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20065ee102_1_102-F06-HW3[1]

# 20065ee102_1_102-F06-HW3[1] - FALL 2006 Put First Letter of...

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FALL 2006 Put First Letter of LAST Name in the corner →→→ Name (LAST, Middle, First):———————— ——————————— EE102: SYSTEMS & SIGNALS HW: # 3 A LATE HW IS NOT A HW! Posted: October 18 Hand In: October 25 Attach This Sheet To Your HW 0. Calculate the Laplace Transforms of those functions on the Table (in the Text) which have NOT been calculated for you in class! In each case clearly indicate the DOC on the PZP of the transform. 1. Find the Laplace Transform of the following signals: (i) e 2 t sinh5 t U ( t ) (ii) e t cos( t π 3 ) U ( t ) (iii) ( e αt sin 2 ω 0 t ) U ( t ) , α > 0 . 2. Show the following properties of L s ( · ): (i) Scaling: F ( s ) = L s { f ( t ) } L s { f ( at ) } = 1 a F ( s a ) , for a > 0 . (ii) Multiplication by t : L s { t f ( t ) } = d ds F ( s ) . 3. Calculate the following integral: y ( t ) = −∞ e ( t σ ) U ( t σ ) σU ( σ ) dσ, t 0 . 1

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Then compute the Laplace transform Y ( s ) of y ( t )—which you just (hopefully “correctly”) found. Next, compute the product: L s { e t U ( t ) } . L s { t U ( t ) } . Finally do you find that: L s { e t U ( t ) } . L s { t U ( t ) } = Y ( s )? ( ) If you don’t, you have done something wrong! If you do, please write down — in plain English — a statement describing the result ( )
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