EE20-2005-10-05-Midterm-01-SOL

EE20-2005-10-05-Midterm-01-SOL - EECS ZDN: Structure and...

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Unformatted text preview: EECS ZDN: Structure and Interpretation of Signals and Systems Department of Electrical Engineering and Computer Sciences DPCL'RL—T— MIUI'ERM l 5 October 2005 UNIVERSITY OF CALIFORNIA BERKELEY Lab Time” - (10 Points) Print your name and lab time in legible, block lettering in the appropriate spaces provided above AND on the back of the last page. This midterm should take you tip to 80 minutes to templete. You will be given at least 30 minutes—mp to a maximum of 90 minutes—to work on the midterm. Aside from a two-sided 8.5 x 11-inch formula sheet for your exclusive use, you may not access, or cause to be used or accessed, any reference in print Or electronic form at any time during the quiz. C‘omputingr communication, and other electronic devices (except dedicated timekeepers] must be turned off. Noncompliance with these or other instructions from the teaching staff— including, for exoiripie, commencing work prematurely or continuing beyond the announced stop time—45 a serious violation of the Code of Student Conduct. The midterm printout consists of pages numbered 1 through 10. When you are prompted by the teaching staff to begin work, verify that your copy of the midterm is free of printing anomalies and contains all of the ten numbered pages. If you find a detect in your copy. notify the staff immediately. Please write neatly and iegibly, because for can? read it, we can't gmdr it. For each problem, limit your work to the space provided specifically for that problem. No other {Mark will be considered in mirroring your midterm. No excep- Hons. a NO PROBLEM ON THIS MIDTERM REQUIRES AN EXPLANATION. Feel free to use for scratch work the blank spaces provided herein. Note, however. that only your final answers are looked at in evaluating your work. 0 We hope you wrestle and beat this midterm to the ground! Problem 1 {42 Points) Consider the set X = [{a,b,c} -r {1,2,3}J . (a) (14 Points) How many elements are in the set X? Explicitly specify one element in X. 3 3 =17 {EX 321'. 1361):} ,fi‘oF—l, ‘Hfiel _ {(13) (1.4 Points) Of the elements in X, how many define, respectively, a graph of'an "on " function f : {a, b, c} —r {1, 2,3}? Explicitly specify one ele- ment in X that defines a graph of an onto function. Eur-fa ti‘CMm-fifl {Ill-2,5: mwd’ lot “mappevl m5 CM linen {1" 051.1; are. unwed To listnti’dmmfg, In 9,9,5}. 1E5 mung fight. MB 3 chokes 'For one, tltmilm indigo}, tin civic!» for nm‘l‘figr 042.] A411. on: oLoILB {1" fit [*3T0": '- it +311»! a‘f’ six sluice.)- Emmi-la: {:6 X, 15-11 16604, H5)“- 1, fit) : 3. “11:15 ‘hwfim I5 tad—0- "(c) (14 Points) Of the elements in X, how many define, 1135pnectimrely'I a graph of. a anew-one function f : {(1, b, c} -+ {1, 2, 3}? Explicitly specify one element in X that defines a graph of a oneto-one function. Eur ‘F'EK To In. onc~Tfi-Mcl c-LoL glamm+fn inning} mtg-(“Lb Mkf’rUl T3 a. Agsfiagj’ekmoi’h 9:1, mimic-5 «Wm fit “jg-1c and? {flat-1‘qu 1-5) C; Oat—heoae {cm m [96; oftflfit. onx. a: ¥@]:I,f(b)=;,f(o)=3 2 hrng (P:Q)A(P=>—IQ). We do not know whether Q is truefitm‘ayor-may not be true. Choose the strongest statement below that'is most nemately réfléttive of the truth or falsehood of P. Circle your choice. (I) P must be true. (II) P could be true, but does not have to be true. (III) P must be false. ) Feel free to use the blank space below for scratch work. There is no partial credit here, so your work below will not be considered in grading this prob- lem. Only the Choice you circle will be considered. If "5 Tait "='*> "1Q ls 'PKLSE If Q. [s Fan-156 ti) fia :5 {~er Ut'rt. ml Lr=~>owk [Fay-1Q} :5 rm. f L HQI'LCL I Ratio. 