15contrp - EECS 501 CONTINUOUS-TIME RANDOM PROCESSES Fall...

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Unformatted text preview: EECS 501 CONTINUOUS-TIME RANDOM PROCESSES Fall 2001 DEF: A continuous-time random process x ( t ) is a mapping x : R R , or: x ( t, ) : ( R ) R where =sample space and R = { reals } . 1. Fix t o R x ( t o , )=random variable indexed by time index t o . 2. Fix o x ( t, o )=sample function=realization (not continuous). 3. Kolmogorov Extension Thm no longer holds: unmeasurable rps. DEF: x ( t ) is N th-order stationary if joint pdfs of order N have: f x ( t 1 ) ...x ( t N ) ( X 1 . . . X N ) = f x ( t 1 + ) ...x ( t N + ) ( X 1 . . . X N ) for any . DEF: x ( t ) SSS strict sense stationary N th-order stationary for all N . Note: iid SSS N th-order 2 nd-order WSS 1 st-order id. DEF: x ( t ) Gaussian { x ( t 1 ) , x ( t 2 ) . . . x ( t N ) } JGRV for all t 1 . . . t N . DEF: x ( t ) WSS wide sense stationary ( t ) = and K x ( t, s ) = K x ( t- s )....
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