6 003 Fall 2007 17 December 2007 Final exam Name Please circle your section number Section Instructor Time 1 Jeffrey Lang 10 2 Jeffrey Lang 11 3 Karen Livescu 11 4 Sanjoy Mahajan 12 5 Antonio Torralba 1 6 Qing Hu 2 Partial credit will be given according to the conceptual features that a proposed 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 35 100 30 Final exam 6 003 Signals and Systems Fall 2007 3 1 Matching time and frequency representations 15 points For each time signal choose the magnitude of its Fourier transform from the four choices below The time signals are zero outside the plotted region Each time and frequency figure has 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 two answers A B C D t t 1 A B C D 1 A B C D t Here are the choices for the Fourier transform magnitude A B C D Final exam 6 003 Signals and Systems Fall 2007 2 Discrete time periodicity 4 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 0 2 4 6 8 10 on the thermometer 12 14 16 18 20 If you are unsure of the correct value you can also mark a second value in which case you will receive the average of the scores for the two values Final exam 6 003 Signals and Systems Fall 2007 3 Find the output signal 5 5 points A given linear time invariant system turns the unit pulse into the triangle 1 1 system 0 0 1 2 0 3 0 1 2 3 The system is given the following input signal 1 0 0 1 2 3 Sketch the output signal on the following graph 2 1 2 1 0 1 2 3 4 5 6 Final exam 6 003 Signals and Systems Fall 2007 4 Frequency response 10 points 6 Final exam 6 003 Signals and Systems Fall 2007 7 a Here is a resistor capacitor circuit 1 1 cos t y t 1 1 1 The input signal is cos t for all time not just for t 0 Choose the output signal from those given on the next page 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 two answers 1 2 3 4 5 6 7 8 9 10 Final exam 6 003 Signals and Systems Fall 2007 8 b Here is an LRC circuit 1 1 y t cos t 1 The input signal is cos t for all time not just for t 0 Choose the output signal from those given on the next page 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 two answers 1 2 3 4 5 6 7 8 9 10 Final exam 6 003 Signals and Systems Fall 2007 9 Here are the possible output signals for the preceding questions 2 1 0 1 2 2 0 2 t 2 1 0 1 2 2 1 2 1 0 1 2 2 0 2 0 2 t 2 1 0 1 2 2 2 0 2 t 2 1 0 1 2 2 2 0 9 0 2 t 0 2 t 6 2 t 2 1 0 1 2 2 7 2 1 0 1 2 t 4 5 2 1 0 1 2 2 2 3 2 1 0 1 2 0 0 2 t 8 2 t 2 1 0 1 2 2 0 10 2 t Final exam 6 003 Signals and Systems Fall 2007 10 5 points 5 Mystery cosine You sample x t cos t with sampling interval T 1 and get this sequence of samples 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 Using an mark all possible values of that lie in the thermometer range 0 1 2 3 4 5 6 7 8 9 10 Final exam 6 003 Signals and Systems Fall 2007 6 Discrete time poles and zeros 10 points A discrete time system with a finite number of poles and zeros turns the causal signal 3n 1 for n 0 x n 0 otherwise into the unit step 1 for n 0 u n 0 otherwise 11 Final exam 6 003 Signals and Systems Fall 2007 12 a What is the smallest number of zeros that the system could have If there are repeated zeros count each 0 1 2 3 3 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 two answers Final exam 6 003 Signals and Systems Fall 2007 13 b What is the smallest number of poles that the system could have If there are repeated poles count each 0 1 2 3 3 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 two answers Final exam 6 003 Signals and Systems Fall 2007 7 Taking medicine 14 15 points Starting at t 0 a patient takes a medicine every 8 hours eating a0 pills at t 0 a1 pills at t 8 hours a2 pills at t 16 hours and so on forever The graph shows the concentration of medicine in the patient s blood as a function of time This concentration is the output of a first order system 3 2 0 0 8 16 24 32 40 t hours Final exam 6 003 Signals and Systems Fall 2007 15 a Mark the average concentration in the region t 0 as an on the thermometer 0 1 2 3 4 5 If you are unsure of the correct value you can also mark a second value in which case you will receive the average of the scores for the two values Final exam 6 003 Signals and Systems Fall 2007 16 b Mark the time constant of the first order system as an on the thermometer 10 min 30 min 1 hr 2 hr …
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