BlankSolutionsEE 350 Exam # 2 16 October 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 100INSTRUCTIONS1. You have 2 hours to complete thi s 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 fol lowing the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NOcredit will be given to soluti ons that do not meet this requirement.5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.6. The quality of your analysis a nd evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you a re doing. To receive credit, you mustshow your work.7. Any student caught cheating on an exam will receive a grade of zero for the exam. Additional sanctions,including assigning an XF grade, will be pursued following university guidelines.1Problem 1: (25 Points)1. (10 points) The network in Figure 1, with i nput f(t) and output y(t), is represented by the ordinary differentialequation¨y +1LCy(t) =1LCf(t).Determine the im pulse response representation h(t) of the network.Figure 1: Passive LC network.232. (7 points) A LTI system, different from the one considered in part 1, has the i mpulse response representationh(t) = δ(t + 1) + e−tu(t).(a) (2 points) Is the system causal or noncausal? In order to receive credit, you must justify your answerusing a short sentence.(b) (5 points) Determine if the system is bounded-input bounded-output stable. Justify you answer.43. (8 points) Figure 2 shows the block diagram of a LT I system with input f(t) and output y(t). Each blockrepresents a LTI system, and the impulse response representations of these systems areh1(t) = 4u(t)h2(t) =12δ(t).Using the properties o f convolution, represent the system by the ordinary differentia l equation˙y + a0y(t) = bof(t)by providing the numeric values of aoand bo.Figure 2: Block diagram representation of a LTI system.5Problem 2: (25 points)1. (15 p oints) A linear time-invariant system with input f(t) and output y(t) is represented by the impulseresponseh(t) = e−2tu(t).Determine the zero-state response of this system to the inputf(t) = e−t[u(t) −u(t − 1)]using the graphical convolution approach. Do not sketch y(t). In order to receive credit, clearly specify theregions of integration and, for each region, provide a sketch of f and h.672. (10 points) The steady-state output of a LTI system is the same form as its input when its input is eωt. Inmathematical terms, eωtis sai d to be an eigenfunction of the system. Show that the zero-state response of aLTI system with impulse response h(t) to the complex exponential inputf(t) = eωtisy(t) = ejωtH(ω),whereH(ω) =Z∞−∞h(t)e−ωtdt.8Problem 3: (25 points)1. (12 points) Consider the RC network i n Figure 3 that has input f(t) and output y(t). Using phasor analysis,determine the frequency response function of the system and express your answer in the standard formH(ω) =bm(ω)m+ bm−1(ω)m−1+ ···+ b1(ω) + b0(ω)n+ an−1(ω)n−1+ ···+ a1(ω) + a0.Figure 3: Passive RC network.92. (13 points) A LTI system, di fferent from the one considered in part 1, has the frequency response functionrepresentationH(ω) =(ω)2(ω)2+ 2(ω) + 100.(a) (3 points) Represent the system as an ODE i n the standard formdnydtn+ an−1dn−1ydtn−1+ ···+ aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ···+ bof.(b) (2 points) Determine the DC gain of the system.(c) (2 po ints) Determine the high frequency AC gain of the system.(d) (6 points) Determine the sinusoidal steady-state response of the system to the inputf(t) = cos(10t)10Problem 4: (25 points)1. (8 points) Determine i f the signalsφ1(t) =√2e−tu(t)φ2(t) =3e−2t−2e−t u(t),defined over the interval t ≥ 0, form an orthonormal set. Justify your answer.112. (9 points) Approx imate a sig nal f(t) as a weighted sum of functions x1(t) and x2(t),f(t) ≈ c1x1(t) + c2x2(t),where the approximation error signal ise(t) = f(t) − c1x1(t) − c2x2(t).The signals f(t), e(t), x1(t), and x2(t) are real-valued and defined on the interva l [t1, t2]. Table 1 lists therelevant inner products. For exam ple, hf, x1i = 6 . Determine the value of the weighting coefficients c1and c2that minimizes the energyEe=Zt2t1e2(t)dtof the approximation error signal.h·, ·i x1(t) x2(t)x1(t) 1 2x2(t) 2 1f(t) 6 9Table 1: Table of inner products.123. (8 points) A p eriodic triangle waveform is approximated asf(t) = 2 +NXn=1ancos(10nt)wherean=16π2n2and N represents the integer number of sinusoids used to generate the approximation. As the value of Nincreases, the energy of the approximation error signal decreases. A pa rtially complete MATLAB function fornumerically generating the signal f(t) appears in Figure 4. The function accepts a vector t containing thetime instants at which to evaluate the function f(t) and an integer N specifying the number of cosine termsto include in the approximation. The function returns a vector f containing the values of f(t) at the timeinstants specified in the vector t. Complete the code in Figure 4.% EE 3 50 Fall 2 014% Exam 2% Problem 4 Part 3%function [ f ] = find_f(t, N)% Function fi nd_f ap proximates f(t ) using N te rms%Figure 4: MTALB m-file for realizing the user defined function find f.13EE 350 Exam # 2 16 October 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 100INSTRUCTIONS1. 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 your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NOcredit will be given to solutions that do not meet this requirement.5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.6. The quality of your analysis a nd evaluation is as important as your
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