ecen314-fall11-hw6

# ecen314-fall11-hw6 - Problem 4 LTI System described by...

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ECEN 314: Signals and Systems, Fall ‘11 Homework #6 Homework Assignment #6 Due date – Nov. 10, 2011 (Thu), 3:55PM in class. Problem 1. Fourier Transform (16 points) Find the Fourier transform X ( ) of the following signals. [4 pt] (a) x ( t ) = e - 3 t cos ( πt 2 ) u ( t ) [4 pt] (b) x ( t ) = e - 2 | t - 1 | [4 pt] (c) x ( t ) = ± cos ( πt 2 ) , | t | ≤ 2 0 , | t | > 2 [4 pt] (d) x ( t ) = te - 3 t cos ( πt 2 ) u ( t ) Problem 2. Inverse Fourier Transform (16 points) Find the inverse Fourier transform x ( t ) of the following signals. [4 pt] (a) X ( ) = 1 1+4 ω 2 [4 pt] (b) X ( ) = ω 1+4 ω 2 [4 pt] (c) X ( ) = e - 2 | ω | [4 pt] (d) X ( ) = u ( ω + 1) - u ( ω - 3)

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Problem 3. LTI System (12 points) Consider an LTI system whose output is y ( t ) = [ e - t - 3 e - 3 t ] u ( t ) if the input is x ( t ) = [ e - t + 2 e - 2 t ] u ( t ) . [4 pt] (a) Find the frequency response H ( ) of the system. [4 pt] (b) Find the impulse response h ( t ) of the system. [4 pt] (c) Find the linear constant coefﬁcient differential equation that describes the given LTI system.
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Unformatted text preview: Problem 4. LTI System described by differential equation (16 points) Consider a causal LTI system described by the following differential equation: d 2 y ( t ) dt 2 + 5 dy ( t ) dt + 6 y ( t ) = dx ( t ) dt-5 x ( t ) [4 pt] (a) Find the impulse response h ( t ) of the above system. [4 pt] (b) Let s ( t ) be the (unit) step response of the above system. Find its Fourier transform S ( jω ) . [4 pt] (c) Find the output y ( t ) of the system when the input is x ( t ) = e-t u ( t ) . [4 pt] (d) Find the output y ( t ) of the system when the input is x ( t ) = e-2 t u ( t ) ....
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ecen314-fall11-hw6 - Problem 4 LTI System described by...

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