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Unformatted text preview: EE 350 EXAM III 10 April 2000 Name: Soiu'honS
ID#: Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO  


 Test Form A Important Guidelines 0 Answer each problem on the exam itself. 0 The quality of your written solutions is as important as your answers.
Your reasoning must be precise and clear. Your grade will reﬂect the
clarity of your presentation. Problem 1: (25 Points)
For a. given signal at {¢1(t), ¢2(t), ¢3(t)} deﬁned over the interval [0, T] <¢h¢j>=0 5%}. <¢21¢2>22
< ¢lr¢1 >21 < ¢3ﬁﬁ3 >: 3. A certain signal f(t) can be exactly represented in terms of the given signal set as f“) = 3915105) + 4452“) + 3953(1) over the interval [0,T]. For t < 0 and t > T, f(t) is zero. 1. (5 points) Is the set signal set orthogonal ? Is the signal set orthonormal ? Justify your
answer in one or two sentences. . TjQ Slgnag. set is orthogonag bacmse. =0 'Er L¢0L «All SiamQ 92:1; :5 not magma mom «was; 7L;
«ma 953,43) i=1. 2. (10 points) Determine the value of for z' = 1,2, and 3. 3. (10 points) Find the energy of the signal f(t) by either directly evaluating < f, f > or using
Parseval’s theorem. b. Because. the. enema Ea 3.3 the. error mam/Q u Ezra) 3
<49, {0 = c7; Mm.» ll <T<¢.,¢.> + C721<¢,,¢;> + <34¢:,¢3>
= (ifl + (W‘ 2. + (3)23 3 ' «39) = 63 Problem 2: (25 points)
A function g(t) is described as 9(1) = cos (3m + + cos (47d) cos (2m + . 1. (5 points) Determine the period To and fundamental frequency we of the signal g(t).
ZII'
50k) = COb(31T'tr ~§) + é" COS(ZIT+ '3‘?) + .212— (93(611£ 1 _ .L
a“: i Q = Co5(3rri:+ 31rTo+ 7—35) + Jacqui + 2W» ’.,"')+ zcdsénbP 611% + 15.) For act)=g{h+ Tax) we, nepl 31TT. : ZFriJ 2W7}: 211,1) 5?; Tc: 27.76 (“here "a m.) Ong P 6W1. minder? to FWU“ S’WV'wﬂebﬁli. eta er) => cur = :2 :5_
The 36%. pg, loaders SwElsiyud the 0‘64: 01540"? "L n 32”) JP 11
M
q
i :1. This ewes To = Z anL We 2. (10 points) Determine the exponential Fourier series coeﬁcients D” of the signal g(t). 3G5 :L J: cos (ZV‘t ‘35) + w(37rt+ + i C95(67F'é '4' = C; Co5(2uo.t +62) 4 C5Ce5(3ua.t' 99 + CfCo5(6ubt +96) arc Thq C 'E hiaonomeﬁric. chhQr SQrHZS Coegicm'hr
Dmpsur Co ‘" 0 Calt) has zero warave, Janine) = =3:JT (=J‘)96$"‘
CL: 1;)913'L;) C3 .383 '31 c’ L v a 2 Co = O
Th9. erponoa'hﬁ FaurlQr SenaS caﬂ‘c‘oaﬁ '2’ alga" %' bo no: 2%. am MO. 13." = 93— e15" . ﬁler/3 _ LeﬁT/‘f 0°: 0 D3 rite / I)“: 'J‘r/‘i
. ' 2.11 5 e bz ‘ #e#W/li D_3 :2 d 3 bé y. b 3. (10 points) A function f(t) has the trigonometric Fourier series representation f(t) : 2 + :2 3 sin sin (4m). 112112 2 The signal f (t) is applied to the system shown in Figure 1, where the frequency response of
the system is speciﬁed by w
H(w) : «28—Jﬁ for 11 5 w S 17 ,
0 otherwise Determine the steadystate response of y(t). f(t) y(t) Figure 1: A system block diagram with input f(t) and output y(t). The. "JpnE S'lgnaﬂ. 1M Jeff”
