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# examIII_f00 - EE 350 EXAM III 20 November 2000 Last Name So...

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Unformatted text preview: EE 350 EXAM III 20 November 2000 Last Name: So First Name: ID#: Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Test Form B Instructions 1. You have 2 hours to complete this exam. 2. This is a. closed book exam. You are allowed one sheet of 8.5” x 11” of paper for notes and a calculator. 3. 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 4.2) Continued”. NO credit will be given to solution that does not meet this requirement. 4. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a. grade of ZERO will be assigned. 5. 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. . .4 “an..wsamgmmmﬂmmWﬂMMMunmannl mind-l. shah; s: ,..- . .- Problem 1: (25 Points) 1. (7' points) Determine all the nonzero coefﬁcient 1:)fl of the exponential Fourier series representation of the signal ﬁt) = 1 + 4cm(3t) + 4 cos (St + + 5 sin(8t). ET __. l g 2. —FC<—_3 = I + ‘fcoa (34;) + '1 {cognac (=55) — 5.0139 517235)} f gsmret) -_~ r+ €0,500 — zrg smash) 4 55m (61:) T T ’1‘ “*4 4‘ T T . 2. (9 points) Consider the fullwave rectiﬁed signal in Figure 1 where A = 3 and T = 21r. Determine its trigonometric Fourier series coefﬁcients do, an and b... Simplify the expresions for an and b“ as much as possible. You may need the following trigonometric identity sin(A) cos(B) = % [sin(A + 3) + sin(A _ 13)] . i f(t) | sin(-) S Q; 11 4F: Figure 1: A fullwave rectiﬁed signal with A = 3 and T = 2r. BecauSQ \$053 '15 rgaﬁ 6mg Queer bn :O anﬂ I. 35 AZ; LETS. «a = "E S #0:»er = \$5 8 “"(gﬂﬁi = 'Z‘ihv “’51” o 0 O _—_ .4: 0-4] a» ‘10:- ? I ’22. W 2 i an: “a oos(nw.~to€-Di4= = —.,:- Qm (\$+)Cos(l.}_ﬁa 1:394: '1‘- x : g-‘E'SZSinCEE + £34191: (Li: + ETﬁ'jZ-Sm ' 2T“? )1: DQ-E, d . o . 3. (9 points) A LTIC system has the frequency response function 10 Jul/100 + 1' Hod) = The sinusoidal steady-state response generated by the input f(t) = Co + 01 coe(wgt + 91) is known to he y(t) = 2 + 10~/§ cos(100t + 45°). Determine the values of mo, 0,, 01, and 01. I o ! Wool: m “M = "W too _. l0 [*(IOO)I -' E o [Hruvlz lo 17-1 4””) :0 ! 1 Problem 2: (25 points) A certain periodic function has the complex exponential Fourier series coefﬁcients shown in Figure 2. For |n| > 3, all coefﬁcients are sero. Figure 2: Fourier series coefﬁcients of a periodic signal 1. (2 points) What is the average value of the signal f(£) ? 3 The, average or DC, vqlcg 7%. "F (f) '5 DO *7 O ' 2. (5 points) Without determining ﬁt), state whether the signal f(t) is real, imaginary or complex valued. If you use a Fourier series property, you must state the property, but you do not need to derive the property. I. Be-CGWSQ— bn 'r" b-n J 4ft) :3 l 3. (5 points) Without determining ﬁt), state whether the signal ﬁt) is an even or odd function of time. i If you use a Fourier series property, you must state the property, but you do not need to derive the g property. Berccwoe bra = ’b-nJ \$4133 :5 I 4. (5 points) Determine the power of the signal ﬁt). a” i :: IL : l- L-l— [~[2'+l 2‘4» 3 2' i Ep 510 n all 54 j} 1/! = 7 _,_ I + } +~ ?