MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 PROBLEM SET 8 Issued: November 4. 2003 Due: November 19. 2003 REMINDER: Quiz #2 will be held from 7: 30-9: 30 p. m. Thursday, November 13 in Walker Memorial. The quiz will cover materials in Chapters 4-7(through Section 7. 4)of O&w, Lectures and Recitations through October 29, Problem Sets #4-6, and that part of Problem Set 7 involving problems from Chapter 7 Reading Assignments Lectures #16-18 PS#8: Section 7.5, Chapters 8&9(through Section 9.6)of Lectures #18-20 &z PS#9: Chapters 9 &e 11(through Subsection 11.3.4 )of O&w Exercise for home study (not to be turned in, although we will provide solutions): O&W8.35 Problems to be turned in Problem 1 o&W 7.34 Problem 2 Consider the following system H(eu) m ye[n -WM 0 WM I m denotes downsampling by m, and t m denotes upsampling by m as illustrated below
� �� × � MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 8 Issued: November 4, 2003 Due: November 19, 2003 REMINDER: Quiz #2 will be held from 7:30 - 9:30 p.m. Thursday, November 13 in Walker Memorial. The quiz will cover materials in Chapters 4 -7 (through Section 7.4) of O&W, Lectures and Recitations through October 29, Problem Sets # 4-6, and that part of Problem Set # 7 involving problems from Chapter 7. Reading Assignments: Lectures #16-18 & PS#8: Section 7.5, Chapters 8 & 9 (through Section 9.6) of O&W Lectures #18-20 & PS#9: Chapters 9 & 11 (through Subsection 11.3.4) of O&W Exercise for home study (not to be turned in, although we will provide solutions): Problems to be turned in: × O&W 8.35 Problem 1 O&W 7.34 Problem 2 Consider the following system: xc[n] � � � H(e � � yc j�) � m [n] H(ej�) cos[�0n] 1 2 3 [n] � �0 = sin[�0n] � −�M 0 �M H(ej�) xs[n] � � m � ys[n] � m denotes downsampling by m, and � m denotes upsampling by m as illustrated below. 1 x
□m一 toutn]=fin(mn] n is an integer multiple of m ↑n 0 otherwise n is a real-valued DT signal whose DTFT for -T<W< T is given by X(e) W0-aM wo+ WM (a) Sketch the DTFT for ac(n] and as(n]for-2r≤u≤2丌 (b) How much can one downsample without aliasing, i. e, what is the maximum integer value of m (c) Design a system which recovers the signal [ n] from ye[n] and ys nl Problem 3 Determine the Laplace transform and the associated region of convergence and pole-zero plot for each of the following functions of time (a)x()=e-t(-t)+2e-2a( (b)a(t)=(e cos t)u(-t)+u(t) Problem 4 Determine the function of time, a(t), for each of the following Laplace trans- forms and associated region of convergence (a)X(s)=x2 s-12 3<e{s}<4 (b)X(s 2s2+7s+9 (s+2)2,e{s
� �� � �� �� �� �� � � xin[n] � � m � xout[n] = xin[mn] � n xin[n] � � m � xout[n] = xin m n is an integer multiple of m 0 otherwise x[n] is a real-valued DT signal whose DTFT for −� < � < � is given by X(ej�) �0 = 2 1 3� , �M = 4 −� −�0 0 �0 � −�0 − �M −�0 + �M �0 − �M �0 + �M (a) Sketch the DTFT for xc[n] and xs[n] for −2� � � � 2�. (b) How much can one downsample without aliasing, i.e., what is the maximum integer value of m ? (c) Design a system which recovers the signal x[n] from yc[n] and ys[n]. Problem 3 Determine the Laplace transform and the associated region of convergence and pole-zero plot for each of the following functions of time: (a) x(t) = e−t u(−t) + 2e−2t u(t) (b) x(t) = (et cost)u(−t) + u(−t) Problem 4 Determine the function of time, x(t), for each of the following Laplace transforms and associated region of convergence: s − 25 (a) X(s) = , −3 < ↑e{s} < 4 s2 − s − 12 2s2 + 7s + 9 (b) X(s) = , ↑e{s} > −2 (s + 2)2 2
Problem 5 O&W9.24(f) Problem 6 O&W 9.26. Also, determine if this system is stable and justify your answer Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests
Problem 5 O&W 9.24 (f) Problem 6 O&W 9.26. Also, determine if this system is stable and justify your answer. Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests. 3