Lecture17 - Properties of autocorrelation function If a...

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•1 ± Properties of autocorrelation function If a random process is WSS, then its autocorrelation function satisfies the following a) (even function) b) c) is periodic if d) When a random process is real valued, continuous time and WSS, its autocorrelation is positive definite . ) ( ) ( τ = X X R R ) 0 ( ) ( X X R R ) ( X R ) 0 ( ) ( / X X R T R T = ² Cyclostationary Process; Definition: A random process is said to be cyclo-stationary if and a constant T. Remark: Similarly to stationarity, may be wide sense cyclostationary. It is WCS if and T>0 we have a) b) )} ( { t X ) ,... , ( ) ,... , ( 2 1 ).... ( ) ( 2 1 ) ( ).... ( ) ( 1 1 2 1 n lT t X lT t X n t X t X t X x x x F x x x F n + + = N l )} ( { t X N l I t t m lT t m X X = + 1 1 1 ), ( ) ( I t t t t C lT t lT t C X X = + + 2 1 2 1 2 1 , ), , ( ) , ( Example: 1. In binary transmission, X(t) takes on with equal probability in each interval we assume the values between two intervals are independent. 1 ± nT t T n T n ) 1 ( : n T t t if t X t X E = 2 1 2 1 , 1 )) ( ) ( ( otherwise 0 Or 2. X(t)=Acos(t)+Bsin(t) A and B are R.V. a) Is stationary? WSS? Cyclostationary? WCS? b) What are the conditions on A and B for all modes of stationarity? )} ( { t X = + ) , ( t t R X nT t t T n < + < ) , ( ) 1 ( 1 w o . 0
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•2 The distribution may be written as: Hence not stationary since the distribution is varying with t. Not W.S.S , since Clearly cyclostationary and WCS.
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Lecture17 - Properties of autocorrelation function If a...

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