Fall07_Exam3soln - ECE 210/211 Analog Signal Processing...

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Unformatted text preview: ECE 210/211 Analog Signal Processing Fall 2007 University of Illinois Levinson, Sarwate, Kudeki Exam 3 Thursday, November 15, 2007 — 7:00-8: 15 PM W. , Section: 9 AM 10 AM 1 PM icircle org) Class: ECE 210 ECE 21 l (circle one) Please clearly PRINT your name IN CAPITAL LETTERS and circle your section in the boxes above. This is a closed book and closed notes exam except for one 8.5 x 11 inch formula sheet. Calculators are not allowed. Please show all your work. Backs of pages may be used for scratch work if necessary. All answers should include units wherever appropriate. Problem 1(25 points) Problem 2 (25 points) Problem 3 (25 points) Problem 4 (25 points) Total (100 points) Problem 1 (25 points) Circle Lug or false for each of the following statements. (a) [12 pts.] (i) If Hon) 2 Mos), then f(t) 2 0 for all t. (t) [@w —— L. ’L ' ' Aw (-9 J—SQM. 44m (2 41‘ (ii) If C(t) = u(t) * u(—t), then c(—l) = 0. (iv) If u(t) >1: f(t) = g(t) and g’(t) = $14? , then 5(t — 1) * f(t) = g'(t + 1). % l V rue ' 53) new) x: a’twb) T E ‘ L; a (+—«>4<1u+>= g’(t-I) #6/C'Efl) f(t) = cos (toot) H(co) = ejwto ? (b) [4 pm] What is the phase shift in y(t)? 136:) = lleo)| c.» (mfi [.Htwa)) La Nana JLM‘F-r : out.» .. % if Problem 1 (continued) (c) [4 pts.] If x(t) is a real—valued signal with Fourier Transform X((o) with X(O) qt 0, what is z X(O)? >k 761+) «we max =; X(-~)=X(w) => X(o)= ><*(v) => XM an. % 4Y7 (d) [5 ptS.] Suppose a filter has frequency response H __ 0 [03142 I (0»!— 1 [mIZQ A signal x(t) with bandwidth < Q is put into the filter. What is the energy in the signal output by the filter? Justify your answer. I 11:1!) WM“ tomato.)in «L I ______.._—————-I- Problem 2 (25 points) Letf<t>=sinc<t). e 1: (to) = 174m (217: 7 (a) [6 pts. ] Sketch F(co) and label the axes carefully. F (co) 11 l w - a +1 Amy“ , e w" 1 u" (b) [7 ptS.] If H(co) = , what is output y(t) in integral form? Do not perform the integral 1+jm but simplify its integrand and integration limits as much as you can. no .w , we amez—‘him F(w)H(:ZéJ M sit-J. 9l+)=MJc)* mafia; K('C)c-Tsiac(t-'C)att =0! 3‘ neuron/fl; - - t . L (c) [6 ptS. ] At what frequency is the power of the output, from filter H(co), —3 dB? .1. ._ L; .L W 5-) HUN” 1 /)/l{ W (Mal~ dad/Lac. (d) [6 pts. ] Suppose instead H(co) is an ideal low-pass filter with cutoff frequency Q, Q < 1. Whatis t? a fjfil, YO ‘ lI-uw)__ Au, (2,” =2 YM : Fa») mozmwg) => 3H) .~ :2 Mae) /. WA wax-f lHl‘O)‘: Problem 3 (25 points) Let a(t) <—> A(co) = , and consider the following system description: a(t) b(t) ca) d(t) Haws» Where H(co) : rect[m_210)+rect[w:10) cos (5t) cos (mot) (a) [4 pts.] Sketch B(03) vs. 0), labeling both axes carefully. BUD) i '2; 03 15- i 5 1—3 7- 5;? (b) [3 pls.] Select a value for (no so that signal a(t) can be recovered from signal d(t) without any distortion. E00: ‘7’ fwd/:44. (c) [8 pts. ] Sketch C(co) and D(0)) for the value of (00 from part (b). C(60) 13(0)) u L .. . L in. ‘1 a) CO 6-"? m '9 ‘19 v ‘i" 6;? (d) [3 pts.] Express D(0)) in terms of A(03). Db E co = fiMw-toyikM—fwzl (e) [3 pts.] Express d(t) in terms of a(t). ——> Nb «:1 F-T- Problem 3 (continued) (I) [4 ptS.] Describe briefly a method which can be applied to signal d(t) in order to convert it back to signal a(t). You will be graded for the clarity of your description. alto = i-a (b00006) H W a ('67 fldw limo“) €12“ Le Aggx 4 as (1%): Z a (4') 5910”) w? %C+)/_l;<((+0>z 4:)» 150.» L91: M4 H0»): xw.+(g;1) I e.c‘), / Problem 4 (25 points) t+l, —1 s t < 0, Let f(t) = rect (t), h(t) = t— l, 0 < t S 1, and y(t) = f >2: h. 0, It! 21, W.W,WWW.N.an.M....l4vm,.,.aaW“mm”.WMWWNTAM~W~MmaW (a) [3 pts.] Sketch h(t). {(+5 : mated ; i g l l, l. t z i i i l E ,1 Z, 7.. (b) [8 pts.] Evaluate y(O), y(—‘/2), y(l) and y(2). 4.3 étoho 3(4):; l _ 2' Z 30):":i . BOAPO (c) [3 pts.] What is the smallest to such that y(t) = 0 for all t > to? tug Problem 4 (continued) i Recall that: g H 1, —1$t< 0, Let f(t) = rect (t), h(t) = t— 1, 0 < t s 1, and y(t) = f * h. 0, [t] 21, (d) [4pm.] Sketch 2(t) = (5(t)—5(t—l)) * h(t). 3 (H ; Jack) _ 4; CL-“ z(t) t (e) [4 pts. ] Find the Fourier Transform of g(t) = Nd; ( 4—5“? Sqm<9g 633103: 1 40¢ rl<C<O ,« o éts) obi MAC“ 2 — 25cm a): t;o dd: gm : mtg}— 28Ct\<+ 60% = ZSMCCM e?- (t) [3 pts.] Is it possible to reconstruct f(t) from its samples f(nT) taken with some appropriate nonzero sampling interval T? Explain your reasoning. NO. is ho'l— a {oqukiW\“i€£ SijVO~9~ QMAL Camvuo’i’ le—a Wow 504433114 gnw i’i‘s Saw) AA mwlvucflfidflr V)? WWW gawk? CM com Cap’louv. ’itu {scorer Lam'de AA ’Tlm. fimwgkm L‘vx 1C gem \ffixfiVULO +3 \Ialm\ QWQMCLVe/{g’fi ...
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