16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde (=d(1)2-o( which is convenient for the calculation of e(0) Also since o(0o=d(o-e(0, this says the optimum mean squared output always less than the mean squared desired output. Autocorrelation functions We have arrived at an extended form of the Wiener-Kopf equation which defines the optimum linear system under the ground rules stated before Recall that. R (r)=r()+R(r)+r(r)+r(t) R2(r)=R2(r)+Rn() The free configuration problem is a specialization of the semi-free configuration In this expressic would take F(s)=l, or w(0)=8(0). In that case we have ∫d:d()Jdrw(x)4;(r,R(r r3-4 dz2(2)dr3w(3)R2(x1+22-3)= Wo(T),(T-T)dr, -wp(I,)R (T-T3)dr=0 for T, 2016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 5 2 22 0 0 et dt ot () () () = − which is convenient for the calculation of 2 0 e t( ) . Also since 2 22 0 0 ot dt et () () () = − , this says the optimum mean squared output is always less than the mean squared desired output. Autocorrelation Functions We have arrived at an extended form of the Wiener-Kopf equation which defines the optimum linear system under the ground rules stated before. Recall that: () () () () () () () () ii ss sn ns nn is ss ns RRR RR RRR τ ττττ τττ =+++ = + since i sn = + . The free configuration problem is a specialization of the semi-free configuration. In this expression we would take F s() 1 = , or () () wt t F = δ . In that case we have 2 2 30 3 4 4 1 2 3 4 22 3 3 123 03 1 3 3 3 1 3 3 1 () () () ( ) () () ( ) ( ) ( ) ( ) ( ) 0 for 0 ii D is ii D is dd dw d R d dw R wR d w R d τ τ τ τ τδτ τ τ τ τ τδτ τ τ τ τ τ τ τττ τ τττ τ ∞∞ ∞ −∞ −∞ −∞ ∞ ∞ −∞ −∞ ∞ ∞ −∞ −∞ +−− − +− = −− −= ≥ ∫∫ ∫ ∫ ∫ ∫ ∫