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 problem 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 properties and basic transform pairs are supplied with this booklet No calculators or cell phones are permitted NAME Check your section Section 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 Time 10 11 11 12 12 1 1 2 11 12 12 1 11 12 12 1 Room 36 112 36 112 36 112 36 112 36 144 36 144 34 304 34 304 Rec Instr Prof Freeman Prof Freeman Prof Adalsteinsson Prof Adalsteinsson Prof Baldo Prof Baldo Prof Daniel Prof Daniel Please leave the rest of this page blank for use by the graders Problem 1 No of points 35 2 30 3 35 4 35 5 30 6 35 Total 200 Score Grader TA Belal Belal Karen Ian Karen Ian James James Ruby Ruby PROBLEM 1 35 points Consider the following causal CT feedback system x t s2 Ks 2s 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 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 s t pole at 3 Part c Determine the value of K if any for which the impulse response h t of the closed loop system is h t A cos 0 t u t where u t is the step function If such value of K exists calculate 0 K 0 4 Spring2005 Final Exam NAME Part d Determine the value of K if any such that the closed loop frequency response H j has the following straight line approximation for the Bode magnitude plot 20 log H j 20dB dec 20dB dec 1 3 K 5 3 log scale Part e Sketch the root locus of the feedback system as K varies from 0 to Label your axes m s e s 6 Spring2005 Final Exam NAME Work page for problem 1 7 PROBLEM 2 30 points Consider the following system r n x n x t T1 H ej 3 C D 1 2000 where x n y n s 2n s n 2 S I Hlp j y t T2 sec x n 3 r n 0 3 n 0 3 6 otherwise The CTFT X j of the input the DTFT H ej of the DT filter and the CTFT of the lowpass filter Hlp j are shown below H ej X j Hlp j 1 2000 2000 0 0 T2 T2 Part a Sketch the DTFT of x n X ej and the DTFT of r n R ej Label your axes X ej R ej 8 Spring2005 Final Exam NAME Work page for problem 2 9 Part b Determine a value of 0 the cutoff frequency of the DT filter H ej such that s n x nT0 for some nonzero constant and some positive T0 What is T0 0 T0 10 Spring2005 Final Exam NAME Part c Using the value of 0 from the previous part b determine T2 such that y t x t for some nonzero constant T2 11 PROBLEM 3 35 points The following CT signal x t 1 2 cos t 4 cos 2t is passed through an LTI filter with impulse response h t to generate an output y t as shown below X t A t nT sin h t t 4T n x t 1 2 cos t 4 cos 2t 1 Part a Calculate the energy of the input signal over a period 2 1 2 Z 2 x t 2 dt 0 12 Z 0 2 x t 2 dt y t Spring2005 Final Exam NAME Part b Determine if it is possible to choose parameters A and T of the filters so that the output is y t 2 cos t 4 cos 2t Indicate your answer in the box YES or NO If yes determine values of A and T If there are multiple possible solutions you only need to specify one combination of A and T that works A T If no briefly explain why 13 Part c Determine if it is possible to choose parameters A and T of the filters so that the output is y t 1 4 cos 2t Indicate your answer in the box YES or NO If yes determine values of A and T If there are multiple possible solutions you only need to specify one combination of A and T that works A T If no briefly explain why 14 Spring2005 Final Exam NAME Work page for problem 3 15 PROBLEM 4 35 points Consider the following communication system x1 t v1 t cos c t cos r t A x2 t v2 t cos 2 c t x3 t cos 2 r t v3 t lp lp lp 2 15kHz cos 3 c t cos 3 r t where throughout this entire problem c 2 1 MHz Signals x1 t x2 t and x3 t are bandlimited to b X1 j X2 j X3 j b b b b 16 b b y t Spring2005 Final Exam NAME Part a Find the largest value of b such that the CTFT of v1 t v2 t and v3 t do not overlap b 17 Part b Assume b 2 15KHz Find ALL positive values of r if any and the corresponding LPF gain A such that y t x2 t r A 18 Spring2005 Final Exam NAME Work page for Problem 4 19 Part c Consider the following variation of the system in the previous page where all three modulators have the same input x t bandlimited to b 2 15KHz cos c t cos r t A x t cos 2 c t cos 2 r t cos 3 c t cos 3 r t lp lp 2 15kHz Find the value of r if any and the corresponding LPF gain A such that y t x t and …
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