SOL01 - Signals and Systems Fall 2012 Solution to Homework...

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Signals and Systems Fall 2012 Solution to Homework Assignment #1: 1. ) ( ) ( s X s Y = 1 4 2 3 5 . 2 2 3 + + s s s s 3 2 3 ( ) 2 ( ) 4 ( ) ( ) 2.5 ( ) s Y s s Y s sY s Y s sX s + + = By taking the inverse Laplace transform of the above equation, we obtain: 3 ( ) 2 ( ) 4 ( ) ( ) 2.5 ( ) y t y t y t y t x t + + =   2. The input and output of the considered system are related by: = + t d x t x t y t y 0 ) ( ) ( ) ( 3 ) ( τ τ By taking the Laplace transform, we have 0 ( ) ( ) (0) 3 ( ) ( ) X s sY s y Y s X s s = + = ( ) ( ) 1 3 ( ) ( ) s s Y s X s s + = It then follows that: ( ) 2 1 ( ) ( ) ( ) 3 s Y s H s X s s s = = + 3. ( ) 2 ( ) ( ) 2 ( ) y t y t y t u t = +  ( ) 2 ( ) ( ) 2 ( ) y t y t y t u t + =  (*) We first consider the following problem: ( ) 2 ( ) ( ) 0 y t y t y t + =  (**) Observe that the corresponding characteristic equation is 2 2 1 0 k k + = , giving the double root 1 k = . Hence, the solution of (**) has the following form: 1 2 t t H y C e C te = +
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