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Unformatted text preview: Lecture Notes 9 Stationary Random Processes StrictSense and WideSense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Response of LTI System to WSS Process Input Linear Estimation: the Random Process Case EE 278: Stationary Random Processes 9 1 Stationary Random Processes Stationarity refers to time invariance of some, or all, of the statistics of a random process, such as mean, autocorrelation, nthorder distribution We define two types of stationarity: strict sense (SSS) and wide sense (WSS) A random process X ( t ) (or X n ) is said to be SSS if all its finite order distributions are time invariant, i.e., the joint cdfs (pdfs, pmfs) of X ( t 1 ) , X ( t 2 ) , . . . , X ( t k ) and X ( t 1 + ) , X ( t 2 + ) , . . . , X ( t k + ) are the same for all k , all t 1 , t 2 , . . . , t k , and all time shifts So for a SSS process, the firstorder distribution is independent of t , and the secondorder distribution the distribution of any two samples X ( t 1 ) and X ( t 2 ) depends only on = t 2 t 1 To see this, note that from the definition of stationarity, for any t , the joint distribution of X ( t 1 ) and X ( t 2 ) is the same as the joint distribution of X ( t ) = X ( t 1 + ( t t 1 )) and X ( t 2 + ( t t 1 )) = X ( t + ( t 2 t 1 )) EE 278: Stationary Random Processes 9 2 Example: The random phase signal X ( t ) = cos( t + ) where U[0 , 2 ] is SSS We already know that the first order pdf is f X ( t ) ( x ) = 1 p 1 ( x/ ) 2 , < x < + which is independent of t , and is therefore stationary To find the second order pdf, note that if we are given the value of X ( t ) at one point, say t 1 , there are (at most) two possible sample functions: x 1 x 21 x 22 t t 1 t 2 EE 278: Stationary Random Processes 9 3 The second order pdf can thus be written as f X ( t 1 ) ,X ( t 2 ) ( x 1 , x 2 ) = f X ( t 1 ) ( x 1 ) f X ( t 2 )  X ( t 1 ) ( x 2  x 1 ) = f X ( t 1 ) ( x 1 ) ( 1 2 ( x 2 x 21 ) + 1 2 ( x 2 x 22 ) ) , which depends only on t 2 t 1 , and thus the second order pdf is stationary Now if we know that X ( t 1 ) = x 1 and X ( t 2 ) = x 2 , the sample path is totally determined (except when x 1 = x 2 = 0 , where two paths are possible), and thus all nth order pdfs are stationary IID processes are SSS Random walk process is not SSS (in fact, no independent increment process is SSS) The GaussMarkov process (as we defined it) is not SSS. However, if we set X 1 to the steady state distribution of X n , it becomes SSS (see homework exercise) EE 278: Stationary Random Processes 9 4 WideSense Stationary Random Processes A random process X ( t ) is said to be widesense stationary (WSS) if its mean and autocorrelation functions are time invariant, i.e., E( X ( t )) = , independent of t R X ( t 1 , t 2 ) is a function only of the time difference t 2 t 1 E[ X (...
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This note was uploaded on 04/07/2010 for the course EE 278 taught by Professor Balajiprabhakar during the Spring '09 term at Stanford.
 Spring '09
 BalajiPrabhakar

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