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lecture15

# lecture15 - 16.322 Stochastic Estimation and Control Fall...

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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 j π − ∞ = Cauchy-Residue Theorem ( ) 2 (residue at enclosed poles) F s ds j 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 ( ) 0 F s 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 c s c s I ds j d s d s π − ∞ = Refer to the handout “Tabulated Values of the Integral Form”. Roots of ( ) c s and ( ) d s

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lecture15 - 16.322 Stochastic Estimation and Control Fall...

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