6.003 (Fall 2007)Final exam 17 December 2007Name:Please circle your section number:Section Instructor Time1 Jeffrey Lang 102 Jeffrey Lang 113 Karen Livescu 114 Sanjoy Mahajan 125 Antonio Torralba 16 Qing Hu 2Partial credit will be given, according to the conceptual features that aproposed answer shares with the correct answer.Explanations are not required and do not affect your grade.• You have three hours. Have fun!• 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).• No calculators, computers, cell phones, music players, or other aids.1. /15 ( ) 5. /05 ( ) 9. /10 ( ) / 30 ( )2. /05 ( ) 6. /10 ( ) 10. /10 ( ) / 25 ( )3. /05 ( ) 7. /15 ( ) 11. /10 ( ) / 30 ( )4. /10 ( ) 8. /05 ( ) / 15 ( )/35 ( ) /35 ( ) /30 ( ) /100 ( )Final exam / 6.003: Signals and Systems (Fall 2007) 31. Matching time and frequency representations [15 points]For each time signal, choose the magnitude of its Fourier transform from the four choicesbelow. The time signals are zero outside the plotted region. Each time and frequency figurehas its own scale, with the origin where the axes intersect. If you are unsure of the correct answer,you can also select a second answer, in which case you will receive the average of the scores for the twoanswers.tA B C DtA B C Dt1 1A B C DHere are the choices for the Fourier transform magnitude:ωωA BωωC DFinal exam / 6.003: Signals and Systems (Fall 2007) 42. Discrete-time periodicity [5 points]Here are two periodic discrete-time signals:x3[n] = . . . , 0, 0, 1, 0, 0, 1, 0, 0, 1, . . .x5[n] = . . . , 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, . . .Mark the period of their sum x3[n] + x5[n] as an on the thermometer:0 2 4 6 8 10 12 14 16 18 20If you are unsure of the correct value, you can also mark a second value, in which case you will receivethe average of the scores for the two values.Final exam / 6.003: Signals and Systems (Fall 2007) 53. Find the output signal [5 points]A given linear, time-invariant system turns the unit pulse into the triangle:system010 1 2 3010 1 2 3The system is given the following input signal:010 1 2 3Sketch the output signal on the following graph:-2 -1 0 1 2 3 4 5 612Final exam / 6.003: Signals and Systems (Fall 2007) 64. Frequency response [10 points]Final exam / 6.003: Signals and Systems (Fall 2007) 7a. Here is a resistor–capacitor circuit:1111 1cos ty(t)The input signal is cos t for all time (not just for t ≥ 0). Choose the output signal fromthose given on the next page. If you are unsure of the correct answer, you can also select asecond answer, in which case you will receive the average of the scores for the two answers.1 2 3 4 5 6 7 8 9 10Final exam / 6.003: Signals and Systems (Fall 2007) 8b. Here is an LRC circuit:111cos ty(t)The input signal is cos t for all time (not just for t ≥ 0). Choose the output signal fromthose given on the next page. If you are unsure of the correct answer, you can also select asecond answer, in which case you will receive the average of the scores for the two answers.1 2 3 4 5 6 7 8 9 10Final exam / 6.003: Signals and Systems (Fall 2007) 9Here are the possible output signals for the preceding questions:-2-1012t2π0−2π-2-1012t2π0−2π1 2-2-1012t2π0−2π-2-1012t2π0−2π3 4-2-1012t2π0−2π-2-1012t2π0−2π5 6-2-1012t2π0−2π-2-1012t2π0−2π7 8-2-1012t2π0−2π-2-1012t2π0−2π9 10Final exam / 6.003: Signals and Systems (Fall 2007) 105. Mystery cosine [5 points]You sample x(t) = cos ωt with sampling interval T = 1 and get this sequence of samples. . . , 1,12, −12, −1, −12,12, 1,12, −12, −1, −12, . . .Using an, mark all possible values of ω that lie in the thermometer range:0 1 2 3 4 5 6 7 8 9 10Final exam / 6.003: Signals and Systems (Fall 2007) 116. Discrete-time poles and zeros [10 points]A discrete-time system with a finite number of poles and zeros turns the causal signalx[n] =3n + 1 for n ≥ 0;0 otherwise.into the unit stepu[n] =1 for n ≥ 0;0 otherwise.Final exam / 6.003: Signals and Systems (Fall 2007) 12a. What is the smallest number of zeros that the system could have? (If there are repeatedzeros, count each.)0 1 2 3 ≥ 3If you are unsure of the correct answer, you can also select a second answer, in which case you willreceive the average of the scores for the two answers.Final exam / 6.003: Signals and Systems (Fall 2007) 13b. What is the smallest number of poles that the system could have? (If there are repeatedpoles, count each.)0 1 2 3 ≥ 3If you are unsure of the correct answer, you can also select a second answer, in which case you willreceive the average of the scores for the two answers.Final exam / 6.003: Signals and Systems (Fall 2007) 147. Taking medicine [15 points]Starting at t = 0 a patient takes a medicine every 8 hours, eating a0pills at t = 0; a1pills att = 8 hours; a2pills at t = 16 hours; and so on forever. The graph shows the concentrationof medicine in the patient’s blood as a function of time. This concentration is the output of afirst-order system.023t (hours). . .0 8 16 24 32 40Final exam / 6.003: Signals and Systems (Fall 2007) 15a. Mark the average concentration (in the region t ≥ 0) as an on the thermometer:0 1 2 3 4 5If you are unsure of the correct value, you can also mark a second value, in which case you willreceive the average of the scores for the two values.Final exam / 6.003: Signals and Systems (Fall 2007) 16b. Mark the time constant of the first-order system as an on the thermometer:10 min30 min1 hr 2 hr 5 hr 10 hr 1 day 3 days 1 weekIf you are unsure of the correct value, you can also mark a second value, in which case you willreceive the average of the scores for the two values.Final exam / 6.003: Signals and Systems (Fall 2007) 17c. Mark the ratio a2007/a0as an on the thermometer:0 0.1 0.2 0.3 0.5 0.7 1 1.5 2 3 5If you are unsure of the correct value, you can also mark a second value, in which case you willreceive the average of the scores for the two values.Final exam / 6.003: Signals and Systems (Fall 2007) 188. Maximizing output samples [5 points]A discrete-time system has system functional 1/(1 +3√10R3) with input signal X and outputsignal Y [the corresponding system function is H(z) = z3/(z3+3√10)]. The input signal hastwo nonzero samples: x[0] and x[1]. Choose x[0] and x[1] from the range [−1, 1] in order tomaximize y[2007].Mark the maximum value of y[2007] as an on the
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