ECE130A-F10-HW07 - ECE 130A Signal Analysis and Processing...

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ECE 130A: Signal Analysis and Processing Prof. Michael Liebling Fall 2010 TAs: Jingning Han, Hari Sivakumar Distribution: Monday, 15 November 2010 Electrical & Computer Engineering Dept. Homework Due: Tuesday, 23 November 2010 at 6:00 p.m. University of California, Santa Barbara Homework 7 Problem 1: Inverse Laplace transform (20 points) Determine the function of time, x ( t ), for each of the following Laplace transforms and their associated regions of convergence: (a) X ( s ) = s s 2 + 9 , ROC: e{ s } < 0 (b) X ( s ) = s + 1 ( s + 1) 2 + 9 , ROC: e{ s } < - 1 (c) X ( s ) = ( s + 1) 2 s 2 - s + 1 , ROC: e{ s } > 1 2 (d) X ( s ) = 2( s + 2) s 2 + 7 s + 12 ROC: e{ s } > - 3, given that: e - at u ( t ) L ←→ 1 s + a ROC: e{ s } > -ℜ e{ a }, Problem 2: LTI Systems and Laplace Transform, Causality, and Stability (20 Points) Consider and LTI system for which the system function H ( s ) has the pole- zero plot shown on the right. (a) Indicate all possible ROC’s that can be associated with this pole-zero plot. (b) For each ROC identified in (a), specify whether the associated system is stable and/or causal.
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