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()
PROBLEM 2 (21%0 The following images labelled A, B, and C are step responses corresponding to three of the four frequency responses labelled 1, 2, 3 and 4. For each of the step responses, please fill in the number of the corresponding frequency response. For this problem, no explanation or derivation is required Step Response Frequency Response A 4 B
Fall 2003: Quiz 2 NAME: Solutions H1(u) Huju) HgW) MNAIA HaG Additional work space on following pages
Work Page for Problem 2 Before matching the step rAnces with the Corres Pending Arei hency resen ces, let's list few wfe/u] expressions 5(t) (z)d乙=klt)为lS(∽) HC)SC 5()=°(x)x=HCj) xx()=31)<7 c bex in w T e, sinc t Sin wt 米 Iu< x(w / The reas9 y be hind ech matched pai“bheh∥wn SB(t Jooks line a box cr Convolve with u(t), hence th resPonding h(t)i a bexar. Therelerc, the cer HCJu)i a sinc fanction = H() 及H,( seems Pa(h2arat→Cics(∞)=2 Thercforc, the Correponding HCju) has a ierdC gain (ic. H(o)c) H(j∽)
Fall 2003: Quiz 2 Name: Solutions Work Page for Problem 2 There i alse anether was te Jct that Sl+)& H(jw) are a pair lt)i are HC H()(t) HCjw)must be ral and even & H(jo)=o H(Ju) i the Correponding freuency pence SACt&H(u) sCt) has twc Properties比 hat wil ena ble us to块 HCw) ∞)≠ H(j0)≠ s()“ oScillating→k(t) must be oscillating bed Essible causes for oscillation in hl+)arc PeaKing at a nan -zer frency(for eyunplc a BPF) Discen ti'no1nn7derh上vaH(小)(比hkab the Gibbs Phenomena yer abent an impale in the friven Complex ey i'n the ti s anon-2er DC-3 上 cont be the Cer/ef 凵 ch has a nan-2ere nd its Corresponding m% c responce么a5inC(;,hot川lhim~小 →H()达 the Correspo1e2 respond
PROBLEM 3 (25 %) Part a. Determine the Fourier transform R(e)of the following sequence r=(1,0≤n≤M, M is an even integer greater than 1 otherwise R(ew) /w(A+ Part b. Consider the sequence 2丌n 0≤≤M otherwise where M is as defined in Part a. Express w(eju), the Fourier transform of w(n] in terms of R(e3), the Fourier transform of r/n] above R(e")-本R(e LRU J(w+2下