ee20-05-Quiz-03-Sol-2006-11-16 - EECS ZON Structure and...

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Unformatted text preview: EECS ZON: Structure and Interpretation of Signals and Systems QUIZ 3 Department of Electrical Engineering and Computer Sciences 16 November 2006 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name / e lfi FIRST Name L o ('5 Points) Print your name and lab time in legible, block lettering above. a This quiz should take up to 20 minutes to complete. You will be given at least 20 minutes, up to a maximum of 30 minutes, to work on the quiz. 0 This quiz 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 elec- tronic form at any time during the quiz. 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 cumin-iced stop time—is a serious violation of the Code of Student Conduct. 1- We will pr0vide you with scratch paper. Do not use your DWn. - The quiz printout consists of pages numbered 1 through 6. When you are prompted by the teaching staff to begin work, verify that your copy of the quiz is free of printing anomalies and contains all of the six numbered pages. If you find a defect in your copy, notify the staff immediately. 0 Please write neatly and legibly, because {fuse 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 quiz. 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. - We hope you do a fantastic job on this quiz. (23.1 {40 Points) A causal discrete-time LTI filter H : [Z —> IR] —> [Z —> R] has an input-output behavior that is at least partially characterized by the linear, constant- Coefificient difference equation +ay(n— 1) : ruin) +5501 — l)1 where the parameter a e R, 0 < loaf < l, is adjustable, but is otherwise constant (with respect to n). Each value of n represents a distinct filter. The input and output signals are denoted by I- and 3;, respectively. Explain your response to each of the questions below succinctly, but clearly and convincingly. I (6 Points) Provide a well-labeled delay-adder-gain block diagram represen- tation of H. Yoru- diagram must have the minimal number of storage elements (delays) necessary to represent the-filter. 3(a) : at Ma} 1— Mimi) — m3 (rt-1} II (7 Points) Determine the filter’s impulse response h : Z —> R. That is, deter- mine an Expression for Mn), Vn E Z. —. 1%“: °< Sb» “r gem—Q -—o(lf1(n-\\ b C‘MH‘JK ——-—\> h[n\:0 . w<©.. \q-Lowzotg(o\-TS O —M= Dd. Max—A3 drum—wow :1-°< Hg): azimys mlnLQ : —o<. k '0 m {WMA :— 90 0—053 .2 III {'7 Points} Determine the filter’s frequency response H : 1R —r C. That is, de» termine an expression for H (w), Vw E R. You should be able to solve this part independently of your answer to Part (II . _ (‘u’ r ) m and} Hm em km “3:,le m 4&4.er H(~a):Zm\e Inc—0Q --——.-....—— Ingl’ma‘) looL’ at "lat cllgkrancfi (ii‘rkw—fiofl anal Malta: final- il no“: C(w‘n) fie“ 6(n\: Ema} \a‘u‘a>-TEQ.3Q ml?) lee ETvCfion To (ye-T: Ls Mol— lfie hngVWeE-xjdro CF. ' ‘ _,!l in“ LN“ Vthl\ ‘ we“! x eWG] \st : me- + e . seals Mir than 550M Sick”. S) CNS/WA. gouaarkna—lér‘mgl wt "(wk Aur— Q—rsagl‘WV) Hod]: 04-1—6 :3 HQ”: 0L+e A 3 \+a(e-tud‘ IV (7 Points) Determine a simple expression for, and provide a well-labeled sketch of, the filter's magnitude response |H(w)|: —7r s w < w. ..‘\__ ‘ LWF' mid-v 6““ = 5““ ‘fle \+ “he:le \ 1— % e_['u_:.-— LI-d'" NOTE. \jQ-Cmmge. 0L6\K) Tat: num‘efi 9(— l+°¢C.-®— l3 ‘ —-l . fie comrlex CorawaflTfl 0‘5: fie olenommacl—TW‘ l‘i-‘OLE, ’ e) l+uetm : 0+U~eriur> =‘>l\+oce..iub\= l1+ cued-“fl 1:1__> Choose one of the descriptions below that best characterizes the filter H: (a) H is a low-pass filter. (b) H is a band-pass filter. (c) H is a high-pass filter. (d) H is an - -pass filter. I (e) H does not belong to any of the filter categories above. Provide a one-sentence explanation (not exceeding two lines) to support your choice. TIE HEW \mxg 0x Cryostat Moxaflii—‘ljkie OBSTD‘NSQ “Nani, 59 fl— mwsTlSE (Ru-AFKSS. 1 V (7 Points) Su ose o = tan 3: = —-—, and the in ut si 31 is characterized b PP 6 P 8“ 3’ J3? Mn) 2 cos (an) , Vn E Z. .a Without complicated mathematical manipulations, determine a simple ex- pression for the filter ’5 output sample values gm), Vn E Z. Thg COI’Y‘QSFOK‘I’LA oak/lilav'j‘ 3 : Co §<§n —1- 2g i We kflovd ) keel. quNerig Jag figkq‘ge 99 five ‘flr NEWS freer-tencg (‘CSFOOSC air )9": ?_~t€.nc Mtg/Z"? Lu} H (U) : CAN.“ Camr‘tx Congw}%l{8 U6 1% {bat—r92 2fio+xe ) _ ‘1'“. e (Tr— : —u_)-—- + EC'H‘W‘) t ":‘taéghirhC/Rl ' ' 4“ ‘1 =E— AFR l “cl: —’___ 2r/‘H‘ (El: Egraltfl‘ :JrEE —‘fi‘ : 2F£+fit) ‘2 “12:4 (a 3 . VI (6 Points) Consider each of the following two limiting cases: (a) o: —r O. (b) o: —-> 1. For each case, provide a reasonable limiting approximation to the impulse response h- and the frequency response H. Briefly explain, for each limiting case, what the filter does to the input signal :3. “Hair M's-'50 04-50 ‘L. a 4L." “ '_-_'_ K + _ _'_-> e .7— H \ "hone-fue— 3 H e 1 '_B As Mo; was} wax—s («em int “1 HM H's We“ -. —"—.¢‘ LPG—L 5 A—s o,__a> i j hfifi —-——=~> 70% Can 0150 See. l-EQSQ fit (I‘Mll’g 9"" TEE 7!?!" SI-Ofllfij pf— #5139 Q‘Tle'I‘OA- Ear Exam ‘2? RS m—Fbol‘rfiE LCCDE [oaks HQ =X(r'L-l>, You may use this page for scratch work only. Without exception, subject matter on this page will 1101‘ be graded. ...
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