examIII_f03 - EE 350 EXAM III 17 November 2003 Last Name 5...

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Unformatted text preview: EE 350 EXAM III 17 November 2003 Last Name: 5 Qigg'LIOQ First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Wei5ght Score The Blue Test Form Instructions 1. You have two hours to complete this exam. #0310 . This is a closed—book exam. You are allowed one 8.5” by 11” note sheet. . Calculators are not allowed. . Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number, for example, “Problem 2.1) Continued.” No credit will be given to a solution that does not meet this requirement. . Do not remove any pages from this exam. Loose papers will not be accepted and a grade of zero will be assigned. . The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1. (12 points) Find the fundamental period To and trigonometric Fourier series coefficients of the periodic signal f(t) shown in Figure 1. fit) Figure 1: Periodic signal f(t). ° 9d. Inspection To ‘= 2.0 _ o Becqugg {3015) 7.5 an even ‘Qflc'Etm % 'bmg.) ans— 1:. l ‘ no = $58 Halit— = Ila—[2.4.1: = 5:44 :— § 0 ° 0 2. (13 points) The periodic signal g(t) = a + ,6 cos(41rt) cos(21rt) + 'y cos(31rt) + 6 sin(31rt) has the exponential Fourier series coefficients Du specified in Figure 2, where a, fi, 1, and 6 are constants that are not necessarily real-valued. Re{Dn} . Im{Dn} Figure 2: Specrfication of the exponential Fourier series coeffic1ents D" for the periodic signal g(t). o (3 points) Find the fundamental frequency w, of the periodic signal g(t). o (10 points) Specify the value the constants a, fl, 7, and 6 that appear in the expression for g(t), and also specify the integer values a, b, and c, that appear in Figure 2. Usmé “Um. ‘lnra o nomz‘lzflc. idlen'ln‘g/ cos xfcsoa ‘—‘ AZ “’5 (9°95 "' Jitfiuy) : + 1.5:. Ca; (GIT-D ‘l' é‘CQ‘CZlT‘t) *f" 7605(37-9 4— 53/0 (37ft) gC-B = cc 2 tlrufié To insure. “1.:th altl=flft+73b we, Maul/‘9— ‘; _ \z GTTTO :: nZ‘II‘ > ‘3: ‘ n 2 —- P3”; n'zé 2r 7" _ :1 =1 ITT m n 2 3m :— 2P > ’Bg3 31T7‘0 “F21! éluen 07,6) To —— Egg 3 2 é 5'»ij 6‘ °° + ~94- z. ’— 2" L 5stch admit 3124”“: ~ ”374: + fab—fl 4— ‘/ {GB/f 1/ Problem 2: (25 points) 1. (13 points) A periodic signal y(t) has the complex exponential Forier series coeflicents specified in Figure 3. Figure 3: Specification of the exponential Fourier series coefficients D" for the periodic signal y(t). o (3 points) Is the signal y(t) real, imaginary, or complexed value ? Justify your answer in a single sentence in order to receive partial credit. BeCowse. lbnlf— lb-n\ chL an = - Lil—”J Do a bra TRQ, 0054: about)? bows cob. I‘Fyé'b) 15 reuQ’valvefl. o (3 points) As a function of time, is the signal y(t) either even or odd ? Justify your answer in a single sentence in order to receive partial credit The “Function 0510061: be. 6&1 bQC‘quJQ. Do *6. (Odlfl £7,550") #ttfl" cannot AM cu Dc valvz, 7R2 ~94; Ian cmmaé 52. even begun. “lama“ 5-06) is Mag-Vol.0! tn. DA owe. Complex 'VaJvQQ. o (2 points) What is the DC value of the signal y(t) ? Do " 17‘ 13(— Valve = lbolééx' : We.+ =— "f. o (5 points) Determine the power Pll of the periodic signal y(t). Be (Aden. &(fi is muQauwlveflJ L — lb 1“ + 2 3.11m" = IbJ‘+ warm-A) Pa - = M1" + 1((20‘ + ml) :16 +1e+8 I‘m 2. (12 points) A periodic signal with Fourier series representation fa) : 5+ 2 £31634OOnt n¢0 is passed through an ideal lowpass filter with cutoff frequency we and passband gain J2 as shown in Figure 4 to produce a periodic output signal y(t). Find the range of values of we, in rad/sec, that yields the largest output power Pit of the filter without violating the constraint P, < 125. HO 0)) \13 "(D c we Figure 4: Frequency response function of the filter. Do 4301- $053 bn %r 30:) r3331 30:) = Z“ #90) + Z “(a-Won) 3m:- e. l l ns-OO ‘ I be five: "1” 5% ha #0.. Hi) F~| __ co Lil-2. *VOOnb (t) — +— ’— 0" 5E E-” n e b?)