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**Unformatted text preview: **Digital Signal Processing I (ECE 410, Summer II 2009) University of Illinois at Urbana-Champaign Electrical and Computer Engineering Midterm 1 1. Answer the following questions. Justify your answer briefly. (a) (5 points) Is this system causal? y [ n ] = x [- n ] . (1) (b) (5 points) Is this system time-invariant (or shift-invariant)? y [ n ] = h [ n ] * x [ n ] , (2) where * is a linear convolution. (c) (5 points) Is this system stable? H ( z ) = 1 ( 1 + 1 100 z- 2 ) (1- 2000 z- 2 ) , anticausal (3) 2. (20 points) The discrete sequence x [ n ] is given by x [ n ] = 1 2 n n = 0 , 2 , 4 , 6 , 8 , , n 4 n =- 1 , 1 , 3 , 5 otherwise (4) Let X d ( ) be the DTFT of x [ n ]. (a) Find X d (0). (b) Find X d ( ). (c) Find R - | X d ( ) | 2 d (d) Consider an LTI system whose impulse response h[n] is x [- n ]. Let y [ n ] be the output of the system for the input x [ n ]. Find R - Y d ( ) d . 3. (20 points) A block diagram of a causal LTI system is depicted below;3....

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