102_1_hw07

102_1_hw07 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #7 Due: Tuesday, June 1, 2010 5PM 1. Find the Laplace transform of the following signals: (a) f ( t ) = (1 - t 2 ) e - 2 t . (b) One cycle of a sinusoid, f ( t )= ± sin(2 πt )0 t< 1 01 t which is plotted below: 0 1 2 t f ( t ) 1 - 1 Hint: Use the delay theorem, and the fact that a delayed signal is padded with zeros. (c) Find the Laplace transform of the following signal, t t 2 1 1 0 2 3 4 5 f ( t ) Hint: Differentiate once or twice, and then use the integral theorem. (d) Find the Laplace transform of the following signal t 1 2 0 3 f ( t ) e - t 1 This is a cosine with an envelope that bounds the cosine below by 0 and above by e - t . Combine terms over a common denominator, and simplify your answer. Hint: Express the signal as a sum of a decaying exponential, and an exponentially decaying cosine.
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2 2. Find the Inverse Laplace transforms of the following functions (a) 4 s 3 +4 s (b) s +3 ( s + 1) 2 ( s + 2) (c) 10 ( s + 1)( s 2 s + 13) (d) s 2 +8 s s 2 s + 25 3. Solving Differential Equations A system is described by the differential equation
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102_1_hw07 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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