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 : Ω → R R , or: x ( t, ω ) : ( R × Ω) → R where Ω=sample space and R = { reals } . 1. Fix t o ∈ R → x ( t o , ω )=random variable indexed by time index t o . 2. Fix ω o ∈ Ω → x ( t, ω 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 + τ ) ...x ( t N + τ ) ( X 1 . . . X N ) for any τ . 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 ⇔ μ ( t ) = μ and K x ( t, s ) = K x ( t- s )....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

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