6.003 (Fall 2009)Quiz #3 November 18, 2009Name:Kerberos Username:Please circle your section number:Section Instructor Time1 Marc Baldo 10 am2 Marc Baldo 11 am3 Elfar Adalsteinsson 1 pm4 Elfar Adalsteinsson 2 pmPartial credit will be given for answers that demonstrate some but not allof the important conceptual issues.Explanations are not required and will not affect your grade.You have two hours.Please put your initials on all subsequent sheets.Enter your answers in the boxes.This quiz is closed book, but you may use three 8.5 × 11 sheets of paper (six sides total).No calculators, computers, cell phones, music players, or other aids.12345Total/12/20/18/25/25/100Quiz #3 / 6.003: Signals and Systems (Fall 2009) 21. Impulsive Input [12 points]Let the following periodic signalx(t) =∞Xm=−∞δ(t − 3m) + δ(t − 1 − 3m) − δ(t − 2 − 3m)be the input to an LTI system with system functionH(s) = es/4− e−s/4.Let bkrepresent the Fourier series coefficients of the resulting output signal y(t). Deter-mine b3.b3=Quiz #3 / 6.003: Signals and Systems (Fall 2009) 3Worksheet (intentionally blank)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 42. System Design [20 points]Design a stable CT LTI system H with all of the following three properties:• the impulse response h(t) has the formh(t) = Cδ(t) + De−2tu(t)where C and D are real-valued constants,• the angle of H(jω) has the following straight-line approximation0−π/2−π1 10 100 1000ω [log scale]∠H(jω) [rad]• if the input x(t) is 1 for all time, then the output y(t) is 1 for all time.Determine the system function H(s) that is consistent with these design specifications.If no such a system exists, enter none.H(s) =Quiz #3 / 6.003: Signals and Systems (Fall 2009) 5Worksheet (intentionally blank)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 63. Input/Output Pairs [18 points]The following signals are all periodic with period T = 1.1x1(t)= x1(t + 1)t01411x2(t)= x2(t + 1)t01411x3(t)= x3(t + 1)t0π101Indicate which of the systems on the next page could/could not be linear and time-invariant.Grading: +3 for each correct answer; −3 for each incorrect answer; 0 for blank or ?.Quiz #3 / 6.003: Signals and Systems (Fall 2009) 7System #1x1(t) x2(t)System #1 could be LTI? (yes/no):System #2x1(t) x3(t)System #2 could be LTI? (yes/no):System #3x2(t) x1(t)System #3 could be LTI? (yes/no):System #4x2(t) x3(t)System #4 could be LTI? (yes/no):System #5x3(t) x1(t)System #5 could be LTI? (yes/no):System #6x3(t) x2(t)System #6 could be LTI? (yes/no):Quiz #3 / 6.003: Signals and Systems (Fall 2009) 84. Fourier Transforms [25 points]The magnitude and angle of the Fourier transform of a signal x(t) are given in thefollowing plots.1−1 2−2ω|X(jω)|1π−π1−1 2−2ω∠X(jω)Five signals are derived from x(t) as shown in the left column of the following table.Six magnitude plots (M1-M6) and six angle plots (A1-A6) are shown on the next page.Determine which of these plots is associated with each of the derived signals and placethe appropriate label (e.g., M1 or A3) in the following table. Note that more than onederived signal could have the same magnitude or angle.signal magnitude angledx(t)dt(x ∗ x)(t)xt −π2x(2t)x2(t)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 91−1 2−2ωM11π−π1−1 2−2ωA11−1 2−2ωM2112π−π1−1 2−2ωA21−1 2−2ωM31π−π1−1 2−2ωA31−1 2−2ωM4112π−π1−1 2−2ωA41−1 2−2ωM5114π−π1−1 2−2ωA51−1 2−2ωM613ππ−π1−1 2−2ωA6Quiz #3 / 6.003: Signals and Systems (Fall 2009) 105. Feedback and Control [25 points]Consider a causal LTI system described by F (s) as follows:F (s) =s2+ 2s + 100s2.a. Sketch the impulse response f(t) for this system on the axes below. Label the axesand indicate the important features of your plot.tf(t)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 11b. Sketch the magnitude and angle of F (jω) on the following axes. Notice the log axesfor ω and for the magnitude. Indicate the important features of your plots, includingextreme values.ω [log axis]|F (jω)| [log axis]10−210−110010110210310−210−1100101102ω [log axis]∠F (jω) [radians]10−210−1100101102103−2π−π0π2πQuiz #3 / 6.003: Signals and Systems (Fall 2009) 12Now consider a feedback system containing F (s) as follows.+F (s) =s2+ 2s + 100s2X Y−c. Let H(s) =Y (s)X(s)represent the closed-loop system function. Sketch the magnitudeand angle of H(s) on the following axes. Notice the log axes for ω and for themagnitude. Indicate the important features of your plots, including extreme values.ω [log axis]|H(jω)| [log axis]10−210−110010110210310−210−1100101102ω [log axis]∠H(jω) [radians]10−210−1100101102103−2π−π0π2πQuiz #3 / 6.003: Signals and Systems (Fall 2009) 13Worksheet (intentionally blank)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 14Worksheet (intentionally blank)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 15Worksheet (intentionally blank)Quiz #3 / 6.003: Signals and Systems (Fall 2009) 16Worksheet (intentionally
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