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314-Quiz2_sol - ² u t 2 A linear time-invariant system has...

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ECEN 314 Solution to Quiz # 2 Georghiades October 17, 2008 Note: Quiz is open-book, open notes, 15 minutes long. Problems carry equal weight. The use of calculators is NOT allowed. 1. Find the inverse Laplace transform of the following functions: (a) X ( s ) = ( s +1) ( s +2)( s +4) . (b) X ( s ) = ( s +4) ( s +2) 2 +4 Solution (a) Using partial fraction expansion: X ( s ) = - 1 / 2 ( s + 2) + 3 / 2 ( s + 4) . Therefore, x ( t ) = bracketleftbigg - 1 3 e - 2 t + 3 2 e - 4 t bracketrightbigg u ( t ) . (b) We have X ( s ) = ( s + 4) ( s + 2) 2 + 4 = ( s + 2) ( s + 2) 2 + 4 + 2 ( s + 2) 2 + 4 . Using the frequency translation property: x ( t ) = bracketleftBig e - 2 t cos(2
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Unformatted text preview: ) ² u ( t ) . 2. A linear, time-invariant system has the following transfer function: H ( s ) = ( s + 1) ( s + 2)( s + 3) . Find the steady-state response when the input is x ( t ) = 2cos( t + π/ 6) . Solution The output will be another sinusoidal signal at the same frequency as the input ( ω = 1). We have H ( jω ) = ( j + 1) ( j + 2)( j + 3) = 1 5 . Therefore: y ss ( t ) = 2 5 cos( t + π/ 6) . 1...
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