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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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