MIT信号与系统试题quiz2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Quiz 2 Thursday, November 13, 2003 Directions: The exam consists of 6 problems on pages 2 to 19 and work space on pages 20 and 21. Please make sure you have all the pages. Tables of Fourier series properties as well as CT Fourier transform and DT Fourier transform properties and tables are supplied to you at the end of this booklet. Enter all your work and your answers directly in the spaces provided on the printed pages of this book­ let. Please make sure your name is on all sheets. You may use bluebooks for scratch work, but we will not grade them at all. All sketches must be adequately labeled. Unless indicated otherwise, answers must be derived or explained, not just simply written down. This examination is closed book, but students may use two 8 1/2 × 11 sheets of paper for reference. Calculators may not be used. NAME: Check your section Section Time Rec. Instr. � 1 10-11 Prof. Zue � 2 11-12 Prof. Zue � 3 1- 2 Prof. Gray � 4 11-12 Dr. Rohrs � 5 12- 1 Prof. Voldman � 6 12- 1 Prof. Gray � 7 10-11 Dr. Rohrs � 8 11-12 Prof. Voldman Please leave the rest of this page blank for use by the graders: Grader 1 15 2 20 3 25 4 25 5 15 100 Problem No. of points Score Total PROBLEM 1 (15%) Consider the following system depicted below: x(t) z(t) y(t)SYSTEM A SYSTEM B Overall System The input-output relation for SYSTEM A is characterized by the following causal LCCDE: dz(t) dx(t) + 6z(t) = + 5x(t), dt dt and the impulse response hb(t) for SYSTEM B is defined as: hb(t) = e −10t u(t). Part a. What is the frequency response of the complete system ? That is, given F F x(t)←→ X(jω) and y(t)←→ Y (jω), determine H(jω) = Y (jω) . X(jω) H(jω) = 2 Fall 2003: Quiz 2 NAME: Work Page for Problem 1 3 Problem 1 continues on the following page. Part b. What is the impulse response, h(t) of the complete system ? h(t) = Part c. What is the differential equation that relates x(t) and y(t) ? 4 Fall 2003: Quiz 2 NAME: Work Page for Problem 1 5 ω ω PROBLEM 2 (20%) Part a. Match the step response s(t) below to the correct frequency response and give a brief justification to your answer in the space provided in the next page. s(t) A t 7pi 7pi 900 300 H1(jω) 1 2 ω −ωc ωc ω 102 + ωc−(10 2 + ωc) −(102 − ωc) 102 − ωc 1 H3(jω) H4(jω) 1 3 pi 1 pi √ 2 3 4 −ωc 0 ωc −ωc 0 ωc 0 H2( ) 2 pi 2 jω 6 Fall 2003: Quiz 2 NAME: SYSTEM Brief justification (You can show why your answer is correct or show why the other three systems are not correct) : 7 Problem 2 continues on the following page. Part b. Find ωc and A. ωc = A = 8 Fall 2003: Quiz 2 NAME: Work Page for Problem 2 9   PROBLEM 3 (25%) Part a. Determine the Fourier transform R(ejω) of the following sequence: { 1, 0 ≤ n ≤ M, M is a positive even integer r[n] = 0, otherwise. R(ejω) = Part b. Consider the sequence ( ( )) 1 2pin  1 − cos , 0 ≤ n ≤ M w[n] = 2 M 0, otherwise, where M is as defined in Part a. Express W (ejω), the Fourier transform of w[n] in terms of R(ejω), the Fourier transform of r[n] above. W (ejω) = 10 Fall 2003: Quiz 2 NAME: Work Page for Problem 3 11 Problem 3 continues on the following page. Part c. Is there a positive even integer M that will make W (ejω) real ? If so, find the values of M that satisfy this constraint. If not, explain why. YES NO Values of M Explanation: 12 Fall 2003: Quiz 2 NAME: Work Page for Problem 3 13 ∞ PROBLEM 4 (25%) Part a is independent of the other parts in this problem. Part a. Consider the following system: x(t) × H ( ) −ωc 0 ωc T lp jω y(t) ∑ 2pi p(t) = δ(t − kT ), ωs = T k=−∞ For this part, suppose ( )( ) sin(4pit) sin(2pit) x(t) = (−1)t , pit pit and p(t) is an impulse train of frequency ωs. Hlp(jω) is a lowpass filter whose gain is T and cutoff frequency is ωc. Determine the cutoff frequency ωc and a frequency ω0 such that y(t) = x(t) for any ωs > ω0. ω0 = , ωc = 14 Fall 2003: Quiz 2 NAME: Work Page for Problem 4 15 Problem 4 continues on the following page. ∞∑ ∑ ∞ ∑ ω ω For the rest of this problem, let’s consider the following system: xp(t) yp(t)x[n] x(t) × Impulse to Sequence Sequence to Impulse H ( ) −2pi 0 2pi T1 lp jω y(t) p1(t) = δ(t − nT1) p2(t) = δ(t − nT2) n= n=−∞ −∞ p1(t) is an impulse train whose fundamental period is T1 and p2(t) is another impulse train whose fundamental period is T2. Hlp(jω) is a lowpass filter whose gain is T1 and cutoff frequency is at ωc. Note that x[n] = x(nT1) and yp(t) = ∞ x[n]δ(t − nT2). n=−∞ The input x(t) is a band limited real signal whose Fourier transform is shown below: X(jω) −2pi −pi pi 2pi ω 1 Part b. Let’s define x1[n] = x(nT1 ), where T1 = 1, x2[n] = x(nT1 ), where T1 = 1 3 . In the given axes below and on the top of the next page, provide the labeled sketches of X1(e jω) and X2(ejω), Fourier transforms of x1[n] and x2[n] respectively. ejω) ejω)X1( X2( −2pi −pi pi 2pi −2pi −pi pi 2pi 16 Fall 2003: Quiz 2 NAME: Work Page for Problem 4 17 Problem 4 continues on the following page. ω 1Part c. Suppose T1 = 1 and T2 = 2 . Provide a labeled sketch of Y (jω), Fourier transform 3 of the overall output y(t). Y (jω) 18 Fall 2003: Quiz 2 NAME: Work Page for Problem 4 19 PROBLEM 5 (15%) We have a cascade of two stable CT LTI systems as shown below: x(t) 1 z(t) y(t) 1 + jω SYSTEM G Overall System The straight line approximation of Bode´ plots of the overall system, H(jω) is shown in the next page. Find the frequency response, G(jω), of SYSTEM G. G(jω) = 20 Fall 2003: Quiz 2 NAME: 20 0 20 log 10 |H(jω)|- dB −20 −40 10−2 10−1 100 101 102 103 104 105 Frequency ω- rad/s 45 0 −45 −90 ∠H(jω)- deg −135 −180 −225 −270 10−2 10−1 100 101 102 103 104 105 Frequency ω- rad/s 21 Fall 2003: Quiz 2 NAME: Work Page 22 Fall 2003: Quiz 2 NAME: Work Page 23