MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 Thursday, November 13. 2003 Directions: The exam consists of 5 problems on pages 2 to 19 and additional work space on pages 20 and 21. Please make sure you have all the pages. Tables of Fourier series properties as well as CT Fourier transform and DT Fourier 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 book- let. 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 two 8 1/2 x 1l sheets of paper for reference. Calculators may not be used NAME: So/u tions Check your section Section Time 110-11 Prof. Zu 口口口口口口口口 11-12 Prof Zue Prof. Gray 4 l1-12 Dr. Rohrs 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 Problem No of points Score Grader 21 4 100
PROBLEM 1(17%) Consider the following system depicted below: Overall Syste 2 SYSTEM A SYSTEM B yt) The input-output relation for SYSTEM A is characterized by the following causal LCCDE dz(t) dt +6x() dt 5a(t) and the impulse response ho(t) for SYSTEM B is defined as hb(t) Part a. What is the frequency response of the complete system That is, given r()→→Xa)andy(t)+Y(u), determine H(juY(ju) XO +5 (+6)(J+
Fall 2003: Quiz 2 NAME: Work Page for Problem 1 Wt Can find H() by inspe∠比ih,Orby认5 24(08W,233) w+ e can find Hs(jw)using the Basic CTFT Pairs kable where at (),1253 ∠) Hb(Ju) 之H()=H(w)H(J) HC ∽+6)(j+10) 3 Problem 1 continues on the following page
art b. What is the impulse response, h(t) of the complete system? lot ut h(t)= Part c. What is the differential equation that relates a() and y(t)? delt
Fall 2003: Quiz 2 NAME: Solutions Work Page for Problem 1 H() A B jw+b)( jw+lo +6Jw+|° A=H(j)(+0 6+5 A B:HCj +5 HCJ)=-1/ -10+b 5/ Jw+o k)=/21 he pair -+ ot)认Lb at et),c>0≤> OHjw)2 y( +5 X(Jw ()2+1Jw+ g Inspection na+16%t) 60 y(E duct +5x()