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# session8 - EE278 Review Session 8 Han-I Su June 1 2009...

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EE278: Review Session # 8 Han-I Su June 1, 2009 Han-I Su () EE278: Review Session # 8 June 1, 2009 1 / 16

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Outline 1 Stationary random process 2 Autocorrelation function and power spectral density 3 LTI system 4 Homework hints Han-I Su () EE278: Review Session # 8 June 1, 2009 2 / 16
Stationary Random Process Strict-sense stationary: finite order distributions are time invariant F X ( t 1 ) , X ( t 2 ) ,..., X ( t k ) ( x 1 , x 2 , . . . , x k ) = F X ( t 1 + τ ) , X ( t 2 + τ ) ,..., X ( t k + τ ) ( x 1 , x 2 , . . . , x k ) for all k , for all t 1 , t 2 , . . . , t k , and for all τ Wide-sense stationary: mean and autocorrelation functions are time invariant E( X ( t )) = μ is independent of t R X ( t 1 , t 2 ) is a function of t 2 t 1 E( X ( t ) 2 ) < Han-I Su () EE278: Review Session # 8 June 1, 2009 3 / 16

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Stationary Random Process Strict-sense stationary: finite order distributions are time invariant F X ( t 1 ) , X ( t 2 ) ,..., X ( t k ) ( x 1 , x 2 , . . . , x k ) = F X ( t 1 + τ ) , X ( t 2 + τ ) ,..., X ( t k + τ ) ( x 1 , x 2 , . . . , x k ) for all k , for all t 1 , t 2 , . . . , t k , and for all τ Wide-sense stationary: mean and autocorrelation functions are time invariant E( X ( t )) = μ is independent of t R X ( t 1 , t 2 ) is a function of t 2 t 1 E( X ( t ) 2 ) < SSS WSS; WSS notdblarrowright SSS Gaussian: WSS SSS Han-I Su () EE278: Review Session # 8 June 1, 2009 3 / 16
Example Random frequency and random phase Consider the random process X ( t ) = a cos( Wt + Θ), where W Exp(1) and Θ U[ π, π ] are independent. Han-I Su () EE278: Review Session # 8 June 1, 2009 4 / 16

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Example Random frequency and random phase Consider the random process X ( t ) = a cos( Wt + Θ), where W Exp(1) and Θ U[ π, π ] are independent. What is the mean function of X ( t )? Han-I Su () EE278: Review Session # 8 June 1, 2009 4 / 16
Example Random frequency and random phase Consider the random process X ( t ) = a cos( Wt + Θ), where W Exp(1) and Θ U[ π, π ] are independent. What is the mean function of X ( t )? μ X ( t ) = E( a cos( Wt + Θ)) = E[E( a cos( Wt + Θ) | W )] = 0 where the last equality follows by E( a cos( Wt + Θ) | W = w ) = E( a cos( wt + Θ) | W = w ) = E( a cos( wt + Θ)) = 0 Han-I Su () EE278: Review Session # 8 June 1, 2009 4 / 16

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Example Random frequency and random phase Consider the random process X ( t ) = a cos( Wt + Θ), where W Exp(1) and Θ U[ π, π ] are independent. What is the mean function of X ( t )? μ X ( t ) = E( a cos( Wt + Θ)) = E[E( a cos( Wt + Θ) | W )] = 0 where the last equality follows by E( a cos( Wt + Θ) | W = w ) = E( a cos( wt + Θ) | W = w ) = E( a cos( wt + Θ)) = 0 The above equation holds for all W F W ( w ) independent of Θ Han-I Su () EE278: Review Session # 8 June 1, 2009 4 / 16
Example - Conditional Expectation Review ( X , Y ) f X , Y ( x , y ), X f X ( x ) and P( A ) > 0 f X | A ( x ) = braceleftBigg f X ( x ) P { X A } if x A 0 otherwise E( g ( X ) | A ) = integraldisplay -∞ g ( x ) f X | A ( x ) dx = integraltext A g ( x ) f X ( x ) dx integraltext A f X ( x ) dx f X | Y ( x | y ) = f X , Y ( x , y ) f Y ( y ) , if f Y ( y ) negationslash = 0 E( g ( X , Y ) | Y = y ) = integraldisplay -∞ g ( x , y ) f X | Y ( x | y ) dx = integraltext -∞ g ( x , y ) f X , Y ( x , y ) dx integraltext -∞ f X , Y ( x , y ) dx Remark: E( g ( X , Y ) | Y = y ) = E( g ( X , y ) | Y = y ) is always true, but in general E( g ( X , y ) | Y = y ) negationslash = E( g ( X , y )) Han-I Su () EE278: Review Session # 8 June 1, 2009 5 / 16

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Example - Iterative Expectation Review ( X , Y ) f X , Y ( x , y ) E( g ( X , Y )) = E[E( g ( X , Y ) | Y )] Proof: Han-I Su () EE278: Review Session # 8 June 1, 2009 6 / 16
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• Spring '09
• BalajiPrabhakar
• Autocorrelation, LTI system theory, Stationary process, Wiener–Khinchin theorem, Han-I Su, Iterative Expectation Review

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