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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems MIDTERM 2
Department of Electrical Engineering and Computer Sciences 24 October 2006
UNIVERSITY OF CALIFORNIA BERKELEY LAST Name L FIRST Name ML Lab Time min/43L— . (10 Points) Print your name and lab time in legible, block lettering above (5
points) AND on the last page (5 points) where the grading table appears. 0 This exam should take up to 70 minutes to complete. You will be given at
least 70 minutes, up to a maximum of 80 minutes, to work on the exam. o This exam is closed book. Collaboration is not permitted. You may not use
or access, or cause to be used or accessed, any reference in print or electronic
form at any time during the exam, except two double—sided 8.5” x 11” sheets
of handwritten notes having no appendage. Computing, communication,
and other electronic devices (except dedicated timekeepers) must be turned
off. Noncompliance with these or other instructions from the teaching staff“ ,including, for example, commencing work prematurely or continuing beyond the
announced stop time—is a serious violation of the Code of Student Conduct.
Scratch paper will be provided to you; ask for more if you run out. You may
not use your own scratch paper. o The exam printout consists of pages numbered 1 through 12. When you are
prompted by the teaching staff to begin work, verify that your copy of the
exam is free of printing anomalies and contains all of the twelve numbered
pages. If you find a defect in your copy, notify the staff immediately. 0 Please write neatly and legibly, because if we can’t read it, we can’t grade it. o For each problem, limit your work to the space provided specifically for that
problem. No other work will be considered in grading your exam. No exceptions. 0 Unless explicitly waived by the specific wording of a problem, you must ex—
plain your responses (and reasoning) succinctly, but clearly and convincingly. 0 We hope you do a fantastic job on this exam. Basic Formulas: Discrete Fourier Series (DFS) Complex exponential Fourier series synthesis and
analysis equations for a periodic discretetime signal having period p: 3301,) = Z Xk,eikw°n +—} Xk = .5 Z $071) ewikwnn 1
k=(p) n=(z:) 2
where = l and p denotes a suitable discrete interval of len h p (i.e., an
19 (do '. p—1
interval containing 17 contiguous integers). For example, 2 may denote Z k=<p> k=o
P
or Z .
k=1 You may use this page for scratch work only. Without exceEtion, subject matter on this Rage will not be graded. MT2.1 (20 Points) Consider a continuous—time system F : [IR —> C] —> [R —> C]
having input signal as and output signal y, as shown below: *1 y This system takes the real part of its input signal:
3,] = F(a:) = Re(:c). In other words,
Vt 5 IR, y(t) = Re(z(t)), where Re() denotes taking the real part of a number. For each part below, you
must explain your reasoning succinctly, but clearly and convincingly. (a). Select the strongest true assertion from the list below. (i) The system must be memoryless. (ii) The system could be memoryless, but does not have to be. (iii) The system cannot be memoryless. 6“) Aefem’k 911} °"‘ *6)" A km L‘TlON T: (Ci 6‘
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fatet 1C: 6% 6‘ War, freq was» (b) Select the strongest true assertion from the list below. (1) The system must be causal. (ii) The system could be causal, but does not have to be.
(iii) The system cannot be causal. ., A mmqnaItSS 365th MWSTWISQ Le Caxutsmo
a AHErnmﬁutl ) 60w Cm“ ska}; X‘(T):X&(T I V‘Z’gl‘ Tﬁen a m: Ream) ‘2 R: (xgtﬂ)=§1(i’) 1 ‘7‘ '7va \ 3 So’Texe S’aﬁ’ewx mws'l' ‘06 wa‘ml. (c) Select the strongest true assertion from the list below. i (i) The system must be time invariant. 1 (ii) The system could be time invariant, but does not have to be.
(iii) The system cannot be time invariant. ”L'd’mna e We knew) LXIQ) ‘15 q ‘oC‘an‘or 9'? F i Let '2 he 435—.er wk their £Lr)=x(ht,) VtatﬂK 76:: Coi‘OESYonAina rtsxaonsts is charmc‘ﬁr‘kerl ‘05 3M: R W“) > R6(x(t«m)=a(t7;). =5 9—— T. l3 Thth {anr'x‘mvj'.
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(d) Select the strongest true assertion from the list below. {rim ‘0“! (i) The system must be linear.
(ii) The system could be linear, but does not have to be. We sailem is awUH’iwe) \oﬁl’ flees ml” Scihs‘ga hoggeneng
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@M‘Tﬁ MT2.2 (25 Points) The unitstep response1 .9 of a discretetime linear, time~invariant
system is given by:
V71 6 Z, s(n) = (n + 1) u(n), where u is the unitstep signal characterized as follows: 0 n<0
1 n20. Vn E Z, u(n) ={ Explain your reasoning for each part succinctly, but clearly and convincingly. (a) Determine and provide a welllabeled sketch of h, the impulse response of
the system. TE DT lm K\SQ Coux ‘03 “FREE“ 0‘5 "\RQ Alf’kt‘ﬁncﬂ
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mm} ‘5? _ . at“) .14“; \srnt + + ““3 39 o r\ an)» 5% Fake 9 Q —————'5 ‘  M st ‘22 M a HAL“ \ o \ 9‘ 3‘ \f\(n\=\\(n) 1Recall that the unitstep response of a system is, as the name suggests, the response of the
system to the unit—step input signal. (b) Select the strongest true assertion from the list below. (i) The system must be memoryless.
