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sp2002_exam3soln - ECE 210/211 Analog Signal Processing...

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Unformatted text preview: ECE 210/211 Analog Signal Processing Spring 2002 University of Illinois Hiskens, Kudeki, Ma Exam 3 Thursday, April 18 — 7:00-8:15 PM Name: Section: (circle one) Please clearly print your name and circle your section in the boxes above. This is a closed book and closed notes exam. Calculators are not allowed. Please show all of your work. Backs of pages may be used for scratch work if necessary. Good luck! Problem 1 (25) Problem2(25) Problem3 (25) Problem4(25j Total Score (100) {25 points) 1(3) {4 points) 2 The normalized energy spectrum mfg)! for a particular signai is given by the following figure. The figure shows the normalized energy spectrum in decibels versus frequency, f = ZED—7t Only the positive frequency axis is shown; the spectrum is the mirror image for negative frequencies, of course. Provide an estimate of the approximate bandwidth for this signal. Justify your answer. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency in Hertz ggfldagdfl a: 2.000 H: 11‘; 5:3an kn: Widthflll’lt “arty-3 L’Llcw HIODOHI ““4 flLOU‘. N “000 He . Bast am .12.. 2. «coo—um ‘ ”W “a. 1(b) Given: {10 points) Hm) f(t) <—> 13(0)), and fit) y(t) = f(t) cos(2t) cos(2t) (i) (5 points) Sketch Y(m), the Fourier transform of y(t). (ii) (5 points) Determine y(t) explicitly from its Fourier transform Y(uJ) sketched in part (i). flea ; gawowmfi + z; 15(w'3)*g(m3>3 (fir a‘t)*%0°>(t)+¥°$(30 W. 1(c) (1} points) Let ya) = e‘mua — 1) be common for the following three questions: (i) (3 points) Compute the derivative g(t) 2 Hdt'ym and simplify your answer as far as possible. - 2 f I - 2"? I 3%):- {e )uH-l) + 6 MN“) —2+ 7..“ ~26” ate") -l- E Elf-*1) .. m'E' -Z '- eze‘ «MU-f) + e Joe-4) 20 (ii) (4 points) Compute the convolution I0 y('t)5(5 — 1:) d1. [Tammy—rm: = 3U") :2 e -I0 (iii) {4 points} If y(t) is also the response of some LTIC system to the input 2110 — 1), what is the impulse response h(t) of the LTIC system? 31%) :- ,gt-s) 4* 2mm...” grog):- $r-é)*25l~t-I) .: 2£H-r) zl‘u-i) :: -z (1-2+un +e'zfz+_,) _ -t- _ in H4) ~.: *5 7' ut-e-r) 4- 3%- 6 15“-.) 13-”) :3 --ef2H+0M{+) + afie—ISH) -2 wu— v-2 51—6. e M£+)+=‘£e Jh‘r) ._ -2 s "‘ e ~62 MU-ré-é—m) ( 25 points) 2. Signal fit) with the Fourier transform 13(0)) is the input of the system shown below. cos(10t) 005(50 Given that and (3) Sketch R(t0). f? (b) Sketch Pan). H~r-——w-~M--——§ W ,— if '- I'D " S, g— !o if, (c) Sketch Y(m). Yow) é; 'léttt); “10096-0 You) = H00) PM 1/2 »— g "7 fifimmw> 0" ((1) Express y(t) in tenns of f(t). 3“) = J6me» (5%) (e) What is the envelope of y(t) if f(t) > O for all t? gamma ._., Web-{m @m. {36070 97 we) (f) When f(t) is replaced by g(t) cos(wct), it is observed that y(t) = g(t) cos(5t). What are the possible values for (Dc? my...“ “mm“ "My Mar—“IO 9M0” wmk; beamga (Furs-f” Mix-u hm th)¢0cm¢) M’l‘vm £01 m MK? MN (5L 9M? 5‘50 Old—mm 19M 04 r (w) ( 25 points) 3. (a) (6 points) Determine from first principles the Fourier transform of fit) 2 e'a‘u(t). fl -— 6 .‘..;f- 3§€0~H ._ SJ“ “ “034.6 .Ll- _ .. - d1.” '6 -8 4C a) 9M- 0 d) 1 ~04“)!- I d~$w "2' 6 I #L-— (Cl—I.» .‘3 al>o 2) 32606X‘ Q45!” SI 1 SM.) (b) (4 points) Differentiate fit) from Part (a) to obtain g(t) = 9% :— ....r we am: {if “a e. are} H. 50») (c) (5 points ) Determine the Fourier transform of g(t) from Part (b), from first principles. (You may use your answer to Part (a), but do N! 1T use the time~derivative property.) -ué‘ “Kaf- :Qsmi “die “0‘32 + 8 30):} an» ="°‘('.T'JLF + 1 _. __.....:-—fl--"""‘ -r w: .- d4&” flan-b 1-. "$— dab” (d) {3 points) Show that your answers to Parts (3) and (c) are consistent with the time- derivative property. Eff 60-)\ a PC”) T.) mg; ‘5” For» fimt- Jewivq‘i‘u‘fifi. FIDfit-‘J‘I-a '. “gig AW.“ 2 g» :rgitmg u-ti‘h" "‘ CranSn‘S‘t-fi" w|+L +t‘mg-‘Je-v’t‘Uq4iv-c Prom/{“5 .— (e) { 7 points ) The signal fit) 2 sincz(t) has Fourier transform FUD) = n A (%). Determine the total energy of this signal. Siam-Art barn-3 ‘ w I fig [PO-a)! tin.» ‘ *r(z—L«'?‘;} =a%°=-1'°‘i¢m (25 points) 4. Two functions f(t) and g(t) are defined according to the Figure 1. 1'03) 1 t —00 0 1 2 3 4 5 +00 W) 2 t Figure 1: Definition of two fimctions f (t) and 90$). (3) (5 points) Draw together f(‘c) and g(t — 1:) as functions in 1:, mark clearly the points where g(t w T) transits from 0 —> 2 and 2 u) 0. (b) (10 points} Compute the convolution c1(t) = f(t) * g(t). Draw (in a separate figure) the function C1(t). 4, ts: Clt‘é) :: 9-2+, t$+sz 2, 75+“ “'9‘; 3 5+5? 4,. fair See next page for part (c). 10 (c) (10 points) Compute the derivative % g(t * 1) and then compute the convolution c2 = f(t) * figfi — 1). Draw (in a separate figure) the function 02(t). A um- ?«t+—l) a 25W) "* ZSH’Z) 11 ...
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