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S11_202Hhwk5 - ECE 202H Spring 2011 Questions that are...

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ECE 202H Spring 2011 Page 1 of 6 ECE 202H: Homework #5 Due: 10 March 2011 at 1:30 PM (5 Points) Questions that are marked with a circled asterisk are to be submitted at the beginning of class on the specified due date. All other problems are provided for practice and are representative of problems that may be included on exams. Solutions to all problems will be posted after the due date. Laplace Transforms 1. Accurately plot the following functions of time: a. 𝑎 ( 𝑡 ) = 𝑢 ( 𝑡 ) ( 𝑡 − 2) 𝑢 ( 𝑡 − 2) + 3 𝑢 ( 𝑡 − 3) ( 𝑡 − 4) 𝑢 ( 𝑡 − 4) b. 𝑏 ( 𝑡 ) = [ 𝑢 ( 𝑡 ) − 𝑢 ( 𝑡 − 1)] + ( 𝑡 − 2)[ 𝑢 ( 𝑡 − 1) − 𝑢 ( 𝑡 − 2)] + ( 𝑡 − 5)[ 𝑢 ( 𝑡 − 3) − 𝑢 ( 𝑡 − 4)] c. 𝑐 ( 𝑡 ) = 2 𝛿�𝑡 − 𝑘 4 20 𝑘=0 d. 𝑑 ( 𝑡 ) = sin(2 𝑡 ) [ 𝑢 ( 𝑡 − 1) − 𝑢 ( 𝑡 − 3)] e. 𝑒 ( 𝑡 ) = 3 𝑢 ( 𝑡 ) + 2 𝑒 −5 ( 𝑡−3 ) 𝑢 ( 𝑡 − 3) 2. If 𝑥 ( 𝑡 ) has Laplace transform 𝑋 ( 𝑠 ) , prove that the following are Laplace transform pairs. a. 𝑥 ( 𝑡 − 𝑡 0 ) ⇔ 𝑒 −𝑠𝑡 0 𝑋 ( 𝑠 ) (assume 𝑥 ( 𝑡 ) is causal) b. 𝑥 ( 𝑡 ) ⇔ 𝑋 ( 𝑠 ) (assume 𝑥 ( 𝑡 ) is causal) c. 𝑥 ( 𝑎𝑡 ) 1 | 𝑎 | 𝑋 � 𝑠 𝑎 (assume 𝑥 ( 𝑡 ) is causal) d. 𝑑 𝑑𝑥 𝑥 ( 𝑡 ) ⇔ 𝑠𝑋 ( 𝑠 ) − 𝑥 (0 ) e. 𝑥 ( 𝛾 ) 𝑑𝛾 ⟺ 1 𝑠 𝑋 ( 𝑠 ) + 1 𝑠 𝑥 ( 𝛾 ) 𝑑𝛾 0 −∞ 𝑡 −∞
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