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# zz-6 - ECE 6010 Lecture 7 – Analytical Properties of...

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Unformatted text preview: ECE 6010 Lecture 7 – Analytical Properties of Random Processes Let X t be a function of time, and let h ( t ) be the impulse response of a (continuous- time) linear time invariant system. If X t is the input to the system, then the output is Y t = X t * h ( t ) = Z ∞-∞ h ( t- τ ) X τ dτ. We will consider such operations when X t is a random process. This will require us to develop some additional analytic properties of random processes. Continuity Let T = ( a, b ) . Recall that a function f : T → R is continuous at t if lim t → t f ( t ) = f ( t ) . f is continuous if it is continuous for every t ∈ ( a, b ) . Definition 1 For a random process { X t , t ∈ T } , we say X t is continuous with probability 1 if P ( { ω ∈ Ω : X t ( ω ) , t ∈ T is continuous } ) = 1 . 2 Continuity w.p. 1 is usually quite strong, in fact stronger than is typically needed for analysis. We say that { X t , t ∈ T } is continuous in probability at t if X t → X t (i.p.) as t → t . That is, lim t → t P ( | X t- X t | > ² ) = 0 for all ² > . We say that { X t , t ∈ T } is mean-square continuous at t if X t → X t (m.s.) as t → t . That is, lim t → t E [ | X t- X t | 2 ] = 0 . An important property: A second-order random process is mean-square continuous at t = t if and only if R X ( t, s ) is continuous at t = s = t , that is lim t → t ,s → t R X ( t, s ) = R X ( t , t ) . Proof (If) Suppose R X ( t, s ) is continuous at t = s = t . Then E [ | X t- X t | 2 ] = E [ X 2 t ]- 2 E [ X t X t ]+ E [ X 2 t ] = ( R X ( t, t )- R X ( t, t ))+( R X ( t , t )- R X ( t, t )) Continuity of R X implies that lim t → t R X ( t, t ) = lim t → t R X ( t, t ) = R X ( t , t ) . So lim t → t E [ | X t- X t | 2 ] = 0 ....
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zz-6 - ECE 6010 Lecture 7 – Analytical Properties of...

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