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FinalF09wsol copy

# FinalF09wsol copy - McGill University Fall 2009 ECSE 303...

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McGill University Fall 2009 ECSE 303: Signals and Systems 1 FINAL EXAMINATION PROBLEM 1 Consider a Linear Time-Invariant (LTI) system with impulse response h ( t ). Let x ( t ) be the input signal and y ( t ) the corresponding output signal. In the following, ’prime’ ( 0 ) will denote the derivative of a signal, i.e., x 0 ( t ) is the derivative of x ( t ), h 0 ( t ) is the derivative of h ( t ), etc. (a) Show that y 0 ( t ) = x ( t ) * h 0 ( t ) = x 0 ( t ) * h ( t ) , where ’*’ denotes convolution. (b) For this part only, assume that the response of the system to x ( t ) = e - 5 t u ( t ), where u ( t ) = 1 , t 0 0 , t < 0 is the unit step function, is sin( ω 0 t ) for some ω 0 > 0. Find the impulse response of the system h ( t ). (c) Let s ( t ) be the unit step response of the system, i.e., the output of the system when the input is the unit step function u ( t ). Show that the response y ( t ) of the system to an arbitrary input x ( t ) is y ( t ) = x 0 ( t ) * s ( t ) . (d) For this part only, assume that the unit step response of the system is s ( t ) = e - 2 t u ( t ). Find the response of the system to the input x ( t ) = 1 t 0 e t t < 0 (e) Is the statement below correct? Justify your answer.

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FinalF09wsol copy - McGill University Fall 2009 ECSE 303...

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