d$\alpha$ = $\int$h$\left(\vphantom{ \alpha }\right.$$\alpha$ $\left.\vphantom{ \alpha }\right)$$\mathcal {R}$_xy$\left(\vphantom{ \tau +\alpha }\right.$$\tau$ + $\alpha$ $\left.\vphantom{ \tau +\alpha }\right)$d$\alpha$ = $\mathcal {R}$_xy$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*h$\left(\vphantom{ -\tau }\right.$ - $\tau$ $\left.\vphantom{ -\tau }\right)$, in cui

$\mathcal {R}$_xy$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$ = $\mathcal {R}$_x$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*h$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$ è l'intercorrelazione tra x$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ ed y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$.

Scritto in altra forma: $\mathcal {R}$_y$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$ = $\mathcal {R}$_x$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*h$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*h$\left(\vphantom{ -\tau }\right.$ - $\tau$ $\left.\vphantom{ -\tau }\right)$, e dunque antitrasformando si ottiene $\mathcal {P}$_y$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ = $\mathcal {P}$_x$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ ^. H$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ ^. H^*$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ = $\mathcal {P}$_x$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ ^. $\left\vert\vphantom{ H\left ( f\right ) }\right.$H$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ $\left.\vphantom{ H\left( f\right) }\right\vert^{2}_{}$ = $\mathcal {F}$$\left\{\vphantom{ \mathcal{R}_{x}\left( \tau \right) *\mathcal{R}_{H}\left( \tau \right) }\right.$$\mathcal {R}$_x$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$*$\mathcal {R}$_H$\left(\vphantom{ \tau }\right.$$\tau$ $\left.\vphantom{ \tau }\right)$ $\left.\vphantom{ \mathcal{R}_{x}\left( \tau \right) *\mathcal{R}_{H}\left( \tau \right) }\right\}$.