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Solution to HW 13 - Solution to EE351k fall 2007 hw 13 1...

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Solution to EE351k fall 2007 hw 13 1. ) 2 cos( )) ( 2 2 cos( )) ( 2 sin( ) 2 sin( 0 )) ( 2 cos( ) 2 cos( ))} ( 2 sin( ) 2 sin( ))] ( 2 cos( ) 2 sin( )) ( 2 sin( ) 2 [cos( )) ( 2 cos( ) 2 cos( { ))]} ( 2 sin( ))) ( 2 cos( )][ 2 sin( ) 2 cos( {[ )] ( ) ( [ ) , ( 2 2 2 2 2 2 τ π σ τ π π σ τ π π σ τ π π σ τ π π τ π π τ π π τ π π τ π τ π π π τ τ f t f ft t f ft t f ft t f ft Y t f ft t f ft XY t f ft X E t f Y t f X ft Y ft X E t W t W E t R W = + - = + + + + = + + + + + + + = + + + + = + = 0 ) 2 sin( ) ( ) 2 cos( ) ( )] 2 sin( ) 2 cos( [ )) ( ( = + = + = ft Y E ft X E ft Y ft X E t W E π π π π A stochastic process is wide-sense stationary if its mean is constant and its autocorrelation depends only on τ . So W(t) is wss. 2. a. There are at least three approaches we can follow to compute the mean. We can use the definition of the expectation of a random variable X whose PDF is computed in last homework, or use the definition of a function of a random variable W whose PDF is given here, or use the linear property of a expectation operator.
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