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0 stem (to —n s) + ems) 30:) = 5135*) "' as“) am = — .15 Comzt 1 n5) Problem 3: (25 points) 1. (9 points) A LTI system is described by the impulse response function (I./ h(t) = 2 er” u(t h 4). Find the frequency response function of the LTI system by direct integration of the Fourier transform integral. No credit will be given for other methods. I “M = i w“ (*3 e”“"£+ = arc3+ mm H
m «f w+3){7 t... gwﬂbt
:: 2] e d =  i Q
q d‘ ‘1
O Phase in Degrees 50 100
—1 The magnitude of H(m). The phase 01H(co)r —10 —5 0 5 ID 15
min radisec Figure 2: The frequency response function, H (w), of a LTI system. 2. (4 points) Char acterize the system described by the frequency response function shown in Figure 2 as being either a lowpass, highpass, bandpass or bandstop ﬁlter. b Beewan. any Furcage"? wmr9n0n£ cure, paJSﬁg the add tens Ts N jakpMS 79 Hush 3. (4 points) A signal f (t) is applied to the system shown in Figure 3, where H(w) is speciﬁed in
Figure 2. Will the characteristics of the ﬁlter allow an arbitrary signal f(t) to pass through the system with out distortion ? Explain your answer. Figure 3: System block diagram with input f(t) and output y(t). ' 81wa £142. pinup. anﬂ. m w.” 4.5:;ch we), 75$ 15 re‘sfonse ARC”) ‘3 m4: £11.54»? 7» ‘4’) weapon” was); Is radé Constan'éJ the. {Hei admin/Q.
? in) g: on“? (e ~70) J 4. (8 Points) If f(t) is a realvalued function, show that F(w) = F‘(—w). Hint: It may be
helpful to sisalt with the expression for f(t) given by the inverse Fourier transform. .. .1. ‘° J"t C
FUD _ M {wage 09M; I)
Became, Ts an. Lunc‘EIOrI) «FCt):F*'fb) when
“° 4:
°° wt 1“ —w i
W =+"'(a = (ll—T, Lam? ﬁt) = gag:ﬁwm i Problem 4: (25 points) The demodulation system shown in Figure 4 uses a lowpass ﬁlter with the frequency response
function b 10] OSwao
H(w)= —103 —w,, 5 w < 0
0 [ml >wo cos(wst) 6 Km 11(0)) Y0) sin((no t) Figure 4: Demodulation system. 1. (6 points) Find an expression for the Fourier transform X (m) of 3(t), and express your result as
the sum of impulse functions (your expression should not contain the convolution operator). X09 = costSE) San (wot) ,._/
Sri‘QSCwsﬁ Si‘c‘votﬂ = '5'”; UL {G’swﬁj a»: sm 004:} X ('5 Ex
H II ‘ ffgcwngq SCI» w5)]*d21r [—Stw+wo)—8(w~Uoﬂ 3’ 7L1 Scw— «15):: Siwrwa + S’CW‘PWQ‘: A'LWWaD
L I: . 3 Cw ws\\l: ['(W'Wo) " 5(WW5\ té'tuIwa1 ":5 [8(w4wa_wg+ soulwe fws)  8(u—kbw3) —§(lu—w¢ «us)1 b 10 2. (4 points) Sketch X (m) using the graph provided in Figure 5. Clearly label the location and
weights (areas) of each impulse function. X ((0) Figure 5: A blank graph for sketching X 3. (9 points) Sketch the Fourier transform Y0») of y(t) in Figure 6. Clearly label the location
and Weights (areas) of each impulse function. Y(w) Figure 6: A blank graph for sketching Y0»). 11 b b 4. (6 points) Find y(t) by direct integration of the inverse Fourier transform integral (no credit
will be given if you use a. known transform pair) From 95 mm. C You}: 511 S—(w+w°w_s) + $11 9(W‘Wa1'u’5) a) ad.
0109: mm? w 9° m on .Ul: W15
‘= S—Cw +wb'ug)ev‘ Jay + % g &'(“V’Wo "Wﬁ Q'OL 0L
'4? 0O S. acwo *WJ)‘L"
"7: Q + 52: 
. ‘ (“0 ‘WJ) ‘5
ei yCE) = .5' cos (moow )1: l2 ...
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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