blank examsolutionsEE 350 EXAM II 13 October 2011Last 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. (8 points) Assume that the operational amplifier i n Figure 1 is ideal and represent the circuit, with inputvoltage f(t) and output voltage y(t), as an impulse response signal h(t).Figure 1: Active circuit with input f(t) and output y(t).22. (7 points) The system shown in Figure 2 is composed of two LTI systems with impulse response functionsh1(t) = δ(t + 4)h2(t) = δ(t −2).Figure 2: System comprised of two subsystems specified by their impulse response function.(a) (4 points) Determine the impulse response function h(t) so thaty(t) = f(t) ∗ h(t).(b) (3 po ints) Is the system in Figure 2 a causal system? In order to receive credit, justify your answer in asingle sentence.33. (10 points) A Linear time-invariant system with input f(t) and o utput y(t) is represented by the ordinarydifferential equation¨y − ˙y + aoy = 2˙f −4f.It is known that the DC gain of the system is two and that the impulse response representatio n of the systemish(t) = 2e−tu(t).(a) (5 points) Is the system BIBO stable? In order to receive credit, you must justify your answer by showingan appropriate mathematical analysis.(b) (5 points) Is the system asymptotically stable? In order to receive credit, you must justify your answerby showing an appropriate mathematical analysis.4Problem 2: (25 points)1. (12 points) The signalf(t) = 2 [u(t) − u(t − 2)]is applied to a system whose impulse response function ish(t) = et[u(t − 1) − u(t − 3)] .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.562. (5 points) Once again consider the system in part 1 where the impulse response function representation ish(t) = et[u(t − 1) − u(t − 3)]and the input isf(t) = 2 [u(t) − u(t −2)] .(a) (3 points) Determine an expression forg(t) =dfdt∗ h(t)(b) (2 points) What is the relationship between the signal g(t) and the response y(t) found in part 1?73. (8 points) Consider a linear time-invariant system whose impulse response representation h(t) is an aperiodicfunction of time. Show that if the input f(t) is a periodic function with fundamental period To, then thezero-state response y(t) of the system is also periodic with fundamental period To, that isy(t) = y(t + To).8Problem 3: (25 points)1. (12 points) Consider the filter network in Figure 3 with input voltage f(t) and output voltage y(t).Figure 3: Filter network with input f(t) and output y(t).(a) (2 points) By inspection, determine whether the network is a low-pass or high-pass filter. Justify youranswer in two or three sentences.(b) (2 points) By inspection, determine the DC gain of the filter network. Justify your answer in two or threesentences.9(c) (8 points) Once again consider the filter network shown in Figure 4. Determine the frequency responsefunction 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.Figure 4: Filter network with input f(t) and output y(t).10112. (13 points) An engineer uses an oscilloscope and function generator to measure the frequency and magnituderesponse of a filter network. Table 1 shows data acqui red by the engineer.ω [rad/sec] |H(ω)| [V/V]6H(ω)| [deg]0 2 -1801√2 -22510 0.2 -270100 0.02 -270Table 1: Measured data .(a) (6 points) Suppose that the inputf(t) = −10 + 100 sin(10t + 135◦)is applied to the network. D etermine the sinusoidal steady-state response y(t).(b) (7 po ints) Based on the data in Table 1, the engineer suspects that the frequency response function hasthe formH(ω) =boω + ao.Using the data in Table 1 and the assumed form of the frequency response function, write MATLAB codethat estimates the values of the parameters boand ao.12Problem 4: (25 points)1. (15 points) The set of Walsh functions are useful in electrical engineering applications such as digital signalprocessing. Defined over the interval −1/2 ≤ t ≤ 1/2, the first three Walsh functions areφ1(t) = ut +12−ut −12φ2(t) = ut +12−2u (t) + ut −12φ3(t) = ut +12−2ut +14+ 2ut −14− ut −12..(a) (6 points) Sketch the Walsh functions in Figure 5.Figure 5: The first three Walsh functions.(b) (9 points) Determine whether or not the signals φ1(t), φ2(t), and φ3(t) are mutually orthonormal. Inorder to receive credit, you must present a mathematical analysis that justifies your answer.13142. (10 points) Defined over the interval t ∈ [−1, 1], the real-valued Legendre polynomialsp1(t) =1√2p2(t) =r32tP3(2) =r58(3t2− 1)are mutually orthonormal.(a) (4 points) A signal s(t) is exactly represented ass(t) = 2 p1(t) − 3 p2(t) + 4 p3(t).Determine < s, s >.(b) (6 points) Approximate the signalf(t) = t3over the interval t ∈ [−1, 1] as˜f(t) = c1p1(t) + c2p2(t) + c3p3(t)by determining the coefficients cithat minimize < e, e >, where e(t) = f(t)
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