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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems MIDTERlVi 1
Department of Electrical Engineering and Computer Sciences 26 September 2006
UNIVERSITY OF CALIFORNIA BERKELEY U ' —  F ._ « LAST Name iii/"15V LKC for FIRST Name awry?) it, XI ,: t 5/
Lab Time “faith/spat \J o (10 Points) Print your name and lab time in legible, block lettering above
AND on the last page where the grading table appears. a 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. 0 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 one doublesided 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—
inciading, for example, commencing work prematurer 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.  The exam printout consists of pages numbered 1 through 8. When you are
prompted by the teaching staff to begin w0rk, verify that your copy of the
exam is free of printing anomalies and contains all of the eight numbered
pages. If you find a defect in your copy, notify the staff immediately. 0 Please write neatly and legibly, because {fare 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 afantastic job on this exam. You may use this page for scratch work only.
Without exception, subject matter On this page will not be graded. MTIJ. (25 Points) Consider a function f defined as follows:
f : R —> C
v: e 1e} 1%) = (—1)“. Each of the following parts can be solved independently of the other. (a) Determine an expression for, and provide a clear sketch of the graphs of,
[ f(t) and A f (t), the magnitude and angle, respectively, of function f, where
ﬁt) 2  f(t) 831;“). Be sure to label all the salient features of your graphs. qt— ”‘htt’n— '*——"'7 (‘lnﬂ = [QTYm = {WM “=3: Ht) ATM '. ‘ i ... e v 0 '12:; _‘ o I 1.
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03) Let fe and f0 denote the even and odd components of f, respectively, where,
WEIR,
magnum), fe(r)=—f‘”+2f‘“”. and fo(:)=———f(“'2f"”. Determine fe and f0. You may do this by showing how each of the compo—
nents is related to f, or providing the graph of each component fa and f0. Moth ﬁat {LAAm, .ue made tad ﬁstt (ﬂ, VF
WA ﬁﬂ') : 0 MT1.2 (30 Points) You can tackle the two parts of this problem independently.
Explain your responses succinctly, but clearly and convincingly. (a) Albert attends the concert only if Sally attends the concert. If Blake attends Le‘l' H: “1 kanlt: TE seeml masta'hq 3': eTwlth‘l' To the concert, then Sally does not attend the concert. \
Albert is attending the concert. Is Blake attending the concert? . mﬁ'ﬁniS 5'. dinls (b) Determine whether the following argument is valid. La” W PC D '— ‘ﬁEICIS news cal—economic; A‘QFFVCESWW
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0 Monday, 2 October 2006: If there is no news today of a looming
economic depression, nor any revelation of a political scandal in the
executive branch, the prime minister will complete the remaining
portion of her term in office, and the parliament will pass her
educatiOn overhaul bill into law at the end of this week. Tuesday, 3 October 2006: The prime minister announced her resig
nation at 8am today. Therefore, her education overhaul bill will never be passed by the
parliament. a S; lhmli (icicle/lien o n " 5 ﬂuvial
poll" (“Q C 3 ﬁt. N}. Cairn.fo Q, Luz4" ‘l’epm in aFPI‘CQ
fa“ Tﬁ'c 9W5 ﬁllACCCEOA E 3 The _ Fairlie«newt will urgekr s (and . at. be A as} is . if:ka Pt :3 S (Pills: r'l' A‘l’ﬁz «is ml 2'9 3"“ “’{EVA‘\ ~
‘I % —.=$ —13 C55" Ahqén'ﬂc“. 49:3» “Ii—Rf”) Say—16. Charla, 1:) —:[S dials MT1.3 (25 Points) Consider a function G deﬁned as follows: ‘V’uJElRE G(w)= Iw, where can has a ﬁxed positive real value. Determine an expression for, and provide a clear sketch of the graphs of, the mag
nitude [G(w) and angle éGﬁu) of function G, where G(w) = G(w) e‘ml‘”). For what value(s) of w does the function G(w)[ attain a maximum? What is the
value of ]G(wﬂ)? What are the values of AGO») for w = —wo, 0, +wo? Determine
the limits: lim law. lim raw, ljm. Law, and ﬁrm may). w—~—oc w—vioo w—>—oo w—~+oo You may express your answers to these questions by placing appropriate labels on
your sketches. lei: “ : ‘ agate)1Mxla10ua MT1.4 (25 Points) Consider the discretelime signal f : Z —> IR, characterized as
follows: 1 m = 0, 1, 2
0 elsewhere. VmE Z, f(m) ={ You can tackle the two parts of this problem independently. (a) A related signal p : Z —+ R results from modulating f , as follows: Vm e z, p(m) = % [1+(—1)m] f(m). Provide a welllabeled sketch of the signal p. (b) A related signal q : Z ——) R results from the convolution of the signal f with
itself; this is written as q = f a: f, or g(n) = (f =x f)(n),Vn E Z. In particular, the signal q satisﬁes the following convolution sum: vn e 2, gen) = Z f(m)f(n — m). m=—oo Provide a welllabeled sketch of graphm). (This would be a stem plot, that is, a “lollypop” plot.) Hint: Discretetime convolution is generally simpler than continuoustime
convolution. Start by sketching the signal f as a function of m. Also, plot the
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