20102ee113_1_hw5_sol_2010

20102ee113_1_hw5_sol_2010 - EE113: Digital Signal...

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Unformatted text preview: EE113: Digital Signal Processing Spring 2010 Prof. Mihaela van der Schaar Homework #5 Solutions 10.2 We re-write ? ( ? ) as ? ( ? ) = 6 ? + 3 − 2 ? − 1 6 ? 2 − 5 ? + 1 = ? − 1 6 ? 2 + 3 ? − 2 6 ? 2 − 5 ? + 1 = ? − 1 (1 + 8 ? − 3 6 ? 2 − 5 ? + 1 ) We let ? ′ ( ? ) = 8 ? − 3 6 ? 2 − 5 ? + 1 , and it can be written as ? ′ ( ? ) = 1 3 ? − 1 + 2 2 ? − 1 Hence, ? ( ? ) = ? − 1 (1 + 1 / 3 ? − 1 / 3 + 1 ? − 1 / 2 ), and ? ( ) is as following: (a) ROC: ∣ ? ∣ < 1 / 3 ? ( ) = ( − 1) − 1 3 ( 1 3 ) − 2 ( − + 1) − ( 1 2 ) − 2 ( − + 1) = ( − 1) − ( 1 3 ) − 1 ( − + 1) − ( 1 2 ) − 2 ( − + 1) (b) ROC: 1 / 3 < ∣ ? ∣ < 1 / 2 ? ( ) = ( − 1) + 1 3 ( 1 3 ) − 2 ( − 2) − ( 1 2 ) − 2 ( − + 1) = ( − 1) + ( 1 3 ) − 1 ( − 2) − ( 1 2 ) − 2 ( − + 1) (c) ROC: ∣ ? ∣ > 1 / 2 ? ( ) = ( − 1) + 1 3 ( 1 3 ) − 2 ( − 2) + ( 1 2 ) − 2 ( − 2) = ( − 1) + ( 1 3 ) − 1 ( − 2) + ( 1 2 ) − 2 ( − 2) 11.2. Correct answer is (a). Recall that an LTI system is causal if and only if the ROC of its transfer function includes the exterior of a disc for some radious < ∞ , including ? = ∞ ....
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This note was uploaded on 06/06/2010 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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20102ee113_1_hw5_sol_2010 - EE113: Digital Signal...

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