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hw1_sol - EML 4312 Fall 2007 Assignment 1 Partial Fraction...

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EML 4312 Fall 2007 Assignment 1: Partial Fraction Expansion The due date for this assignment is Wednesday 9/12. 1. (50 points)Determine f ( t ) given F ( s ) for the following problems. (a) (10 points) F ( s ) = s 2 + 2 s + 2 ( s + 1)( s + 2) F(s) is not a strictly proper function, so we must divide. F ( s ) = s 2 + 2 s + 2 s 2 + 3 s + 2 = 1 s ( s + 1)( s + 2) = 1 + 1 ( s + 1) + 2 ( s + 2) F ( t ) = δ ( t ) + e t 2 e 2 t (b) (10 points) F ( s ) = 2 s 2 6 s + 13 s 1 s 2 2 s + 2 F ( s ) = 2 ( s 3) 2 + 2 2 s 1 ( s 1) 2 + 1 2 F ( t ) = e 3 t sin 2 t e t cos t (c) (10 points) F ( s ) = s + 10 s 2 + 6 s + 34 F ( s ) = s + 3 + 7 ( s + 3) 2 + 5 2 = s + 3 ( s + 3) 2 + 5 2 + 7 5 5 ( s + 3) 2 + 5 2 F ( t ) = e 3 t cos(5 t ) + 7 5 e 3 t sin(5 t ) (d) (20 points) F ( s ) = 1 ( s + 1) 3 ( s + 2) 1 ( s + 1) 3 ( s + 2) = A s + 1 + B ( s + 1) 2 + C ( s + 1) 3 + D s + 2 1 ( s + 1) 3 | s = 2 = A ( s + 2) s + 1 + B ( s + 2) ( s + 1) 2 + C ( s + 2) ( s + 1) 3 + D ¸ | s = 2 1 = D 1 ( s + 2) | s = 1 = A ( s + 1) 3 s + 1 + B ( s + 1) 3 ( s + 1) 2 + C ( s + 1) 3 ( s + 1) 3 + D ( s + 1) 3 s + 2 ¸ | s = 1 1 = C 1
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B = 1 1! μ d ds ( s + 1) 3 1 ( s + 1) 3 ( s + 2) ¸¶ | s = 1 = 1 ( s + 2) 2 | s = 1 = 1 A = 1 2!
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