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Exam1SP2011Soln

Exam1SP2011Soln - SQLOTeow EE301 Signals and Systems...

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Unformatted text preview: SQLOTeow EE301 Signals and Systems In-Class Exam Exam 1 Tuesday, Feb. 22, 2011 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 1,2 Open Book but Closed Notes. One 8.5 in. X 11 in. crib sheet Calculators NOT allowed. This test contains two problems. All work should be done on the sheets provided. You must show all work for each problem to receive full credit. Plot your answers on the graphs provided. Prob. No. Topic(s) Points 1. Continuous Time Signals and System Properties and CT Fourier Series 50° S7 “S. 2. Discrete Time Signals and System Properties 50' 4S 5:3; Problem 1. [5’0 points] Consider two LTI systems connected in parallel (as in Fig. 2.23 (a) on text page 105) , where each of the two systems in parallel are respectively characterized by the following input—output relationships: System 1: y1(t) : 2ftt_4 :1:(T)dT System 2: y2(t) 2 L574 :1:(T)dT As is the case with systems in parallel, the two systems have :v(t) as a common input and their i‘esective outputs are summed to yield the overall output y(t) : y1(t) + y2(t). 'S (a) Determine and plot the impulse response of System 17 denoted h1(t). m" (b) Apply a test to System 1’s impulse response, h1(t), to determine if System 1 is stable or not. V (0) Determine and write a closed-form expression for the output of System 1, y1(t), when l O the input to this system is the exponential si nal :13 t : e‘2tu t . g (d) Now, determine and plot the impulse response of the OVERALL system, denoted h(t). I 0 Plot in the indicated spot on the sheets attached and show as much detail as possible. . (e) Determine and plot the output y(t) when the input to the overall system is the rect- / ‘ angular pulse: :v(t) = rect (%) = u(t) — u(t — 2). (f) Determine and plot the output y(t) when the input to the overall system is the rect- 5 angular pulse: :v(t) 2 rect ($) : u(t — 2) — u(t — 4). (g) Determine and plot the output y(t) when the input to the overall system is the rect— l O angular pulse: :v(t) = 2rect (%) = 2{u(t) — u(t — 4)}. Hint: You should be able to use your answers to the last two parts, (e) and (f). Problem 2. [45 points] (a) "I (D W 4% ”L "— g: 4%113 “(j 7< g “LIZ/C) Consider a system whose impulse response is h[n] Mn] u[n — 4]. Determine and n plot the output yjn] when the input is xjn] = 8 (é) {u[n] — u[n — 4]} Do a stem—plot in the space provided on the sheets attached. An signal Mn] is a sum of two DT sinewaves with frequencies 37r / 8 and 77r/8, respec— tively. . . Mn] 2 367?” + 267?“ (1) Consider this signal as the input to each of the four systems described below. System 1: y[n] : m[n] + (—1)”_1x[n — 1] (2) System 2: y[n] : ( —j)“m[n] (3) System 3: y[n] = m[n]:r[n — 1] (4) System 4: y[n] = m*[—n] (5) W For EACH of the four systems above, you must answer EACH of the following THREE questions in the Table provided in the sheets attached. NOTE: you do not have to determine the numerical values of any multiplicative scalars in the output ~ just determine what are the frequencies of the complex sinewaves present in y[n]. (i) Is the system linear? Yes or No (don’t need to justify your answer.) (ii) Is the system time—invariant? Yes or No (don’t need to justify your answer.) (iii) Determine the frequencies in the output y[n] of each system given the input in Equation 1 above. Each answer should be in the range [—7T, 7T]. Plot your answer to Problem 1 (a) here. M Na) . Show your work and write your answers to Problem 1, parts (b) and (c) on this page. 00 ' 7 ~— (t)*m(t“4} my) EWNQWC 2 00 “may ) , ' 0° €ys¥em ’8. @ 0° \kJC —_ 4 (ft) 1. 8 < 60 1'; ‘ 9mm Sh“ (“k3 le'ZtM(§\¥ If OLCJYA _, u(t-4§ ?\ : (26°Hu(t)>aruct§ — (29' at“(fl)*“k‘9 -11: ~16 “'26 a“) "18 (new (ACE-4) 34mm) r—WM er Exmw 2 6 6‘“: «(£3 “£09 \AKJEB 1 \r\\('€\ + L‘ICk\ \1\ Qt}: (A k'E"4>”‘(/L(£"7> 1 Plot your answer to Problem 1 (d) here. 496.13%“? Plot your answer to Problem 1 (e) here. \(‘0 '\ :7 awsmer Jro (Eb gk‘msfek "To Urgb‘a— ha“ 1" Plot your answer to Problem 1 (f) here. u m ~w m- «d 59) «02): 1(760 5r3<4;(£3 wker', ’X €(t‘) is \'v\pq‘5' flag“ (6") V; K‘EB )l$ "KPM+ "Por (75-) ’r‘rws: Ag (H: Z &92(‘c> + 13+(03 ((2.) an; (+1») a“; $iVS*>\ek‘9 Ad Age (tymgflt) 9(a) Plot your answer to Problem 1 (g) here. A—L—L—L—L—L—L N—LO—LNOO-PU‘IO‘JNCDCOO—LNCD-PO‘IO‘J _—2—1'o1léé4ééféé1lo11 1:6 14 13‘ 12* 11‘ 10‘ <0 I ANCH—Pmmflm \\\\\>I\ Plot your answer to Problem 2 (a) here. Ch) (‘7 10 Show your work and write your answers to Problem 2, part (b) on this page. QYslrevw Ll Lhaew/ =5 5° clu Oack €4than 5epan’wka'. _ 3%“ near <3w 8w W 633,6”.3‘56 .) 2%.“. '3‘?!“ 3?(V\-)) .3- ‘5 J . 21:- “I “ J a J Te“ ~ 61 Q +- , + c><L 6 Jr as Q HDTV 5T 8““ T!" “‘9—5— —:"3_‘_": Same ’1'” 8T’~__ System Linear? Time—Invariant? Frequencies in y[n] in range [—7r,7r] 1 “165742” No “'8 >‘S‘VB CW8 )‘“{8 g 2 in“, No ~11 37mg 5' 3 No 'L ”L ‘(es ...3w14j-3rr/q “‘14 s 4 Numb No 3w18> WM?) ‘3 , Ewan SYshm 4 how” "r ”((9:5de -5 (“Wagon 3mm 9 Mr- o<* E ”(e =>FVQQKQV~CC€S MDCKQWCQA $V55'9M I'} no! \(AQAM 8‘48 % homOiQWO‘.B ’09; n5 V)'o\&\-eé 1-) Supo Vpo§3171f0& Ski“ [fields ...
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