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Unformatted text preview: EE 350 EXAM III 17 November 2003 Last Name: 5 Qigg'LIOQ First Name: ID number (Last 4 digits): Section:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
Problem Wei5ght Score
The Blue Test Form
Instructions
1. You have two hours to complete this exam. #0310 . This is a closed—book exam. You are allowed one 8.5” by 11” note sheet.
. Calculators are not allowed. . Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number, for example, “Problem 2.1) Continued.” No credit will be given to a solution that does not meet
this requirement. . Do not remove any pages from this exam. Loose papers will not be accepted and a grade of zero will be assigned. . The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing.
To receive credit, you must show your work. Problem 1: (25 Points) 1. (12 points) Find the fundamental period To and trigonometric Fourier series coefﬁcients of the periodic
signal f(t) shown in Figure 1. ﬁt) Figure 1: Periodic signal f(t). ° 9d. Inspection To ‘= 2.0 _
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no = $58 Halit— = Ila—[2.4.1: = 5:44 :— § 0 ° 0 2. (13 points) The periodic signal
g(t) = a + ,6 cos(41rt) cos(21rt) + 'y cos(31rt) + 6 sin(31rt) has the exponential Fourier series coefﬁcients Du speciﬁed in Figure 2, where a, ﬁ, 1, and 6 are
constants that are not necessarily realvalued. Re{Dn} . Im{Dn} Figure 2: Specrﬁcation of the exponential Fourier series coefﬁc1ents D" for the periodic signal g(t). o (3 points) Find the fundamental frequency w, of the periodic signal g(t). o (10 points) Specify the value the constants a, ﬂ, 7, and 6 that appear in the expression for g(t),
and also specify the integer values a, b, and c, that appear in Figure 2. Usmé “Um. ‘lnra o nomz‘lzflc. idlen'ln‘g/ cos xfcsoa ‘—‘ AZ “’5 (9°95 "' Jitﬁuy) : + 1.5:. Ca; (GITD ‘l' é‘CQ‘CZlT‘t) *f" 7605(379 4— 53/0 (37ft) gCB = cc
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4— ‘/ {GB/f 1/ Problem 2: (25 points) 1. (13 points) A periodic signal y(t) has the complex exponential Forier series coeﬂicents speciﬁed in
Figure 3. Figure 3: Speciﬁcation of the exponential Fourier series coefﬁcients D" for the periodic signal y(t). o (3 points) Is the signal y(t) real, imaginary, or complexed value ? Justify your answer in a single
sentence in order to receive partial credit. BeCowse. lbnlf— lbn\ chL an =  Lil—”J Do a bra TRQ, 0054: about)? bows cob. I‘Fyé'b) 15
reuQ’valveﬂ. o (3 points) As a function of time, is the signal y(t) either even or odd ? Justify your answer in a
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13(— Valve = lbolééx' : We.+ =— "f. o (5 points) Determine the power Pll of the periodic signal y(t). Be (Aden. &(ﬁ is muQauwlveﬂJ L
— lb 1“ + 2 3.11m" = IbJ‘+ warmA) Pa  = M1" + 1((20‘ + ml) :16 +1e+8 I‘m 2. (12 points) A periodic signal with Fourier series representation fa) : 5+ 2 £31634OOnt n¢0 is passed through an ideal lowpass ﬁlter with cutoff frequency we and passband gain J2 as shown in
Figure 4 to produce a periodic output signal y(t). Find the range of values of we, in rad/sec, that
yields the largest output power Pit of the ﬁlter without violating the constraint P, < 125. HO 0)) \13 "(D c we Figure 4: Frequency response function of the ﬁlter. Do 4301 $053 bn %r 30:) r3331
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b?) (1 +0 (1119 Iwon i (‘09 P} ‘18?!” zigmmz 2 §0+ £21 lWon < WC, [WMICWU : 50 + 6‘! +31+§i+. (an) ('1: 2.) C3133) To keep 9 < [253 we, (Tan oil; fad! cf 153 0:2] “Mme 7i 0% ‘t> 5W W (25 Points) 1. (13 points) An energy signal f(t) has the Fourier transform F(w) = e_2“’l. o (3 points) Without determining the inverse Fourier transform, what conclusions can be drawn
regarding the function f(t) ? In order to receive credit, justify your answer with a short sentence. BQC‘MQ PI“? 75 {W Wild reag— arrIQ an even
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RugEton «%— ”Elma. ‘ o (10 points) Determine f (t) by direct integration, and express your result in a form that validates
the conclusions reached in the previous question. For example, if you determined that f(t) is
realvalued, then your expression for ﬁt) should not involve complex exponentials or factors of J. «7 ‘1'. w °° O
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The signal m(t) is a voltage derived from Neo’s central nervous system, while the output voltage s(t)
is the signal patched into the matrix. Given the Fourier transform representation of m(t) in Figure 5(3), sketch the Fourier transform 5(w) of the signal 9(t). M(m)
4
3 cos(500t)
. m(t) . g 8“) co
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(B) (A) Figure 5: (A) Partial block diagram of the communications system aboard the hovercraft Neb
uchadnezzar. (B) Fourier transform of Neos’ neural activity. 869 :: 3 mC—Ih Cos (Soct) 4. 3 m (a) sin (lfoo‘t) 05m; Q’L {ta as (wear) _: F (w+w°3 +_ pa” We) OWL 2. 2—
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Z 2. 30v): %_M(w+so® +.§:m(w5OO} *a’l'ii’M‘VHSO‘D 7%le—lfm) 1800 $co 1100 2. (13 points) Agent Smith destroyed the receiver in the machine world that recovers the signal m(t)
from a(t). Trinty and Morpheus solicit your help in designing a new receiver that recovers m(t) from
3(t). With the Oracle’s help, Trinty and Morpheus can provide you with the signals cos(500t) and
sin(1500t). Draw a block diagram of your receiver, and, using appropriate sketches in the frequency
domain, demonstrate that your receiver correctly extracts m(t) from s(t). If you use an ideal ﬁlter,
you must clearly indicate the cutoff frequencies and passband gain in your block diagram. The quality
of your analysis and evaluation is as important as your answer ! In order to receive credit, your block
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