final_s97 - EE 3973 FINAL EXAM 6 May 1997 Name Sea iJ‘Lie...

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Unformatted text preview: EE 3973 FINAL EXAM 6 May 1997 Name: Sea iJ‘Lie as ID#: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Score - - = - 100 --20 --2o —-20 --20 --2o - This test consists of five problems. Answer each problem on the exam itself; if you use additional paper, repeat the identifying information above, and staple it to the rest of your exam when you hand it in. 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. Problem 1: (20 points) The electrical network in Fig. 1 has an input :c(t) and an output y(t). xfl) Fig. 1. Electrical network with input voltage 2(t) and output voltage y(t). (i) (3 points) Find the transfer function Y(s)/X(s). (ii) (2 points) From the transfer function, obtain an ordinary differential equation that relates the input 3U) to the output 110:). (iii) (7 points) Suppose that :c(t) = 0 for all t > 0. The initial capacitor voltage and inductor current are y(0) = l V and iL(0) :2 11 mA respectively. Solve for the output voltage y(t), for t > 0, using the Laplace transform method. (iv) (8 points) Determine the frequency response function H ( 1w) and sketch its Bode magnitude and phase plots using the graphs provided in Fig. 2 on page 5. In the magnitude plot, clearly specify the dB level at each corner frequency and the slope (in dB / decade) of each straight line segment. In the phase plot, clearly label the phase at each corner frequency and the slope (degrees/ decade) of each straight line segment. Y0 3i?“ s R m x—‘(SD -- 453+9~ a sac. +1 _ R 51,-!- it“ 5!. + .L stun. 1-31.. ‘14; JPc-H 32+ R loco i000 M 32'4" Nos 4- loan (8+ 10361-109) (it) 51 Rs) + no 31’“) + 1000 YCS): nomxm " ' = e xL-e.) Ltd + no (-9 + [0:30 (4-,) i co (“4) With )L (t) = O) (J ii'gt—m 1- Heath) klOOoai-h)1 :0 5?‘ Yes} - Sofia) ~30» + no 5 rcs) 4:05.09) + 1000 v0) =0 ((5)151 + ”05 1- love} ‘1— 5 alto) 4- 3(0} *ley(o) w “W =m4=mm we. Y5) =1 deal vJv) «1103(9). (8 1-10) (Sq-led) Fran- the prolo‘em a'EuJLeme/y‘ll: a £0) = IV- PM Fig. l. (L (03 z: LCCO) + (41(0) me) = 93““ an») R. . IV (07:.- J— tr~"° = L“ 1:0 (.1— a“ alt-Qgr “f” air“- USHg‘ 3.05) = O nan-4L a (0) 2.] a\€«[(05 ch) _—_ ——t—-——-‘ "° = A 4' ”6"" (s +lo)( s-Hoo) 5+": 5+ we __ (8+Ilo) -== __L°.9.. —- 4°— .. (S B. Magousuaa LIN” 70 ‘7 1 J... 6: (syn) ————————1(S+"°) == ’ ‘7: .. ' 7 aye-m (517(96- 5= -I°° Slate. ... .19.... —[—- -- 'L“ I {(5) —- 7 5+“, 9 .S-Huo A: on. ckeJ‘J no‘LQ l'l'.‘ —\-oo’\'.‘ thy;- we): [1% 6' ° * "7" e J‘U’e) 3m :— 1 can (5+no)(s+ loo) _l (4:? +1) ( {a H) mazes . lo (.ch 3.5 .— — I ._.—_————.-—p—\ -f’—°-+a)(;§.:+l) |H(jm)l (dB) Cl) (log scale) __ d B H 0 deg Z HUm) (degrees) 180 90 a) (log scale) «ht . - .. {-fiun . fl wsv/fl-a I ‘ 70712. Fig. 2. Bode magnitude and phase plots of the network frequency response function. Problem 2: (20 points) (i) (10 points) Consider a linear time invariant system whose impulse response h(t) is W) = 2[u(t+1)~ nu — 1)]. Find the response y(t) of the system to the input :1:(t) = u(t — 1) — u(t — 2) using convolution. (ii) (10 points) In response to the input z(t) = e“u(t), a. causal linear time invariant system produces the output signal y(t) = te-ztufl) + e—“um. Assuming that the system is initially relaxed, compute the impulse response function h(t) of the system. a) h (1.») S< It) #m = Problem 3: (20 points) Consider the signal w(t) which is periodic for t 2 0. 2(t) = :(wl)nu(t— 1— 271.). (i) (5 points) Sketch the signal z(t) and find its period T . (ii) (5 points) Find the Laplace transform of 2:(t). If possible, place your answer in closed form. The following identity may be of use °° 1 Z a." = l , lal < 1. n=0 - a Now consider the signal y(t) = Z (—1)“u(t — 1 -— 211). (iii) (5 points) Compute the Fourier series coefficients Yfl of the periodic signal y(t). (iv) (5 points) Compute the Fourier transform of the periodic signal y(t). (‘0 sins) = «(+4) —— path-33 + th-s) - (LL-bu}, + .- . mm: gm" tam-mm no 0 - +2-n 5 —S -25 - °° " 4.. u 3 s... 200"?- ) '" EC") 5 e —.-.- _5 you "=9 lime}: 3:... Z “a 7‘“ we Problem 4: (20 points) Consider the causal linear time invariant system represented by the block diagram shown in Fig. 3. Y(s) Fig. 3. Linear time invariant system with input ::(t) and output y(t). (i) (7 points) Using the block diagram, find a. differential equation relating the input z(t) and output y(t). { Hint: Use the fact that the output of the summer, which is equal to y"(t), can be expressed as a. function of 2(t) and y(t).} (ii) (3 points) Find the transfer function H(s) = Y(s)/X(s). (iii) (2 points) Determine whether or not the system is bounded-input bounded-output stable. (iv) (8 points) In order to change the transient response of the system the following feedback law is used EU) = TU) ~ ayfi), where a is the feedback gain and r(t) is an external input signal that represents the desired value of y(t). Find the range of values of a for which the system described by the transfer function Y(s)/R(s) is bounded-input bounded-output stable. ('1!) sins): xfs) - 2511(5) adv/(S) (LL) [31 + 2.3 *33‘1’63 = 9%) = xfie‘) - 25 (4.) +8595) "9 (9+2; 1 (iii) Since 44(5) -'- CS-l-HXJ'Z) haul. a. pale af: 5-“— ’Une s rhea is m4: 131-30 3—ng Problem 5: (20 points) The system shown in Fig. 4 has one input 2(t) and tw0 outputs y(t) and z(t). x(t) cos(5 (not) cos(3 mot) Fig. 4. Partial block diagram of a communications receiver. The Fourier transform of the input z(t) and the filters represented by HIUw) and 1120:») are shown in Fig. 5. X0 (0) H10 0)) HzU 0)) o) a) _5% .3m0 3mo 5% -3coo 3% Fig. 5. Fourier transforms of the input signal 2(t) and filters. (i) (10 points) Determine and sketch the Fourier transform of y(t). (ii) (10 points) Determine and sketch the Fourier transform of z(t). 15 (a) Li 3MB r. x05) CoSCS‘wot) b We». ’\\’ (dub 2‘ ff {“4935 (“9‘58 3 szgwyfwo) +— 2: Xé/wym .. [1L ( gag/31d 0”) cued ob‘éoub Sung 2500 .. 4/ a“ J 17...
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