zz-32 - Utah State University ECE 6010 Stochastic Processes...

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Unformatted text preview: Utah State University ECE 6010 Stochastic Processes Homework # 8 Solutions 1. Suppose { X t , t } is a Wiener process. Define a process { Y t , t } by Y t = X t + D- X t for a fixed positive number D . (a) Find the mean and autocorrelation functions of { Y t } . Mean : Y ( t ) = E [ Y t ] = E [ X t + D ]- E [ X t ] = ( t + D )- ( t ) = D Autocorrelation : R Y ( t, s ) = E [( X t + D- X t )( X s + D- X s )] = E [ X t + D X s + D ]- E [ X t + D X s ]- E [ X t X s + D ] + E [ X t X s ] = [ 2 min( t + D, s + D ) + 2 ( t + D )( s + D )]- [ 2 min( t + D, s ) + 2 ( t + D ) s ]- [ 2 min( t, s + D ) + 2 ( S + D ) t ] + [ 2 min( t, s ) + 2 st ] = 2 [min( t + D, s + D )- min( t + D, s )- min( t, s + D ) + min( t, s )] + 2 [ ts + Dt + sD + D 2- st- sD- st- Dt + st ] = 2 [ s + D- s- min( t, s + D ) + s ] + D 2 t s 2 [ t + D- t- min( t, s + D ) + t ] + D 2 t < s = 2 D 2 t s , t- s- D 2 [ s- t + D ] + 2 D 2 t s , t- s- D < 2 D 2 t < s , t- s + D < 2 [ t- s + D ] + 2 D 2 t < s , t- s + D = 2 D 2 | t- s | D 2 [ D- | t- s | ] + 2 D 2 | t- s | < D So R Y ( t, s ) = 2 ax (0 , D- | t- s | ) + 2 D 2 . (b) Show that { Y t } is a stationary and find its spectrum. y ( t ) is a constant and R Y ( t, s ) = R Y ( t- s ) W.S.S....
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This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.

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zz-32 - Utah State University ECE 6010 Stochastic Processes...

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