quiz2bsu09_soln

quiz2bsu09_soln - GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of...

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Unformatted text preview: GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL 85 COMPUTER ENGINEERING QUIZ #2 DATE: June 26, 2009 COURSE: EOE} 2025 NAME: ‘3 I) is. " -' 3T3“- " STUDENT as; LAST, FIRST Recitation Section: Circle the day 85 time when your Recitation Section meets: (1 point) L01: M 2:40-4:25 (Taylor) L03: T 10—11245 (Chang) L05: T 2—3:45 (Rom‘oerg) L02: M 4:40-6:25 (Verriest) L04: T 1100-1315 (Chang) LOG: T 4-5315 (Romberg) 0 Write your name on the front page ONLY. DO NOT unstaple the test. 0 Closed book, but a calculator is permitted. However, one page (8%” x 11”) of HAND- WRITTEN notes permitted. OK to write on both sides. 0 Unless stated otherwise, justify your reasoning clearly to receive any partial credit. Explanations are also required to receive full credit for any answer. 0 You must write your answer in the space provided on the exam paper itself. Only these answers will be graded. Circle your answers, or write them in the boxes provided. If space is needed for scratch work, use the backs of previous pages. Problem 02.1: In parts (a)-(d), suppose that when the input to an FIR filter is 5M, the output is —5[n — 1] + 36[?t — 4]. (a) (5 pts) Draw a stem plot of Mn], the impuise response. {15 I; l i i { "'-"'#'-:-fll- '5 '—€§"-df) new ‘2- .09“ ca _..-’ =- h (b) (5 pts) Write the diflerence equation for the filter relating the output yin] to the input fits/5 2 FAKE-Vie!) + 3 «Met-l (c) (5 pts) Suppose we want to filter a signal xx in MATLAB with the FIR filter described above. What should hh be in the command conv(hh,xx), i.e., hh = [something goes here] '? hi”: {01'1-10101 3-1 f’: (d) (5 pts) Suppose a signal a:[n] = 3E1)“ is input to the FIR filter. Give a formula for the output y[n] which is valid for all n. h V1”! h-I’L I"! h+ $3,135 '3'." . — r’Bf-Fl 1—71") = 3“ ) ”' gm- t. l Elwin (e) (5 pts) Consider a cascade connection of two systems; i-e., the output of the first system is the input to the second system, and the overall output is the output of the second system. [ 1 Suppose System 1 is described by the input /output relation to n = xfn] + 3x[n— 1] —2a:[n—3], and System 2 is described by the input /output relation: yin] yin] nvlnl In ECE2025, we’ve learned that we can often change the order of systems in a cascade without changing the output, i.e., putting System #2 in front of System #1 doesn’t change y[n]. My won’t that work here? Be brief! ' r“ 750%”: «(r-j? '7' Pvt-4'5 H1 Vfi‘ft WWI-ll 6 17 t? d f . . .. .- -. e71; , H.) {D .P {laqn‘ _-’ j (L. ' I J _ 1| ' -- fflfis‘af l (his?! 3‘4" " / \ -U.1n Now suppose g[n] e' u[n]. (f) (5 pts) Simplify the convolution fin] :2 g[n] * 6[n — 5]. .- 6 MEHAQJ (g) (5 pts) Suppose an LTI system has the output e—U'1(“+1}u[n + 1]. when g[n] is the input. Is this system causal? Explain why or why not. W i 3 {n3 (5'4";:T1-_}l__+5 n : a r Problem 02.2: f (u g,“ (a) (8 pts) Suppose 11:6) is given by 1,! {I DO 20 _ ' ma) = Z G (2u__40)e—3(0.1)1rkudu} ej(0.l)7rkt kf—OO Sketch 95(t) fort between —20 and 40. Explain your reasoning. (Hint: Do not try to integrate; the form of the equation should be enough.) 