SOL15 - Signals and Systems Fall 2012 Solution to Homework...

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Signals and Systems Fall 2012 Solution to Homework Assignment #15 1. (a) ( ) ( ) ( ) 2 2 2 2 2 3 3 2 7 4 ( ) 3 6 25 4 3 4 3 4 s s X s s s s s + + = = × × + + + + + + We know that: ( ) 2 2 sin ( ) L at s b e bt u t s a b ←→ + + ( ) 2 2 cos ( ) L at s s a e bt u t s a b + ←→ + + Therefore, we obtain the following signal in time domain: ( ) 4 4 7 3 cos3 ( ) sin3 ( ) 4 t t s s x t e t u t e t u t = × × (b) 4 4 1 1 1 ( ) 2 2 2 s s e X s e s s s = = × + + + We know that ( ) 1 1 L at s e u t a ←→ + and ( ) ( ) ( ) , 0 L as s x t a u t a e X s a ←→ > Therefore, we obtain the following signal in time domain: ( ) ( ) ( ) ( ) 2 4 2 4 t t s s x t e u t e u t = 2. ) 4 ( 2 ) ( ) ( ) ( 2 . 0 ) ( + = + + t x t x t y t y t y Taking the Laplace transform of the above equation: ( ) ( ) ( ) ( ) 3 2 4 0.2 2 s Y s s s s X s s e + + = + We can write the input as below: < < = 2 2 0 0 0 ) ( 2 t e t e t t x t .

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