A1 - Signals and Systems Fall 201 5 Issued 2 7 August 201 5...

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Signals and Systems Fall 201 5 Issued: 2 7 August 201 5 Due: 2 September 201 5 Homework Assignment #1 1. Find the differential equation relating y ( t ) and x ( t ) if Y ( s ) X ( s ) = s 2 + 2 s 3 s 2 + 4 s + 3 . 2. Find the system transfer function ) ( ) ( ) ( s X s Y s H = if the input x ( t ) and the output y ( t ) of the system are related by the equation = + t d x t x t y t y 0 ) ( ) ( ) ( 3 ) ( τ τ . Assume y (0) = 0. (Hint: Take the Laplace transform of the equation.) 3. Solve the differential equation ) ( ) ( 9 ) ( t f t y t y + = with initial conditions y (0) = 0, ) 0 ( y = 6, and with f ( t ) = 0. 4. For the RC network shown, find the transfer function V out ( s )/ V in ( s ) and sketch the frequency response magnitude plot. + v in + v out 10 k Ω 50 μ F
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Signals and Systems Fall 2013 Solution to Homework Assignment #1 1. Find the differential equation relating y ( t ) and x ( t ): 3 2 2 2 ( ) ( ) 4 ( ) 3 ( ) ( ) 2 ( ) s Y s s Y s sY s Y s s X s s X s + + = + Taking inverse Laplace transform of the above equation, leads to: ( ) ( ) 4 ( ) 3 ( ) ( ) 2 ( ) y t y t y t y t x t x t + + = + ±±± ±± ± ±± .
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