examIII_f07

examIII_f07 - BB 350 EXAM III 15 November 2007 Last Name...

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Unformatted text preview: BB 350 EXAM III 15 November 2007 Last Name (Print): S plu‘é I o n S First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Test Form A w 1. You have 2 hours to complete this exam. This is a closed book exam. You may use one 8.5” X 11” note sheet. Calculators are not allowed. 159°?” 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 1.2 Continued. NO credit will be given to solutions that do 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. (7 points) Consider the periodic signal A = I n =2- f'vfl ...-————-. f(t) = a +ficos(2 t + 1(1) + 'ycos(4 t + 45), where the parameters a, [3, 1/}, 'y, and 45 are real-valued and constant. Figure 1 shows the amplitude and phase spectra of the periodic signal f(t). C n Figure 1: Amplitude and phase spectra of f (t). (a) (2 points) Determine the fundamental period of the signal f (t). {ICE}:— -$C:E +70) Cebuth 11', = 1111‘ car-.1 *4 To == 27"!” e ud/I'él‘eJ To = "'0 5 Elm .) n: [J "’32. Smul‘lbf‘ urbtyed Sniff-.59]? 6 To =T 1-9 No = E 1‘. 2. To 2. (8 points) Find the trigonometric Fourier series coefficients a0, an7 and bn of the periodic signal g(t) in Figure 2. Note that over the interval 0 S t S 277, the signal g(t) has the representation _67z —47z —27I 0 27: 47; 6,, Figure 2: Periodic signal g(t). 7; 2.77 ‘4‘ ‘J‘ ‘ ‘ ———""—-' - - Go Tojgétbl-E - ng - 2 (—ZTJYL ’_ L w 0 o - 13 7'". use 6.7’7 an " %Sa&3603(nub'k># = jig: Q05 (“+30Qé- 1 l o o 211 4—“ , S, c a) I} 1‘??— i’i (Cosérb) -r M: 5m (nt)\o = 20,01. lacs/742w?» n ll ur (Q J use, 307—7 T "' L I; “£7 5" c %J ogC-D sf» (hWot)JfE ‘ %-SO 1.". 5 (ma 0 o :17“ ’0 | -zrrncqs(zm0 ] =-' .L- .L [Sm (.n—D - a“? Cosmvbl o -.: 2-5;); [5m 42/?) 211" n‘ C. \ L,— —_L os/Vr‘rn) E=--,,‘-fi~ all! “" nrr 3. (10 points) Figure 3 shows all the nonzero magnitude and phase spectra components of a periodic signal h(t) whose fundamental period To is 7r sec. Find an expression for h(t) in terms of sinusoidal functions; do not leave your answer as the sum of complex exponential terms. (I ID,,| 4 D Figure 3: Magnitude and phase of all the nonzero exponential Fourier series coefficients. ' Bemusz anl= lD-nl anoL AD“ 3 — 4b,") “the 53mg bH-D 15 - From Dede-.493 Do fl 5’ m 9.. 9 uni '50 = — AB— CD = 3 D-n‘ “n2. 6/) 2 A—DI) n -F,.o,,‘ the. speafim above. Co=ba= lDoleAbnz“! c, = 2.15).): '1 6,: lb.=1T/v (LL: 2.11324 =7— 6;: lbL= 77/2, t 10.5 To = W"_’ (U0 3 277/70 :2” - usury these, fejults q Co‘wao’t‘f‘ 9‘) 4- C2, CasCLle: +62) gC-OQ :2 Co '1' q + L: 0.05 CZt+U/~1) + zcosmt+ 77/2.) Problem 2: (25 points) 1. (6 points) Figure 4 shows the magnitude and phase spectra of a periodic signal f(t) whose fundamental frequency wo is 10 rad/ sec. [D II Figure 4: All nonzero terms of the magnitude and phase spectra of f (a) (2 points) What is the average value of f(t)? 3-00 0° [The «warm value. JP {(-0 (5 Do’ mole} = 2.9, :2“, (b) (2 points) Is the signal f(t) real, imaginary, or complex valued? To receive credit7 you must justify your ansgfrc'ausz lbnl= lb—ol +(t) 3.5 fee-Q- VGA/99. Q01- 3-D“ = ' LD'FJ (c) (2 points) As a function of time, is the signal f(t) even, odd, or neither? To receive credit, you must justify your answer. Not; that bl“ [Dth- 3 l9] = Baggy f}; the, ho are, [lather pvr¢£i reoQ 0r qu mar/U ‘§:('b) 78 {)9th am am of Own 'Fvnc‘éwn inme- 2. (12 points) Once again consider the signal h(t) with magnitude and phase spectra shown in Figure 4. The periodic signal h(t) is passed through a LTIC system with the frequency response function w —Jfifi H(jw) 2 we 5 g |w| g 15 0 otherwise to produce a steady—state response y(t). The filter parameters a and fi are real-valued and satisfy a > 0 and 0 S fl S 7T. (80 (6 points) Specify all nonzero exponential Fourier series coefficients D3 of y(t) in terms of the parameters a and 5. H pale} 1s onl} non zero gun n =- IJ 9.5 we 1:: Lo raga/Sec, _- g 4.17/1 gar/we) U- — ' ‘F' - c. a l ' e ‘—‘ 0(— 9- nm. - Home» 