examIII_s02

examIII_s02 - EE - 350 Exam 3 1 April 2002 Last Name Sol...

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Unformatted text preview: EE - 350 Exam 3 1 April 2002 Last Name Sol V'Elg 05 First Name Student # Section DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO so Problem 1 2 3 Total .EI Test Form B Instructions You have 2 hours to complete the exam. Calculators are not allowed. This exam is closed book. You may use one 8.5" x 11" note sheet. Solve each part of the problem in the space following the question. If you 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. If you introduce a voltage or current in the analysis of a circuit, you must clearly label the new variable in the circuit diagram and indicate the voltage polarity or current direction. If you fail to clearly define the voltages and currents used in your analysis, you will receive ZERO credit. 7. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. PWNI" Problem 1 (25 points) Define g(t) =eM [u(t+1)—u(t—1)] and let f(t)= 2 30-2")- "3—90 1.1) (5 points) Determine any existing symmetry of the signal f(t) that implies that any trigonometric Fourier series coefficients are zero. 1.2) (10 points) Compute the trigonometric Fourier series coefficients of the signal f(t). 1:" even 19’ 2 0.0“ t;— zflfizflfi “r... 3.28 {431,5 31322 _% Too 1 I \ '1: «a: gem..- e-n O 1'2. 'Ht) Ll OQ even an :3 $(QCOMWJEOQA; 2:: fSO-FCQCosnwd: {2 Us”? Taub‘Q 337—7 'E: qn 3 ii...— (Cosn'lT-l: + MTSM Wt '2. ., l 1- (011‘) O (-0” O 7' 1.. (anTr-v‘nn/‘Jznnfl- “+01 “(on)” . o n a 2., H, 6,... 1.3) (10 points) Consider a periodic signal x(t) that has the trigonometric Fourier series x(t) = a0 +21%l cos(n%t). X n=l Suppose fit? is passed through a causal low — pass filter with impulse response h(t) = 3e'2'u (I). Find the compact trigonometric Fourier coefficients of the (zero — state) output signal, y(t), in temis of the trigonometric Fourier series coefficients of x(t) and the frequency response function of the filter. 710“ crave“: ares/Jame 'Flnc'Li'O/I “(ft-Q 3! the- C'HSGV‘ 7‘ the. Fayr\.¢r transgrm # impvlfl (arc/5e £45498le th), 03m Table. |-l.| a -m— .__3___. “(a-W3 = if {36. “('93 = 2 «lye» .___3._._— 21 “(W5 2 — Tan" m male—sent. mpg/m to the be compofinoniémoi fl the mp3: me) is ac [Waléww = %QO. S‘kebl‘k— s‘L—uii Pgspgm to “0 CO3 (00.00%) ; s an Coanwot) “(dub an! H(+nwo’)| (as (I)th 4- 4 H ldnwo» = .._..——-3“~ mom — w" $95)) V“! +(flwo " 1‘": ‘C‘ o“w5 vi Problem 2 (25 points) 2.1) (7 points) Consider a periodic signal f(t) where f(t)=e’2’, —1$t<1 and is found elsewhere by periodic extension. Determine the complex exponential Fourier series of f(t). Tye math CH» 7: Ital-Hue» even 0” 0001, 13:7. -—--> "‘"°= "‘—“§= “7" 2.2) (8 points) Consider a periodic signal f (t) = Z D.e’"”"’ where Dn as a function of n is shown in the figure below. Determine whether the signal f(t) is even, odd or neither and whether f(t) is real valued, imaginary valued or complex valued. Justify your answer using the symmetry properties of the Fourier series. Dn -1 ' Observe. .thmt (I) by, Is {tomb mag anno- C23 Dn =-b-n is 0‘" oolOQ— cmp‘blw} % fl. ' 95 L;- 4: I2. 7:. " “W9 3- 2' bn = J“ Wm it = —‘- 4cocosnwgcJ+s - ‘—‘—~ flosmmfifi TO To ; To -7. . ‘2‘ DI— ‘a. v . 