# exam2 - Exam 2 EE 223—04 jgh Name 30)cmwx'ms closed book...

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Unformatted text preview: Exam 2 EE 223—04 , jgh 5/14/06 Name 30 )cmwx'ms closed book with no calculator; one sheet of tables provided (50 min.) I have neither given nor received unpermitted assistance during this test. (signature) 1. The transfer function for a LTI system is given by H(s) = (s + 1) / (s2 + 53 + 6). Showing all steps in your solution, determine: i (4)a) pole/zero plot _ _ gm 0“) 3 L; - \ 8—H (8 +1) ts + 3) \Ats) : 3 “ ‘ ‘ P 7‘ ‘2) n 3 (10)b) impulse response of the system, h(t) ham} 4—) VH3} ‘4‘ + ‘47.. \$+l 5+3 Algebra (9 -'l 2:; +3 ., ‘ ._, \ 1 M 2 ‘4‘ ‘ LS'X'ZJHQSWv-z s+&\‘.s=—2 HLS\ : 5 l i z 2 i 5 g , l i 14L: (snowy,th -; 5—H s,l : 2 \$34. ‘35:»3 L— H A», (2) 3+2, “*5 ll\ : -7, 3-6 , i with, ~ 3-6 fut-+5 4. 2e’ W“) L ‘ I” i i l i l i (12)c) step response of the system, s(t) war-3 : 1,, La 4—.) A a keg) \(m “a mums : Jig-w a: 2.4.9 + \<\ + ‘42 SL\$+ 23mm 5 8+7. 5*3 \éo—a 5 Hails, : 2 ,_L 'o \$k+w55+g Q, I 5:0 Olgy'vim Q rs ‘4‘ : {an W5) is” l 3 5+» l i 13.2-2 + 1 x4L:(srar3)Hz5)lj~ 3: 3‘ \ ; -3 " SC\$+13 S~-3 ; . LVQ L 3-3 <_. 7—13\ \E - __ 7— a ‘f‘ (53’ S 3&1 A" ﬁ—y's f _, x -3-4— *c ‘Stﬁ - Em?) 3,, ALe_L+ M“); alga aw, i; ’C ‘t 73 34” ~ «31' 9.9 we : 3 ht’wc‘Y : LEE—6 um +26. “Mk” 3? at __ ﬁx; ‘1’? It J‘s/(aw ‘_ e - I? \t‘ ’— n; 8" \o “ 50 <~ Q )0"? + O 2 l 1 a 3 . ii (4)d) differential equation that relates the input x(t) to the output y(t) - 5“ Yrs) - 2 - WS — -——-—-—-—- “A .—— : S 4-5‘ Yrs) - (5%))(15) i s 2 +§5 +c MS) 3 ( 5 w) ‘ ,3"(<—) 4- Saves) +4, 2 kw”; ‘Hd'H (4)e) frequency response of the system, H(a)), i.e., the Fourier transform of the impulse response J (ad—H \ e)- on Maw-50mm ‘ é-w’M—JS‘Q ‘ 9953‘ 2 H1W\ '3 5=JLo ‘ (4)f) if the system is BIBO stable with justiﬁcation \—\£53 \3 9% WW W Pg log 6. Lute :3 V5ng (4)g) if the system is causal with justiﬁcation M09; 2: o ,éoir + «0" => causal (12)2. State the definition of the Fourier transform, and then use the properties of the Fourier transform provided in the tables to determine the Fourier transform X((o) of the signal x(t) shown. You must show all steps in your solution. I 5L Lt) - but cl (1'1 xug) 2 goo—Xi-H € 4: ’4’ i 43 y \rec'rUcB 4—) SWCUEFB' ‘ x 2 3 ‘t 4 SW9 Gilt: p to i Q. {is chLm(t\\= ﬁSw‘cirtl 3L.) 4' rear (A w» 4 m3: (41+) <—> 1 a k 4V‘ec’r(";(‘l7‘ll)4‘5 '6‘ 95mm 2. Au 8%LL9r—r-l men ': ‘3 Q ‘- Jvﬁ‘z M+-Z @sl (10)3. State the deﬁnition of the Laplace transform, and then use the deﬁnition to determine the Laplace transform of the signal x(t) = 46“ u(t); don’t forget that you must specify the region of convergence (ROC). 