examIII_f00

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

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

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