1. a . Bacwose ha is an «UL ‘rvflc‘klufl % n} the, firs-l: theft vanish, amQ. So 'Fl'a V'WS‘fi be. an oflj. 4:.deth ‘élma. Becwusq, Do is pure—9L TORQ'J 'FL‘E) Maj-l: be we Inns”? so that ff”) 35 he -va.alvee£. ’ 2.3) (10 points) A real periodic signal is applied to an ideal high pass filter with the frequency response function H(w)={1, lw|21.57r 0, otherwise. Given that the power of f(t) is Pf = g , determine the power of the filter’s output y(t). ‘ From the, axpoflaflé Lag FM\€Y‘ [e vies perks-20 nwo :1 run ) anoQ. So (Do:- 11". o kcwun the ‘cmbu-EflCy. % fikQ 35 L5" 11', the. erter sear-um ComeflQfl‘AS a. 41*) a’é n: vi) 0 \ our-Q. y the. Qléer, 1L"; (:oi‘owi note. «Lth the cum/1’th and pit“!- n: #92. [tab a mum Man» as" Im=uw=° m the. (liberpoquuMQ, ' U§m¢ paneud‘? Wynn P—F- = Z nan-O [1: —09 D: 0th? rwl'SC) gs E o n =-1,0, I Problem 3 (25 points) 3.1) (13 points) Consider the signal f<r)=jrect[;). a) (6 points) Determine, without computation of the spectra, whether F(0)) is even function, odd function, or neither and whether F( (0) is real valued, imaginary valued, or complex valued. Justify your answer using the properties of the Fourier transform. b) (7 points) Compute the spectrum of f(t) using direct integration of the Fourier transform integral. (a) .FGQ Fit) "5 f‘reia. Madmen; anti qn . Qvnn aniswn fi/ 4: «.5 ‘F’l-b3= ¥C"+) ‘ BCCAUSQ. ‘5 4n even Quickly“ % flung the $6le MiG/rag) f':f'(t3$mw‘kJ-&J Vantskei. Eeccwsg +L~kb is pure} (Imaginary) “’50 PAH) ‘5 15 Uan‘ I au 0 p I I43 red: 6? CSmC ,t 'Cououfi a 2/ SMCW 3.2) (7 points) Given cos(a)0t) <—> 7r[6(a)— 04,)+ 6(w+ 6%)] , use the time shift property of the Fourier transform to determine 3 {cos (wot + 6)} . Simplify your expression. 1 Let 301:): (oS(%t)) qni abSevve, £110.14 cos[w.t +6] = Cos[‘*b(t+§;)] = a(*=+ gt.) . Usiy the ewe s)“ng property («Cy-om Tabb ~11) W'Lo a_(-l:+-h5 e—a Sofie?" 3 to= G/wa 3.3) (5 points) Given f (t) <—-> F (m) , show that dfdgt) (—) ij '9‘bar‘l: UH'U Itkq, gig-Cm, m % thQ, mver’se. FbvnQr- framwcorm 00 «A; 1cm: j; FMJ Jul ' bt££eren£tw£fi£ bo'bh 9.0005 .44th raved»; '50 fiume °° 4: (H: 3 I 8 0w! PM ed‘W in» 37‘: ET? ,w W4 n a, u 'b -@ 95 gas-3 é—é scab, ,t $6M“; we éif a éiw PCW) art 10 Problem 4 (25 points) Consider a real-valued signal f(t), with Fourier transform F( (0), that is band — limited as shown in the figure below. F(w) A "Dr 0% 4.1) (5 points) Find and sketch the Fourier transform of the modulated signal m(t) = f (t)cos(a)ct) for me >> 20);. Q) { m («:33 = 3'; (KNEW 3’ {cos wc'ir] .- ‘ 51} Flu) a: [If Hurt-MA +17 SCw—waj Mtuh = J; F(w+w°) + "z‘ F-(w— wc.) 11 Assume that m(t) = f (t)cos(a)ct) from part 1 is used as the input to the system shown below where cos (coat-F6) 4.2) (8 points) Assume that 9 = 0, sketch 0(0)), the spectrum of g(t), and find an expression for y(t) in terms of f(t). xcm (76+) = x (+3 C05 (wot) .. ,L, 4% % é: 6cm - szwm + ,_ m» ) —w¢, '00; We M. “’ (m3 a,qu cm. W4: m) +w$ u»; 50*” Ym = “Lth 6.0») :1. EH: L: A]? -W to; w W '2“ 13'; ""4 “4 «5‘33; 2”“ no.5. we 12 4.3) (6 points) Assume that 6? %, determine the phase of G(co) in terms of the spectrum F(c0). 30:) = Me) (.05 (wt: + 1,5) = —x(-t> sin (Wat) {Eye} = “ a}? fixmlx JisiMwLflB : g; mm a‘ 0:117‘E8Cw-ch) —~ sow—web] 2' XCw— 000) A j: X (W + Wc.) 6 (M M/v 13 4.4) (6 points) Assume 9 = 0 and Find the phase of Y(co). Fv‘om Part Ll-3 #210060») ‘1an 50 2Hwa s 2wth + 2mm 0 ); gt“) :- 0 199.60wa Gav) 2C) ‘C‘IY‘ 992 W, 1)- H’L‘W3 7- 30%- o lW\ >Wc. 14 ...
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examIII_s02 - EE - 350 Exam 3 1 April 2002 Last Name Sol...

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