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 you answers directly in the spaces provided on the printed pages of this booklet

REMINDER: The 6.003 Final Exam will he held on december 16th. The final will cover all the material covered during the term, including the material or z transforms and DT feedback systems, the subjects of this problem set Reading Assignments

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

Problem set SolutioN Exercise for home study o&W8.35 (a) From the system diagram, we see that a(t)=a(t)cos(wet) Using the multiplication property ZGu)= o(X(u)* FT(cos wct)) FT of cos wt is two impulses with area T at +wc. Therefore, Z(w)is the spectrum

Reading Assignments Lectures #18-20 PS#9: Chapters 9 1l(through Subsection 11.3.4 )of O&W Lectures #21-22 PS#10: Chapters 10& 11(through Subsection 11. 3.4)of O&w Exercise for home study(not to be turned in, although we will provide solutions)

The lowpass filter H(u) has a cutoff frequency wc=205T rad/ sec. Thus, c(t)is r(t) where all terms with frequency above we are removed by the lowpass filter. The terms which are kept have kwol 205T rad /sec k|< 10.25, so the output, ac(t),is r(t)= To obtain n, we sample c(t) every T=5 10-3 seconds with an impulse train The sampling frequency is 400T=2 x maximum frequency in c(t). Therefore

Exercise for home study O&W4.47 tea this problem examines the Fourier transform of a continuous-time LTI system with a real, causal impulse response, h(t) (a) To prove that H(w) is completely specified by eh(u) for a real and causal h(t) lore the even part of a function, he(t)

REMINDER: Quiz 1 will be held from 7: 30 to 9: 30 p. m. Tuesday, October 14 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

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

Reading Assignments Lectures #5-6& PS#2: Chapter 3 of Oppenheim and Willsky(O& w) Lectures #3-4 &z PS#3: Chapters 3&4 of Oppenheim and Willsky(O&w) Exercise for home study (not to be turned in, although we will provide solutions)