MT2SOL - ECE 301 Midterm#2 6:30—7:30pm Thursday LYNN 1136...

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Unformatted text preview: ECE 301, Midterm #2 6:30—7:30pm Thursday, February 25, 2010, LYNN 1136, . Enter your name, student ID number, e—mail address, and signature in the space provided on this page, NOW! . This is a closed book exam. . This exam contains multiple choice questions and work—out questions. For multiple choice questions, there is no need to justify your answers. You have one hour to complete it. The students are suggested not spending too much time on a single question, and working on those that you know how to solve. . There are 11 pages in the exam booklet. Use the back of each page for rough work. . Neither calculators nor help sheets are allowed. Name: Student ID: E—mail: Signature: Question I: [35%, Work—out question] Consider two LTI systems with impulse responses being h1(t) = 6(t ~ 2) and 1 fi—1§t<0 } h2(t)= e-t ifogt , (1) 0 otherwise respectively. Answer the following questions. 1. [10%, Outcomes 2 and 3] Consider the following parallel combination of the two systems: i . :1 1 [\{L U) ' w... 1 Plot the overall impulse response hpmuefit) for the range t = ~2 to 4. 2. [10%, Outcomes 2 and 3] If I combine the two systems serially: Mai his) man its) a“; what is the overall impulse response hseriadt)? Plot hserial(t) for the range t = —2 to 4. 3. [15%, Outcomes 2 and 3] For an input $05) = etM(—t), compute the output y(t) of the serially concatenated system. If you do not know the answer for the serially concatenated joint system, you can compute the output 3/205) when passing through a single LTI system with impulse response hg(t). You can still get 12 pts if your answer is correct. 1, AMI/4(6) = LIMHHIt/ A t 1"0‘13‘)’ 1. [um (I?) = Axe) Half) : (376-1) fluid ; sz/ iét<2 I : (8&4) t 22 0 w L l: ‘1‘10/L37 3. W) = et‘m-t) [/e) : 453,“ (6/ 4’ KM = j” W) Anny/162w»: -o¢ Cue 1 t‘l(0, 6(l H H H H H A < 0 2' SA? t4 év/ tv—Z - ~zi~z hi 2’ (t / z + ' I) L c e 0/; £26 C J? («I Question 2: [20%, Work—out question, Outcome 4] ,, 1. [10%] Consider a continuous—time signal 3t 90(15) 2 cos( 3—?) +sin< Find its Fourier series representation 2. [10%] Consider a discrete—time signal mm] = cos (375%) +3111 < Find its Fourier series representation. .1 :1 T’:‘I/'!Y p'_£; 7' , - mew -'(%é+’f/ + -r 0 Mr): luau ‘) +0) ) 1M6— JT '1” -'J‘-’ '1 ~3e 2 ‘é‘cdlddz +Eyedzer() .1 - , fl arm , J. #25112: W +353 (1 J6 d W a : Z k 25. U4 2 i (32517 sz zJ6: l,-L2’ #:40 ._. J} 1J6 0w ”:5 Vfiz"? 19’ L117 ‘I -- h ’ w" __.2 _L J 5” ‘_ l): X[h:f€JJ+e)’¢Efl) +2.;(C e / Jr 24?." .1 J—mn - MM" "il[\)5(’hf'CJ‘r-[l))+zj(€s C _I .L :I in; k" “u k 2. J : 4—1. k=7 4k: L4 k=‘/ ’~ ‘3 7- 1*) 0w 0 Wn/ 0 27f 5t —— +3 ) . [56+ 5?) ~ Question 3: [25%, Work—out question] Consider the following continuous time signal 33(t) with period 4: Let (uk, to, = 2—1) denote the Fourier series representation of x(2€) 1. [5%, Outcome 4] What is the value of a0? 2. [5%, Outcome 4] What is the value of 2:: 00 (1k? Hint: Use the synthesis formula. 3. [5%, Outcome 4] What is the value of 221—00 ak(~1)"‘? Hint: (—1)’“ 2 e77”. 4. [5%, Outcome 4] What is the value of 2:: [ak[2? —OO 5. [5%, Outcome 4] Consider another signal y(t) that has period 2: Let (bk,wy == 221') denote the Fourier series representation of y(t). Write down the relationship between bk and (1],. l -%,(—-{+z) = 7/ 3 2 Z X&0):k%. Qk : 1‘ 3 «1/0: 4%: tr 1W .‘ ,L ‘7 "Jt + “L (”2“ _ 711(“0 7; .- - = V . 1 Lfl‘ e "’ Jar/4am) 3 ’53“ /U.)=Z “(c ‘2’ “kc 3’ kZ‘od [(3‘05 W LR ‘slr[£,a(e*+zt+l)«"‘ + Jo, W] (it3+t1+t)lf + t/OIJ Question 4: [20%, Multiple Choices] Consider the following two systems, System 1 and System 2: System 1: y(t) = / :t(s)2t_SI/{(t — s)ds (4) x[n — 2] if n is odd System 2: y[n] = a:[n/ 2] if n is a multiple of 4 (5) 1 otherwise (n is even but not a multiple of 4) 1. [5%, Outcome 1] Are Systems 1 and 2 memoryless? 2. [5%7 Outcome 1] Are Systems 1 and 2 causal? 3. [5%, Outcome 1] Are Systems 1 and 2 stable? 4. [5%, Outcome 1] Are Systems 1 and 2 linear? 5. [5%, Outcome 1] Are Systems 1 and 2 time—invariant? 5pm“ 1: y/é) = Marthe) Alt): Ztiilé) (V07 A€fl1¢7,(cy; A/OT Ant/“(17 [EU Cawh! NUT cqma/ NOT :hl/e «S'fa K/e ltlfled/ ”67‘ “near +l‘m1— ianrl‘flhl' {Vdr fi'fiQ ’(‘any‘Iu/gf ...
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