MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.003: Signals and Systems—Spring 2005Final ExamTuesday, May 17, 2005Directions:• 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 sketchesmust 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 pagesof this booklet. Additional work spaces are supplied from pages 38 through 40. Thesepages 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 ofpaper 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. TA2 1 10-11 36-112 Prof. Freeman Belal2 2 11-12 36-112 Prof. Freeman Belal2 3 12- 1 36-112 Prof. Adalsteinsson Karen & Ian2 4 1- 2 36-112 Prof. Adalsteinsson Karen & Ian2 5 11-12 36-144 Prof. Baldo James2 6 12- 1 36-144 Prof. Baldo James2 7 11-12 34-304 Prof. Daniel Ruby2 8 12- 1 34-304 Prof. Daniel RubyPlease leave the rest of this page blank for use by the graders:Problem No. of points Score Grader1 352 303 354 355 306 35Total 200PROBLEM 1 (35 points)Consider the following causal CT feedback system.x(t)−++Kss2− 2s + 1y(t)Part a. Determine the range of K for which the closed-loop system is a stable LTI system.range of K :2Spring2005: Final Exam NAME:Part b. For a particular value of K, the closed-loop system has two real poles, one of whichis at s = −1. Determine the other pole and the step response, s(t) of the closed-loop systemcorresponding to such value of K.pole at, s(t) =3Part c. Determine the value of K, if any, for which the impulse response h(t) of the closedloop system ish(t) = A cosω0t u(t)where u(t) is the step function. If such value of K exists, calculate ω0.K =, ω0=4Spring2005: Final Exam NAME:Part d. Determine the value of K, if any, such that the closed-loop frequency responseH(jω) has the following straight-line approximation for the Bode magnitude plot.ω [log scale]133−20dB/dec20dB/dec20 log |H(jω)|K :5Part e. Sketch the root-locus of the feedback system as K varies from 0 to +∞. Label youraxes.ℜe{s}ℑm{s}6Spring2005: Final Exam NAME:Work page for problem 17PROBLEM 2 (30 points)Consider the following systemx(t)C/DT1=12000secx[n]↑ 3r[n]H(ejΩ)s[n]↓ 2y[n] = s[2n]S/IT2Hlp(jω)y(t)wherex[n] ↑ 3 r[n] =(x[n/3], n = 0, ±3, ±6, · · ·0, otherwiseThe CTFT X(jω) of the input, the DTFT H(ejΩ) of the DT filter, and the CTFT of the low-pass filter Hlp(jω) are shown below.ω−2000π2000π1X(jω)Ω−Ω0Ω0−ππH(ejΩ)ω−πT2πT2Hlp(jω)Part a. Sketch the DTFT of x[n], X(ejΩ), and the DTFT of r[n], R(ejΩ). Label your axes.X(ejΩ)ΩR(ejΩ)Ω8Spring2005: Final Exam NAME:Work page for problem 29Part b. Determine a value of Ω0, the cutoff frequency of the DT filter H(ejΩ), such thats[n] = αx(nT0)for some nonzero constant α and some positive T0. What is T0?Ω0=, T0=10Spring2005: Final Exam NAME:Part c. Using the value of Ω0from the previous part b, determine T2such thaty(t) = βx(t)for some nonzero constant β.T2=11PROBLEM 3 (35 points)The following CT signal:x(t) = 1 + 2 cos t + 4 cos 2tis passed through an LTI filter with impulse response h(t) to generate an output y(t) asshown belowx(t) = 1 + 2 cos t + 4 cos 2th(t) =+∞Xn=−∞Aπtsinπt4Tδ(t − nT )y(t)Part a. Calculate the energy of the input signal over a period:12πZ2π0|x(t)|2dt.12πZ2π0|x(t)|2dt =12Spring2005: Final Exam NAME:Part b. Determine if it is possible to choose parameters A and T of the filters so that theoutput isy(t) = 2 cos t + 4 cos 2tIndicate your answer in the box: YES or NO.If yes, determine values of A and T . [If there are multiple possible solutions, you only needto specify one combination of A and T that works.]A =T =If no, briefly explain why.13Part c. Determine if it is possible to choose parameters A and T of the filters so that theoutput isy(t) = 1 + 4 cos 2tIndicate your answer in the box: YES or NO.If yes, determine values of A and T . [If there are multiple possible solutions, you only needto specify one combination of A and T that works.]A =T =If no, briefly explain why.14Spring2005: Final Exam NAME:Work page for problem 315PROBLEM 4 (35 points)Consider the following communication systemx1(t)×cos ωctv1(t)x2(t)×cos 2ωctv2(t)x3(t)×cos 3ωctv3(t)+×cos ωrt×cos 2ωrt+×cos 3ωrtAωlp−ωlpωωlp= 2π · 15kHzy(t)where throughout this entire problem ωc= 2π · 1 MHz.Signals x1(t), x2(t), and x3(t) are bandlimited to |ω| < ωbω−ωbωbX1(jω)ω−ωbωbX2(jω)ω−ωbωbX3(jω)16Spring2005: Final Exam NAME:Part a. Find the largest value of ωbsuch that the CTFT of v1(t), v2(t) and v3(t) do notoverlap.ωb=17Part b. Assume ωb= 2π · 15KHz. Find ALL positive values of ωr, if any, and the corre-sponding LPF gain A such that y(t) = x2(t).ωr:A:18Spring2005: Final Exam NAME:Work page for Problem 4.19Part c. Consider the following variation of the system in the previous page, where all threemodulators have the same input x(t), bandlimited to ωb= 2π· 15KHz,x(t)×cos ωct×cos 2ωct×cos 3ωct+×cos ωrt×cos 2ωrt+×cos 3ωrtAωlp−ωlpωωlp= 2π · 15kHzy(t)Find the value of ωr, if any, and the corresponding LPF gain A such that y(t) = x(t) and AIS MINIMIZED.ωr=A:20Spring2005: Final Exam NAME:Work page for Problem 4.21Part d. Consider another variation of the original system, where all three modulators havethe same input x(t), bandlimited to ωb= 2π· 15KHz, and the receiver’s demodulators aresubstituted with impulse trains:x(t)×cos ωct×cos 2ωct×cos 3ωct+×∞Xn=−∞δ(t − nT )×∞Xn=−∞δ(t − nT )+×∞Xn=−∞δ(t − nT )Aωlp−ωlpωωlp= 2π · 15kHzy(t)Find THE LARGEST sampling period T , if any, and the corresponding A such thaty(t) = x(t).T =A =22Spring2005: Final Exam NAME:Work page for Problem 4.23PROBLEM 5 (30 points)The following two DT LTI systems are connected in cascadex[n]System
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