EE 350 Exam 2 Spring 2019 Last Name (Print): _______________________________________ First Name (Print): _______________________________________ PSU User ID (e.g. map244): _______________________________________ Recitation Section: _______________________________________ Problem Weight Score 1 25 2 25 3 25 4 25 Total 100 Instructions: You have 2 hours to complete the exam. You are permitted to have one 8.5” x 11” note sheet and your writing implements; otherwise, the exam is closed book and closed notes. Complete the solutions in the space provided. If you need additional space, indicate that your work is continued on the back of the page and finish your work there. Circle or otherwise enclose your final solutions. The clarity of your mathematical analysis is an important part of your work; unclear analysis can lead to loss of credit even if your solution is correct. Academic Integrity Statement: The work in this exam represents my own efforts. During the exam I did not refer to any material except my note sheet. From the start of the exam to the end of the exam, I did not communicate with any student or outside party. I started the exam when told to do so and did no further work on the exam after the exam proctor called time. I understand that exams without a signed academic integrity statements will receive a grade of zero. Signature: _________________________________________________ Problem 1: (25 points) 1. (5 points) Using the block diagram provided in Figure 1, determine an expression for ℎ() such that () ∗ ℎ() = () Your solution for ℎ()should be expressed in terms of the four impulse response functions shown in the block diagram. Use properties of the convolution to simplify the expression as appropriate. Figure 1: A system block diagram. 2. (5 points) Use the integral definition of convolution to prove that convolution is commutative. Problem 1 (cont.) 3. (15 points) Given () = − 1 2 ( − 5)(( − 1) − ( − 5)) and ℎ() = 4−(), sketch the graphical regions of the convolution () = () ∗ ℎ(). Use () as your stationary function. Determine the boundaries for each region and set up the integral for the region. Do not solve the integrals! Figure 2: The input function (). Additional region graphs available on the next page. Problem 1 (cont.) Problem 2: (25 points) 1. (8 points) Determine () = () ∗ () for () = ( + 3) + ( − 3) and () shown in the graph of Figure 3. Sketch the result in the graph provided for () below. Figure 3: The function (). Some grids for sketching intermediate calculations. 2. (9 points) The impulse response of a system is given by ℎ() = −2−3( + 1) a. (3 points) Is the system causal? Justify your answer. b. (6 points) Determine if the system is BIBO stable by showing the impulse response is absolutely integrable. 3. (8 points) A system has the input-output relationship 24() = (3 + 12)() and its step response is () = 2(1 − −4)(). Determine, if possible, the impulse response of the system. Problem 3: (25 points) 1. (15 points) Given 1() = , 2() = and 3() = (3 2 − 1). a. (5 points) Show that 2() and 3() are orthogonal on the interval −1 ≤ ≤ 1. b. (5 points) Determine the value of so that 2() is orthonormal on the interval −1 ≤ ≤ 1. c. (5 points) Let () = () − ( − 1) and assume that = 1 2 so that 1() = 1 2 . Determine the value of 1 such that the energy of the approximation error () = () − 11() is minimized on the interval −1 ≤ ≤ 1. Problem 3 (cont.) 2. (10 points) The function () can be approximated by the expression () ≈ 5 + ∑ 20 sin ( 2 ) cos(2) =1 Complete the Matlab code on the following page so that the code plots the approximation of the function ()as shown in Figure 4. Your code should include the necessary commands to label the plot as shown and to compute the approximation of (). Figure 4: Matlab graph showing the approximation of (). Problem 3 (cont.) t = linspace(-1,1, 10001); N = 250 f = _______________________________________ plot_______________________________________ ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________ function f = f_N(N, t) f = 5*ones(size(t)); ___________________________________________ ___________________________________________ ___________________________________________ end Problem 4: (25 points) 1. (12 points) Use phasor analysis on the circuit in Figure 5 to show that the frequency response function () = ̃ ̃ = 1 ()2 + 1 + 1 Figure 5: A RLC circuit. Problem 4 (cont) 2. (13 points) Given that the frequency response of a system is: () = 23 + 230 ()2 + 10 + 100 a. (3 points) Determine the DC gain of the system. b. (3 points) Determine the high frequency gain of the system. c. (7 points) Let the input to the system be () = 10 sin(10 + 15°); determine the sinusoidal steady state output of the system ().
