16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde For ergodic processes E[x(o]=limx(dt x2=E[x()]=lim[x(dt R(r)=E[(ox(+r]=lim x(Ox(t+r)dt →2T R,()=E[(On(+D)]=lim ,T ]x(O)y (+rdr a time invariant system may be defined as one such that any translation in time le input affects the output only by an equal translation in time System This system will be considered time invariant if for every r, the input u(t+r) causes the output v((+r). note that the system may be either linear or non- linear It is proved directly that if u(t) is a stationary random process having the ergodic property and the system is time invariant, then y(() is a stationary random process having the ergodic property, in the steady state. This requires the system to be stable, so a defined steady state exists, and to have been operating in the presence of the input for all past time Example: Calculation of an autocorrelation function semble: x(0=Asin(at +0) B, A are independent random variable 0 is uniformly distributed over 0, 2T This process is stationary(the uniform distribution of 0 hints at this)but not ergodic. Unless we are certain of stationarity, we should calculate: Page 4 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 For ergodic processes: [ ] [ ] [ ] 22 2 1 ( ) lim ( ) 2 1 [ ( ) ] lim ( ) 2 1 ( ) ( ) ( ) lim ( ) ( ) 2 1 ( ) ( ) ( ) lim ( ) ( ) 2 T T T T T T T xx T T T xy T T x E x t x t dt T x E x t x t dt T R E x t x t x t x t dt T R E x t y t x t y t dt T ττ τ τ τ τ →∞ − →∞ − →∞ − →∞ − = = = = = += + = += + ∫ ∫ ∫ ∫ A time invariant system may be defined as one such that any translation in time of the input affects the output only by an equal translation in time. This system will be considered time invariant if for every τ , the input u t( ) +τ causes the output y t( ) +τ . Note that the system may be either linear or non￾linear. It is proved directly that if u t( ) is a stationary random process having the ergodic property and the system is time invariant, then y t( ) is a stationary random process having the ergodic property, in the steady state. This requires the system to be stable, so a defined steady state exists, and to have been operating in the presence of the input for all past time. Example: Calculation of an autocorrelation function Ensemble: xt A t ( ) sin( ) = + ω θ θ, Aare independent random variables θ is uniformly distributed over 0,2π This process is stationary (the uniform distribution of θ hints at this) but not ergodic. Unless we are certain of stationarity, we should calculate: