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Unformatted text preview: EE 313 Fall 2010 Solution to MW (16195) Exam #2 Indicated below is the regrade policy as stated in the course syllabus. ‘ Since, the Exam # 2 was returned to the class Wednesday, December
ist, no request for regrades will be considered after Wednesday, December 8th. You must clearly indicate in your written statement where you
believe an error was made in grading. Staple your written statement to your exam, put it in an envelope with my name on it and slide it
under the door of my office, ENS 342A. Thefollpmwinsis aneeixserptrfrom the Feurrse Syllabus made in writing within one week of their return to you. No verbal submissions will be
considered. The purpose of a regrade is to correct any error that was made in grading; therefore,
in your written statement you must clearly indicate where you believe an error was made in grading the problem in question. Please note that if, in the instructor’s opinion, a regrade of a problem solution is warranted, the
entire problem solution will be regraded, not just the part in question. Regrade prolicyerllrequests for regrade of a homework or iiiclass exam problem must be 7 7 , 7 ‘I’Wrin 376a; ‘ Page} ofr6v , same” (Last name) (First name) My signature certifies that the work on this exam is strictly my own and that I have not received nor utilized any outside information beyond the allowed formulae sheets. Furthermore, my signature certifies that I have read and understand the instructions given below. i
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i
\ Your Signature THE UNIVERSITY OF TEXAS AT AUSTIN
Department of Electrical and Computer Engineering
EE 313 °MW (16195)  Fall 2009
Exam # 2, November 22, 2010 explanation of your approach. Instructions You must c1rcle your ansWers. 1.
,, , , , , , , , , , , ,,,,,, 2,
All problems count equally. 3.
N 0 credit for answers for which no supporting work is shown, 1
or for which the supporting work is inconsistent with the answer given. 4.
To maximize the'probability of receiving part credit, show
your work and provide a brief (a few phrases will usually do) 5. 1/6 % 7185 115 18. To .receivelcredit for your anSwe . r, you must indicate, on your plot, the numerical vafué of
c[n] for each value of n for 485 115 18. '  , . ’ § hawk LN} f A ‘34be (MAJLX
, +4., 0... .. ., 7 ‘ mam (mtble—‘IIl
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2.l 9 6  L73 45 (5?3‘91 \o F mm; ~xza+sc?s~c‘\ou. ‘Ce¢‘:‘ (*m +13 \ 411.00 @otu‘ﬂ‘ avatr‘urs ICY3'1=\XI+17“= 2. ’IY‘ 4: ﬂ '/ +‘X‘a3‘.
“F‘s CL47=3 , .
\n b ‘q__ +uJﬁ {30?wa 0V2Pla13 d1]: iﬂ‘i‘VK‘SZ‘
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m 300a. " H 3c (K #‘Aliimkﬁ ' "*"¢\’~*‘H43“° “5‘1— “.“ Duff. ‘3, VCLYmel‘a
“9“;  , .__ _ . ,, L V v . , , V '92‘7 F2." L. . V~HM3I~ rm~~~~4~~¥m~~~~~~4 47379~——#~v1\roiGEM+1:a»~Jree'vw\mAg'v3—‘éQn?4q31‘grtDu‘A w~—  ‘ 1(3) ' Problem (2) Please note that parts (a) and (b) are independent of each other.  ' _
Part (a) Consider a series connection of hire LTI CT Systems as indicated belowfwith each being characterized by
their respective transfer functions HMS) and H2{s) also described below. 9 . _ _ p _ s i: 4 . .7
It is desired to replace the series combination by a single LTI CT system characterized by a transfer . ‘
“function H369) and an impulse response h3(t). .. , ~ 1 ’ “ ’ ’ ‘ " ' "‘ ’ ' ' ' ’ ‘ ' ' " ” 32(5) =' . Hx£$3 ~A\‘2(S\ ““9 _ => ‘ “a ‘43(S3 !* 5. > ’ .
