6.003 (Fall 2007) 17 December Final exam. Name: Please circle your section number:

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1 6.3 (Fall 27) Final exam 7 December 27 Name: Please circle your section number: Section Instructor Time Jeffrey Lang 2 Jeffrey Lang 3 Karen Livescu 4 Sanjoy Mahajan 2 5 Antonio Torralba 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 sheets of paper (six sides). No calculators, computers, cell phones, music players, or other aids /5 ( /5 ( /5 ( ) ) ) /5 ( / ( /5 ( ) ) ) 9... / ( / ( / ( ) ) ) / 3 ( / 25 ( / 3 ( ) ) ) 4. / ( ) 8. /5 ( ) / 5 ( ) /35 ( ) /35 ( ) /3 ( ) / ( )

3 3. Matching time and frequency representations [5 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. t A B C D t A B C D A B C D t Here are the choices for the Fourier transform magnitude: A ω B ω C ω D ω

4 4 2. Discrete-time periodicity [5 points] Here are two periodic discrete-time signals: x 3 [n] =...,,,,,,,,,,... x 5 [n] =...,,,,,,,,,,,... Mark the period of their sum x 3 [n] + x 5 [n] as an on the thermometer: 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.

5 5 3. Find the output signal [5 points] A given linear, time-invariant system turns the unit pulse into the triangle: 2 3 system 2 3 The system is given the following input signal: 2 3 Sketch the output signal on the following graph:

6 6 4. Frequency response [ points]

7 7 a. Here is a resistor capacitor circuit: cos t y(t) The input signal is cos t for all time (not just for t ). 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

8 8 b. Here is an LRC circuit: cos t y(t) The input signal is cos t for all time (not just for t ). 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

9 9 Here are the possible output signals for the preceding questions: π 2π t π 2π t π 2π t π 2π t π 2π t π 2π t π 2π t π 2π t π 2π t π 2π t 9

10 5. Mystery cosine [5 points] You sample x(t) = cos ωt with sampling interval T = and get this sequence of samples...,,,,,,,,,,,, Using an, mark all possible values of ω that lie in the thermometer range:

11 6. Discrete-time poles and zeros [ points] A discrete-time system with a finite number of poles and zeros turns the causal signal { 3n + for n ; x[n] = otherwise. into the unit step { for n ; u[n] = otherwise.

12 2 a. What is the smallest number of zeros that the system could have? (If there are repeated zeros, count each.) 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.

13 3 b. What is the smallest number of poles that the system could have? (If there are repeated poles, count each.) 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.

14 4 7. Taking medicine [5 points] Starting at t = a patient takes a medicine every 8 hours, eating a pills at t = ; a pills at t = 8 hours; a 2 pills at t = 6 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 t (hours)

15 5 a. Mark the average concentration (in the region t ) as an on the thermometer: 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.

16 6 b. Mark the time constant of the first-order system as an on the thermometer: min 3 min hr 2 hr 5 hr hr day 3 days week 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.

17 7 c. Mark the ratio a 27 /a as an on the thermometer: 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.

18 8 8. Maximizing output samples [5 points] 3 A discrete-time system has system functional /( + R 3 ) with input signal X and output 3 3 signal Y [the corresponding system function is H(z) = z 3 /(z + )]. The input signal has two nonzero samples: x[] and x[]. Choose x[] and x[] from the range [, ] in order to maximize y[27]. Mark the maximum value of y[27] as an on the thermometer: 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.

19 9 9. Musical timbre [ points] Time waveforms for three musical instruments each playing the same musical note are shown below (left panels, each with same time scale). Determine which Fourier series coefficients (right panels) correspond to the bassoon and the oboe. bassoon time A oboe time B violin time C 4 ms 2 harmonic #

20 2 a. Fourier series coefficients for the bassoon. A B C 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.

21 2 b. Fourier series coefficients for the oboe. A B C 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.

22 22. Lightbulb flicker [ points] Here is a model for variations in the brightness of a lightbulb. Resistive heating pours heat into the lightbulb filament at a rate proportional to V(t) 2, where V(t) is the line voltage oscillating at the line frequency f = 6 Hz. The filament acts as a leaky tank with time constant τ 3 ms (in a typical W bulb), turning the varying heat input into a varying temperature T filament. The brightness of the lightbulb is proportional to T 4 filament. V(t) ( ) 2 heat T filament input ( ) 4... ω brightness

23 23 a. Fractional ripple in the quantity γ is defined by maximum value of γ minimum value of γ fractional ripple in γ. average value of γ Mark the fractional ripple in T filament as an on the thermometer: 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.

24 24 b. The brightness of a bulb is proportional to T 4. What is the ratio filament fractional ripple in brightness? fractional ripple in T filament Mark the ratio as an on the thermometer: 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.

25 25. Half the unit step [ points] Consider a discrete-time signal f [ ] that has no negative samples and satisfies ( f f )[n] = the unit step u[n] =,,,... for n (and zero otherwise).

26 26 a. Mark f [] as an on the thermometer:

27 27 b. Mark f [] as an on the thermometer: 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.

28 28 c. [Extra credit worth 5 points] Give a closed form for f [n]. f [n] = (extra credit)

29 MIT OpenCourseWare Signals and Systems Spring 2 For information about citing these materials or our Terms of Use, visit:

6.003 (Fall 2011) Quiz #3 November 16, 2011

6.003 (Fall 2011) Quiz #3 November 16, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 3 1 pm 4 2 pm Grades will be determined by the correctness of your answers (explanations