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lecture13 - 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 8 Lecture 13 Last time: ( ) ( ) 2 2 2 1 , ( ) , a xx a T T R a T τ σ τ τ τ + = > ( ) 2 2 2 2 0 2 2 2 2 2 2 ( ) 1 2 ( ) 2 1 cos 2 2 ( ) 1 cos sin 2 2 ( ) 2 T j j xx a T T a a a S a e d e d T a d T a T T T a T T ωτ ωτ τ ω τ σ τ τ π δ ω σ ωτ τ σ π δ ω ω ω ω π δ ω σ ω −∞ = + = + = + = + Amplitude of xx S falls off, but not very rapidly.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 Use error between early and late indicator to lock onto signal. Error is a linear function of shift, within the range ( , ) T T . Return to the 1 st example process and take the case where the change points are Poisson distributed. 2 2 2 2 ( ) a xx S λσ ω ω λ = + Take the limit of this as 2 a σ and λ become large in a particular relation: to establish the desired relation, replace 2 2 2 2 2 2 ( ) ( ) a a a xx k k k k S k σ σ λ λ λ σ ω ω λ = + and take the limit as k → ∞ . ( ) 2 2 2 2 2 lim ( ) lim 2 a xx k k a k S k λσ ω λ σ λ →∞ →∞ = = Note this is independent of frequency.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 This is defined to be a “white noise” by analogy with white light, which is supposed to have equal participation by all wavelengths.
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