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final_F09_sol

# final_F09_sol - 6.003(Fall 2009 Final Examination Name...

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2 Final Examination / 6.003: Signals and Systems (Fall 2009) 1. CT System with Feedback [15 points] Let G represent a causal system that is described by the following differential equation dy ( t ) + y ( t ) = dx ( t ) x ( t ) dt dt where x ( t ) represents the input signal and y ( t ) represents the output signal. a. Determine the output y 1 ( t ) of G when the input is x 1 ( t ) = e t ; t 0 0 ; otherwise Enter your result in the box below. y 1 ( t ) = (1 2 t ) e t u ( t ) ( s + 1) Y = ( s 1) X Y = s 1 s + 1 X Y 1 = s 1 s + 1 X 1 ; X 1 = 1 s + 1 ; Re s > 1 Y 1 = s 1 ( s + 1) 2 = 1 s + 1 2 ( s + 1) 2 y 1 ( t ) = e t u ( t ) 2 t e t u ( t )
3 Final Examination / 6.003: Signals and Systems (Fall 2009) Now consider a feedback loop that contains the G system described on the previous page. 1 + K G w ( t ) x ( t ) y ( t ) b. Determine a differential equation that relates w ( t ) to y ( t ) when K = 10 . The differ- ential equation should not contain references to x ( t ) . Enter the differential equation in the box below. 11 dy ( t ) 9 y ( t ) = 10 dw ( t ) 10 w ( t ) dt dt Y W = K s 1 s + 1 1 + K s 1 s + 1 = K s 1 s + 1 + Ks K = K s 1 ( K + 1) s ( K 1) = K s 1 11 s 9 1 The minus sign near the adder indicates that the output of the adder is w ( t ) y ( t )

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4 Final Examination / 6.003: Signals and Systems (Fall 2009) c. Determine the values of K for which the feedback system on the previous page is stable. Enter the range (or ranges) in the box below.
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final_F09_sol - 6.003(Fall 2009 Final Examination Name...

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