Lecture21.pdf - ESE 520 Probability and Stochastic Processes Lecture 21 ”An application of WSS processes White noise representations of signals.”

Lecture21.pdf - ESE 520 Probability and Stochastic...

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ESE 520 Probability and Stochastic Processes Lecture 21 ”An application of WSS processes: White noise representations of signals.” Assume that ( X t ) , t IR is a WSS process with mean μ ( t ) and covariance function R ( τ ). Let us consider a deterministic linear dynamical system described by the following ODE system: x 0 ( t ) = Ax ( t ) + Bu, y ( t ) = Cx ( t ) where x ( t 0 ) = x 0 is the initial state of x ( t ) (which we can assume for simplicity to be zero). In a general case: the coefficients A, B, C are matrices; in case of n = 1 they are just constants. Here u can be seen as a ”control parameter”. It can be directly verified that the solution of the above first ODE is x ( t ) = e A ( t - t 0 ) [ x 0 + Z t t 0 e A ( t 0 - s ) Buds ] . If we let the control parameter to be u = X t , then we obtain as the ”output” process for the system described Y t = C Z t t 0 e A ( t - s ) BX s ds + x 0 e A ( t - t 0 ) C = [ we use for simplicity x 0 = 0] = C Z t t 0 e A ( t - s ) BX s ds. What can we say about the process ( Y t )? Define h ( τ ) := Ce B, τ 0 0 , τ < 0 . 1
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called the impulse-response of the system. Then Y t = Z -∞ h ( t - s ) X s ds. Formally (integrals need to converge), for any t 0, we obtain E ( Y t ) = Z -∞ h ( t - s ) E ( X s ) ds = [ E ( X s ) = μ = const ] = μ Z -∞ h ( t - s ) ds = [ τ := t - s ] = μ Z -∞ h ( τ ) = μL where R -∞ h ( τ ) = L . Similarly, for all s, t 0, we have R Y ( s, t ) = E [( Y t - EY t )( Y s - EY s )] = E [( Z -∞ h ( t - τ )( X τ - μ ))( Z -∞ h ( s - σ )( X σ - μ ))] dτdσ = Z -∞ Z -∞ h ( t - τ ) h ( s - σ ) E ( X τ - μ )( X σ - μ ) dτdσ. Now, by definition of the covariance function, R X ( τ, σ ) = E ( X τ - μ )( X σ - μ ), and since X is a WSS process, R X ( τ, σ ) depends only of the difference τ - σ so that we can write R X (
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  • Spring '14
  • Arthur
  • Stochastic process, Yt, Wiener process

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