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 yFsFsSsd jπ ∞ ?∞ =? ∫ Cauchy-Residue Theorem ( ) 2 (residue at enclosed poles)F s ds jπ= ∑ ∫> 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 Fsds ∞ ?∞ ∫ by integrating around a closed contour enclosing the entire left half plane, note that if () 0Fs→ faster than 1 s for large s , the integral along the curved part of the contour is zero. If ()~ n k Fs s as s →∞, (1) semi-circle () 0 as if 1 n n k Fsds R k R R n R ππ ?? ≤ =→→∞> ∫> 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 jdsdsπ ∞ ?∞ ? = ? ∫ Refer to the handout “Tabulated Values of the Integral Form”. Roots of ()cs and ()ds in left half plane only. Should check the stability of the solution. 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 ()yt and process that data. [] 111 1 11 00 11111 00 () ()() ()() () ()()()() ()()( ) ()()( ) xd xy xd ytwxtd wdtd R Extyt xtyt wxtytdwxtdtd τττ τττ τττ τ τττ τ τττ ∞∞ ∞∞ =?+? =+=+ =+?++? ∫∫ ∫∫ If (), ()x tdtare independent and at least ()x t is zero mean, then () 0 xd R τ = . 111 0 () ( ) ( ) xy x xx R wR dτ ττττ ∞ =? ∫ If ()x t is wide band relative to the system, approximate it as white. () () xx x RSτ δτ= 111 0 () ( ) ( ) () xy x x xx R wS d Sw τ τδτττ τ ∞ =? = ∫