MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 Final Exam Tuesday, December 16, 2003 Directions: The exam consists of 7 problems on pages 2 to 33 and additional work space on pages 34 to 37. Please make sure you have all the pages. Tables of Fourier series properties, CT and DT Fourier transform properties and pairs, Laplace transform and z-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 booklet. 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 three 8 1/2 x ll sheets of paper for reference. Calcula- tors may not be used NAME Check your section Section Time Rec. Instr Prof. Zu 12345678 l1-12 Prof, Zue 1-2 Prof. gray l1-12 Dr Rohrs 12-1 Prof. Voldman 12-1 Prof. gray I0-11 l1-12 Prof. voldman Please leave the rest of this page blank for use by the graders: Problem No of points Score Grader 15 4 30 6 Total 200
PROBLEM 1(30 pts) Let h(t)be a right sided impulse response of a system and its Laplace transform is given by H(s) 10(-8+1) (s+10)(s+ Part a. Find the differential equation describing the system 29+)td) His loS+ (+|)(5+)52+|s+10 Y1)(s2+15+10)X(9)(-10s+|0) K3sY)+sY5)+10V(9)=-0s×5)+10×(}= Part b. Is the system causal? YES or No es Brief explanation ince Hs)) (+边成-siox K ct) io rgrt-nided po yeo i is caee
Fall 2003: Final Exam NAME Work Page for Problem 1 3 Problem I continues on the following page
Part c. The response of this system to a positive step starts off in a negative direction before turning around. Show this by finding limt-o+ ds(t), Justify your method lim )电:h)) he Itae value theore9yesH) 七20 1(+):o段七 wulst af Wxen ade ∝t=9. Since c(+) causae aw+(N)∠C(0)M use the IVT 10(-s+1)s Q42=M1=.4)=M 6+0)(+1 tao Part d. Let H, (s)be the transfer function of a stable but noncausal inverse system of H(s) i.e., HI(S)H(s)=1. Find Hr(s)and its region of convergence (5+1)(s+|o) |O(s-() HI(s ROC H9) T。be5ale,kRC如 indude the rw a人S tte c erfeurdo leftwar减A2m0+民 人
Fall 2003: Final exam NAME: Work Page for Problem 1
PROBLEM 2 (15 pts) Consider the dT lti system shown below: H(eu) The input sequence is r[n]=cos(o as sketched below an 23 6 Determine and sketch yn if the magnitude and the phase of H(eju) are given below: ∠H(e) 2 6
Fall 2003: Final Exam NAME n v n ee xIn\>co FT.4心Ak+ aM waing te te X(ev):e S(w-7. 22)+e Siw, 2+ 2uQ) Y(em)=lxlen)(H(e")I 4 Xle)+X Her) +2 8W-;2)e 20):nSw,号,2A) f cee ofte Mle). 5T Slw-+ 2u8)-T'slw,2+2uA)>yIn=m i n
Work Space for Problem 2
Fall 2003: Final exam NAME PROBLEM 3 (35pts) Consider the following system cos wbt cost ae(t) ge(t) HGw) HGu) a(t) y(t) H(w) sin wbt sin wct The Fourier transform of r(t), XGu)has real and imaginary parts given below Re(xGu)) SmX(w) -wb For your convenience, the identical figures above are attached along with the transform tables
Part a. Provide labeled sketches of the real and imaginary parts of Xs (jw) Re(xs w)) 3m{X。(ju)}