lecture11

# lecture11 - 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 9 Lecture 11 Last time: Ergodic processes An ergodic process is necessarily stationary. Example: Binary process At each time step the signal may switch polarity or stay the same. Both 0 x + and 0 x are equally likely. Is it stationary and is it ergodic? For this distribution, we expect most of the members of the ensemble to have a change point near t=0.

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 9 [ ] 12 1 2 2 34 3 2 43 21 0 (, ) ()() 0 () xx xx R tt Extxt xt xt Rt t x t ttttx = = −=−= Chance of spanning a change point is the same over each regular interval, so the process is stationary. Is it ergodic? Some ensemble members possess properties which are not representative of the ensemble as a whole. As an infinite set, the probability that any such member of the ensemble occurs is zero.
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 9 All of these exceptional points are associated with rational points. They are a countable set of infinity which constitute a set of zero measure. The complementary set of processes are an uncountable infinity associated with irrational numbers which constitute a set of measure one.

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 For ergodic processes: [] 22 2 1 ( ) lim ( ) 2 1 [()] l im () 2 1 ()( ) l ) 2 1 ) l ) 2 T T T T T T T xx T T T xy T T xE x t x t d t T x t x t d t T R Extxt xtxt d t T R Extyt xtyt d t T ττ τ →∞ →∞ →∞ →∞ == =+ = + = + A time invariant system may be defined as one such that any translation in time of the input affects the output only by an equal translation in time.
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lecture11 - 16.322 Stochastic Estimation and Control Fall...

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