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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 twosided 8.5 x 11inch 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. Eurfa ti‘CMmfiﬂ {Ill2,5: 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!»
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‘hwﬁm I5 tad—0 "(c) (14 Points) Of the elements in X, how many define, 1135pnectimrely'I a graph
of. a anewone function f : {(1, b, c} + {1, 2, 3}? Explicitly specify one
element in X that deﬁnes a graph of a onetoone function. Eur ‘F'EK To In. onc~TﬁMcl cLoL glamm+fn inning} mtg(“Lb
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hrng (P:Q)A(P=>—IQ). We do not know whether Q is truefitm‘ayormay 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
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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.
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_ ‘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 ofX x Y.
Circle your answer: False You will receive 0 credit for an incorrect choice, 8 points for nochoice (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 ﬁreu: iﬁx,3.)\ xexharE—‘f A 3: {Mg
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Rﬁpkulaa c r1 W'kul int\c Xx‘f elf?! PORT) “las {it A} Loam  5 Problem 5 (30 Points) Consider two finitestate machines A and B having input
charac and output alphabets I... = 0;. = 15  O  huth
' ‘ — a — {0,1, absent and
tenzed, respechvely, by the statetransition 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 ﬁnitestate machine C having input and on t
alphabets 33%" by 'C = 0C = {0: 1.ab'semt}. The stateRaﬁsi'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 ﬁnitestate 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
selfloops, if any), etc. For this problem, ignore stuttering. ...
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 Spring '08
 EdwardA.Lee

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