# ECE 4213/5213 Test 1. SHOW ALL OF YOUR WORK for maximum partial credit! GOOD LUCK!

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1 ECE 4213/5213 Test 1 Fall 2018 Dr. Havlicek Monday, October 15, :30 PM - 5:45 PM Name: S_O_L_U_'T_I Q_t\l Student Num: Directions: This test is open book. You may also use a calculator, a clean copy of the course notes, and a clean copy of the formula sheet from the course web site. Other materials are not allowed. You have 75 minutes to complete the test. All work must be your own. Students enrolled for undergraduate credit: work any four problems. Each problem counts 25 points. Below, circle the numbers of the four problems you wish to have graded. Students enrolled for graduate credit: work all five problems. Each problem counts 20 points. SHOW ALL OF YOUR WORK for maximum partial credit! GOOD LUCK! SCORE: 1. (25/20) 2. (25/20) 3. (25/20) 4. (25/20) 5. (25/20) TOTAL (100): On my honor, I affirm that I have neither given nor received inappropriate.aid in the completion of this test. Signed: Date: 1

2 1. 25 /20 pts. A discrete-time LTI system H has impulse response h[n] given by h[n] = (1)n u[n-l]. The system input is given by x[n] = (i) n (u[n] - u[n - 6]) = { (¼6,n' 0 ~ n ~ 5, otherwise. ~(k.1 ~)"" ~k \ h Cl<--1'\1-:. hf")~ l c~)ti.jck fbrrrr.~ \-'r\ ----=-=-=-=-==--,,_,_-=--=-~=:::-=~o~,,,,_,,,> V\="-l t~)~ -l ~) :)} \ ~ W\ ' t., ~ (l)v\ ) 1/\ 71 b 2

3 0Tl+~vt WA'l ~ More Workspace for Problem ~['A):: r: ~ [\(.J1lC~ --~ J \c: ~ -IX> --'-d I) '-.S- I} 1 b,,, l -x (t<.) lt)~ _Lhfn ~k. () s 0 \ V\<.\ '2-((~)~-t~)l\1 J, ~ ~<.b b-ll)~ 1 "'.,, b '= ~ ~--_,_, ---=1> 3

4 2. 25 /20 pts. The input x[n] and output y[n] of a discrete-time system H are related by y[n] = x[2 - n] + 1. (a) 6/5 pts. Is the system H linear? Justify your answer. Le+ '1--t t' ;() -;. 0 -v V\,_,. ~ t t't"1 ~ 1- V V'f L-e, \-- 1t 2 t\-\) := o 'QI/\ _., i1-t)1),-:-, :fl.. YI r-1 Le~ ~= 2 ~~ b ~ '3. Le-\- ~ 3 t,a ')-=:- ~ ~1. ~1/\."1--\- l1 ~z 'C" 1.:::. () J,I V\ ---,. ~~LIA)-=- i \-I II\. 5~\A.(e a.~tl"-jt \o11:cta1 t- ~,; rta1; -rke '5'f~etAA tr t0dt L-f N~L :pt,an- n,eo_ a.. 4

5 Problem 2, cont... (b) 6/5 pts. Is the system H causal? Justify your answer. LeJ 61t:t'n:) ~~ ~e.. ~"'- \>'~-:t f,jlf\."- \, \J "'-e"' vy ==--0 J we.. ~ut. \J e., :f Co 1 = --x C: -z.1 -r- i.. w"'itvl- J-e,~.e11dt, M +1tt~ -Gv,"h-t\ff - \l/\~l.il~ "'/. C"2-?, (c) 6/5 pts. Is the system H BIBO stable? Justify your answer. Let -x tla) be. tt t~ej., \f\.\>v---\- s,~ Ill~\. T~eV\ 3'5 ~ t\'l 1 B-;, 0 J S\.,d.A ~(d\._..\ \""'t-(v\'1 \ ~ ~ ~ "'E ~.. NDWJ \ ~ tvt~ \ -:::. \ -X C 1.-Y\1 -t<l \ ~ \1<..t1.,II\)\ 4r 1- th~" e.jore., '1 c ~ ~-ti. <.-:-:::,,... >, 1 ~ti-ea by 'B +.i. P,,,..). e" ery 1ovV\kd l"'-\)ia.t sl~m\ 1V'11t\J c0-s a. bo~ c;u,\~t" 5t~V\~\. \ i J: 5 1? \Bb Sf/1..'e Lf l!i- (/' e ~Wl~--) ~ S'f f r ---:::-..:.,.

