SolutionsBlank ExamEE 350 EXAM III 11 November 2010Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOProblem Weight Score1 252 253 254 25Total 100Te st Form AINSTRUCTIONS1. You have 2 hours to complete this exam.2. This is a closed book exam. You may use one 8.5” × 11” note sheet.3. Calculators are not allowed.4. Solve each part of the problem in the space following the question. If you need more space, continue yo ur solutionon the reverse side labeling the page with the question number; for example, Pro blem 1.2 Continu ed. NOcredit will be given to solutions that do not m eet this requirement.5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be a ccepted and agrade of ZERO will b e assigned.6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you mustshow your work.1Problem 1: (25 Points)1. (12 points) Figure 1 shows the exponential Fourier series spectra of a si gnal x(t) with fundamental p eriod ωo.Figure 1: Exponential Fourier sp ectra for x(t).(a) (6 points) Determine the compact trigonometric Fourier series coefficients of the signal x(t).(b) (6 points) Determine the trigonometric Fourier series coefficients of the signal x(t).22. (13 points) Consider the full-wave rectified signaly(t) = |sin(ωot)| ,where the parameter ωois a real-valued constant.(a) (2 points) Sketch y(t), and from your sketch, determine the fundamental frequency of y(t) i n terms of theparameter ωo.(b) (4 points) Determine the average value of y(t).(c) (7 points) Derive the exponential Fourier series representation of y(t). To receive full credit, evaluatethe integral(s) and simplify your answer to a single fraction whose numerator is a constant and whosedenominator is a function of n.34Problem 2: (25 points)1. (8 points) Figure 2 shows the frequency spectra of a periodic signal s(t).Figure 2: Fourier series spectra of s(t).Answer the following questions using properties of the Fourier series sp ectra; no credit will be awarded if youranswer i s obtained by determining s(t). You must ju stify each answer with o ne or two s hort sentences.(a) (2 points) What is the average value of the signal s(t)?(b) (2 points) Is the signal s(t) real or complex valued?(c) (4 points) Is the signal s(t) either an even or odd function of time?52. (8 points) A system with frequency response function H(ω) is driven by a periodic input signal f(t) toproduce a periodic output y(t). The f undamental frequency ωoof the input f(t) is 1 0 rad/sec. Figure 3 showsthe frequency response function H(ω) and the spectra Dynof the system output.Figure 3: System frequency function and output sp ectra.(a) (5 points) Determine the spectra Dfnof the periodic input f(t).(b) (3 points) Write an expression for the input signal f(t) in terms of a sinusoidal function of time. Minimalpartial credit will be awarded if your answer is left in terms of exponential functions.63. (9 points) Figure 4 shows a MATLAB m -file tha t approximates a periodic function f(t).t = linspace(0, 10, 1000)’;f = 1;for k = 1:2:6f = f + (1/k) * cos(5*k*t);endFigure 4: MATLAB co de for a pproximating a periodic signal.(a) (1 point) W hat is the average value of the periodic signal f(t)?(b) (1 point) What is the fundamental frequency ωoof f(t)?(c) (1 point) What is the highest harmonic of the fundamental frequency used in approximating f(t)?(d) (2 points) Specify the compact trigonometric Fourier series representation of f(t) up to the hig hest har-moni c used in the m-file code.7(e) (4 points) The signal f(t) is passed through a passive RC filter whose frequency response function isH(ω) =2ω/10 + 1.Write an m-file that approximates the sinusoidal steady-state output of the filter using the same timevector and number of harmonics as the code in Figure 4.8Problem 3: (25 points)1. (10 points) Figure 5 shows the Fourier transform of a signal f(t).Figure 5: Fourier transform of f(t).(a) (2 points) Without determining f(t), use the properties of the Fourier transform to determine if f(t) isreal or complex valued. Justify your answer in one or two sentences.(b) (2 points) Without determining f(t), use the properti es of the Fourier transform to determine if f(t) isan even or odd function of tim e. Justify your answer in o ne or two sentences.(c) (2 points) Based on the graph of F (ω), is f(t) a periodic or aperiodic function of time. Justify youranswer in one or two sentences.9(d) (4 points) Fi nd f(t) using direct integration.102. (9 points) The relationship between the input f(t) and o utput y(t) of a linear time-invariant system isy(t) = f(t − 2) + f(t − 5).The system can b e represented by an impulse response function h(t) so that y(t) = f(t) ∗ h(t)(a) (2 points) Determine the Fourier transform Y (ω) in terms of F (ω).(b) (2 points) Using your result from part(a), determine the Fourier transform H(ω) of the impulse responsefunction.(c) (5 points) Using the Fourier transform pairejωot↔ 2πδ(ω − ωo)and appropriate Fourier transform properties, determine the impulse response h(t) from H(ω) determinedin part (b). In order to receive full-credit, you must use the provided Fourier transform pair.11(d) (6 points) Let F (ω) represent the Fourier transform o f a signal f(t). Show that the Fourier transform off(−t) is F (−ω).12Problem 4: (25 points)Suppose thatf(t) = sin (20t) + 2 sin (40t)andg(t) = f(t) sin (40t) .The signal g(t) is passed through a linear-time invariant system, with impulse responseh(t) =60πsinc (30t)to produce the outputy(t) = g(t) ∗ h(t).1. (5 points) Sketch the Fourier transform F (ω). To receive credit, carefully label important features of thegraph, in particular, the location and area of Dirac delta functions.2. (5 points) Sketch the Fourier transform of the signal g(t). To receive credit, as in part (1), carefully label allimportant features of the graph.133. (5 points) Sketch the Fourier transform of the signal h(t). To receive credit, as in part (1), carefully l abel allimportant features of the graph.4. (5 points) Sketch the Fourier transform of the signal y(t). To receive credit, as in part (1), carefully label allimportant features of the graph.5. (5 points) Determine the signal
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