solutionsblank examEE 350 EXAM III 10 November 2011 Last Name (Print): .sold::.; onS First Name (Print): ID number (Last 4 digits): _ Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO --Problem Weight Score --1 I------25 2 25 3 25 4 25 Total 100 --Test Form A INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. This is a closed hook exam. You may use one 8.5" x 11" note sheet. 3. Calculators are not allowed. 4. Solve each part of the problem in the space following the question. Ifyou need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your rea.<;oning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your wo~Problem 1: (25 points) 1. J(1(2)points) ~ig.ure 1 shows a periodic signal J(t). Determine the trigonometric Fourier series representation of t by speclfymg the parameters (ao ' an, bn). f(t} To.::: 2:tT ~ WIQ::: ~ ::=. J rD Figure 1: Periodic signal J(t). Bect<l..v~e. .f(-t:)::. -f(--t:') IS afl oJl.J--{")L~On I .f:,meJ i:J,e.. <A.V"-,"~ .,....I_a. [~o -01 <frtJ)L fin ~OJ f<r f)~I)L,~,--. • f. T", ~o I:+(",).1'/. (nwot) -= ¥:rt-(.f;) SIf){_.-t)£L -;: 0 v~ P.>.7-7 1JV --~ J.2.'!: Sl~ (n -/;).P.b ~ .;,:- [s''ICOt) -n~co5(n-l.) L 232. (13 points) Consider the periodic signal f(t) in Figure 2. As indicated in the figure, the shape of f(t) in the interval -n/4 < t < n/4 is cos(at). J(f) SJr 4 3Jr 4 Jr 4 o cos(at),/ 4 3Jr 4 rr 5Jr 4 Figure 2: Periodic signal f(t). (a) (2 points) Determine the fundamental frequency of the periodic signal f(t). To =11' => tV€) = ~::::. "2.. .,.-i.t~e;... To of (f:.) = fC-t:) (a) (b) (3 points) What is the average value of f(t)? (c) (8 points) Determine the complex exponential Fourier series coefficients Dn for f(t). 4VSlif &.7-b C.O~ "f. C OSJ = "i: COJ(7'~) -4-~ C¢~ (;<~)) Do":: ;n: of f 1~ co>fcIM)<."] J-! () + +Problem 2: (25 Points) 1. (6 points) Consider the following three periodic si n I . gas represented uSlllg an exponential TO" •" rouner senes 100 k JklOtXl(t) ~(D e100 X2(t) 2: cos(kJr) t?k20t k=-lOO 100 X3(t) = 2: Jsin (k1r) eJk30t. k=-IOO 2 (a) (3 points) Which signals, if any, are real-valued? Justify yOU . . r answer III one or two short sentences ')«(:1:.) rec.Q-.,-~ ~ ~~ -(b!n)~ . D~' ::: (1:.)' orJL ~o &, l-t:.) IS not rf2j)-v~ToJ((J~.) (D~()'Z..)~:= (Co~ (-I)1r»)~:= CDSVJ.,,-).= D:~ (b) (3 points) Which sigr I of sentenceso la s, r any, are an even function of time? J ustify your answer in one or two short fl (.0j c..o~ en (1-) an Q-<!n .c....~.["u\ I "J C:<.{) '0 t~·":r;..·I-h,(r'Q. J Cl,,{l roSI" ("'E) ~" () Q~ {...,. ~t'W'\ 62. (10 points) A periodic signal f(t) with fundamental frequency Wo and the exponential Fourier series represen-tation JAfD = -[1 - cos(mr)]n n7T is passed through a linear time-invariant system with frequency response representation 27T WQ < Iwi < 4wo { o otherwise W .". -4WD < w < 4woWo 2"{o otherwise to produce a periodic response y(t). (a) (2 points) What type of filter is the linear time-invariant system? Justify your answer in one or two short sentences. n~ +,Ife.r ~~ bc:a.J.ptAS -h Ikr~ ·Ii. p~re.s +-,.eP"'S!.r-C(e~ J}o~~ be...