102_1_FinalSol-05[1]

102_1_FinalSol-05[1] - UCLA Dept. of Electrical Engineering...

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UCLA Dept. of Electrical Engineering EE102: Systems and Signals Final exam solutions 1. The output of a linear, time-invariant, causal system is y ( t ) = e - 2 t sin( t ) U ( t ) when the input is x ( t ) = δ ( t ) + U ( t ) . (a) Find the transfer function H ( s ) of the system. Y ( s ) = 1 / [( s + 2) 2 + 1], X ( s ) = 1 + 1 /s , H ( s ) = Y ( s ) /X ( s ) = s [( s +2) 2 +1]( s +1) , with ROC = <{ s } > - 1. (b) Find the frequency response H ( ) of the system. Is H ( ) = H ( s ) | s = ? Why? The ROC of H ( s ) includes the imaginary axis, therefore H ( ) = H ( s ) | s = = [( +2) 2 +1]( +1) . (c) The input x 1 ( t ) = 5 + cos( t ) + e - t U ( t ) is now applied to the system. Compute the corresponding output, y 1 ( t ) . We can rewrite x 1 ( t ) = x 11 ( t ) + x 12 ( t ) + x 13 ( t ), where x 11 ( t ) = 5, x 12 ( t ) = cos( t ), x 13 ( t ) = e - t U ( t ). The output is the equal to y 1 ( t ) = y 11 ( t ) + y 12 ( t ) + y 13 ( t ). For x 11 ( t ) and x 12 ( t ) we can use the fact that the output corresponding to cos( ω 0 t ) is equal to | H ( 0 ) | cos( ω 0 t + θ ( ω 0 )), where H ( 0 ) = | H ( 0 ) | e ( ω 0 ) . We use this expression with ω 0 = 0 for x 11 ( t ) and ω 0 = 1 for x 12 ( t ), which yield
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102_1_FinalSol-05[1] - UCLA Dept. of Electrical Engineering...

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