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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 Final Exam Solution 1. (a) (5 points) Advantage: easy to achieve a linear phase. Guaranteed to be stable. disadvantage: larger transition band for the same filter length or smaller length of filter (less computation) for the same transition band. Poor approximation to a desired magnitude response. (b) (5 points) Zero-padding does not change the frequency response and leads to a densely sampled frequency response when the DFT is applied. Truncation leads to the convolution with sync function so that frequency component spreads and ripple effect occurs). (c) (5 points) X d ( ), X [ k ], and X ( z ) are related by the following three equations X d ( ) = X ( z ) | z = e jw (1) X [ k ] = X d ( ) | = 2 N k (2) X [ k ] = X ( z ) | z = e j 2 N k (3) 2. (a) (5 points) No. For the bounded input | x [ n ] | < B , the magnitude of the output | y [ n ] | will diverge for n = 0. (b) (5 points) Stable. Since [ n ] + 10 9 [ n- 1] is the impulse response of FIR system, it should be stable. In addition, 2 n u [- n- 1] 1 1- 2 z- 1 , | z | < 2 so that the ROC includes the unit circle....

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