12discrp - EECS 501 DISCRETE-TIME RANDOM PROCESSES Fall...

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Unformatted text preview: EECS 501 DISCRETE-TIME RANDOM PROCESSES Fall 2001 DEF: A discrete-time random process=random sequence x ( n ) is mapping x ( n, ) : ( Z ) R where =sample space and Z = { integers } . 1. Fix n o Z x ( n o , )=random variable indexed by n o . 2. Fix o x ( n, o )=sample function=realization. 3. Can think of x ( n ) as a random vector of infinite length. THM: Kolmogorov Extension Thm.: Discrete-time random process x ( n ) is completely specified by its joint pdfs f x ( i 1 ) ...x ( i N ) ( X 1 . . . X N ). EX: An iidrp (independent identically distributed random process) has f x ( i 1 ) ...x ( i N ) ( X 1 . . . X N ) = f x ( X 1 ) f x ( X 2 ) f x ( X N ) for any i 1 . . . i N . DEF: x ( n ) is N th-order stationary if joint pdfs of order N have: f x ( i 1 ) ...x ( i N ) ( X 1 . . . X N ) = f x ( i 1 + j ) ...x ( i N + j ) ( X 1 . . . X N ) for any j . Means: Shifting time origin does not affect marginal pdfs of order N . EX: 1 st-order stationary f x ( i ) ( X ) = f x ( j ) ( X ) x ( n ) idrp (not iidrp)....
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12discrp - EECS 501 DISCRETE-TIME RANDOM PROCESSES Fall...

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