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Unformatted text preview: 1 2 3 4 5 6 7 Total / 15 / 10 / 14 / 14 / 15 / 16 / 16 / 100 6.003 (Fall 2009) Final Examination December 17, 2009 Name: Kerberos Username: Please circle your section number: Section Instructor Time 1 Marc Baldo 10 am 2 Marc Baldo 11 am 3 Elfar Adalsteinsson 1 pm 4 Elfar Adalsteinsson 2 pm Partial credit will be given for answers that demonstrate some but not all of the important conceptual issues. Explanations are not required and will not affect your grade. You have three hours. Please put your initials on all subsequent sheets. Enter your answers in the boxes. This quiz is closed book, but you may use four 8 . 5 11 sheets of paper (eight sides total). No calculators, computers, cell phones, music players, or other aids. 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 ) 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)...
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This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Spring '11 term at MIT.
 Spring '11
 Freeman

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