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20115ee102_1_hw6_soln

# 20115ee102_1_hw6_soln - 1 EE102 Fall Quarter 2011 Systems...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #6 Solution 1. Response of compound systems H 1 (jω) H 2 (jω) G 1 (jω) G 2 (jω) F 1 (jω) F 2 (jω) + - - + - + + + α 1 α 2 x(t) y(t) Derive the overall system response H ( ) in frequency domain. Express it in terms of α 1 , α 2 , H 1 ( ) , H 2 ( ) , G 1 ( ) , G 2 ( ) , F 1 ( ) , and F 2 ( ) . Solution We deﬁne intermediate signals v ( t ) and w ( t ) (input and output of small feedback loop). H 1 (jω) H 2 (jω) G 1 (jω) G 2 (jω) F 1 (jω) F 2 (jω) + - - + - + + + α 1 α 2 x(t) y(t) v(t) w(t) We can write: W ( ) = F 1 ( ) V ( ) - F 1 ( ) F 2 ( ) W ( ) W ( ) = F 1 ( ) 1 + F 1 ( ) F 2 ( ) V ( ) Also, V ( ) = X ( ) - α 1 V ( ) H 1 ( ) - α 2 V ( ) H 2 ( )

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2 V ( ) = X ( ) 1 1 + α 1 H 1 ( ) + α 2 H 2 ( ) Write the expression for output: Y ( ) = W ( ) + α 1 G 1 ( ) V ( ) + α 2 G 2 ( ) V
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20115ee102_1_hw6_soln - 1 EE102 Fall Quarter 2011 Systems...

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