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EE210_F11_HW2

# EE210_F11_HW2 - EE210-2 Fall 2011 Prof Essam Marouf Linear...

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EE210-2, Fall 2011 Prof. Essam Marouf Linear Systems (Transform Theory & Applications) ---------------------------------------------------------------------------------------------------------- Homework #2 To be Completed by Wednesday September 21, 2011 ------------------------------------------------------------------ 1- a) Evaluate and sketch the convolution x ( t ) = g ( t ) h ( t ) , where g ( t ) = Π ( t 1 2 ) and h ( t ) = Λ ( t ) u ( t ) b) For the two signals g ( t ) and h ( t ) shown below, use graphical construction to identify the distinct time intervals over which analytical evaluation of the convolution x ( t ) = g ( t ) h ( t ) is to be performed. Give numerical values for x (t) at times t corresponding to the boundaries of these time intervals. [You need NOT evaluate x ( t ) for all t ]. ----------------------------------------------------------------------------------------------------------- 2- a) Show analytically, graphically, or both, that i) u ( t ) [ x ( t ) u ( t )] = x ( t ) dt 0 t ii) u ( t ) u ( t + 1) u ( t ) u ( t 1) = Λ ( t ) b) Evaluate and sketch i) x 1 ( t ) = Λ ( t ) I I ( t ) , where I I ( t ) is the even impulse pair. ii) x 2 ( t ) = 1 2 III( t 2 ) [ Π ( t 1 2 ) −Π ( t 3 2 )] ---------------------------------------------------------------------------------------------------------- 3- a) Calculate the autocorrelation function Γ xx ( t ) of the energy signal x ( t ) shown. Check that your answer satisfies all properties of a valid autocorrelation function.

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EE210_F11_HW2 - EE210-2 Fall 2011 Prof Essam Marouf Linear...

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