- (1 +0 (1119- Iwon i (‘09 P} ‘18?!” zigmmz 2 §0+ £21 lWon |< WC, [WMICWU : 50 + 6‘! +31+§i+--. (an) ('1: 2.) C3133) To keep 9 < [253 we, (Tan oil; fad! cf 153 0:2] “Mme 7i 0% ‘t>- 5W W (25 Points) 1. (13 points) An energy signal f(t) has the Fourier transform F(w) = e_2|“’l. o (3 points) Without determining the inverse Fourier transform, what conclusions can be drawn regarding the function f(t) ? In order to receive credit, justify your answer with a short sentence. BQC‘MQ PI“? 75 {W Wild reag— arr-IQ an even 4w", [em W) 4:06) Ts pJNz 2d mag 0mg even Rug-Eton «%— ”Elma. ‘ o (10 points) Determine f (t) by direct integration, and express your result in a form that validates the conclusions reached in the previous question. For example, if you determined that f(t) is real-valued, then your expression for fit) should not involve complex exponentials or factors of J. «7 ‘1'. w °° O .. I w 42(43) " E{F(W3€é’ o‘w = 3% FuQmsw'Lflw ’fi Pl muffle” -6, 217 Pm .' w In'lue’rana l5 4" m'hefirafln ,‘5 _2w {won £nc‘hon # at all! 'C—nc’éloflfl w u I Q $3 3 b E -2w - (”zap 2. (12 points) Using the Fourier transform pair rect (é) H 1' sine (951:) , and appropriate Fourier transform properties, find the Fourier transform of f(t) = 12 sinc(2t — 6). USM the, re .5, fa,“ ,._ red: (39;) qnfl J S me F Far w: p .a at #21 PCM :: L. Smc. 7) awe; Fm . é——-—> , zwr Hwy TSth(tt) E 3 277‘ rec+(’¥‘) 7- ZFWot(%) 2. (feat: 75 an Quefl \fMC‘é/on) Se‘Hzmfi 'C -.-. 12 away- (Li [Z Smc. (61:) H 277 rev": (T5: USO. the. Sammy~ prop-arty.- 1:0»an 701: F(%) (with o..-:.-JS 12. .5m<.(2.{:) e—a en‘ reoL(—‘e;_.) 7M... 032 fLQ. tUWQ. ‘5th Proper? £66 ‘hn H F(w\€ with Problem 4: (25 points) 1. (12 points) Figure 5(A) shows a partial block diagram of the system that connects Neo to the Matrix. The signal m(t) is a voltage derived from Neo’s central nervous system, while the output voltage s(t) is the signal patched into the matrix. Given the Fourier transform representation of m(t) in Figure 5(3), sketch the Fourier transform 5(w) of the signal 9(t). M(m) 4 3 cos(500t) . m(t) . g 8“) co ' ‘ ‘3‘si'n'(1500t)” " ’ ’ ' ‘ -300 -3 3 300 (B) (A) Figure 5: (A) Partial block diagram of the communications system aboard the hovercraft Neb- uchadnezzar. (B) Fourier transform of Neos’ neural activity. 869 :: 3 mC—Ih Cos (Soc-t) 4. 3 m (a) sin (lfoo‘t) 05m; Q’L {ta as (wear) _-: F (w+w°3 +_ pa”- We) OWL 2. 2—- g‘éwasmtwaag : 0,1 POM—kw.) .. J POM-We) ) we obh‘m Z 2. 30v): %_M(w+so® +.§:m(w-5OO} *a’l'ii’M‘V-HSO‘D 7%le—lfm) 1800 -|$co -1100 2. (13 points) Agent Smith destroyed the receiver in the machine world that recovers the signal m(t) from a(t). Trinty and Morpheus solicit your help in designing a new receiver that recovers m(t) from 3(t). With the Oracle’s help, Trinty and Morpheus can provide you with the signals cos(500t) and sin(1500t). Draw a block diagram of your receiver, and, using appropriate sketches in the frequency domain, demonstrate that your receiver correctly extracts m(t) from s(t). If you use an ideal filter, you must clearly indicate the cutoff frequencies and passband gain in your block diagram. The quality of your analysis and evaluation is as important as your answer ! In order to receive credit, your block diagram and sketches must be neatly drawn and labeled. 7:ka «fa Severe-IQ {”561th Sole-E1605) one. % wln'ch 75 be. mol‘t’npla' sl—b) bfik (as (Soot) So tk iLz. COMFOfiefi‘L‘é letyoo) 5“") are. resh’faoa ‘50 MU»), In ’U‘lS arr/f re main we, mas-E Lou/pa); 147% S (t) coSCSOO'b) 1‘22 alt/maven unwanEcQ, by}! "9%va Confonen‘ét: n Hz) CcSCSoo a H 9”,) Note, thgi: ““3: Sit) CosCSoot) é——-> XIWW‘; 5(0):”) + 5"” +57”) 2. SCW'Soo) L urn-«v. 'I300 d‘m -709 S um m “5L {kg 04543 ‘Bwo flue-Eckc‘s V'L’Qy plus amps/Ion t5 btlow ~300 PM Comps/heme; above, 3’00 '30:) '3 +3 +300 w TL- ?‘QQaQ ”00‘” “-55 £l'éek' Mil-fl ”SAMMIE Q1 2/3 afla ((1th 7Cr /) F a! n aivencr. Cue = 300 remove) fits. Componan‘b) alum/Q anoQ 56/0w £0 {0qu- ( 300) (“3°03 y X(w) H-(Vlw) W‘HJ‘ is Qbulu¢‘0w£ “to M660) ' 11 12 ...
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