BlankSolutionsEE 350 EXAM III 13 November 2014Last 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. Tables for indefinite integrals and trigonometric identities are provided.4. Calculators are not allowed.5. 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.6. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be a ccepted and agrade of ZERO will b e assigned.7. 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. (10 points) Figure 1 shows all the non-zero complex exponential Fourier series coefficients for a periodic signa lf(t) with fundamental frequency ωo= 10 rad/s. Determine an expression for f(t) in terms of cosine and sinefunctions. No credit will be awarded for answers in expressed in terms of com plex exponentials.Figure 1: Complex exponential Fourier series coefficients for a periodic signal f(t).22. (15 points) Consider the full-wave rectified signalf(t) = 2| cos(4πt)|(a) (3 points) Neatly sketch three periods of f(t) in Figure 1, and determine the numeric value of the funda-mental Period To. To receive credit, you must label the axes with the numeric va lue of the peak amplitudeof f(t) and the ti me instants at which f(t) is zero.Figure 2: Sketch of the periodic signal f(t).(b) (12 points) Determ ine the Fourier series coefficients ao, an, and bnof the periodic signal f(t). If applicabl e,state any properties that all ow you to state the value of a coefficient without evaluating an integral.34Problem 2: (25 Points)1. (13 points) Fig ure 3 shows all the non-zero complex exponential Fourier series coefficients for a periodic signalf(t). Answer the following questions using the properties of the Fourier series spectra; no credit will be awardedif your answer is obtained by determining f(t). You must justify each answer with one or two shortsentences.Figure 3: Complex exponential Fourier series coefficients for a periodic signal f(t).(a) (2 points) Determine the average value o f the signal f(t).(b) (2 points) Determine if the signal f(t) is real or complex valued.(c) (2 points) Determine if the signal f(t) is either an even or an odd function of time.5Figure 3: Complex exponential Fourier series coefficients for a periodic signal f(t).(d) (3 points) Determine the power metric Pfof the signal f(t).(e) (4 points) Determine the compact trigo nometric Fourier series co efficients (C0, Cn, θn) of the signal f(t).62. (12 points) A system with the ODE representation˙y + 100y = 200fis driven by a periodic input f(t) with a fundamental period of 2π × 10−2s and the Fourier spectra shown inFigure 4.Figure 4: Fourier sp ectra of the input signal.(a) (4 points) Determine the frequency response function representation of the system, and place your answerin the formH(ω) =Y (ω)F (ω)=Kω/ωc+ 1by specifying the numeric values of the constants K and ωc.(b) (4 points) Determine the complex exponential Fourier series coefficients Dynof the output y(t) in terms ofthe parameters K and ωc, as well as numerically.7(c) (4 points) Write an expression for the output signa l y(t) in terms of the param eters K and ωc, as well asnumerically. Express y(t) using a cosine function, rather than a complex exponentia l function; minimalpart ial credit will be awarded for answers left in terms of complex exponential functions oftime.8Problem 3: (25 points)1. (10 points) Using an appropriate Fourier transform pair and property, determine an expression for the Fouriertransform of the signalf(t) =160πsinc(40t) cos(100t),and carefully sketch your result in Figure 6. In order to receive full credit for your sketch, add appropriatenumeric labels to the axes. For example, the maximum value of F (ω) and the frequencies at which F (ω) g oesto zero.Figure 5: Fourier transform of the signal f(t).92. (7 points) By direct integration, determine the Fourier transform of the signalf(t) = etu(−t).Carefully specify the limits of integration and show how you evaluated the definite integral at the limits ofintegration.103. (8 points) Suppose that f(t) is a real-valued function of time with Fourier transform F (ω).(a) (4 points) Using a ppropriate properties, o r the definition of the Fourier transform pair, show that themagnitude of the Fourier transform is an even function of ω,|F (ω)| = |F (−ω)|.(b) (4 points) Using appropriate properties, or the definition of the Fourier transform pair, show that thephase of the Fourier transform is an odd function of ω,6F (ω) = −6F (−ω).11Problem 4: (25 points)In order to avoid unauthorized eavesdropping on an open communications channel, the system in Figure 6 scramblesthe speech signal m(t) to pro duce an output s(t) that is applied to a transmitter input. Figure 6 shows the Fouriertransform of the speech input m(t) and the frequency response of the filters H1(ω) and H2(ω). The speech signalm(t) is band limited to B rad/s, a nd the frequency ωcis much larger than B.Figure 6: Speech scrambler.Figure 7: Fourier sp ectra of the input speech signal and filters.1. (7 points) Determine an expression for X(ω) in terms M (ω), and sketch X(ω) in Figure 8. To receive credi t,carefully label important features of the graph.Figure 8: Fourier sp ectra of x(t).122. (3 points) Sketch the Fourier transform Y (ω) in Figure 9. To receive credit, carefully label important featuresof the graph.Figure 9: Fourier sp ectra of y(t).3. (7 points) Determine an expression for Z( ω) in terms Y (ω), and sketch Z(ω) in Figure 10. To receive credit,carefully label important features of the graph.Figure 10: Fourier spectra of z(t).134. (3 points) Sketch the Fourier transform S(ω) of the output si gnal in Figure 11. To receive credit, carefullylabel important features of the graph.Figure 11: Fourier spectra of s(t), the
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