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Unformatted text preview: BB 350 EXAM III 5 April 2004 Last Name: 3 u '
First Name: ID number (Last 4 digits): Section:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
Problem Weight Score
1 25
2 25
3 25
4 25
Total 100
Test Form A
Instructions 1. You have two hours to complete this exam.
2. This is a closedbook exam. You are allowed one 8.5” by 11” note sheet. 3. Calculators are not allowed. 4. 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. 5. Do not remove any pages from this exam. Loose papers will not be accepted and a grade of
zero will be assigned. 6. 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. (10 points) Find the trigonometric Fourier series coefﬁcients (a0, an, bu) for the periodic signal f(t)
shown in Figure 1. f(t) Figure 1: Periodic signal f(t). 1; 
4°: #:Sﬁttwrl’w 4: (351,») = 150,} ‘0 =7 qo— 7—
oweq of anc. truly); To 3 We. TuHe.
an: %L Hacosnwgtii a 3%.: .'%‘=_cas(l—g.!LQi£ 5.77 _ 8 7 211’» 277 in 3
Cos—t + ___’2.+_ .rm 0
7 ‘m‘n‘l 3 3 3 1' 6
l
 2.
‘ ——~ 2170 5m v 
"Lnl— [NSC 0) + I 6 J a 0
an :0 n: IJ z" 3 ’
— we Table 3
2. 2.. '1 21M
5.. = 17:36 Fle)s.n Mbaﬂe = "3'1; 5' sm( 3 1)“: 6’7_7 2. (9 points) Determine the trigonometric Fourier series coefﬁcients (up, an, bu), compact trigonometric
Fourier series coefﬁcients (Co, Cm 9,1), and the complex exponential Fourier series coefﬁcients (Dn)
for the periodic signal n =‘ n=L n a 1 w. = z
1 1 ur 1—
h(t) : 3 + 5 cos(2t — 45°) + cos(4t) + E cos(8t + 90°)_ To :2 7:; _. I 8& 'mf’e’i‘mb hit) :5 when»? errveJJOQ. as a. Caﬂ’au‘t ingonome'tno Rumer Series w (UH all other C“) en are. are Ji COS (70°) =0 " "i‘. 5"" (70‘) =" 0: my 3. (6 points) The following MATLAB code generates the triangle wave 9(1) in Figure 2 using 100 sample
points E triangle 1, = [0 : a : b]; D = [0 : c : d]; g : pulstran(t, D,’ tripuls’, 1);
p10t(t,g) (spa(€09 wcl‘ﬂn %
inunale is athl— What must be the values of a, b, c, and d ?
g0)
1 0 1 2 3 4 Figure 2: Triangle wave g(t) generated by MATLAB. Each trwuvle RG5 OﬂeJ and, there are, loaf:
()0? triangle. These. memo the. time fem£3 are. Span: 5}. k = LE—e—Ez 3 Co’
loopts & 05,4: 2 '1 Chemo.
b = H.O Beagng there, )5 4.. ﬁrmniﬂ, OGﬂ'beraQ 01‘: Ever} ‘6QConsg‘
Q a 1 Finally) LC—CMJQ, there am. ‘FIVQ. hwyales) i=w. (I): E01 [1‘1] :. f0, 1,2.)3)‘1J3 *Hlvo. “Enqny’e;
Copierao— “76 i=0) {1.2. a) «ma/'1 4 Problem 2: (25 points)
A periodic signal f(t) has the following complex exponential Fourier series coefﬁcients Dn for n > 0 D0 = 1677r
D1 = 2e?”
D3 = 36—Ja
D5 = 16—Jﬂ, where a > 0 and ,3 > 0 are realvalued phase angles in units of radians.
1. (2 points) What is the DC value of f(t) '?
11'
DC VW‘VQ = 0., z c'e' : 2. (9 points) Given that f(t) is real—valued and an even function of time, specify the magnitude and
phase of the complex exponential Fourier series coefﬁcients for n < 0. The. Drs and: In. real ana— an M Rnc‘fuon a. n bacan
~FCbX is real“Avail ang— an even {LogEm; ﬁlma. 5'3 ‘53 = 32.1"“. = +3.” 3 63.00 mg t
66  +l gf‘ ' W 090.“ «33:713.; ﬂIlor 20's... 3. (4 points) Given that f(t) is realvalued and an even function of time, specify the power P of the
periodic signal f(t). +33 Ls Wwlwﬂ) 3664M"!
