lecture13

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: () 22 2 1 , a xx aT T R τ στ ⎛⎞ +− ⎜⎟ = ⎝⎠ > 0 2 2 2 2 1 2( ) 21c o s 2 ) 1 c o s sin 2 ) 2 T jj xx a T T a a a S a ed T ad T T T T ωτ ω τσ π δω σ ωττ πδω −− −∞ =+ ∫∫ 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 (, ) TT . Return to the 1 st example process and take the case where the change points are Poisson distributed. 2 22 2 () a xx S λσ ω λ = + Take the limit of this as 2 a σ and become large in a particular relation: to establish the desired relation, replace 2 2 aa a xx k k kk S k σσ λλ = + and take the limit as k →∞ . 2 2 2 lim ( ) lim 2 a xx a k S k →∞ →∞ = = 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. Can shape () x t to the correct spectrum so that it can be analyzed in this manner, by adding a shaping filter in the state-space formulation. Definition of a white noise process White means constant spectral density. 0 ( ) , constant xx SS ω = 0 0 1 2 j xx R Se d S τω τ π δτ −∞ = = White noise processes only have a defined power density. The variance of a white noise process is not defined.
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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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lecture13 - 16.322 Stochastic Estimation and Control, Fall...

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