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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 + τ ) , X ( t 2 + τ ) , . . . , X ( t k + τ ) are the same for all k , all t 1 , t 2 , . . . , t k , and all time shifts τ • 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 τ = 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 ) = α cos( ωt + Θ) where Θ ∈ U[0 , 2 π ] is SSS ◦ We already know that the first order pdf is f X ( t ) ( x ) = 1 πα 1 ( x/α ) 2 , α < x < + α 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 δ ( x 2 x 21 ) + 1 2 δ ( 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 )) = μ , independent of t ◦ R X ( t 1 , t 2 ) is a function only of the time difference t 2 t 1 ◦ E[ X ( t...
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This note was uploaded on 11/28/2009 for the course EE 278 taught by Professor Balajiprabhakar during the Fall '09 term at Stanford.
 Fall '09
 BalajiPrabhakar
 Signal Processing

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