102_1_hw07_sol

102_1_hw07_sol - 1 EE102 Spring 2009-10 Systems and Signals...

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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 . Solution: F ( s ) = L e - 2 t - L t 2 e - 2 t = 1 s + 2 - 2 ( s + 2) 3 = s 2 + 4 s + 4 - 2 ( s + 2) 3 = s 2 + 4 s + 2 ( s + 2) 3 (b) One cycle of a sinusoid, f ( t ) = sin(2 π t ) 0 t < 1 0 1 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. Solution: We can write this signal as f ( t ) = sin(2 π t ) - sin(2 π ( t - 1)) u ( t - 1) The Laplace transform is then F ( s ) = 2 π s 2 + 4 π 2 - 2 π e - s s 2 + 4 π 2 = 2 π (1 - e - s ) s 2 + 4 π 2
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2 (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. Solution: If we differentiate the signal once, we get t 1 1 0 2 3 4 5 2 t f ( t ) 2 and a second time t 1 1 0 2 3 4 5 2 f ( t ) - 2 ! ( t - 1 ) Then f ( t ) = 2( u ( t ) - u ( t - 1)) - 2 δ ( t - 1) The Laplace transform of f ( t ) is L [ f ( t )] = 2 s - 2 e - s s - 2 e - s The Laplace transform of f ( t ) is then L [ f ( t )] = 1 s 2 L [ f ( t )] = 1 s 2 2 s - 2 e - s s - 2 e - s = 2 - 2 e - s - 2 se - s s 3 (d) Find the Laplace transform of the following signal
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3 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. Solution: First write an expression for the function, f ( t ) = 1 2 e - t + e - t cos(4 π t ) Then take the Laplace transform F ( s ) = 1 2 1 s + 1 + s + 1 ( s + 1) 2 + (4 π ) 2 = 1 2 ( s + 1) 2 + (4 π ) 2 + ( s + 1) 2 ( s + 1) 3 + ( s + 1)(4 π ) 2 = s 2 + 2 s + 1 + 8 π 2 s 3 + 3 s 2 + 3 s + 1 + 16 π 2 s + 16 π 2 = s 2 + 2 s + (1 + 8 π 2 ) s 3 + 3 s 2 + (3 + 16 π 2 ) s + (1 + 16 π 2 ) 2. Find the Inverse Laplace transforms of the following functions (a) s 2 ( s + 1) 2 Solution:
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