final_6.003_2005

final_6.003_2005 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Spring 2005 Final Exam Tuesday, May 17, 2005 Directions: PLEASE WRITE YOUR NAME ON THIS COVER SHEET NOW! Please also write your name on all remaining sheets. Unless indicated otherwise, answers must be derived or briefly explained. All sketches must be adequately labeled. There are a total of 6 problems in this booklet from pages 2 through 37. Enter all your work and answers directly in the spaces provided on the printed pages of this booklet. Additional work spaces are supplied from pages 38 through 40. These pages will NOT be graded unless you specify that you are continuing a particular prob- lem on a particular continuation page. The bluebook is for your scratch work. We will NOT grade anything in your bluebook. This quiz is closed book, but students may bring and use three 8 1/2 × 11 sheets of paper as reference. Tables of Fourier Series, Fourier transform, Laplace transform, and Z transform prop- erties and basic transform pairs are supplied with this booklet. No calculators or cell phones are permitted. NAME: Check your section Section Time Room Rec. Instr. TA a 1 10-11 36-112 Prof. Freeman Belal a 2 11-12 36-112 Prof. Freeman Belal a 3 12- 1 36-112 Prof. Adalsteinsson a 4 1- 2 36-112 Prof. Adalsteinsson a 5 11-12 36-144 Prof. Baldo James a 6 12- 1 36-144 Prof. Baldo James a 7 11-12 34-304 Prof. Daniel Ruby a 8 12- 1 34-304 Prof. Daniel Ruby Please leave the rest of this page blank for use by the graders: Problem No. of points Score Grader 1 35 2 30 3 35 4 35 5 30 6 35 Total 200
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PROBLEM 1 (35 points) Consider the following causal CT feedback system. x ( t ) + + Ks s 2 2 s + 1 y ( t ) Part a. Determine the range of K for which the closed-loop system is a stable LTI system. range of K : 2
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Spring2005: Final Exam NAME: Part b. For a particular value of K , the closed-loop system has two real poles, one of which is at s = 1 . Determine the other pole and the step response, s ( t ) of the closed-loop system corresponding to such value of K . pole at , s ( t ) = 3
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Determine the value of K , if any, for which the impulse response h ( t ) of the closed loop system is h ( t ) = A cos ω 0 t u ( t ) where u ( t ) is the step function. If such value of K exists, calculate ω 0 . K
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final_6.003_2005 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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