13, _ o e - - n ' r “TN—6) + r; 9 /“l 3 - a; e i’ l . a ace I)if —— tic—04w) bf * h='l (b) (6 points) If the signal y(t) is real—valued and an even function of time, and the power Py of y(t) is 32, what must be the numeric value of the parameters a and fl? m&-4Juaa dug, an even ‘RM0 00 % ‘b‘m‘J the” 1-? éC-D 3:. we, mast ham. 5?; = 03;) ana. we, bfi‘ Must be. reg. 7M: rorvmms the}: out = anew/1'9) = bit-I =wéécw/PG) in mg ma 5? so mfi Dné‘. —-v Dné-l =09 :- 9 on: —I 5°C) 3- _ as t)‘e. on}? nonzero Dn are. Oh ,i ‘9: Be cwusfi— 09 70) 11)::ch im‘ = Zoo ’- 3. (7 points) A student generates a function g(t) that approximates a periodic signal f(t) using the m—file in Figure 5. g = 3; x = 5*pi; t = linspace(-1e1, 1e1, 2e4); for n = 1 : 50 g = g - (2 / n) * (—1)“n * cos(n*x*t); end; Figure 5: MatLab m—file for generating a function g(t) that approximates a periodic signal f(t). (a) (2 points) What is the average value of g(t)? W0 Nut: that-l; 50 r__' Lt): '3 +- 2. (-13-)(7 I)" (as (511' 0 —£ D 3 U In” I I $9 an T’RQ “VET device. Ct) 75 3. (b) (2 points) What is the fundamental period of g(t)? From pfif“: cubed?) 600: {ITO “ma 60 To: A" = 355% “la O (c) (3 points) As a function of time, is the signal g(t) odd, even7 or neither? To receive credit, you must justify your answer. From Par-k Coo) abwa note, thaé amfl 5° g Problem 3: (25 points) 1. (10 points) By direct integration determine the Fourier transform of the aperiodic signal f(t) : 2 [u(t + 4) — u(t — 2)]. Express your answer in terms of the sinc function. {L457 i 4: -1 7.- 2. Z —J,\JL w — wt " “I: — _.E-—[ F092] +0966 4:1: = 23:1} ii: ‘ otw e’ —1 —w - - 3"“ Sm 3W W .— —_-. "led .sm3u/ —— Lie. 3 /’ W 2. (8 points) Using the Fourier transform pair 1 e_atu(t) H , a + ]w a>07 and appropriate Fourier transform properties, determine the Fourier transform of g(t) = 36—2t [u(t — 1) — u(t — 4)]. - -z.£ 3H) = 39 “1 bLL‘t‘D - 3e web—w) 3. (7 points) Figure 6 shows the Fourier transform of an aperiodic signal f(t) in terms of a positive real—valued parameter a. Determine the value of a so that the energy of the signal f (t) is 20. F(w) Figure 6: Fourier transform of an aperiodic signal f 6) E4: = i7;— chmILlw ’W Pa») 75 an 9w“ Rno‘lfim % W) Because, 2. a) 7" 54 = 27-5 Ile‘flw = #F‘lea: z‘flw o o co 200 OC’ _‘j__ { cc. Vac ‘53: : w W := _...- 'l' fl -— M :ln-"f lo + 77 06 TV 77 W = 5:5: E4 77. "'77- 1 5 oar—HF 10 Problem 4: (25 points) Figure 7 shows a square-law modulator. The input to the modulation system is a signal f (t) whose spectra is band limited to B as shown in Figure 8. The signal 5(t) applied to the square—law device is 5(t) ——— f(t) + cos(wct), where the carrier frequency we is significantly larger than the bandwidth 5 of the signal The output of the square—law device is passed through a bandpass filter with center frequency wc, bandwidth B, and amplitude A as shown in Figure 7 to produce the output signal Bandpass Filter F 2 m “3:2? m a) cos(wk.t) —fl )8 Figure 7: Square—law modulator. Figure 8: Fourier transform of 1. (6 points) Sketch the Fourier transform of the signal 3(t) in Figure 9. 3 (+3 s 4H.) + Cos (Wot) = 1.10;» + 1’” + 71' Figure 9: Sketch of .7-"{s(t)}. 11 2. (13 points) Sketch the Fourier transform of the signal 32 (t) in Figure 10. 2- 320;) :- f -F L13 + cos (we-DJ = 43711:) + 2 +55) coslwo-Lj + 008‘ prt) {ac-n +— z-Ft-aciouwot) + i- + 45¢“sz \l QviSZHOB —_-, 3: Flow) + F(w hug} +FCw-Wc.) + $34". 3' Lou) 2 TT' 4' E 4"ch) 4’ £22, g’cw— 2. pate, 1&3» PO“) F cw) PM 86! -20. 749- Figure 10: Sketch of .7-'{s2 12 3. (6 points) Specify the filter amplitude A and the smallest value of the filter bandwidth B so that y(t) = f(t) cos(wct). F‘V‘J‘t Dehrmm-Oa Yb”) +rcm the. “have. l‘e{u:Eam)1tf) > Fat! +Wo) + PlW' We) 2' 2. H” 71w) 2b. l l w .w‘_ 0 WV t 4; sa: 9: J5: anfl. 822$ Jo Tide 'U'e—o‘epv % it». badpr may Wales ‘F‘t)@$(%15) . 13 ...
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examIII_f07 - BB 350 EXAM III 15 November 2007 Last Name...

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