9% aux-l Pfl) “162 M t "T-LQ I TILT Ia, 1h ¢Mfit P Mail'k ‘FJSE' Nut: This fume.va Ham I ‘ Pioblem 3 (14 Points) Comider the [allowing System F: F:[R—+R}—r[R-—rlll]suchthatv:ce [R—Ivfl]andWER, (FWDU) = (t + Di“) 1 where R denotes the set of real numbers. Choose the strongest statement below that most accurately reflects one or more properties of the system F. (I) The system F is memoryless and causal. (II) The s stem F is memo less but it is not causal. (IV) The system F is neither memoryless nor causal. There is no partial credit here. Accordingly, no explanation will be consid- ered in grading your selection among the statements above. _ F .3 (‘ch L/c, wreaks MIC} at is) to Name gm. Align-117368“ [75' Tva‘ Mind" sfaguufls K‘ mi K; art: {4th “‘P 1’" “hart-‘1'“? “s t,» ‘16-, If wer) wage; 1&6?! film“ Caches.qu lingo, °~+fvdz5 {FLX:))(+): [ti-01,“) “I! (mum: Chaim mt Le can»? mu: m est-g f5 1’ch :sfml'll [triage K‘fifl 414?. 3‘1”) Inge Nah/tLM‘L“ C 9+(Fen)c+1-.gmxm,fwcxwflc- 7U , _ F is no?” mm 1e55, hwewr‘, btcmst‘. at it 9266'!“- ‘ (Hi) '31 I left FM fir ‘x‘fifki “M; Er “'1 tuft-1' 5‘ fit 632:; TM“ (it’llzdfi) fir 5m: Tn”; , WEEK. (Laird: 0:14) xcm .—. amongst Lynda). 1i}, tildes _ ‘L/J a Hbtt‘ijnra mitt.“ fir Natalee.“ as. ,il’roblem 4 (14 Points} Determine whether the following statement is true or false: {gly = graphtf) A f = X —* Y} C W x Y). where P(X x Y) denotes the owe: set of-X x Y. Circle your answer: False You will receive 0 credit for an incorrect choice, 8 points for no-choice (i.e.,‘for_ not answering the problem), and 14 points (full credit) for a correct choice. This is to discourage selection based on a coin toss! gm, gimme §a\a=wt~o‘m£m§ is 6F fit fire-u: ifix,3.)\ xexharE—‘f A 3: {Mg 77:: fewer svT' PIC XLYJ tie-I. an”) (em gins awash eawh afJIJ-Ju ,3 « sei‘a‘; Marcel fl';P5!EE*41x:§/Tof' Mimi (3.0105 .‘n. Pad) 40%“??? Mina-[Sane dome,qu ganglia £9097), however; AQJQ’IE JCMCJEW5 fur-’7- _./\—"' d’f‘f’” "F mwfiffii Tfifi «SM—Fauna :fi #P-‘LE - “Vt—G: fig ff‘a‘n‘Qm .15 ac Varthj/gf’ flwctse 3‘00) ad" @mk OQL c; of fl fad'ank/ whim-r1 &0U\.¢ WWW Rfipkulaa c r1 W'kul int-\c Xx‘f elf?! PORT) “las {it A} Loam - 5 Problem 5 (30 Points) Consider two finite-state machines A and B having input charac- and output alphabets I... = 0;. = 15 - O -- hu-th ' ‘ -— a — {0,1, absent and tenzed, respechvely, by the state-transition diagrams shirwn belower The input and output signal spaces for each machine are identical, i.e., 2:. y 6 [No —> {U.1,absent}], where Nu = {0.1,2,...}. (a) (15 Points) Which one of the following four statements is true? Circle yam- choice. (I) A is not memoryless, and B is not memoryless. (III) A is memoryless, but 8 ' not memoryless. (IV) A is memoryless, and B is memoryleSS. 6 (b) (15 Puma) (Panama a finite-state machine C having input and on t alphabets 33%" by 'C = 0C = {0: 1.ab'semt}. The state-Rafisi't'i'” gram for C 15 shown below: on - G g four statements is true? Cirdeyfiur'ctmice. Which one 0! the followin (I) C is bis'uznilar to A and is bisimilar to B. (11) C is bisinu'lar to A but is not bisimiia: to B. (III) C is not bisimilar to A but'is bisimilar to BI (IV). C is bisimilar neither to A no: to C {S 1?: {Jeni}??? fish, (nae—Hi. Problem 6 {20 Points) Consider a system with an input si . gnal space X = N {0, 1I 2, 3}] and an output Signal space Y identical to the input signal signage.-i For any input signal :1: E X, the system produces an output signal y E Y as follows: w e No, an) = max(z(n).a=(n — 1)) . That is, the output at instant n is the larger of the input signal instant at n and at the prior instant n — 1. The following skeleton finite—state machine, if properly defined, implements the system. Define the finite-state machine by labeling all the salient fea- tures, including, for example, drawing the appropriate arcs, specifying all state transition parameters (guards and outputs on the arcs, including on self-loops, if any), etc. For this problem, ignore stuttering. ...
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This note was uploaded on 01/23/2012 for the course EE 20 taught by Professor Edwarda.lee during the Spring '08 term at University of California, Berkeley.

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EE20-2005-10-05-Midterm-01-SOL - EECS ZDN: Structure and...

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