Midterm 2 - EECS 20N Structure and Interpretation of...

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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 discrete-time 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‘Tl-ON T: (Ci 6‘ EXlSTS vac\f\ fih.\' afifl\:¥(yxl+\\) . 7K? gunmen '8: 15,} l/\ fate-t 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 Caxutsm-o 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 {an-r'x‘mvj'. 3 AWtfncL-hye‘dxevgra "\Uflor‘aNCSS 365E“, ACCQFAMB"; our is "Kent WW mm , (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 . I‘ll Q3133 “ml (3139.23 Mt Lekmv‘gF3 0‘s: TEE gas em) (in a\(fl: Re 9‘6“) MA 2:28): Re (Kola—W . Ckmrla‘ ~ 3\(r\+ 32M __ RE (amy- Re (5.8%} = Re (nth hit” ; “\‘k’x‘ muons QR‘+XR)3\+3¢1\ is a behavlqr‘. . . “TE Alsawgs \’\°W\0&CV\€;_\3) \a, . (x) x ‘03 a. behavwr‘pfi. Lei g‘mcx , :3er , "wig“ 3&3: RCG‘XG‘A)’ Ike riak’r’lxml siAQ is 33‘: eqrvch o 0t R664)“ gxéfi'} “Ages OCE‘K, AIR“ 4 eomYlex K m‘ms Tag @M‘Tfi- MT2.2 (25 Points) The unit-step response1 .9 of a discrete-time linear, time~invariant system is given by: V71 6 Z, s(n) = (n + 1) u(n), where u is the unit-step 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 well-labeled sketch of h, the impulse response of the system. TE DT lm K\SQ Coux ‘03 “FREE“ 0‘5 "\RQ Alf’kt‘fincfl kelweerx SLCH’C: “AR-r STE? slung/hang 'fi‘mfi' .15) “(“3 gm: «m- ils-m Jill—— > _\(?)ll-L 3“ . REE} Stet: W: 5831‘,“ 13 LTT) mas n Mb}: Sme- SCn-\) PLA' 3(“\ fa $31 Move finv‘as NOC\L 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 unit-step 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 Write) ‘M “(Glam—Wm?“ =1 HWER", The ”fix 55‘s SWC\'\ TR: slow 3% 503 + sens -- - Hermfi’fie‘t Sag—EM Camq— \oe memora\€.$3. g - ' \35‘3 runs Q( t £\° («(370 ~ onr a k W LTl 3 {EM \‘5 mEmoPa ‘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\\‘€\€ MW 34%.? Irma‘riang «Mex fin.“ VASE TCSY 3e “\n': Von {5 L S 2 “Wu. “refs-Bf?) smzwwv wam i=5 MZ‘OQ 4 \Ova utn-m\ —. 0 7 its? ”‘7“ 1‘ §qérmgn n 5 GA :_ Z \an\ firs—.5 £1 \qufi\ _—___ SG\\:(V\-\'\> “6'“ m: '00 MT2.3 (25 Points) Consider a discrete-time system F : [Z —> C] ——> [Z ——> C] having a periodic input signal a: and a corresponding periodic output signal y, as shown below. -4-3-2-101234567 n W) ---(1) (1) (1) (1) (1) (1) -6—5-4-3-2—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 A ' gr ”=(Pu) — Lk —V\ ): 36k g Wit-0'2 On\a "two firms) 3Q» AKA 3(1‘\ )_ N‘t nonIfif‘O - TE: know! [kg "kaé——]_9~C52Wq -—%’—- 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: sax-Ems ca“, 0&6 «\mb‘f’é (6.8.)m-nf‘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 discrete-time LTI filter A : [Z —> C] —:> [Z ——> C] hav- ing impuLse response a and frequency response A. The figure below is a graphical, input-output 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 . E ;_ =6 13" -—-‘> Look Q DC 8mm 14(0) 'l'c. SCEIVDL‘J’L‘MWQnS ‘l? 1% zero frequnr'xcalmn TronEn/‘l’ A(o)e VS: 3 v - ll 9 CQELLnUC'Cfi—f H'Llill—n>___a LookG Ill—kid) anal A(%)A W l— ”(E—)Ermme TEE Cof‘f‘ES-‘ji’karOfioll «LVN-l: 7=A(L:I)—: A(.0€\\)=H(OYT1))= (3?) mft‘hfe) fit. C°m\\>—xfiofl 08,87 C93 (Ii—“5h fifi PQ5Y°~2$Q \$ ClCoS(i—V\3 ‘ mtg“ :8 mar mat +3... 3s myth": la (it ‘ tn: at?“ 1:5,} 0W+r¢ caml'nlw-ton I5 Eta-l“ :. 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 discrete-time LTI system is given by: 1 n . Vn e Z, h(n) = 5 Mn), MAM less cm cl where u is the unit-step 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(fl-W\\ =- :olNQYQKQm—m\ 3M: Um mam Mal—Q Wall wt Cmvx whl‘l’fim i T‘ras, so "am" We ol— lie {its ~3(“l7i“°“l‘l*“°““h ilk Venus ‘ t Exp? ssfcm,e.a.) ER cu- t no xé'vrfi, X‘mrg\). - - TEN/x) :LB $61515,“ ‘ 5 (Nu _ (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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