6 Problem 2, cont... (d) 7 /5 pts. Is the system H time invariant? Justify your answer. --- 'T~tll\ y, t'i\')-=,( [ '2.--nj ; u (1'2. - V\) -t' 1-. Le.+ ~" =- 1,. T~el/\ ult ~-'nb 1 :: \j ~ rl'\-f) ; U[ 2-eJ l Ti. j j g-;~f Now \e:t ')(, lv\1 = % t t'4\-'ao1 =:.. 1<.t r~-r) = IAX'A-r) u V',):: "'l r 2.-'f\ ::: U. Le-, 1 l., Jl- L {:).:. '2-Y\ '::;. \A. ( '2. - \I\.- \ 1 ~ 1- ::; vl'c \-1/\"")-t-1. -= Lt t_-i. - LV\ -1) J + i. ::;;. U'C. 3-~j -t ""'111 '--.1 ( ~iii 3 I 2 --l ~ !!u"p- ti\ 6

7 3. 25/20 pts. The input x[n] and output y[n] of a discrete-time LTI system Hare related by y[n] = 4 x[n] + 2 x[n - 1) + x[n - 2] + 2 x[n - 3] + 4 x[n - 4]. Find the system group delay r 9. :lrn===---r : y { d"') ::: X ( t'.,lw) [ ¾- t- "i e.-j""' -1- e.-\ '2.-;,..I- ~ e.,-.i,...,-\- ~ e J -l '1-w H lei"'):: YLe:i") = L-..L +½e_-i~+ e-~i~~½.e..-33~ ~ke-.i~ '1 X(-eS'--') A: ~ ) -\ jt.w l ~~ A \ e-j'~ l. e-j1.wj e-12~ L - 4e 4--i:.e ~ L 1-1- COS'-> +- ~ Cos '2.w] e- ~ '2.~ ~ arj l-l le jv) SfQr"'1rt'll\ ~"'use.~ ar-i ~lei'-')) == 9 (Lv) ::. - '2 ~ e "r>vt d.e¼y -= l:j :: - JwGltw) ~.i ' 7

8 4. 25/20 pts. The input x[n] and output y[n] of a discrete-time LTI system Hare related by 10 1 y[n] - 3 y[n - 1] + y[n - 2] = x[n] - x[n - l]. 2 (a) 5/4 pts. Find the transfer function H(z) and give a pole zero plot. l - ½=e.-1 ( l - ½ 2-, ) L.1-3 z-~... ierc?'> ~ ~.::. 0, :z -:::. { -rolh ~ t=-\, ~ ~ 3 ff~', ' \.,LG ' l+lb) = fl= 1e)(t~30) f,-=-,- ~e -t-'38 l0:::i = a 1~56 ~ \ - -=. -, - ~ \ \ J.. - ls> ~ B +_ 1-19 _ 1~ \ \ l:, - -f " 8

9 Problem 4, cont... (b) 10/8 pts. Assume that the system His causal. Specify the ROC of H(z) and find the impulse response h[n]. For this assumption, is the system H BIBO stable?!-\ ca.u.s(.(_.l Wttv\S '"t1ae R,l'L \~ ev...-kbttlvr -h i~e. \&t~ei..\ ~e,\e, -- - fu>l-:. l't.\ '7 3 S'l'l\{e.. +ke \U)L doe.~ -- ~{ft" Cc,t\'t~ill\ ~e \Ali\l"\7 t':t\j'"'-l e, t...\ g ~,rt ~\~0 S7k~ t,..i;: I =>--= ~-t lr) ::: \/l 6 l S /i,ja + - l -j l"l l-,rt ~ ~ l'l' \-, \~ \1:\-,3 - g

10 Problem 4, cont... (c) 10/8 pts. Now assume instead that the system His BIBO stable. Specify the ROC of H(z) and find the impulse response h[n]. For this assumption, is the system H causal? 5',"<e -.+"'-.e. RDL 1s ttf/\v\\j--\~ I/' 1 ~(vl) \s \\,..,p-s, tled ttv\d. ~e s '1 ~keva \s ~ 1. at I.As~ \.,, u t~\ =:: \Jtt~ \--~ :e--\ + \~~b \ - "3 ~\ v-v----) ~ \~\ ') 1 \~\ ~ 3 10

11 5. 25/20 pts. The continuous-time LTI system Ha(D.) shown in the figure below is implemented using an ideal A/D converter, a discrete-time LTI system Hd(eiw), and an ideal D / A converter. r f=r ~ 44-1 l ~i-\1 == lot> H ~ xact) Ha(ejro) yact) -fl' -:: '2.. 'Tf F-r - 8~, iuo-rr r.d: ' - \ sec. '2."if - T.L ""' ::..JlT - F,. - 4~> '" ~c. As in professional compact disc (CD) audio, the sampling frequency of the A/D and D/ A converters is Fr= 44.1 khz. So Or= 88, 2001r rad/sec. All input signals xa(t) are bandlimited to IOI < Or/2 (this simply ensures that the overall structure shown in the figure will be an LTI system). The impulse response of the continuous-time filter Ha is given by ha(t) = sin 70001rt. 1rt (a) 8/6 pts. Find the continuous-time frequency response Ha(O). TtthLe ~ s\~~!. 6-=) \ i :] :1-,.{).. w::; -tooo,c, -1/J C vi - -,()Q()'7l (b) 8/7 pts. Find the discrete-time frequency response Hd(eiw)...n. i ) \w\~ r I ~ l r.,.j\ < 1t \ - 70()01f i ooo 1( 44J \ob 0-1' ,t(JP 44-J lfjo 0 l i!!f!!jj I) (..J 1\

12 Problem 5, cont... (c) 9/7 pts. Find tlie discrete-time impulse response hd[n]

### ECE 4213/5213 Test 2

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