an L z o. 1
PF = 5010") _, [)0 + 2 31an
= lbol" 4 719.17" + 7l'33l1 + z/DJI‘
= 0137' + 2.12.11 +2.13;‘ +242)"
= I + e + 18’ +7 Another periodic signal g(t), with fundamental period T}, has the complex exponential Fourier series coef
ﬁcients D51. Consider a new periodic signal
T9
h(t)=g(t—Ta) . 4. (2 points) Find the fundamental period Tah of h(t), and express your answer in terms of Tag. Because. hlt) is Ova£ a. +»mo5luP'EJ replun. 716.1%),
hit) qnﬂ, mJJt' Imwe. the Samer penocg‘. 5. (8 points) Determine an expression for the complex exponential Fourier series coefﬁcients D5; of the
periodic signal h(t) in terms of the coefﬁcients D51. 3’" n l: h 2. — Wu 1),, = 3‘; J Mae d (LL
1: 3. Problem 3: (25 points) 1. (8 points) Consider the signal f(t) = 3 rect sin(12t). The Fourier transform of f (t) exists and is denoted as F o (4 points) Determine, without computation of the spectra, whether the Fourier transform F(w)
is an even function of w, an odd function of w, or neither. 86 c «~50 ﬁnk) = 3mt('3;) Sudn.4,) = '3Y‘Oot(:§) Sm (Iae) =.  {1 a) {4.6) B an em ‘Fuﬂu'élun % “Ema. If 'FOHOM F0») musl.— bo. cm 0.901, ‘Fvnc‘évon i w:
'9 .. “A: a» O on
FCW) = I. “4:369 if: f—Flb3éwLlé —/fFlt)$mw‘b Jé
” 0:9. 4” o (4 points) Determine, without computation of the spectra, whether the Fourier transform F (w)
is realvalued, imaginaryvalued, or complexvalued. 655w”, .Fce) :  {445) 75 am am ﬁanc‘ém ;{ fume.) °° {Tum «IOU12.
Pa”) = _7 [moH £35m «viralE 1 401 .So be. Cause. ‘FCb) 7.5 «.150 Y‘QGJVO—IVJJ F’th mu)“: bﬁ ImaZlnaVi I 2. (9 points) By direct integration of the Fourier transform integral, ﬁnd the Fourier transform of the
signal 9(t) = 6(t + T) + 6(t — T), where T is a realvalued positive parameter. Simplify your answer so that it does not contain complex
exponential terms. on _ Wt O
604') = f SOL+17 ed a“: 4' $(tﬂedwBJ‘ _—————‘ acacia“. git) 1S fear—Qvwlvcg— anoa an even Lno‘ém
% 'lslme:
alt) a) a) T T \ *V u L
We" Sl‘oJ‘Q appeal: that 610.4;5 us rewl col 2g an an QUGA ‘p/ho‘em % W. 3. (8 points) Given the Fourier transform pairs f1(t) H F10”)
f2(t) H F20”): show that
f1(t) * f2(t) H F1(w)F2(w)
3Y4: (tax F m} = a 'éwb
. t 56. m ﬂu; t\ic e OH: = [Em 'F;(tT)€:#V:L—b} £1: = gm [IiCltk‘CV} net on  (gt
= 4:, (c3 lime? it
 °° 1W ( 3
— g 41. (t\ e, c FL W
¢7 1 a '1qu gliﬁtqkﬁta} = FUN3 F71“) Problem 4: (25 points) Because analog multiplication is difﬁcult to implement over a wide dynamic range, the switching scheme in
Figure 3 is used to obtain a DSBSC modulation. x(t) y(t) bandpass ﬁlter m(t) k m(t) cos (not centered at i we switch Figure 3: Implementation of DSBSC modulation using a switching scheme. The periodic switching signal 2(1) in Figure 4 has a fundamental period of To = 21r/wc and the complex sin (mr)
___2_. 7L7I' exponential Fourier Series D" = x(t) —To _IQ 0 In. T
4 4 Figure 4: Switching waveform 1. (9 points) Given that the Fourier transform of the modulation signal m(t) is M(w), and that the output of the switch is
y(t) = m(t) 3(1), ﬁnd an expression for the Fourier transform Y(w) of y(t) in terms of M(w) and the expression given
for D". \ Bach“. xtﬁ Is av riflev.99 ‘va'lzmn’ Thurs} = 217’ E Dn 5(W'%n> Ira—ﬂ USN; the. "Nbvcnca convolv‘bvn Pro/hora Y0”) : MUM) a: XIW) a
2 mm.) at 271‘ “gym 5(W"7'7Ec) 10 co Ym = 217 Z on mama—g") ll 2. (9 points) Consider the DSB—SC demodulation scheme shown in Figure 5(A), where the input signal
m(t) has the spectra shown in Figure 5(B). M((1)) In(t) cos((0ct) 8%» 6(0 f8
cos((0ct) (1)
—100 0 100
(A) 03) Figure 5: (A) Demodulator with output e(t) and (B) the spectrum of Find an expression for the Fourier transform E(w) of e(t) in terms of M (w), and sketch E(w) assuming
that we >> 100. 80:) = m(e36o.s7'(wc;b) = mCt) + i; Cos (Zn/0+6 em = 12: awe) + true) 60422099 3. (7 points) A signal f(t) with spectrum F(w) is passed through a LTI ﬁlter whose impulse response
h(t) has the Fourier transform H(w), where F(w) and H(w) are shown in Figure 6. F(co) H(£0)
4
2
—2 o 2 m —3 —1 1 3 m
(A) (B) Figure 6: (A) Fourier spectra of the input signal f(t) and (B) the impulse response h(t). Determine the energy E, of the response nu.) .: F(w) H’lw) M
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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