‘ p; Ho % t 5. (8 points) If a. periodic signal is expressed as an exponential Fourier series on f“) = Z Dﬂejmot! n=~m then the exponential Fourier series for N) = ﬂ1L + T), where T is a constant parameter not equal to the period of f0), is given by no fat) = E Dne3m0t. ﬂ=-OO Determine a (4 points) the relationship between |D,,| and |Dn|, and o (4 points) the relationship between 4D,. and 11-3,, CD D egﬂwat \$08) = “2:10 n 7- t a, novon +10 no awe) am {:Ct-I-T 3 =— i ‘30 61‘ ': n2, e W “1“” z-w —-_- ‘Pftl " 1 Problem 3: (25 points) 1. (8 points) By direct integration, determine the Fourier transform of the signal = -|t-4|= e4“): 124 ﬁt) 8 {ct—4, t<4. 09 FM = £4 (1:) 2 with .. ‘i 'b-‘f 7M4: ‘m—(t-v) —w+: reeaae +36 J11;- —-w 1 ‘1 .—.I-f (1* WU'E to "(H’ W)";— =9 S e 64 0Q 4' 6‘43 (2 a if -a, ‘1 co *(H—wﬁ: .q I (“goat ‘1 q I / 3 e 1-: e. + e —(J+dw) y -(l+ou)'1 -1 I (Paw)? 61 I [5 "6 i J : a @[C _]+ 1,047») e 1—fw 1+?” 7"“? __i—l* W + l' w 3: e, l+~w7-' ’"HVL 2. (10 points) Using only the Fourier transform pair rect(t) 4—0» aim: , and Fourier transform properties, determine the inverse Fourier transform of F(w) z rect (ml; 5) e-BJw. BO. evident? Pct? H zrr Sic—“Q, Si‘hc é—‘J 2.7T mcfbu) : 2,11" rpc'bfw) '2. Barswhqyo Hunt) 4—9 7t; F(%—,> am; Emmy in Sync. (5t) é——’D + red: ( --'- 3. (7 points) Determine the energy E! of the signal f(t) whose Fourier spectrum F(w) is shown in Figure 3. F( 0)) 69 E4: 131;. leUvDIZ‘ 00w @ 2- ‘f ngcﬂw + ~71; \zﬁﬂw o 2. :: IQ'Z + 2' V T,- 71' 10 Problem 4: (25 points) The system shown in Figure 4 i used in a communication system to ﬁlter the signal f(t). The frequency response functions of the ﬁlters and input signal ﬁt) are shown in Figure 5. f(t) cos(500 t) Figure 4: Subsystem with input f(t) and output y(t) that implements a. ﬁlter for a. receiver. H 1( 0)) H2( 03) i i (I) 0) 400 400 «600 500 Figure 5: Fourier spectra. F(w) of the input signal ﬁt) and the frequency response functions 31(2)) and ill-2(9)). 11 l. (12 points) Neatly sketch the Fourier transform of the signal g(t) (the output of the ﬁlter with frequency response function 3103)). P/ gm» .1 EM) uL {new (‘05 300.223 _: H,(w) E Ftw—Soou + PM» +9903} Z 2. Gyfwﬁ 12 . 2. (8 points) Neatly sketch the Fourier transform Y(w) of the output signal y(t). WW) —_ Alma ’Cflggmcosroot} :_ Hzav) i (amt—Sou) + G(W+foo)j Z Z. 3. (5 points) Comparing the Fourier transform F(w) of the input to the Fourier transform Y0») of the output, what type of ﬁlter (for example, lowpass, highpass, etc.) does the subsystem shown in Figure 4 implement ? Tﬂ‘Q 5d S‘Lzm /my;/an’) Qn'éf CL, any/3d}; "cﬁ/éﬁngy (\$4.55!? b'E‘VOQh (00 [W 4200) 13 ...
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## This note was uploaded on 03/17/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Penn State.

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examIII_f00 - EE 350 EXAM III 20 November 2000 Last Name So...

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