examIII_s04 - BB 350 EXAM III 5 April 2004 Last Name 3 u...

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Unformatted text preview: BB 350 EXAM III 5 April 2004 Last Name: 3 u ' First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total 100 Test Form A Instructions 1. You have two hours to complete this exam. 2. This is a closed-book exam. You are allowed one 8.5” by 11” note sheet. 3. Calculators are not allowed. 4. 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. 5. Do not remove any pages from this exam. Loose papers will not be accepted and a grade of zero will be assigned. 6. 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. (10 points) Find the trigonometric Fourier series coefficients (a0, an, bu) for the periodic signal f(t) shown in Figure 1. f(t) Figure 1: Periodic signal f(t). 1; - 4°: #:Sfittwrl’w 4: (351,») = 150,-} ‘0 =7- qo— 7— oweq of- anc. truly); To 3 We. TuHe. an: %L Hacosnwgtii a 3%.: .'%‘=_cas(l—g.!L-Qi-£ 5.7-7 -_- 8 7 211’» 277 in 3 Cos—t- + ___’2.+_- .rm 0 7 ‘m‘n‘l- 3 3 3 1' 6 l - 2. ‘ ——-~ 2170 5m v - "Lnl— [NSC 0) + I 6 J a 0 an :0 n: IJ z" 3 ’ ---—-- we Table 3 2. 2.. '1 21M 5.. = 17:36 -Fl-e)s.n M-bafl-e = "3'1; 5' sm( 3 1-)“: 6’7_7 2. (9 points) Determine the trigonometric Fourier series coefficients (up, an, bu), compact trigonometric Fourier series coefficients (Co, Cm 9,1), and the complex exponential Fourier series coefficients (Dn) for the periodic signal n =‘ n=L n a 1 w. = z- 1 1 ur- 1— h(t) : 3 + 5 cos(2t — 45°) + cos(4t) + E cos(8t + 90°)_ To :2 7:; _-. I 8& 'mf’e’i‘mb hit) :5 when»? errveJJOQ. as a. Cafl’au‘t ingonome'tno Rum-er Series w (UH all other C“) en are. are Ji- COS (70°) =0 " "i‘. 5"" (70‘) =" 0: my 3. (6 points) The following MATLAB code generates the triangle wave 9(1) in Figure 2 using 100 sample points E triangle 1, = [0 : a : b]; D = [0 : c : d]; g : pulstran(t, D,’ tripuls’, 1); p10t(t,g) (spa-(€09 wcl‘fln % inunale is athl— What must be the values of a, b, c, and d ? g0) 1 0 1 2 3 4 Figure 2: Triangle wave g(t) generated by MATLAB. Each trwuvle RG5 Ofl-eJ and, there are, loaf: ()0? triangle. These. memo the. time fem-£3 are. Span: 5}. k = LE—e—Ez- 3 Co’- loopts & 05,4: 2 '1 Chemo. b = H.O Beagng there, )5 4.. firmnifl, OGfl'beraQ 01‘: Ever} ‘6QConsg‘ Q a 1 Finally) LC—CMJQ, there am. ‘FIVQ. hwy-ales) i=w. (I): E01 [1‘1] :. f0, 1,2.)3)‘1-J3 *Hlvo. “Enqny’e; Copier-ao— “76 i=0) {1.2. a) «ma/'1 4 Problem 2: (25 points) A periodic signal f(t) has the following complex exponential Fourier series coefficients Dn for n > 0 D0 = 16-77r D1 = 2e?” D3 = 36—Ja D5 = 16—Jfl, where a > 0 and ,3 > 0 are real-valued phase angles in units of radians. 1. (2 points) What is the DC value of f(t) '? 11' DC VW‘VQ = 0., z c'e' :-| 2. (9 points) Given that f(t) is real—valued and an even function of time, specify the magnitude and phase of the complex exponential Fourier series coefficients for n < 0. The. Drs and: In. real ana— an M Rnc‘fuon a. n bacan ~FC-bX is real-“Avail ang— an even {Log-Em; filma. 5'3 ‘53 = 32.1"“. = +3.” -3 63.00 mg t -66 - +l gf‘ -' W 090.“- «33:713.; flI-lor 20's... 3. (4 points) Given that f(t) is real-valued and an even function of time, specify the power P of the periodic signal f(t). +33 Ls W-wlwfl) 3664M"! an L z o. 1 PF = 5-010") _, [)0 + 2 31an = lbol" 4- 7-19.17" + 7-l'33l1 + z/DJI‘ = 0137' + 2.1-2.11 +2.13;‘ +242)" = I + e + 18’ +7- Another periodic signal g(t), with fundamental period T}, has the complex exponential Fourier series coef- ficients D51. Consider a new periodic signal T9 h(t)=g(t—Ta) . 4. (2 points) Find the fundamental period Tah of h(t), and express your answer in terms of Tag. Because. hlt) is Ova-£- a. +»mo-5lu-P'EJ- replun. 716.1%), hit) qnfl, mJJt' Imwe. the Samer penocg‘. 