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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 — 7:008: 15 PM W. , Section: 9 AM 10 AM 1 PM
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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
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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
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53) new) x: a’twb) T E ‘ L; a (+—«>4<1u+>= g’(tI) #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‘Fr : 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
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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 sitJ. 9l+)=MJc)* maﬁa; K('C)cTsiac(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 lowpass ﬁlter with cutoff frequency Q, Q < 1. Whatis t? a fjﬁl,
YO ‘ lIuw)__ Au, (2,” =2 YM : Fa») mozmwg) => 3H) .~ :2 Mae) /. WA waxf 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))
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‘1 a) CO
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(d) [3 pts.] Express D(0)) in terms of A(03). Db E co = ﬁMwtoyikM—fwzl (e) [3 pts.] Express d(t) in terms of a(t). ——> Nb «:1 FT 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 = ia (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
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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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 Spring '07
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 Digital Signal Processing, Signal Processing, pts, Analog signal processing

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