(ii) The system could be memoryless, but does not have to be.
(iii) The system cannot be memoryless. MC”. MAE lad” The wall—5"}? («fir has ﬁne
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‘8‘ K r ‘ “ C “W‘CQ a NSC“— TﬁlQ—erA an\ t l‘l’s lm‘mlse. Feslfonse (C) Determine a sim 1e expression for " )
p n k((\\: Kagcﬁx ) 30L e C 
Z h(m). \ Hint: Your answer will depend on n. You should be able to solve this part
even without knowing the impulse response h from part (a). TE 8": {‘QSTOASE“SH;S nLt:n€:<1\ \Da C0“\F0\Vlr\3\\‘€\€
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5 GA :_ Z \an\ ﬁrs—.5 £1 \quﬁ\ _—___ SG\\:(V\\'\> “6'“ m: '00 MT2.3 (25 Points) Consider a discretetime system F : [Z —> C] ——> [Z ——> C] having
a periodic input signal a: and a corresponding periodic output signal y, as shown
below. 432101234567 n W)
(1) (1) (1) (1) (1) (1) 6—5432—101234567n (a) Determine (13;, com) and (p,,, wy), the period and fundamental frequency of a:
and 3;, respectively.
23? T ﬂit—H. ”:5 ”xi—irza‘  7. r255
$3“; $033” s’ (b) Determine the complex exponential discrete Fourier series (DPS) representa
tion of the output signal y. In particular, determine a simple expression for
the coefﬁcients Y; in the DPS expansion 1 .
Yk = — Z y(n)e“k“‘yr.‘ 77y
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—%’— o —S—_— 7‘5”? [C ‘t'C (c) Select the strongest true assertion from the list below. Explain your reasoning
succinctly, but clearly and convincingly. (i) The system must be LTI.
(ii) The system could be LTI, but does not have to be. {ME Cu» at Con aims ﬁerencies Kai/OWE 00% RV Pnesemfi‘q ﬁ‘i {ﬁlms/l7 Eremefk; “817;: ‘5 AonCXk—‘EN‘V imﬁe mpd’ '5;an X. " _ We \Kﬂod LT: saxEms ca“, 0&6 «\mb‘f’é (6.8.)mnf‘bfg of‘ mﬁEnud’E) ﬁt “enca 8‘ com onCt/J’S Fresefcr in BM
f creed? {\er ﬁreTwenCiU' (“N's 3 'th6L Coxﬂﬁo MT2.4 (20 Points) Consider a discretetime LTI filter A : [Z —> C] —:> [Z ——> C] hav
ing impuLse response a and frequency response A. The ﬁgure below is a graphical,
inputoutput depiction of the ﬁlter: xly Recall that the frequency response and impulse response are related as follows: Vw E 1R, A(w) = i0: a(n) B—M. 11—_ — DO The ﬁgure below depicts A(w), Va) 6 [—7r, +7r]. Notice that for this particular ﬁlter,
A(w) is real—valued at all frequencies. Ps (4T) Frequency Response ROY) 9 z/ ; 2  A62} : 3 N .. lie/1:?)
410$; 1 «0.3 0.6 0. o.4 o.2 o 9.2 0.4 050.6 0.8 1 Normalized Frequency (x 1: radians/sample) The frequency axis in the ﬁgure is normalized by 7r; hence, for example, the nor
malized frequencies 0.5 and 1 refer to w = 1r/ 2 and w _= 7r radians per sample,
respectively. Determine a reasonable and simple (possibly approximate) expression for the out
put y of the ﬁlter, if the input :c is: _~—Note thathhere IS no’ ”u ’inthe ﬁrst term. This' IS notatypographical error
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‘ tn: at?“ 1:5,} 0W+r¢ caml'nlwton I5 Etal“ :. gin lo 01 rcs case :5 Olatifeal L01 “BTW 0" You may use the blank space below for scratch work. Nothing written below this
line on this Rage will be considered in evaluatingy‘our work. 10 MT2.5 (15 Points) The impulse response h of a discretetime LTI system is given
by:
1 n .
Vn e Z, h(n) = 5 Mn), MAM less cm cl where u is the unitstep function. \3 halal?!“ ' (a) Select the strongest true assertion from the list below. ,, G’s«505‘ (72‘)
(i) The system must be causal. (ii) The system could be causal, but does not have to be. (iii) The system cannot be causal. _; Z lq m X(ﬂW\\ = :olNQYQKQm—m\ 3M: Um mam
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(b) Determine a simple expression for the frequency response H of the system.
Recall that the frequency response and impulse response are related as fol—
lows: Va) 6 R, H(w) = i h.(n) 64“". TL: —00 LAST Name Md l_— ' "s‘Eo'r FIRST Name No" l’ ”e“ r Lab Time My]; i“l 12 a EU ...
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