SolutionsBlank ExamEE 350 EXAM II 15 October 2009Last 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 100Test 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 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 b e accepted and agrade of ZERO will be 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. (8 points) A Linear time-invariant system with input f(t) and output y( t) has the impulse response represen-tationh(t)=e−3t− e−4t u(t +1).(a) (4 points) Is the LTI system causal or noncausal? Explain your answer in one or two sentences.(b) (4 points) Is the LTI system BIBO stable? Justify your answer.22. (9 p oints) The system in Figure 1 with input f(t) and output y(t) is comprised of three LTI subsystems whoseimpulse response representations are h1(t), h2(t), and h3(t). It is desired to represent the composite signal bya single impulse response function h(t) such thaty(t)=f (t) ∗ h(t).Figure 1: Block diagram of a system with input f(t) and output y(t).(a) (3 points) Express h(t) in terms of h1(t), h2(t), and h3(t).(b) (6 points) Given thath1(t)=e−6tu(t)h2(t)=u(t)h3(t)=δ(t − 1) ,determine an expression for h(t).33. (8 p oints) Another LTI system, different from the ones considered in parts 1 and 2, has the zero-state responsey(t)=4 − 10e−3(t−1)+6e−5(t−1)u(t − 1).when the input isf(t)=2u(t − 1).(a) (6 points) What is the impulse response representation h(t) of this LTI system?(b) (2 points) What is the DC gain of this LTI system?4Problem 2: (25 points)1. (13 points) The signalf(t)=u(t) − u(t − 2)is applied to a system whose impulse response function ish(t)=t [u(t) − u(t − 4)] .Determine the zero-state responsey(t)=f(t) ∗ h(t)using the graphical convolution approach. Do notsketch y(t). In order to receive credit, clearly specify theregions of integration and, for each region, provide a sketch of f and h.562. (6 points) A LTI system with impulse response representationh(t)=(t +1)[u(t +1)− u(t)] + (1 − t)[u(t) − u(t − 1)]is driven by the inputf(t)=∞Xn=−∞δ(t − 3n).Sketch the zero-state responsey(t)=f(t) ∗ h(t)in Figure 2.Figure 2: Zero-state response y(t) of the system to the input f(t).73. (6 points) In response to the input f(t), a LTI system with impulse response representation h(t) has thezero-state responsey(t)=f (t) ∗ h(t).Let a>0 and define g(t)asg(t)=f (at) ∗ h(at).Show thatg(t)=1ay(at) .8Problem 3: (25 points)1. (8 points) The circuit in Figure 3 has an input voltage f(t) and an output voltage y(t). Assume that theoperational amplifier is ideal and find the frequency resp onse function of the network. Express your answer inthe 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: Active filter circuit with input f (t) and output y(t).92. (9 points) A LTI network, different from the one considered in part 1, is driven by the inputf1(t) = 3 + 5 cos(10t +45◦),and the resulting sinusoidal steady-state response isy1(t) = 12 + 100 cos(10t +30◦).(a) (1 point) What is the DC gain of the system in dB?(b) (4 points) What is the magnitude, in dB, and the phase, in degrees, of the frequency response function at10 rad/sec?(c) (4 points) A new inputf2(t)=−1 + 5 cos(10t +75◦)is applied to the system. Determine the resulting sinusoidal steady-state response y2(t).103. (8 points) Given an insatiable thirst for wickedly fermented, jug busting apple cider, David Salvia recruiteda team of EE 403W students to design an industrial sized fermentor. The input f (t) to the fermentor is theapple mash feed rate in kg/hr, while the output y(t) of the fermentor is liters/hr of apple cider. Dependingon the quality of locally available apples, the feed rate must be varied to maintain a steady flow of apple ciderto David’s office. Rather than regulate the feed rate by hand (and shovel), the students decide to automatethe process. While one student converted a paper shredder into an apple masher, another student designeda sensor to measure the flow rate of apple cider at the fermentor output. A control circuit that uses theoutput measurements y(t) to determine the feed rate f(t) is needed. As a first step in designing this circuit,the students determined a dynamic model of the fermentor by observing its sinusoidal steady-state response.For ten different frequencies fi, the students recorded the input amplitude fi, the output amplitude yi, andthe phase difference pi, in degrees, between the output and input sinusoids. A partial m-file for estimating adynamic model from the experimental measurements is shown b elow.% the juicy detailsfreq_exp = [fr1, fr2, fr3, fr4, fr5, fr6, fr7, fr8, fr9, fr10];y_amp = [y1, y2, y3, y4, y5, y6, y7, y8, y9, y10];f_amp = [f1, f2, f3, f4, f5, f6, f7, f8, f9, f10];p_deg = [p1, p2, p3, p4, p5, p6, p7, p8, p9, p10]; % angle y - angle f[a, b] = invfreqs(H, 2*pi*freq_exp, c, d);Execution of the m-file yieldsa=10 1b=50000 5000 500 1(a) (3 points) The MatLab command invfreqs requires an input vector H. Write a MatLab command line tospecify the vector H in terms of the vectors yamp, f amp, and p deg.(b) (2 points ) What values of c and d did the students use?(c) (3 points) State the ODE representation of the fermentor model. Express your answer in the standardformdnydtn+ an−1dn−1ydtn−1+ ···+ aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ···+ bof.11Problem 4: (25 points)1. (6 points) Consider the real-valued signal setφ1(t)=1φ2(t)=3tφ3(t)=3t2− 1defined over the interval t ∈ [−1, 1]. Determine if the signal set {φi}3i=1is either mutually orthogonal ormutually orthonormal.122. (11 points) Approximate the signalf(t)=35t4over the interval t ∈ [−1, 1] asf(t)=3Xi=1ciφi(t)+e(t)using the signal
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