SolutionsBlank ExamEE 350 EXAM II 14 October 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 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.1Problem 1: (25 Points)1. (15 points) A linear time-invariant system with input f(t) and output y(t) has the ODE representation¨y + a1˙y + a0y(t) = 30 f(t),where a0and a1are real-valued constants. The impulse response of the system has the formh(t) =C1e−2 t+ C2e−3 t u(t),where the coefficients C1and C2are real-valued constants.(a) (5 points) Find the numeric values of the parameters a0and a1.(b) (10 points) Find the numeric values of the constants C1and C2.22. (10 points) Consider another linear ti me-invariant system whose impulse response representation ish(t) = e3 t[u(−3 − t) − u(−5 −t)] .(a) (2 points) Neatly sketch h(t) and appropriately label the ax es.(b) (2 points) State whether or not the system is causal. Justify your answer in one or two sentences.(c) (3 po ints) Determine whether or not the system i s BIBO stable.(d) (3 points) Determine the values of T for whi ch the system with impulse response f unctiong(t) = h(t) ∗ δ(t + T )is causal.3Problem 2: (25 points)1. (15 points) The signalf(t) = e−at[u(t) − u(t −1)]is applied to a system whose impulse response function ish(t) = 2 e−bt[u(t) −u(t −1)] ,where a and b are posi tive real-valued constants with a 6= b. Determine the zero-state responsey(t) = f(t) ∗ h(t)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.452. (10 points) The unit triangle function ∆(t),∆(t) = (1 − 2 |t|)u (|t|) − u|t| −12,is a triangular pulse tha t is centered at the origin. The impulse train,δT(t) =∞Xk=−∞δ(t −kT ),is a periodic signal of period T .(a) (2 points) Neatly sketch ∆(t) and appropriately label the axes.(b) (2 points) Neatly sketch δT(t) and appropriately label the axes.6(c) (4 po ints) Suppose that T > 4, neatly sketch y(t) = ∆(t/4) ∗ δT(t) and appropriately label the axes.(d) (2 points) Determine the largest value of T for which y(t) is a constant f or all values of t .7Problem 3: (25 points)1. (10 points) Humpty Dumpty is a newly hired summer intern at Poached Egg Incorporated. He was taskedwith constructing a low-pass filter, and is considering using the circuit in either Figure 1 or Figure 2. However,Humpty is not sure w hich of the circuits is a low-pass filter. Therefore he decides to subcontract you to helphim with his design task.Figure 1: Prototype Filter 1.Figure 2: Prototype Filter 2.(a) (2 points) In order to bring home the bacon, apply your knowledge of electrical engineering to determine,by inspection, which circuit is a low-pass filter. Justify your answer in two or three sentences.(b) (2 points) For the circuit you have chosen in part (a), specify the DC gain.8(c) (6 points) For the circuit you have cho sen in part (a), determine the frequency response function H(ω)and place 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) + a0.92. (15 points) Because Humpty Dumpty is pleased with the excellent work you did in cracking your first assign-ment, he decides to hire you to analyze another linear time-invariant system whose frequency response functionisH(ω) =˜Y˜F=ω + 1(ω)2+ ω + 6.(a) (3 points) State the ODE representation of this system. Express your answer in the standard formdnydtn+ an−1dn−1ydtn−1+ ··· + aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ··· + bof.(b) (6 points) Determine the sinusoidal steady-state response y(t) of the system to the inputf(t) = 12 + 2√3 sin√3 t + 30◦.Humpty reminds you that tan(45◦) = 1 , tan(60◦) =√3, and tan(30◦) =√3/3.10(c) (6 points) Humpty Dumpty requests that you write a MATLAB m-file that searches for the maximum valueof the magnitude, in dB, of the frequency respo nse function, and the frequency at which the maximumoccurs. Humpty wants you to search from 0.1 Hz to 100 Hz using ten thousand frequencies equally spacedon a logarithmic scale. Store the maximum magnitude, in dB, of the frequency response function in avariable called mag max, and the frequency at which the maximum occurs in a variable called w max.11Problem 4: (25 points)1. (10 points) Consider the real- valued signal setφ1(t) = 1 − αe−2|t|φ2(t) = βe−|t|defined over the interval −∞ < t < ∞.(a) (5 points) Determine the value of the parameter α so that the signals φ1(t) and φ2(t) are orthogonal.(b) (5 points) Using the value of α determined in part (a), is it possible to choose the value of β so that thesignals φ1(t) and φ2(t) are o rthonormal? If so, determine the value of β. If not, explain why in one ortwo sentences.122. (15 points) The real-va lued signals {φi(t)}3i=1defined over the interval t ∈ [t1, t2] are mutually o rthonormal.The signals s1(t) and s2(t) have the exact generalized Fourier series representationss1(t) = a1φ1(t) + a2φ2(t) + a3φ3(t)s2(t) = b1φ1(t) + b2φ2(t) + b3φ3(t)where the coefficients aiand biare real-valued.(a) (3 points) Determine the energy of the signal s1(t) in terms of the parameters ai.(b) (6 points) Determine the energy of the signal g(t) = s1(t) − s2(t) in terms of the parameters aiand bi.(c) (6 points) What relationship must hold among the parameters aiand biso that the signals s1(t) and s2(t)are
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