16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde The Wiener-Hopf equation requires l([)=0 for r,20. Thus L(s)should be analytic in LHP and go to zero at least as fast as for large Isl L(S=F()()F(SS(s)-F-S)D(s)Sis(s) We have solved the problem of the optimum linear filter under the least mean squared error criterion Further analysis shows that if the inputs, signal and noise, are Gaussian, the result we have is the optimum filter. This is there is no filter linear or nonlinear which will yield smaller mean squared error If the inputs are not both Gaussian, it is almost sure that some nonlinear filters can do better than the Wiener filter. But theory for this is only beginning to be developed on an approximate basis Note that if we only know the second order statistics of the inputs, the optimum linear filter is the best we can do. To take advantage of nonlinear filtering we must know the distributions of the inputs Example: Free configuration predictor (real time) t+T S(s) S(s)=s The sn are uncorrelated Page 4 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 7 The Wiener-Hopf equation requires 1 l() 0 τ = for 1 τ ≥ 0 . Thus L s( ) should be analytic in LHP and go to zero at least as fast as 1 s for large s . 0 () ( ) () () () ( ) () () Ls F sH sFsS s F sDsS s =− −− ii is We have solved the problem of the optimum linear filter under the least mean squared error criterion. Further analysis shows that if the inputs, signal and noise, are Gaussian, the result we have is the optimum filter. This is, there is no filter, linear or nonlinear which will yield smaller mean squared error. If the inputs are not both Gaussian, it is almost sure that some nonlinear filters can do better than the Wiener filter. But theory for this is only beginning to be developed on an approximate basis. Note that if we only know the second order statistics of the inputs, the optimum linear filter is the best we can do. To take advantage of nonlinear filtering we must know the distributions of the inputs. Example: Free configuration predictor (real time) 2 2 ( ) ( ) ss nn n A S s a s Ss S = − = The s n, are uncorrelated. ( ) sT Ds e =