16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Lecture 15 ast time: Compute the spectrum and integrate to get the mean squared value F(SF(sS.(s)d Cauch-Residue Theorem 手F(s)d=2/∑ (residue at enclosed poles, lote that in the case of repeated roots of the denominator, a pole of multiple order contributes only a single residue To evaluate F(s)ds by integrating around a closed contour enclosing the entire left half plane, note that if F(s)>0 faster than- for large s, the integral along the curved part of the contour is zero. HF(s)~asl→,∮F(s)≤nR=kxR→0aR→n>1 pplicable to rational functions; no predictor or smoother. Must factor the spectrum of the input into the following form Refer to the handout Tabulated values of the Integral Form Roots of c(s) and d(s) in left half plane only. Should check the stability of the solution Page 1 of 2
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 2 Lecture 15 Last time: Compute the spectrum and integrate to get the mean squared value 2 1 () ( ) () 2 j xx j y F s F s S s ds π j ∞ − ∞ = − ∫ Cauchy-Residue Theorem >∫ F s ds j ( ) 2 (residue at enclosed poles) = π ∑ Note that in the case of repeated roots of the denominator, a pole of multiple order contributes only a single residue. To evaluate ( ) j j F s ds ∞ − ∞ ∫ by integrating around a closed contour enclosing the entire left half plane, note that if F s() 0 → faster than 1 s for large s , the integral along the curved part of the contour is zero. If ( )~ n k F s s as s → ∞ , ( 1) semi-circle ( ) 0 as if 1 n n k F s ds R k R R n R π π − − ≤ = → →∞ > >∫ Integral tables Applicable to rational functions; no predictor or smoother. Must factor the spectrum of the input into the following form. 1 ()( ) 2 () ( ) j n j csc s I ds π j d sd s ∞ − ∞ − = − ∫ Refer to the handout “Tabulated Values of the Integral Form”. Roots of c s( ) and d s( ) in left half plane only. Should check the stability of the solution
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 2
16.322 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 ∞ = − = ∫
