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lecture16

# lecture16 - 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 5 Lecture 16 Last time: Wide band input Notice that 0 ( ) ( ) ( ) ( ) xy R S w x t y t τ τ τ = = + so if t is the current time in a real time situation, we cannot compute ( ) xy R τ for 0 τ > which is necessary since ( ) x w τ is nonzero only for 0 τ > . But we showed earlier that ( ) ( ) ( ) ( ) xy yx R R y t x t τ τ τ = = This can be computed in real time by averaging the product of the current output, ( ) y t , and the delayed input, ( ) x t τ , for 0 τ > . The Shaping Filter A shaping filter is a filter that produces an output process with a white noise input, which has a specified spectrum. If a given system is driven by a colored noise, we can precede it with the appropriate shaping filter and always treat the augmented system as driven by white noise. We usually work with rational spectra – so consider ( ) xx S ω to be a rational function. That is, it is a ratio of polynomials in ω . Since ( ) xx S ω is always an even function , it is a ratio of polynomials having only even powers of ω . 2 4 0 1 2 2 4 0 1 2 ... ( ) ... xx a a a S b b b ω ω ω ω ω + + + = + + + We wish to characterize the filter in terms of the Laplace variable s . The relation between s and ω is s j ω = .

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 5 Rewrite xx S in terms of s . Since only even powers of ω appear there will be no imaginary terms.
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lecture16 - 16.322 Stochastic Estimation and Control Fall...

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