MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 LIVE Tuesday, October 14, 2003 Directions: The exam consists of 5 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 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 booklet. Please make sure your name is on all sheets. DO IT NOW!. 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 one 8 1/2 x 1l sheet of paper for reference. Calculators may not be used NAME: Check your section Section Time Rec. Instr 10-11 Prof. zue 口口口囗□口口 11-12 Prof. zue 2345678 1-2 Prof. g Dr. Rohrs 12-1 Prof voldman 12-1 Prof g Dr rohrs 1-12 Prof voldman Please leave the rest of this page blank for use by the graders Problem No of points Score Grader] 18 4 21 21 Total 100
1 2 3 4 5 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Quiz 1 Tuesday, October 14, 2003 Directions: The exam consists of 5 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 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 booklet. Please make sure your name is on all sheets. DO IT NOW!. 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 one 8 1/2 × 11 sheet 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 18 20 20 21 21 100 Problem No. of points Score Total
PROBLEM 1(18%) For the questions in this problem, no explanation is necessary. Consider the following three systems SYSTEM A: y(t)=r(t+2)sin(at +2), where w+0 SYSTEMB8=()(间+ SYSTEM C:=∑(2k+1-(l where and y are the input and output of each system. Circle YES or NO for each of the following questions for each of these three systems SYSTEMA SYSTEM B SYSTEM C Is the system linear? YES NO YES NO YES NO Is the system time invariant? YES YES YES Is the system causal? YES YES NO Is the system stable YES NO YES NO YES NO
� �� PROBLEM 1 (18%) For the questions in this problem, no explanation is necessary. Consider the following three systems: SYSTEM A: y(t) = x(t + 2)sin(�t + 2), where � �= 0 1 �n SYSTEM B: y[n] = − (x[n] + 1) 2 n 2 SYSTEM C: y[n] = x [k + 1] − x[k] k=1 where x and y are the input and output of each system. Circle YES or NO for each of the following questions for each of these three systems. SYSTEM A SYSTEM B SYSTEM C Is the system linear ? Is the system time invariant ? Is the system causal ? Is the system stable ? YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO 2
Fall 2003: Quiz 1 NAME: Work Page for problem 1
Fall 2003: Quiz 1 NAME: Work Page for Problem 1 3
PROBLEM 2(20%) Consider a DT LTI system, H2 with a unit sample response h2(n=hn] *h(n+l], as shown below, where hn=8n-8n-1. You may remember from one of the lectures that h(] can be viewed as the unit sample response of a dt lti system that acts as an edge detector The purpose of this problem is to develop an edge detector that is robust against additive h 一[一 d(n Part a. Assume that the input to the system, pIn is as shown below, and there is no noise. i.e., d(n]=0 and p[n=an] Provide a labeled sketch of yn, the output of the system 2 pln
PROBLEM 2 (20%) Consider a DT LTI system, H2 with a unit sample response h2[n] = h[n]�h[n+1], as shown below, where h[n] = �[n] − �[n − 1]. You may remember from one of the lectures that h[n] can be viewed as the unit sample response of a DT LTI system that acts as an edge detector. The purpose of this problem is to develop an edge detector that is robust against additive noise. h2[n] 1 1 System H2 x[n] + 0 p[n] h2[n] y[n] n −2 −1 1 2 d[n] −2 Part a. Assume that the input to the system, p[n] is as shown below, and there is no noise, i.e., d[n] = 0 and p[n] = x[n]. Provide a labeled sketch of y[n], the output of the system. 2 2 2 2 p[n] −4 −3 −2 −1 0 1 2 3 n y[n] n −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 4
Fall 2003: Quiz 1 NAME: Work Page for problem 2 5 Problem 2 continues on the following page
Fall 2003: Quiz 1 NAME: Work Page for Problem 2 5 Problem 2 continues on the following page
Part b For the same input signal as Part a, now assume that the noise signal is dn]=-0n+1] Provide a labeled sketch of the output yIn], i.e., the response to r[n]=pIn]+dn
Part b. For the same input signal as Part a., now assume that the noise signal is d[n] = −�[n + 1]. Provide a labeled sketch of the output y[n], i.e., the response to x[n] = p[n] + d[n]. y[n] n −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 6
Fall 2003: Quiz 1 NAME: Work Page for problem 2 7 Problem 2 continues on the following page
Fall 2003: Quiz 1 NAME: Work Page for Problem 2 7 Problem 2 continues on the following page
Part c. In order to use system H2 as a part of an edge detector, we would like to add an LTI system Hs whose unit sample response, hs n is shown below. System Hs smoothes out effect of noise on rn. The overall system can be represented as below 2h] System H System H. hs(nl h2In dn Provide a labeled sketch of the overall output ys[n), when pn] and dn] are exactly the same as in Part b ys[n 1234567
Part c. In order to use system H2 as a part of an edge detector, we would like to add an LTI system Hs whose unit sample response, hs[n] is shown below. System Hs smoothes out effect of noise on x[n]. The overall system can be represented as below: 2 hs[n] −2 −1 0 1 n 1 1 System Hs System H2 p[n] + x[n] hs[n] h2[n] ys[n] d[n] Provide a labeled sketch of the overall output ys[n], when p[n] and d[n] are exactly the same as in Part b. ys[n] n −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8
Fall 2003: Quiz 1 NAME: Work Page for problem 2
Fall 2003: Quiz 1 NAME: Work Page for Problem 2 9
PROBLEM 3 (20%0) Consider the CT lti system whose impulse response is given as h(t) h(t) y(t) The following two parts can be done independently. Part a. The input r(t), an impulse train starting at t= 2, is depicted below: (1)(1)(1 Provide a labeled sketch of the corresponding output y(t) y(t)
PROBLEM 3 (20%) Consider the CT LTI system whose impulse response is given as: h(t) h(t) 1 x(t) y(t) t −1 0 1 The following two parts can be done independently. Part a. The input x(t), an impulse train starting at t = 2, is depicted below: x(t) −1 0 1 2 3 4 5 ����� (1) (1) (1) (1) t Provide a labeled sketch of the corresponding output y(t). y(t) t −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 10