03 x13) : f 1(ny 9" SJ: gt (UhQ-«SK‘MCU O- ‘1’ - -st ‘13 —(s+st x5): f 46 game at“: = f 4-2 .) At 0. 0- c __ s+s")t 0“ t 4. Q. NVHK \$2 quid” (3 +5) 0. Jun-PS ﬂat :1.— :M‘iﬁe(}te ]‘(s«-s) Same {9‘55 : 4 4dr G¥S>O Sk-S' fl; Sat—5‘ Boa 7, (12)4. Sketch the Bode plot on the graphs provided of the transfer function H(s) = (s+1) / (s+10); you must show all steps in obtaining your solution, and provide each axis with a scale, units and 32:3 ‘ label. I \ H 04“ J6 V 7' ' . ( (dog: \\ _ \ aw 4'0 K / H (w : sass -=- x0 / pl 5:»; Jaw u x/ / - .1... \ +» a 6%) , x )Q g, _,A w , “ A'Jt/m) 0‘0' I0 \JOO moo‘dfmiisi ¢QQ¢M§ A20 ’w- k I—- 4— >‘\: ,— W‘“ X z 20 in H» w “in "15x ‘ \QB Cato i 4 3‘ (ud:\o> , a L “ginkgo To to 2 ‘12 l—"Zohcyo LH-Ud) (“at”) 4— 10 new ( mtg/ow l 5. A relaxed LTI system is described by the follovﬂnidifferential equation which relates the output y(t) to the input x(t): ‘6 “’3 COW‘QCi‘e W“ “(L W Y”(t) + 3Y’(t) + 2 = 2 KW + X(t) +3 I ma -- - —%E}r Showmg all work, determine: (10)a) impulse response of the system, h(t) {We +33%; +2‘dé4d : we; 4-th) with 3"”): ‘5”) ‘0 (“Emil :3 (51 +3s+zjvm = (254435445) Yd) ' 2 +4 iZS‘l’l ~ Zr 14,; “‘5‘: w) ' :5 = '+»\( ' “3:” 2:» 3 +35 +2, [3 ).S+z\, 5.. 2 - ZS+\ - LN) +4 _-\ ‘ \4 - - M2 ~ —-—-. : - l \ 5*; iS=-\ tax) “*1 ' t 254'! .4, .. ‘ “53‘5" S=~2 L—u H t 226 ‘- m5) : :43. A, ‘3‘ <~> - a‘iuwt JP .36." Mr} :; \nlﬂ ‘ ‘ SM 3472 (4)b) frequency response function, H((o) ‘ mum HtW}: H“(S)\S:360 w Gwyn!» 3C;w\_i,2 \ Jr 4 2w HIW‘X : z-wl-\- J 500 (10)6.Arelaxed LTI system with the transfer function H(s)-—=s/ (s+7t) has the input signal. .. v . x(t) = 2 cos(1rt) u(t). Determine the steady-state output signal y(t)ss. Show all steps in your solution. J u M V Lam mun : Htswsgm - M “V M . ﬂ» R'l'} ~\~\ siuo‘j—si‘oie 1663 WE <~> 2 l1“ Si “H7” 4’ “gag-m] x? ‘ 3“) Wm) wad) _ ‘uw‘l \l 3: 4m) WW3 - 2- “ {U-J \ 3 J94 *5“ Se +v SU/J'VQ Jéﬂ um) E ‘9 F, )g ‘ Slwm " Stu—“'5 ‘1 I“ “*3 w L”) m 3 “mm 5 It. ' 1-. Home» ‘ ﬁl752\$m »*ﬁr€wm'ﬂ l .h my F—g‘ ». J; 655% “idol-Wt) Mae." 4 n SZUJ-Tll t“ w -- rs: mm + ‘98 ss * 4r . V, W ‘ youth 2lH(W)\(o>(w4-+2;_H<rl) — W r us :1 *Ca( -s+a’v€ LI _ p 5 n cud. :ﬁrcv T‘ ado/1 A. “9+ Km win v:- ﬁgmss “(Q)iu:“ : -g“+“ ._ MWTVEdW/4: W 6 Y5) 2 items} is; ...
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