102_1_Final-W06sol[1]

102_1_Final-W06sol[1] - UCLA Electrical Engineering Dept....

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Unformatted text preview: UCLA Electrical Engineering Dept. EE102: Systems and Signals Final Solutions 1. Consider a linear system with impulse response h ( t ) = e- t (cos( t )- sin( t )) U ( t ) . (a) Find the transfer function H ( s ) of the system. From the tables, H ( s ) = s + 1 ( s + 1) 2 + 1- 1 ( s + 1) 2 + 1 = s ( s + 1) 2 + 1 . (b) Compute the input x ( t ) that generates the output y ( t ) = e- t cos( t- 1) U ( t- 1) . Because y ( t ) = e- 1 e ( t- 1) cos( t- 1) U ( t- 1), Y ( s ) = e- 1 e- st s +1 ( s +1) 2 +1 . From the relation Y ( s ) = H ( s ) X ( s ), we have that X ( s ) = e- 1 e- s s + 1 s = e- 1 e- s 1 + 1 s ¶ . By taking the inverse Laplace transform, we have that x ( t ) = e- 1 ( δ ( t- 1)+ U ( t- 1)). (c) Find the frequency response H ( iω ) of the system. Is H ( iω ) = H ( s ) | s = iω ? The poles of H ( s ) are to the left of the imaginary axis, and h ( t ) = 0, t < 0, therefore H ( iω ) = H ( s ) | s = iω , and H ( iω ) = iω (1 + iω ) 2 + 1 = iω (2- ω 2 ) + i 2 ω . (d) What are the expressions for | H ( iω ) | and θ ( ω ) , where H ( iω ) = | H ( iω ) | e iθ ( ω ) ? Compute | H ( i √ 2) | and θ ( √ 2) . | H ( iω ) | = ω p (2- ω 2 ) 2 + 4 ω 2 = w √ w 4 + 4 , θ ( ω ) =...
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This note was uploaded on 10/31/2009 for the course EE ee102 taught by Professor Levan during the Fall '09 term at UCLA.

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102_1_Final-W06sol[1] - UCLA Electrical Engineering Dept....

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