5. (8 points) Determine an expression for the complex exponential Fourier series coefficients D5; of the periodic signal h(t) in terms of the coefficients D51. 3’" n l:- h 2. -— Wu 1),, = 3‘; J Mae d (LL 1: 3. Problem 3: (25 points) 1. (8 points) Consider the signal f(t) = 3 rect sin(12t). The Fourier transform of f (t) exists and is denoted as F o (4 points) Determine, without computation of the spectra, whether the Fourier transform F(w) is an even function of w, an odd function of w, or neither. 86 c «~50 fink) = 3mt('3;) Sud-n.4,) = '3Y‘Oot(:§) Sm (Ia-e) =-. - {-1 a) {4.6) B an em ‘Fuflu'élun % “Ema. If 'FOHOM F0») mus-l.— bo. cm 0.901, ‘Fvnc‘évon i w: '9 .. “A: a» O on FCW) = I. “4:369 if: f—Flb3éw-L-l-é- —/f-Flt)$mw‘b J-é -” 0:9. 4” o (4 points) Determine, without computation of the spectra, whether the Fourier transform F (w) is real-valued, imaginary-valued, or complex-valued. 655w”, .Fc-e) : - {4-45) 75 am am fianc‘ém ;{ fume.) °° {Tum «IOU-12. Pa”) = _7 [mo-H £35m «viral-E 1 401 .So be. Cause. ‘FC-b) 7.5 «.150 Y‘QGJ-VO—IVJJ F’th mu)“: bfi- ImaZlnaVi I 2. (9 points) By direct integration of the Fourier transform integral, find the Fourier transform of the signal 9(t) = 6(t + T) + 6(t — T), where T is a real-valued positive parameter. Simplify your answer so that it does not contain complex exponential terms. on _ Wt O 604') = f SOL-+17 ed a“: 4' $(t-fle-dw-BJ‘ _————-—‘ acacia“. git) 1S fear—Q-vwlvcg— anoa an even Lno‘ém % 'lslme: alt) a) a) -T T \ *V u L We" Sl‘oJ‘Q appeal: that 610.4;5 us rewl col 2g an an QUGA ‘p/ho‘em % W. 3. (8 points) Given the Fourier transform pairs f1(t) H F10”) f2(t) H F20”): show that f1(t) * f2(t) H F1(w)F2(w)- 3Y4: (ta-x F m} = a 'éw-b . t 56. m flu; -t\ic e OH: = [Em 'F;(-t-T)€:#V:L—b} £1: = gm [Ii-Clt-k‘CV} net on - (gt = 4:, (c3 lime? it - °° 1W ( 3 —- g 41. (t\ e, c FL W -¢7 1 a '1qu glifit-qkfit-a} = FUN-3 F71“) Problem 4: (25 points) Because analog multiplication is difficult to implement over a wide dynamic range, the switching scheme in Figure 3 is used to obtain a DSB-SC modulation. x(t) y(t) bandpass filter m(t) k m(t) cos (not centered at i we switch Figure 3: Implementation of DSB-SC modulation using a switching scheme. The periodic switching signal 2(1) in Figure 4 has a fundamental period of To = 21r/wc and the complex sin (mr) ___2_. 7L7I' exponential Fourier Series D" = x(t) —To _IQ 0 In. T 4 4 Figure 4: Switching waveform 1. (9 points) Given that the Fourier transform of the modulation signal m(t) is M(w), and that the output of the switch is y(t) = m(t) 3(1), find an expression for the Fourier transform Y(w) of y(t) in terms of M(w) and the expression given for D". \ Bach“. xtfi Is av rifle-v.99 ‘va'lzmn’ Thurs} = 217’ E Dn 5(W'%n> Ira—fl USN; the. "Nbvcnca convolv‘bvn Pro/hora Y0”) : MUM) a: XIW) a 2 mm.) at 271‘ “gym 5(W"7'7Ec) 10 co Ym = 217 Z on mama—g") ll 2. (9 points) Consider the DSB—SC demodulation scheme shown in Figure 5(A), where the input signal m(t) has the spectra shown in Figure 5(B). M((1)) In(t) cos((0ct) 8%» 6(0 f8 cos((0ct) (1) —100 0 100 (A) 03) Figure 5: (A) Demodulator with output e(t) and (B) the spectrum of Find an expression for the Fourier transform E(w) of e(t) in terms of M (w), and sketch E(w) assuming that we >> 100. 80:) =- m(e36o.s7'(wc;b) = mC-t) + i; Cos (Zn/0+6 em = 12-: awe) + true) 60422099 3. (7 points) A signal f(t) with spectrum F(w) is passed through a LTI filter whose impulse response h(t) has the Fourier transform H(w), where F(w) and H(w) are shown in Figure 6. F(co) H(£0) 4 2 —2 o 2 m —3 —1 1 3 m (A) (B) Figure 6: (A) Fourier spectra of the input signal f(t) and (B) the impulse response h(t). Determine the energy E, of the response nu.) -.: F-(w) H’lw) M £01 “ 2L; Iwa)iz'jw -.° L _l 4—, j :: .L— 6“! W LIT —L6“ JAM + 27 ’ z 122 111' 13 ...
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