~"'" lAJo Q.~ '(cPO. (b) (4 points) Specify all nonzero exponential Fourier series coefficients D; of the periodic signal y(t). of.D" ..) ,,~ -UI) -r D~ - D~~ J+ fThQ. C'nl1 (I0.,'i>QA coeff,qe..i& o-I{Q.-::3--3 !>I -(c) (4 points) Determine an expression for the response y(t) using real-valued sinusoids. 'i-0-3 € -c;l3w o 'b '+-f )\N;;)~ + D,) e _--~--~::r -} r (~) =-§f!. CUS('3w,,-l)j~ .3 783. (9 points) A periodic signal f(t) with fundamental period Wo = 10 rad/sec and trigonometric Fourier series representation oc 2 f(t) = L ;;:cos(mr)sin(nwot) n=1 is passed through a linear-time invariant system represented by the frequency response function 5H(Jw) =-JW to produce a periodic response y(t). (a) (4 points) Determine the compact trigonometric Fourier series representation of y(t). 11:.. CO~(il1T) cos(nWo-l: -:IE> :. 0 -to ~~ (Os(nL41...,i: +~ -JE)fC-t:')= 0+ "I- ' 11-::1 :zI n ~ J 11::=1 )r(..hal'~ So.ltJ'1 •fJ •..1 c~H~e ......., "" -1 ,.., 0«"" 7T +-2-ct=-o.) c,,: ;lJ e;f .:;. 1J"n - 11/2-14 &"'"J :: s_ ~rJII-' (b) (5 points) Compose a MATLAB m-file that calculates and plots y(t) using the first 500 harmonics over the time interval -2 S t [sec] S 2 with a time vector that contains 500 uniformly spaced points. 1:: -1,1' S~(-e, (-~ 2J S-OO» ~~ ("0.5 (S I =!. ~ (1:::) ) j n::. 1: 5 00 Cn =-Ij,..,"2j the 0,.. _n -::. (\ ~ pL -pI., j d-=, + e" !Ie CoS Cto ... " 1<--t -t Chetu-. -,,)j et"~ plot ( -i:.J Ji) ~ \CA.bQ. \ (\ ~f"'\pl,L~ () A \(.(.bz.\ (' t-tme,.') 9Problem 3: (25 points) 1. (8 points) Determine b dO tOo , y lrec mtegratlOn, the Fourier transform of the signal J(t) = 2e-1t/[u(t +1) -u(t -1)] ° f(-t) f-ft) 1S ('~ Q.~~ CU'l e.v~ +vnchCll"\ ? -1:. r?1 ~~ p- ( 'V) ~J reSt CJ..VI{l al'1 e0J4" ~L -bv... I <-V 0. 2. Jof 1-t:.)co5 r...,i.)P'" I ::: -t f) 2(Jv 2.. e QoS(wt) dt t -o 'i 1~ c Co" (,.ft.)~.£ o 6.7'0 Jeo..~ co{'I;,'1.)O"J = qe,..:J>L (<<CdJ?o. + b.,,,bll\ P{~ _ -'1 [ ~-f= ( cos ("'" -e) + w sin (wI;)~ (I l+-"'".... o = l:~'" ref r-CO)(.....) .... "-'S." r....)5- t-I +-0J] --------------------------, 102. (9 points) For second-order linear time-invariant system, the input f(t) = e-Ztu(t) results in the zero-state response y(t) = (e-3t - e-4t) U(t). (a) (5 points) Determine the frequency response function representation, H(Jw), of the system and express your answer in the standard form +12-(b) (2 points) Determine the ODE representation of the system and express your answer in the standard form (c) (2 points) Is the system asymptotically stable, marginally stable, or unstable? - - ro o-+'> ()I ~ - J CA. ,.J2 r-V -;. -'1 ~~tn I) oJ ~f? -L--b; c~!t.3. (8 points) Suppose that f(t) is a real-valued function whose Fourier transform is F(Jw). Show that 100 00 .J 1 1 2Ef = f-(t)dt = -/F(Jw) I dw, -00 21r -00 which is Parseval's theorem for Fourier transforms. -::. -t"H:.) ((p SeO -t -ir,.
View Full Document
Unlocking...