Calculate, usin La lace and inverse La laCe transforms the impulse response, ham; for the oyerall' eQUivalent
LTI CT system. _ _ _ f _ . _ : ~
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. 2 m = I W :1 > 2 S " l .2: ELM}
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‘13) S: SL‘HZS + 43:1: {as “(1%{93ewn W W.._..,r.....m. aw
V W iiifiékﬂk3; was. .;i_;_1.—_i:+.s 21.35.32 s...X(.3___..m.).. _. ...__
.. éi ”EH—‘25! If) +453H=3= ‘
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i , 73/6, Problem (3) Suppose the input—output relationship of a particular LTl CT system is described by the following
differential equation: ‘ Jﬂ/(f) +22/Wf) 4.720”) =2Xﬂ)
The initial conditions are y(0)— ‘2, MUM)  1.9. The input X()' Is given by X()‘‘ U() I L." I I ' V Llsing Laplace transforms, calculate each of the followmg (a) the zero— —input response as a function of time, and _ , , (b) the zero—state response as a function of time. (To receive credit, you must use Laplace transforms to solve
I . parts (a) and (b)). i i Lia) ‘2E—NPL‘w‘{"w+ masI'IP‘AFL 39l‘0"."‘4I°u{=> RH$1° I SLiIS)—5 3L um *‘éfl’lv\ +7.2. Cs EZM)..:‘{_tcl‘\z 4. (1‘ 133:0 I ' ¥l83C5L+22s +tll1~ 5(2)  (“WA “FINE23 :0
sexism». +m1 ‘ 1‘3 ”I ““4““th ‘LCSZ CS3} 225+ \‘Mj v2.37» 7.5 =19 ‘ ‘  . ~ ..L..._. 4. 7. 4—» I ~ , 1&3} m 2 zg'vnsw “7:: *ﬁjﬂv‘ i1“... '13...) 1:: 3LS+USL L939“) { I Pwu I441; 5““)? , LeiII)?” LET”  _ , 7 7 . . ....  S“  V , v . 3 ~ ‘ ‘ , 'ZS‘LV_ A. §__ + B 3,; 3 ,
 . 3 , ‘ A. 1 L135“, 2. ~LtII3+zs 3 //,XRHsS’ElFﬂM‘LIVIVSVIftkfu Ia+S—>00 3 s=—a ,, 53.. s __ ‘1‘ ‘ ‘ r V w...“ .. H , ___, 0 +4}, => B=L LE) 27.2.m—slmdra Vteépuuu’AL ' H ,. .V . 1  " .
’ ‘ .3 .1. xu\'=““
‘ SLIZSB+2zsilS)+\1I21N3=Zé' xtﬂ w" 5‘ , . I .. _ 43 .  A \3 <2.
: W9 l =~‘’="' k = . = —— ... AAA—m
, )Ls +113+I1l =§ .LIs‘; , ‘1. ‘ S +LSIH‘3'2. (3MB
, Etsmx , ~ A: 2. gtsIII31—X: LIIP 7% 15: —Z‘————\ —— 3.
‘5‘“: ' ' " _' SlLSIIOLI Hf... " s=~II
XLIHé _ ,.
RIJIsbIas 4rk.“\z(~£§o¢: 13. —‘A.§_.+B—§‘\*C§‘ Problem (4) Please note that parts (a) and (b) are independent cf each other.
Part (a) Suppose the inputoutput relationship of a particular LTl DT system is described by the following difference equation .y[n+2]:(2,/5)y[n+1].+ (1/25)y[n].%_9x[n],. wherey[1] = 25, y[2] = 175, and x[n] = (1/6)"u[n] Calculate the zeroinput response yo[n]. " ‘ ‘Pégé'56f‘6 < ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f~~~_~~~~~~~~~~~~~~~~~~~~~~~~~~I~~~~ Part (b) Solve the following difference equation recursively for the ﬁrst four terms (y[U], y/ 1] y[2] and y[3j) subject " to the folloWing initial conditions, y[— 1]= 2, y[2]= 1, and when x[n]= (0+ DUI/7] y[n+2] + 2y[n+1] +y[n] =x[n] .‘gLoLY‘jF. C\:§.3V1:5:CLV\L+5}:~ 13.1435. 7 ’L. § , "‘¢,“E>’.i55..,‘j3W1}j: 9—2 “up: 9(th +' me‘ia' Mm=rl ‘éial :~7%t—L*"3 iLliv/ﬁ ’ _;_(,,x—i= —+\=J_‘3 l ll' M55} 24:»: 4.M—2 2m ELLDY+ XL”? whim‘93 x {331  va ans) MM} 2> (lemu‘gndll (19943 K; Y; :2
, . ' j , V1391: (oll) “[02: \X\\  l u.— ‘3 xC»L'L=(__1+t:}iZ;—1 ”‘9 ' .. "act2:1 ,.~_~ ——v—ca~~~¢~s~>+<~~=~»~:—~*Aé'+3+4: —'~° .: ‘ , . .,, . , . , 7 V . .. 7
“ 313.1 1” . — 2 #L21*%£'1+ x131 “ XQ1(Z " 7‘
4 : 2°~‘3’”" ‘4’ ' " ;  . 5/6}. .. : ' Pagea‘of‘s' “ ' " Problem (5) Using tables for the unilateral Laplace transforms and properties of the unilateral Laplace transform,
determine the Laplace transforms of the following tWO functions of time (To receive credit, you must brieﬂy
, explain your approach and you must simplify your answer.) , . ' I (a) my = e6fcqsﬁr(f8))u{z‘8) , ' ' (b) yﬂ) = 93(f2)u(2‘2) Veal/(f) (The asterisk denotes convolution.) (0‘) :54:— , _'e\s__I,(J:—&3 _ , ..
, iLXLHze u IJ:— —83lule 8415.1: 6 a,” y mmz:—2\,\ua:— ‘33 i
1 ~48’ ‘éCtQ) ’ __4.g: “<93 “(at .1
: ixe‘ (49.901 (A: .m inf—H: %\ =6 6. i6 Twunlrtuuzx
1“ _. ‘—o(,"L‘ ﬁi—ot “=1: 13:"
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Ute e.“ ‘ms l3. (ginatsmﬁ—ﬁl, ,. ‘ yTHE'END—i "1:616 ..V ...
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