Random processes The auto-correlation of a random process x(t is defined as Rxx(t, t2=E[x(t x(t2) A random process is wide-sense-stationary (WsS) if its mean and auto-correlation are not a function of time that is m()=E【(=m Rx(1,+2)=R(T, whereτ=t少t2 If x(t)is WSS then: RA(T=R-可 IRx )<=Rx(ol(max is achieved at t=0) The power content of a wSS process is: cT/2 T/2 P= ellit x(tdt= lim R2(0)dt=R2(0) 1→∞TJ-7/2 1→∞TJ-7/2τ τ τ τ τ 0 τ Random Processes • The auto-correlation of a random process x(t) is defined as – Rxx(t1,t2) = E[x(t1)x(t2)] • A random process is Wide-sense-stationary (WSS) if its mean and auto-correlation are not a function of time. That is – mx(t) = E[x(t)] = m – Rxx(t1,t2) = Rx(τ), where τ = t1-t2 • If x(t) is WSS then: – Rx(τ) = Rx(-τ) – | Rx(τ)| <= |Rx(0)| (max is achieved at τ = 0) • The power content of a WSS process is: 1 T / 2 1 T / 2 Px = E[lim 2 ( ) t→∞ T ∫−T / 2 Rx (0)dt =Rx (0) t→∞ T ∫−T / 2 x t dt = lim Eytan Modiano Slide 3