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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
cuminiced 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 discretetime LTI ﬁlter H : [Z —> IR] —> [Z —> R] has an
inputoutput 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 welllabeled delayaddergain block diagram represen
tation of H. Yoru diagram must have the minimal number of storage elements
(delays) necessary to represent theﬁlter. 3(a) : at Ma} 1— Mimi) — m3 (rt1} 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<©.. \qLowzotg(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 (yeT: Ls Mol— lﬁe hngVWeExjdro 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]: 041—6 :3 HQ”: 0L+e A
3 \+a(etud‘ IV (7 Points) Determine a simple expression for, and provide a welllabeled
sketch of, the ﬁlter's magnitude response H(w): —7r s w < w. ..‘\__ ‘ LWF'
midv 6““ = 5““ ‘ﬂe
\+ “he:le \ 1— % e_['u_:.— LId'" NOTE. \jQCmmge. 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 lowpass ﬁlter. (b) H is a bandpass ﬁlter.
(c) H is a highpass filter. (d) H is an  pass ﬁlter. I (e) H does not belong to any of the ﬁlter categories above. Provide a onesentence explanation (not exceeding two lines) to support your
choice. TIE HEW \mxg 0x Cryostat Moxaﬂii—‘ljkie OBSTD‘NSQ
“Nani, 59 fl— mwsTlSE (RuAFKSS. 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
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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 0450 ‘L. a 4L."
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Without exception, subject matter on this page will 1101‘ be graded....
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