《信号与系统 Signals and Systems》课程教学资料（英文版）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 1/2x ll sheets of paper for reference. Calculators may not be used NAME: Check your sectionSection Time 110-11 Prof. zue 口口口口口口囗 l1-12 Prof. Zue 2345678 1-2 Prof. Gray l1-12 Dr rohrs 12-1 Prof. voldman 2-1 Prof. Gray 10-11 Rohrs l1-12 Voldman Please leave the rest of this page blank for use by the graders Problem No of points Score Grader 20 15 Total 100

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 Overall System a(t) SYSTEM A SYSTEM B y(t) The input-output relation for SYSTEM A is characterized by the following causal LCCDE dz(t) d c(t 2+62(t) dt and the impulse response hb(t) for SYSTEM B is defined as hb(t)=eu(t) Part a. What is the frequency response of the complete system? That is, given z(t)+→X(ju)andy(t)4 Y(u), determine H(w) Hg

PROBLEM 1 (15%) Consider the following system depicted below: x(t) z(t) SYSTEM A SYSTEM B y(t) 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(j�) and y(t) �� Y (j�), determine H(j�) = Y (j�) . X(j�) H(j�) = 2

Fall 2003: Quiz 2 NAME: Work Page for problem l 3 Problem continues on the following page

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 r(t)and y(t)?

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 l

Fall 2003: Quiz 2 NAME: Work Page for Problem 1 5

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 A H1( cOD0D人 Hsw) HgW)

� � 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 7� 7� 900 300 H1(j�) 1 2 � −�c �c � 10 − 2 + �c (102 + �c) −(102 − �c) 102 − �c 1 H3(j�) H4(j�) 1 3 � 1 � �2 3 4 −�c 0 �c −�c 0 �c 0 H2( ) 2 � 2 j� 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

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 w, and A

Part b. Find �c and A. �c = A = 8

Fall 2003: Quiz 2 NAME: Work Page for problem 2

Fall 2003: Quiz 2 NAME: Work Page for Problem 2 9

PROBLEM 3 (25 %0) Part a. Determine the Fourier transform R(eu)of the following sequence 0≤n≤M, M is a positive even integer 0. otherwise R(eu) Part b. Consider the sequence 0<n<M otherwise where M is as defined in Part a. Express w(eu), the Fourier transform of w[ n] in terms of R(eu), the Fourier transform of r[n] above W(e

� � PROBLEM 3 (25%) Part a. Determine the Fourier transform R(ej�) of the following sequence: ⎩ 1, 0 � n � M, M is a positive even integer r[n] = 0, otherwise. R(ej�) = Part b. Consider the sequence ⎧ ⎧ ⎨⎨ � 1 2�n 1 − cos , 0 � n � M w[n] = 2 M 0, otherwise, where M is as defined in Part a. Express W(ej�), the Fourier transform of w[n] in terms of R(ej�), the Fourier transform of r[n] above. W(ej�) = 10