15 {f a ‘ I“ J. I“. a? G )1 (41,535 filiggwg/ gig ‘ £2795 u” :- we? (b) (8 pts) Find the nonzero Fourier coefficients ak of 32(3) 2 5 — 5 cos(3007rt + ff/3) + 83in(7007rt) 5 ’5' 15 or _ r f:- 6 CL 0’3 .3 g. ,- .- ..u w! , 5f 5? 1A?) 1“ if.) Hz"! 3 Ar“; (c) A real signal, w(t), has period 0.01 and these known Fourier series ooefl‘icients: an 2 1.5, {13:4 2e'j’r/2, and a7 : —3ej’r/5. a ... “ is} 2;"!er t} «- 3 (i) (8 points) Find a_3 and a_7. w r” .«s- . ‘L £43 “1': '" 2“? 9 ' . «n (“if "" 5' 'éufl’if v- "2 first 3"; " "26 ~ J‘- (11) (8 points) Assuming all other FS coeffients are zero, express w(t) as a sum of cosines and/or sines with zero phase in each term. - 4 ...---~ mrlwr -‘.=I - '1‘ me) +— 7 5/ 1;.) ,:";:s( w '4:- .’ 2. a \. ____J .' «If/15' + 36” C ‘" “er/w ! _ It; glqaiflbfgfifit + U :P‘? ? 1.2 f "II' * gar-ng ‘ 77 L.) é -* ‘5 e 1 .. 4‘ C” f 1" # “gull/FA I: I évfi" l ’7' ' “If *5 ’WMWT ‘ ,fiflmu-s- —- Problem 02.3: Consider the ideal sampling and reconstruction system shown below. a: liki‘d] (fin-E} (Hm-2.11:? .r(r) 3’0“) I '} ‘ It '.--(..7 (.‘mm‘t‘l'lttl' 1 |f3=Ts (a) (8 pts) Suppose that the discrete-time signal who] in the above figure is given by the formula = —2 oos(0.21m + 7:14) If the sampling rate of the 0th converter is fs = 10000 samples/second, many dif— ferent continuous-time signals aft) = 563(3) could have been inputs to the above system. Determine two such inputs with frequency between 3000 and 15000 Hz; i.e., find x1(t) L A1 cos(w1t I- Q51) and m2(t) = A2 606(th + d2) such that who} $101115) mfinTs) if Ts =1/10000 sew. f (a s it” e ’3 {is I. "a: E if» é P i WWW“ -.-,__ .. w.- , f 0+ 2r x; 3 s a Q ,_ ‘ r? J F 14 wk. (/3. J j: “,4 my} {yams J r- a “I (if; :1 w (irfigfij - “we Pass Elks . a . 2.} {J r 97' 2'” 3’ '2 8‘ a. =i ( pm I I r j I «fast: ' 7-: V! 1 “it? (b) (8 pts) Suppose a periodic signal m(t) with a fundamental period 0.01 seconds is input to the system in the figure on the previous page, where x(t) for 0 S t S 0.01, is given by x(t) = { sén(27r100f:) U for 0 S t 5 0.005 for 0.005 S t S 0.01 Does there exist a minimum f5 for which the rectified sine will be reconstructed without aliasing, i.e., for which y(t) = mot)? If there is, give the f3 and explain how you obtained it. r .'i If no such f3 exists, explain why. (Hint: Think back to Fourier series.) * = - J -' .1. \ r" . ‘ -'."..«“ :' f .3. :3 1 a is" t; i: ‘1': {5“ “1“ 0:: (I II I 3"!" Va“, fl K a “ e " ‘ yak Sf“) ". a "-i fax" ’°“" (c) (8 pts) Consider the following piece of MATLAB code: 121.". XX 1: (1/10000) :3; cos(2*pi*(100 tt"2 + 50 tt)); soundsc (xx , fs_playback) ; What choice of faplayback would cause a 12 second long sound to be played? “K- "I LI) if J.» E i? g _.__ rec; (d) (8 pts) Suppose we make a digital recording of a pure tone with a frequency of 256 Hz using a sampling rate of 8,192 Hz . We can make it sound like other notes by changing the playback sampling rate, which will change the frequency. What playback sampling rate should we use to make it sound like a 384 HZ tone? ...
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This note was uploaded on 10/21/2010 for the course ECE 2025 taught by Professor Juang during the Summer '08 term at Georgia Tech.

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quiz2bsu09_soln - GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of...

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