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lecture10 - 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 10 Last time: Random Processes With (,) f xt we can compute all of the usual statistics. Mean value: 11 () x tx f x t d x −∞ = Mean squared value: 22 x f x t d x −∞ = Higher order distribution and density functions. You can define these distributions of any order. [ ] 11 2 2 1 1 2 2 12 ( , ; , ;. ..) ( ) , ( ) ,..., ( ) ( , ; , ;. ..) , ; , ;. .. ... nn n n Fx t x t Pxt x xt x xt x fxtxt Fxtxt xx x =≤ = ∂∂ ∂ F is the probability that one member of the ensemble x satisfies each of these constraints at times t i . But we rarely work with distributions higher than second order . A very important statistic of a random process for the study of random processes in linear systems is the autocorrelation function, R xx – the correlation of 1 x t and 2 x t . [] 1 212 11 22 (, ) ,; , ,() xx R t t dx dx x x f x t x t Ext xt ∞∞ −∞ −∞ = = ∫∫ This could be computed as a moment of the second order density function (as above), but we usually just specify or measure the autocorrelation function directly . Notice that the autocorrelation function and the first order probability density function express different kinds of information about the random process. Two different processes might have the same pdfs but quite different xx R τ s. Conversely, they might have the same xx R but completely different pdfs.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 9 This is called the autocorrelation function for the random process { } () x t . Note some useful properties: [] 22 21 2 1 1 2 12 (,) () () ( , ) ( )() ()( ) (, ) xx xx xx Rt t Ex t x t x t R tt Ex tx t t ⎡⎤ == = ⎣⎦ === Also note that x(t2) is likely to be independent of x(t1) if |t2-t1| is large. For that case: [ ] 1 2 || lim ( , ) ( ) ( ) ( ) ( ) xx tt t x t −→ →= The members of the processes {x(t)} and {y(t)} must be associated as corresponding pairs of functions. There is a particular y which goes with an x .
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lecture10 - 16.322 Stochastic Estimation and Control, Fall...

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