16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Application to the problem of System Identification (t) y(t) Record x(o)and y(o) and process that data y(o=w(,)x(t-r,dr,+w(r, )d(t-T)dr R(r)=E[x()+)]=x()y(+z) w,(t)x(y(+t-t, dt, +wa(r )x(d(+T-T,Xr, If x(o), d(t)are independent and at least x(t) is zero mean, then rd(r)=0 R(r)=w,()R(r-T)dr, If x(t) is wide band relative to the system, approximate it as white Rx()=S6(r) hite app tion 3()=」w:(x1)S列(x-1)dr Sw(T) Page 2 of 216.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 2 Application to the problem of System Identification Record x( )t and y t( ) and process that data. [ ] 1 11 1 11 0 0 1 11 1 11 0 0 () ( ) ( ) ( ) ( ) ( ) () ( ) () ( ) ( ) () ( ) ( ) () ( ) x d xy x d yt w xt d w dt d R E xt yt xt yt w xt yt d w xtdt d τ ττ τ ττ τ ττ τ ττ τ τ ττ τ ∞ ∞ ∞ ∞ = −+ − = += + = +− + +− ∫ ∫ ∫ ∫ If x( ), ( ) t dt are independent and at least x( )t is zero mean, then () 0 Rxd τ = . 1 11 0 () ( ) ( ) Rxy x xx τ wR d τ ττ τ ∞ = − ∫ If x( )t is wide band relative to the system, approximate it as white. () () R S xx x τ = δ τ 1 11 0 () ( ) ( ) ( ) xy x x x x R wS d S w τ τ δτ τ τ τ ∞ = − = ∫