16.322 Stochastic Estimation and Control, Fall 2004 Prof Vander Velde S(o)=li X(OY(o) S_(o)=lim Yr(o)Xr (o) 2T We note from this that 0)= so that the sum of these two as they appear in S(o)is real Also note that S(o)=S(@)=S(o) Examples of Random Processes Analytically Defined Example: Randon step function Amplitude am independent, rando Change points t,, Poisson-distributed with average density a(points per secon (t) P(k)=(x1)e P(0)=e Rx(r)=E[x(1)x(+) P(at least one change point in a+P(no change point in rpa 2a2+e-da a2e-4+a2 Page 5 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 * * () () ( ) lim 2 () () ( ) lim 2 T T xy T T T yx T X Y S T Y X S T ω ω ω ω ω ω →∞ →∞ = = We note from this that: * () () yx xy S S ω = ω so that the sum of these two as they appear in ( ) zz S ω is real. Also note that * ( ) () () xy xy yx S SS −= = ω ω ω . Examples of Random Processes Analytically Defined Example: Random step function Amplitude n a independent, random Change points nt , Poisson-distributed with average density λ (points per second) ( ) 1 ( ) ! (0) k Pk e k P e λτ λ τ λ τ − − = = [ ] ( ) 2 2 2 2 2 2 ( ) () ( ) (at least one change point in ) (no change point in ) 1 xx a R E xtxt P aP a e aea e a λτ λτ λ τ τ τ τ τ σ − − − = + = + =− + = +