MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems -Fal 2003 Final Exam Tuesday, December 16, 2003 Directions: The exam consists of 7 problems on pages 2 to 33 and additional work space on pages 34 to 37. Please make sure you have all the pages. Tables of Fourier serie properties, CT and dT Fourier transform properties and pairs, Laplace transform and z-transform properties and pairs are supplied to you as a separate set of pages Enter all your work and your answers directly in the spaces provided on the printed pages of this booklet. 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 three 8 1/2 x 1l sheets of paper for reference. Calcul tors may not be used NAME: Check your section Section Time Rec. Instr Prof. zue 12345678 Prof zue 1-2 Prof. Gray l1-12 Dr rohrs 12-1 Prof. voldman 12-1 Prof. Gray 10-11 Dr rohrs l1-12 Prof. voldman Please leave the rest of this page blank for use by the graders I Problem No of points Score Grader 30 6 Total 200
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Final Exam Tuesday, December 16, 2003 Directions: The exam consists of 7 problems on pages 2 to 33 and additional work space on pages 34 to 37. Please make sure you have all the pages. Tables of Fourier series properties, CT and DT Fourier transform properties and pairs, Laplace transform and z-transform properties and pairs are supplied to you as a separate set of pages. Enter all your work and your answers directly in the spaces provided on the printed pages of this booklet. 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 three 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 30 2 15 3 35 4 30 5 30 6 25 7 35 200 Problem No. of points Score Total
PROBLEM 1(30 pts) Let h(t)be a right sided impulse response of a system and its Laplace transform is given by H(s)= 10(-8+1) (S+10)(s+1) Part a. Find the differential equation describing the system Part b. Is the system causal YES or no Brief explanatio
PROBLEM 1 (30 pts) Let h(t) be a right sided impulse response of a system and its Laplace transform is given by 10(−s + 1) H(s) = . (s + 10)(s + 1) Part a. Find the differential equation describing the system. Part b. Is the system causal ? YES or NO Brief explanation: 2
Fall 2003: Final Exan NAME: Work Page for problen 3 Problem I continues on the following page
Fall 2003: Final Exam NAME: Work Page for Problem 1 3 Problem 1 continues on the following page
Part c. The response of this system to a positive step starts off in a negative direction before turning around. Show this by finding limt-of as( Justify your method. lim Part d. Let H,(s)be the transfer function of a stable but noncausal inverse system of H(s) i.e., HI (s)H(s)=l. Find H,(s)and its region of convergence ROC
Part c. The response of this system to a positive step starts off in a negative direction before turning around. Show this by finding limt�0+ ds(t) . Justify your method. dt ds(t) lim = t�0+ dt Part d. Let HI (s) be the transfer function of a stable but noncausal inverse system of H(s), i.e., HI (s)H(s) = 1. Find HI (s) and its region of convergence. HI (s) = ROC: 4
Fall 2003: Final Exan NAME: Work Page for problem 1
Fall 2003: Final Exam NAME: Work Page for Problem 1 5
PROBLEM 2 (15 pts) Consider the dt lti system shown below H(e3) ynI e[nl as sketched below: 2-1 6 Determine and sketch yn] if the magnitude and the phase of H(eju)are given below H(eju)l ∠H(e)
� PROBLEM 2 (15 pts) Consider the DT LTI system shown below: x[n] H(ej�) y[n] The input sequence is � � 5� � x[n] = cos 2 n − 4 as sketched below: �2 x[n] 2 −2 −1 0 1 2 3 4 5 6 n �2 − 2 Determine and sketch y[n] if the magnitude and the phase of H(ej�) are given below: |H(ej�)| �H(ej�) −� � � 1 −� � � 2 −� 2 1 6
Fall 2003 Final exam NAME: gn]= n
Fall 2003: Final Exam NAME: y[n] = y[n] −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 n 7
Work Space for Problem 2
Work Space for Problem 2 8
Fall 2003: Final Exan NAME: PROBLEM 3 (35pts Consider the following system cos wbt cos wct H(w) (t) HG The Fourier transform of r(t), X(w)has real and imaginary parts given below Relxljw)l Mix(ju)l - ab For your convenience, the identical figures above are attached along with the transform tables
Fall 2003: Final Exam NAME: PROBLEM 3 (35pts) Consider the following system: cos �bt cos �ct x(t) × H( ) xc(t) × yc(t) −�b �b 1 H( ) � × H( ) xs(t) × ys(t) + y(t) j� j� j� sin �bt sin �ct The Fourier transform of x(t), X(j�) has real and imaginary parts given below: −�b �b � 1 �e{X( )} −�b �b � 1 −1 j� �m{X( ) j� } For your convenience, the identical figures above are attached along with the transform tables. 9
Part a Provide labeled sketches of the real and imaginary parts of Xs u) Re{X。(ju)} m( w)
� � Part a. Provide labeled sketches of the real and imaginary parts of Xs(j�). �e{Xs(j�)} �m{Xs(j�)} 10