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lecture19 - 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 6 Lecture 19 Last time: 0 3 1 3 3 3 1 3 3 1 ( ) ( ) ( ) ( ) 0 for 0 ii D is w R d w R d τ τ τ τ τ τ τ τ τ −∞ −∞ = Solution in the Free Configuration, Non-Real-Time Filter Case The applicable condition in this important case is: ( ) ( ) 0 3 1 3 3 3 1 3 3 ( ) ( ) 0 ii D is w R d w R d τ τ τ τ τ τ τ τ −∞ −∞ = for all 1 τ . [ ] 1 1 0 s e d τ τ −∞ = Since this function of 1 τ must be zero for all 1 τ , its transform must also be zero. ( ) ( ) ( ) ( ) 1 3 1 3 3 3 1 3 0 3 1 3 1 3 3 1 3 ( ) ( ) 0 s s s s ii D is d d w R e e d d w R e e τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ −∞ −∞ −∞ −∞ = 0 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ii is is ii H s S s D s S s D s S s H s S s = = with ( ) D s = desired signal transfer ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) is ss ns ii ss sn ns nn S s S s S s S s S s S s S s S s = + = + + + This is the optimum filter “transfer function.” Since it is non-real-time, it is usually implemented as a weighting sequence in a digital computer. The continuous weighting function is 0 0 1 ( ) ( ) 2 j st j w t H s e ds j π − ∞ = I will give an integral form which is useful in evaluating this integral. 0 ( ) H s typically has poles in both left and right hand planes. This does not imply instability in the case of a two-sided transform.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 6 ( ) 0 ( ) ( ) o t i w t d τ τ τ −∞ = In terms of s , the integrals are written: 1. If the real part of a is positive, ( ) ( ) ( ) ( ) 1 1 0 1 1 ! 2 0 0 n at j st n j t e t e n ds j s a t π − ∞ > = + < 2. If the real part of a is negative, ( ) ( ) ( ) ( ) 1 1 0 1 1 ! 2 0 0 n at j st n j t e t e n ds j s a t π − ∞ < = + > In writing power density spectra, which are rational functions of 2 ω - if rational at all – we note that 2 ω is replaced by 2 s .
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