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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 — 7:008:15 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 “arty3 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 )uHl) + 6 MN“) —2+
7..“ ~26” ate") l E Elf*1) .. m'E' Z
' eze‘ «MUf) + e Joe4) 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%) : ,gts) 4* 2mm...” grog): $ré)*25l~tI) .: 2£Hr) zl‘ui) :: z (12+un +e'zfz+_,) _ t _
in H4) ~.: *5 7' uter) 4 3% 6 15“.)
13”) :3 ef2H+0M{+) + aﬁe—ISH) 2 wu— v2
51—6. e M£+)+=‘£e Jh‘r) ._ 2 s
"‘ e ~62 MUréé—m) ( 25 points)
2. Signal ﬁt) with the Fourier transform 13(0)) is the input of the system shown below. cos(10t) 005(50 Given that and (3) Sketch R(t0). f? (b) Sketch Pan). H~r——w~M——§ W
,— if ' I'D " S, g— !o if, (c) Sketch Y(m). Yow) é; 'léttt); “100960
You) = H00) PM 1/2 »— g "7 ﬁﬁmmw> 0" ((1) Express y(t) in tenns of f(t). 3“) = J6me» (5%) (e) What is the envelope of y(t) if f(t) > O for all t? gamma ._., Web{m @m. {36070 97 we) (f) When f(t) is replaced by g(t) cos(wct), it is observed that y(t) = g(t) cos(5t). What are the
possible values for (Dc? my...“ “mm“ "My Mar—“IO 9M0” wmk; beamga (Fursf”
Mixu hm th)¢0cm¢) M’l‘vm
£01 m MK? MN (5L
9M? 5‘50 Old—mm 19M 04 r (w) ( 25 points)
3. (a) (6 points) Determine from first principles the Fourier transform of ﬁt) 2 e'a‘u(t). ﬂ — 6 .‘..;f
3§€0~H ._ SJ“ “ “034.6 .Ll
_ ..  d1.” '6
8 4C a) 9M
0 d)
1 ~04“)!
I d~$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&” ﬂanb
1. "$—
dab” (d) {3 points) Show that your answers to Parts (3) and (c) are consistent with the time
derivative property. Eff 60)\ a PC”) T.) mg; ‘5” For» ﬁmt Jewivq‘i‘u‘ﬁﬁ. FIDﬁt‘J‘Ia '. “gig AW.“ 2 g» :rgitmg uti‘h" "‘ CranSn‘S‘tﬁ" w+L +t‘mg‘Jev’t‘Uq4ivc Prom/{“5 .— (e) { 7 points ) The signal ﬁt) 2 sincz(t) has Fourier transform FUD) = n A (%). Determine the total energy of this signal. SiamArt barn3 ‘ w I ﬁg [POa)! tin.» ‘ *r(z—L«'?‘;} =a%°=1'°‘i¢m (25 points)
4. Two functions f(t) and g(t) are defined according to the Figure 1. 1'03)
1
t
—00 0 1 2 3 4 5 +00
W)
2
t Figure 1: Deﬁnition of two ﬁmctions f (t) and 90$). (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) :: 92+, 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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