mt-01-2008-02-14-sol-pg1-8 - EECS 20N Structure and...

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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems MIDTERM 1 Department of Electrical Engineering and Computer Sciences 14 February 2008 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name JAM—.— FIRST Name Ame—— LabTime Mart/M ,2. 100% Si: Inche/wlv‘r o (10 Points) Print your name and lab time in legible, block lettering above AND on the last page where the grading table appears. 0 This exam should take up to 70 minutes to complete. You will be given at least 70 minutes, up to a maximum of 80 minutes, to work on the exam. 0 This exam is closed book. Collaboration is not permitted. You may not use or access, or cause to be used or accessed, any reference in print or electronic form at any time during the exam, except one double-sided 8.5” x 11” sheets of handwritten notes having no appendage. Computing, communication, and other electronic devices (except dedicated timekeepers) must be turned off. Noncompliance with these or other instructions from the teaching staff— including, for example, commencing work prematurely or continuing beyond the announced stop time—is a serious violation of the Code of Student Conduct. Scratch paper will be provided to you; ask for more if you run out. You may not use your own scratch paper. 0 The exam printout consists of pages numbered 1 through 8. When you are prompted by the teaching staff to begin work, verify that your copy of the exam is free of printing anomalies and contains all of the eight numbered pages. If you find a defect in your c0py, notify the staff immediately. 0 Please write neatly and legibly, because if we can’t read it, we can’t grade it. o For each problem, limit your work to the space provided specifically for that problem. No other work will be considered in grading your exam. No exceptions. 0 Unless explicitly waived by the specific wording of a problem, you must ex- plain your responses (and reasoning) succinctly, but clearly and convincingly. c We hope you do a fantastic job on this exam. MT1.1 (30 Points) The continuOus-time function 95 characterized by Vt E R, 23(t) = eatemfi‘fi) is ubiquitous in the engineering and physical sciences. The parameters a, w, and d) are all real-valued. Suppose x is the trajectory of a particle on the complex plane. We may think of :r(t) as the instantaneous position of the particle at time t. Each part below specifies a set of values for a, w, and a3. For each part, provide a well-labeled sketch of the particle’s trajectory m in the time interval [0, 00); use an arrow to specify the direction of the particle’s journey as time moves forward. On each sketch, indicate the location of the particle at time instants t = 0, 0.5, 1, 1.5, and 2 seconds, and show where the particle is headed as t —> 00. (a) 0:0,w%—w/2,and¢=0. t: O x“) "-l t>o.s xCon): 6/153in) 7. 13 L0 x(1.0):e (0)1) e“ {1L5 X(I.S)= (13le w—(n, \) Z {12.0, x(2.o)—— eZ(—|,O) (C) 0 = ——1,w= —7r/2,andq§=0. Ym t4) x(0)z(\)o) t‘Vz XCVz)? E7: (WU, I) t) x0): e"(o,«|) t‘ 3/l xCS/zh 3"“(“1 4’ til "((1) : C_Z(‘])O) tzo x63): 5-} CM” 10—; C. ,‘hz t x05),e J’;<I’l) k» tvl-o x0.“ K ‘ 2e 1 (1,1) til-S (1- ¢ 3h ‘ 5) 9. ECU!) Z 1 .0 ‘ t 2 («Czohe’ L: (l, H W12 (30 Points) Consider an N th-order polynomial F (2) in the complex variable 2. Specifically, F(z) zao+alz+a2z2+~-+aNzN, where the coefficients a0, . . . , a N are all real-valued. (a) Show that (z*)” = (z")*, for all n = 0, 1, . . . , N. )e‘l ZZ re J '4) Z” 2 rte—W ”> (L'Y‘ awn . Y I" C Zn rn‘fli—jwu _. fzfl)"’ roe an (b) Use the result of part (a) to show that F*(z) = F(z*). F‘(Z)?<Q +az+ 1 ' 11‘ ' 2 0 i all ‘f—«iqu ) : 0‘2; 4 a. Z) i ngll) +w't +qAI/7M .1 U r Wu“) mm (c) Suppose F(z) has a non-real root zo; that is, F(20) = 0. Show that 23‘ must also be a root of F (2) This means that if a polynomial has real coefficients, then its roots are real-valued or appear as complex-conjugate pairs. HZOH): 0" = (Fézfil‘ : Wu): PCZD‘) MT1.3 (25 Points) The continuous-time signal :3 is characterized by Vt E R, :p(t) = Asin(B cost) whereAER,and0<B<1. The two parts of this problem are mutually independent, so you may tackle them in either order. Of potential use may be cos(0z + B) = cos a cos fl ~ sin a sin fl. (a) Is the signal x periodic? If so, determine its fundamental period p. Explain your reasoning succinctly, but clearly and convincingly. Yes Far «(fl lo R/?a/{oolic thrp>=x(f ) X (t+zfi)< A Sfm (B {.33 (£4237 )3 z A 3‘"? < 8 (£5- (ill 1 nétl Fu “dqmméql WNW :5 7'” (nor§<2'fi Saftsfrcs .x(-r';«‘:5}:x{t)l (b) Recall that cos(wt) has frequencies :|:w. More generally, for a real-valued func- tion 0(t), the instantaneous frequencies of cos[6(t)] at time t are i605). Determine the range of instantaneous frequencies of the signal x. 8m: gar «lace; J: ." “Bsfqt ) Sint fol/veg] {,0,“ [:i) ‘3 gr 5”?an X. (1-5“ ”‘LOl‘ 933ml 3(1” rial Insi- (3m: fraud-(1.350” Vqlkkoé r; ,5 I MT1.4 (20 Points) George and Harry run at a steady rate around a circular track field 'of unit radius. They run independently, so neither makes any strategic ad- justment based on what the other one is doing. ' George completes one lap in Tg seconds. Harry takes Th seconds to complete a lap. Harry is the slower one, so Tg < Th. Naturally, George overtakes Harry periodically. How long does it take George to overtake Harry if they start their jog together at time t = 0? Hint: Let 0905) and 0;,(t) denote the respective angular positions (in radians) of George and Harry on the track field at time t. Then 6 9(t) = 271' /T and 6h(t)— — 271' /Th will be their respective angular frequencies. Let their ”phase difference” be <23 = 09 — 0h. By looking at 43(t), determine the time T0 it takes George to overtake Harry for the first time after their jog begins. Express your answer in terms of T9 and Th. (—760ij ovw’lt’j H‘WY Mechvlcnbj C ,4 ea, l H 1’ 9. 001 filkfilnnéfflj (St/f», Y ZTT oflo, ("AW 4r k. '8 .__.__EL T T 3 h T37h l/Tr‘ , TST t mm.» T J TST“ “'13 You may use this page for scratch work only. Without exception, subject matter on this page will not be graded, LAST Name ‘WCLIQAA | Problem Points 3" -| 30 ...
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