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Unformatted text preview: Midterm Solution Solution by Harvey Thornburg 1. (a) Factor H ( s ) = s +4 ( s +1)( s +2) . There are two poles : s = 1 and s = 2, and only one zero : s = 4. (b) Possible ROC is: Re { s } > 1 2 < Re { s } < 1 Re { s } < 2 As Re { s } > 1 contains the jaxis, the system is stable. (c) As Re { s } > 1 is rightsided, the system is causal. (d) H 1 ( s ) = ( s +1)( s +2) s +4 It seems that H 1 ( s ), with ROC: Re { s } > 4, does work here. However, there is an important subtlety that some of you mentioned. Since the degree of numerator is larger than the degree of the denom inator, we can write: H 1 ( s ) = s 2 + 3 s + 2 s + 4 = s s 2 s + 4 (1) Lets consider the system H ( s ) = s . It is a differentiator, and is differentiation BIBO stable? Consider x ( t ) = sin t 2 bounded x ( t ) = 2 t cos t 2 unbounded So, having all poles in lefthalf plane does not, on its own, guarantee BIBO stability....
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 Spring '08
 thornburg

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