6 003 Fall 2009 November 18 2009 Quiz 3 Name Kerberos Username Please circle your section number Section 1 2 3 4 Instructor Time Marc Baldo Marc Baldo Elfar Adalsteinsson Elfar Adalsteinsson 10 am 11 am 1 pm 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 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 1 12 2 20 3 18 4 25 5 25 Total 100 Quiz 3 6 003 Signals and Systems Fall 2009 1 Impulsive Input 2 12 points Let the following periodic signal x t X t 3m t 1 3m t 2 3m m be the input to an LTI system with system function H s es 4 e s 4 Let bk represent the Fourier series coefficients of the resulting output signal y t Determine b3 b3 Quiz 3 6 003 Signals and Systems Fall 2009 Worksheet intentionally blank 3 Quiz 3 6 003 Signals and Systems Fall 2009 2 System Design 4 20 points Design a stable CT LTI system H with all of the following three properties the impulse response h t has the form h t C t De 2t u t where C and D are real valued constants the angle of H j has the following straight line approximation H j rad 0 2 log scale 1 10 100 1000 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 Worksheet intentionally blank 5 Quiz 3 6 003 Signals and Systems Fall 2009 3 Input Output Pairs 6 18 points The following signals are all periodic with period T 1 x1 t x1 t 1 1 t 0 1 1 4 x2 t x2 t 1 1 t 0 1 1 4 x3 t x3 t 1 1 t 0 10 1 Indicate which of the systems on the next page could could not be linear and timeinvariant Grading 3 for each correct answer 3 for each incorrect answer 0 for blank or Quiz 3 6 003 Signals and Systems Fall 2009 x1 t System 1 x2 t System 1 could be LTI yes no x1 t System 2 x3 t System 2 could be LTI yes no x2 t System 3 x1 t System 3 could be LTI yes no x2 t System 4 x3 t System 4 could be LTI yes no x3 t System 5 x1 t System 5 could be LTI yes no x3 t System 6 x2 t System 6 could be LTI yes no 7 Quiz 3 6 003 Signals and Systems Fall 2009 8 25 points 4 Fourier Transforms The magnitude and angle of the Fourier transform of a signal x t are given in the following plots X j 1 2 1 1 2 1 2 X j 2 1 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 place the appropriate label e g M1 or A3 in the following table Note that more than one derived signal could have the same magnitude or angle signal dx t dt x x t x t 2 x 2t x2 t magnitude angle Quiz 3 6 003 Signals and Systems Fall 2009 9 M1 A1 1 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2 1 2 2 M2 A2 1 1 2 2 1 2 1 1 2 M3 A3 1 2 1 2 1 1 2 M4 A4 1 1 2 2 1 2 1 1 2 M5 A5 1 1 4 2 1 1 2 1 2 M6 A6 1 3 2 1 1 2 2 1 Quiz 3 6 003 Signals and Systems Fall 2009 5 Feedback and Control 10 25 points Consider a causal LTI system described by F s as follows F s s2 2s 100 s2 a Sketch the impulse response f t for this system on the axes below Label the axes and indicate the important features of your plot f t t Quiz 3 6 003 Signals and Systems Fall 2009 11 b Sketch the magnitude and angle of F j on the following axes Notice the log axes for and for the magnitude Indicate the important features of your plots including extreme values F j log axis 102 101 100 10 1 10 2 10 2 log axis 10 1 100 101 102 103 100 101 102 103 F j radians 2 0 2 10 2 log axis 10 1 Quiz 3 6 003 Signals and Systems Fall 2009 12 Now consider a feedback system containing F s as follows X Y F s s2 2s 100 s2 Y s c Let H s X s represent the closed loop system function Sketch the magnitude and angle of H s on the following axes Notice the log axes for and for the magnitude Indicate the important features of your plots including extreme values H j log axis 102 101 100 10 1 10 2 10 2 log axis 10 1 100 101 102 103 100 101 102 103 H j radians 2 0 2 10 2 log axis 10 1 Quiz 3 6 003 Signals and Systems Fall 2009 Worksheet intentionally blank 13 Quiz 3 6 003 Signals and Systems Fall 2009 Worksheet intentionally blank 14 Quiz 3 6 003 Signals and Systems Fall 2009 Worksheet intentionally blank 15 Quiz 3 6 003 Signals and Systems Fall 2009 Worksheet intentionally blank 16
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