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 yF s F s S s d s j π −∞ =− Cauchy-Residue Theorem ( ) 2 (residue at enclosed poles) F s ds j = > 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 Fsd s by integrating around a closed contour enclosing the entire left half plane, note that if 0 Fs faster than 1 s for large s , the integral along the curved part of the contour is zero. If ()~ n k s as s →∞ , (1 ) semi-circle 0 a s i f 1 n n k s 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 jd s d s = Refer to the handout “Tabulated Values of the Integral Form”. Roots of cs and ds in left half plane
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.

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

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