assignment11 - R X τ =(1 | τ |/T ²or | τ |< T and...

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Assignment 11 1. (a) Find the autocorrelation function corresponding to the power spectral density shown in the following ±gure. (b) Find the total average power. (c) Plot the power in the range | f | > f 0 as a function of f 0 > 0. f - f 2 - f 1 0 f 1 f 2 A A B 2. Show that (a) R X,Y ( τ ) = R Y,X ( - τ ). (b) S X,Y ( f ) = S * Y,X ( f ). 3. Let R X ( k ) = 4( α ) | k | + 16( β ) | k | , α < 1, β < 1. (a) Find S X ( f ). (b) Plot S X ( f ) for α = β = 0 . 5 and α = 0 . 75 = 3 β and comment on the e²ect of value of α/β . 4. Let D n = X n - X n - d , where d is an integer constant and X n is a zero-mean, WSS random process. (a) Find R D ( k ) and S D ( f ) in terms of R X ( k ) and S X ( f ). What is the impact of d ? (b) Find E [ D 2 n ]. 1
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5. Let X ( t ) be a diferentiable WSS random process, and deFne Y ( t ) = d dt X ( t ) ±ind an expression ²or S Y ( f ) and R Y ( τ ). Hint : ±or this system, H ( f ) = j 2 πf . 6. Let Y ( t ) be a short-term integration o² X ( t ): Y ( t ) = 1 T i t t - T X ( t p ) dt p . (a) ±ind the impulse response h ( t ) and the trans²er ²unction H ( f ). (b) ±ind S Y ( f ) in terms o² S X ( f ). 7. In problem 6, Let
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Unformatted text preview: R X ( τ ) = (1- | τ | /T ) ²or | τ | < T and zero elsewhere. (a) ±ind S Y ( f ). (b) ±ind R Y ( τ ). (c) ±ind E [ Y 2 ( t )]. 8. The input into a Flter is zero-mean white noise with noise power density N / 2. The Flter has trans²er ²unction H ( f ) = 1 1 + j 2 πf (a) ±ind S Y,X ( f ) and R Y,X ( τ ) . (b) ±ind S Y ( f ) and R Y ( τ ). (c) What is the average power o² the output? 9. (a) A WSS Gaussian random process X ( t ) is applied to two linear systems as shown in the ²ollowing Fgure. ±ind an expression ²or the joint pd² o² Y ( t 1 ) and W ( t 2 ). (b) Evaluate part a i² X ( t ) is white Gaussian noise. h 1 ( t ) h 2 ( t ) X ( t ) Y ( t ) W ( t ) 2...
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assignment11 - R X τ =(1 | τ |/T ²or | τ |< T and...

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