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 txfxtdx ∞ ?∞ = ∫ Mean squared value: 22 11 () (,)x txfxtdx ∞ ?∞ = ∫ Higher order distribution and density functions. You can define these distributions of any order. [ ] () 11 2 2 1 1 2 2 11 2 2 11 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 . () [] 12 1 212 11 22 12 (, ) ,; , (),() 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 τ but completely different pdfs. 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 Rtt Extxt Ext xt R tt Extxt Extxt R tt ??=== ?? === Also note that x(t2) is likely to be independent of x(t1) if |t2-t1| is large. For that case: [ ] 21 12 12 12 || lim ( , ) ( ) ( ) ( ) ( ) xx tt R tt Extxt xtxt ?→∞ →= 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 . To study the statistical interrelation among more than one random process we need to consider their joint distributions: The general joint distribution function for two processes {x(t)} and {y(t)}, defined over the sample space of the same experiment in general: [ ] (,) 11 11 1 1 1 1 ( , ;..., , ; , ;..., , ) ( ) ,..., ( ) , ( ) ,..., ( ) mn xy m m n n m m n n F xtxtytytPxtxxtxytyyty′′ ′=≤ ≤ ≤ ≤ Examples include: Elevation, azimuth of radar tracker. The general joint density function is: (,) (,) 11 11 11 ( , ;..., , ; , ;..., , ) (...) ... ... mn mn mn xy m m n n xy mn fxtxtytyt F xxyy + ? ′′= ???? 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 9 Any of the lower ordered joint or single distributions can be derived from this by integration over the variables to be eliminated. The mathematical expectation of a function of these random variables is: []{} (,) 11 11 ( ),..., ( ), ( ),..., ( ) ... ( ... ) (...) mn mn nnxy Egxt xt yt yt dx dygx y f ∞∞ ?∞ ?∞ ′′= ∫∫ By far the most important statistical parameter involved in the joint consideration of more than one random process is the cross correlation function. [] (1,1) 12 1 2 1 2 (, ) ()() (,;, ) xy xy R t t E x t y t dx xyf x t y t dy ∞∞ ?∞ ?∞ == ∫∫ [ ] [ ] 21 2 1 1 2 12 (,) ()() ()() (,) xy yx R tt Extyt Eytxt R tt=== If (), ()x tyt are statistically independent [ ] [ ] [ ] 12 12 1 2 12 (, ) ()() () () ()() 0 xy R t t Ext yt Ext E yt xt yt== == if either ()x t or ()yt or both is zero mean. If () () () ()wt xt yt zt=++ Then [ ] 12 1 2 12 12 12 12 12 12 12 12 12 (, ) ()() (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) ww xx xy xz yx yy yz zx zy zz R t t Ewt wt R tt R tt R tt R tt R tt R tt R tt R tt R tt = =++ +++ +++ such that if any two of ()x t , ()yt or ()zt have zero mean and if they are all mutually independent, all the cross correlation terms vanish. 12 12 12 12 (, ) (, ) (, ) (, ) ww xx yy zz R tt R tt R tt R tt=++ Thus for independent processes with zero mean, the autocorrelation of the sum is the sum of the autocorrelations. This has special relevance since it implies that for independent processes with zero mean, the mean square of a sum is the sum of the mean squares. This simplifies the problem of minimizing the mean squared error when more than one random process is to be considered. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 [] 2 22 12 1 122 11 12 2 2 12 11 22 12 11 22 22 12 22 12 1 2 () () () 2()() () (,) 2 (, ) (, ) 0 1 (, ) (,) (, ) 2 1 (, ) (,) (, ) 2 1 () () 2 1 (, ) () () 2 xx xy yy xy xx yy xy xx yy xx xt yt xt xt yt yt Rtt Rtt Rtt Rtt Rtt Rtt Rtt Rtt Rtt xt yt Rtt xt xt ±=± + =± +≥ ??≤+ ?? ≤+ ?? ≤+ ?? ?? ≤+ ? m ? Intuitively we feel that if the conditions under which an experiment is performed are time independent, then the statistical quantities associated with a random process resulting from the experiment should be independent of time. Analytically we say that a process is stationary if every translation in time transforms members of the process into other members of the process in such a way that probability is preserved. This could also be stated by the statement that all distribution functions associated with the process ()( ) () () 11 2 2 11 21 2 1 , , , ,..., , , , , ,..., , nn nn n n FxtxtxtFxtxt xtτ τ=++ be functions of the differences in the i t only and independent of the actual values of the i t . Define 1n? i τ . Then ()n f is independent of 1 t . 22 (,) () () () f xt f x xt x xt x ? ? ? 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 9 Still depends on differences between time samples, but does not depend on time at which sampling starts. 11 2 2 2 2 3 3 22 ( , ; , ;..., , ) ( , ; , ; , ;...; , ) nn n n fxtxt xt fxtx x x tt τ ττ τ = =+ 11 2 2 1 2 21 12 (,; ,) (, ,) (, ) () () ()( ) ()() () ( ) () xx xx xx xx fxtxt fxx tt Rtt R Rxtxt x txt xtxt R τ τ τ τ τ τ τ τ ? =? ? =+ =? =? =? A stationary random process is further said to have the ergodic property if the time average of any function over a randomly selected member of the ensemble is equal to the ensemble average of the function with probability 1. This means there must be probability 1 of picking a function which represents the ensemble. Note that a “representative” member of the ensemble excludes any with special properties which belong to a set of zero probability. A representative member must display at various points in time the same distributions of amplitude and rates of change of amplitude as are displayed in the entire ensemble at any one point in time. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 9 This is why no member of the ensemble of constant functions is representative of the ensemble. Autocorrelation of a stationary function is an even function of its time argument. 2 12 11 22 22 2 (0) 1 (, ) (,) (, ) 2 1 () (0) (0) 2 1 2 () (0) xx xy xx yy xy xx yy xx xx xR R tt R tt R tt RRR xy Rx R τ τ = ??≤+ ?? ??≤+ ?? ?? ≤+ ?? ≤ ≤ () ()( ) ()() () ( ) () xy yx Rxtyt x tyt ytxt R τ τ τ τ τ =+ =? =? =? 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 9 Ergodic property A stationary random process is further said to have the ergodic property if the time average of any function over a randomly selected member of the ensemble is equal to the ensemble average of the function with probability 1. This means there must be probability 1 of picking a function which represents the ensemble. Note that a “representative” member of the ensemble excludes any with special properties which belong to a set of zero probability. A representative member must display at various points in time the same distributions of amplitude and rates of change of amplitude as are displayed in the entire ensemble at any one point in time. This is why no member of the ensemble of constant functions is representative of the ensemble. Example: Constants 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 9 Stationary, but not ergodic. Time average = ensemble expectation, with probability 1. Example: Sinusoids ( ) sin( )xt A tω θ=+ , fixed random A ω θ Ensemble expectation of ( )()gxt : [][] 2 0 1 ( sin( )) sin( ) 2 EgA t gA t d π ω θωθθ π += + ∫ Time average of ()()gxt : 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 9 of 9 ()[] ()[] ()[] () 0 0 2 0 1 Ave sin( ) sin( ) 1 Ave sin( ) sin( ) 2 1 Ave sin( ) sin( ) 2 ensemble expectation of ( ) T T gA t gA t dt T t d gA t gA T T gA t gA d gxt ω π ωθ ωθ φω φ ωθ φθ ω ωπ ωθ φθφ π += +?? ?? = += +?? ?? = += +?? ?? = ∫ ∫ ∫ So when ()f θ is uniformly distributed, the process is stationary and ergodic.