MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 PROBLEM SET 4 Issued: September 25. 2003 Due: October 3. 2003 REMINDER: Computer Lab #1 is due on October 8 REMINDER: Quiz 1 will be held from 7: 30-9: 30 p. m. Tuesday, October 14 in Walker Memorial. The quiz will cover material in Chapters 1-3 of o W, Lectures and Recitations through September 26, Problem Sets# 1-3, and that part of Problem Set #4 involving problems from Chapter 3 Reading assignments Lectures#7-8 PS#4: Chapter 3 of O& W Lectures #9-11 PS#5: Chapters 4&5 of O&w, plus begin Chapter 6(through Section 6.2) Exercise for home study(not to be turned in, although we will provide solutions) (E1)O&W3.63 Problems to be turned in: Problem 1 Consider the Lti system with impulse response given in O&w3.34. Find the Fourier series representation of the output y(t)for the following input a (t)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 4 Issued: September 25, 2003 Due: October 3, 2003 REMINDER: Computer Lab #1 is due on October 8. REMINDER: Quiz 1 will be held from 7:30 - 9:30 p.m. Tuesday, October 14 in Walker Memorial. The quiz will cover material in Chapters 1 -3 of O & W, Lectures and Recitations through September 26, Problem Sets # 1-3, and that part of Problem Set #4 involving problems from Chapter 3. Reading Assignments: Lectures #7-8 & PS#4: Chapter 3 of O&W Lectures #9-11 & PS#5: Chapters 4&5 of O&W, plus begin Chapter 6 (through Section 6.2) Exercise for home study (not to be turned in, although we will provide solutions): (E1) O&W 3.63 Problems to be turned in: Problem 1 Consider the LTI system with impulse response given in O&W 3.34. Find the Fourier series representation of the output y(t) for the following input. x(t) −4 −3 2 3 5 6 · · · −5 −2 −1 2 −1 1 4 · · · t 1
Problem 2 The periodic triangular wave shown below has Fourier series coefficients ak sin(kx/2)-kP,k≠0 k=0. Consider the LTI system with frequency response H(w) depicted below c (t) HOw) Ay 923-92-9191929 Determine values of A1, A2, A3, 1, Q2, and Q of the LTI filter H(u) such that y(t)=1-cos Problem 3 Consider a causal discrete-time LTI system whose input an] and output yInI are related by the following difference equation y]-yn-1]=rn]+2an-4 Find the Fourier series representation of the output y[n] when the input n=2+sin(Tn/4)-2 cos(Tn/2)
Problem 2 The periodic triangular wave shown below has Fourier series coefficients ak. x(t) � �2 sin(k�/2)e−jk�/2 ⎧ , k =� 0 1 ak = j(k�)2 ⎧� 1 , k = 0. 2 · · · · · · −4 −2 0 2 4 t Consider the LTI system with frequency response H(j�) depicted below: H(j�) x(t) H( ) j� y(t) −�3 −�2 −�1 �1 �2 �3 � A1 A2 A3 Determine values of A1, A2, A3, �1, �2, and �3 of the LTI filter H(j�) such that ⎨ ⎩ 3� y(t) = 1 − cos t . 2 Problem 3 Consider a causal discrete-time LTI system whose input x[n] and output y[n] are related by the following difference equation: 1 y[n] − y[n − 1] = x[n] + 2x[n − 4] 4 Find the Fourier series representation of the output y[n] when the input is x[n] = 2 + sin(�n/4) − 2 cos(�n/2). 2
Problem 4 Specify the frequency response of a discrete-time LTi system so that if the e/n=2+ cos(rn)-sin(Tn/2)+2 cos(Tm/4+T/4) then the output is yn=4-2 sin(n)+2 cos(Tn/4) Problem 5 Compute the Fourier transform of each of the following signals (b)The signal r(t) depicted below 2 Problem 6 Determine the continuous-time signal corresponding to each of the following (a)X(ju)=j(u+1)-6(u-1)-3{6(u-丌)+6(u+丌) (b)X(w)=2 sin(2w-T/2)
Problem 4 Specify the frequency response of a discrete-time LTI system so that if the input is x[n] = 2 + cos(�n) − sin(�n/2) + 2 cos(�n/4 + �/4) then the output is y[n] = 4 − 2 sin(�n) + 2 cos(�n/4). Problem 5 Compute the Fourier transform of each of the following signals: (a) x(t) = e−|t| cos 2t (b) The signal x(t) depicted below: x(t) · · · · · · − t −5 3 −2 0 1 3 4 −4 −1 1 2 5 −1 Problem 6 Determine the continuous-time signal corresponding to each of the following transforms: (a) X(j�) = j [�(� + 1) − �(� − 1)] − 3 [�(� − �) + �(� + �)] (b) X(j�) = 2 sin(2� − �/2) 3
Problem 7 Answer the questions asked in O& w 4.24(a) for each of the following signals (t) 2(t) 1 0123 3(t) 56 1234 Problem 8 O&W.25. Do parts(b)-(f) plus the following new part(a) (a)xu)can be written as A(jw)eje(u)where A(jw)and e(w) are real. Find e(ju) 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 fo tests
Problem 7 Answer the questions asked in O&W 4.24 (a) for each of the following signals: x1(t) 1 −1 0 1 t −1 x2(t) t 1 −3 −2 −1 0 1 2 3 x3(t) t −3 −2 −1 1 2 3 4 5 6 1 −1 Problem 8 O&W 4.25. Do parts (b) - (f) plus the following new part (a) (a) X(j�) can be written as A(j�)ej�(j�) where A(j�) and ω(j�) are real. Find ω(j�). 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. 4
