HW09s09_soln

# HW09s09_soln - EOE 2025I HWﬁq Soluh’ons , SE'OQI (Ll...

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Unformatted text preview: EOE 2025I HWﬁq Soluh’ons , SE'OQI (Ll (a) LTI gys+¢m 1's. Ct SI'X Pofnf' running Sum “Hit?! By sfem ﬁne/ﬂow. HCz) = (—2”4 1—2" . I‘ _, We. Evalua+c 4h: Frequcncy response 0+. w - 1T K % -Z-For {SJ—=17: z=ch =6J7T="{ “To de+armme ~Hae Value. 0+ Mme, frequean response, a+ each Frequﬁncv) we evafuai-a (+£2.32 Fitz): 25—l 2" —z‘5 HCZ=-I) (—,)6_l = 1—1 = o = O (-0 5 -(—I)5 1-c—a) ‘2." Huhﬁjlazﬁzo “(71“J)= (j)6‘l _ -|—{ t -2 _ z (De—(j); ~l-(J‘) ""J “U =Tr = .2... = __ |__ = Z e—J‘I}: 2' I+j T ‘ J “0 De+ermmc -Hne 'Frequcncq response and hmce. yC-b) “POT “b149— sam?|ccl inPo-l". )(C-b)-.: awsozomrt—TT/‘Q 1. 3w5(300017t jg?) ‘00 CoM'd 1 £5 = 2400 Hz X131]: .ZcoSC IZOO'ITn -I[) ‘1' 3Cos(30001rn *1) 4- 3 2.400 “m XEn]: 2Cos(_27In EYE) + 3cos(_z5r_rrn —£) 3 \L all'ases +0 XII]: 3 cos (%0 ‘21:)1- 3cos ( gun +§I -Hrom Par-I' (at), we have Maia)!" :11: J we need +0 evalua+ﬂ 2 'H‘e' PreﬁMf/Y’ICY response 61\$ E 4 H o '——-. 2: f. 1- " "0.70714- 0.707;J = ——{§+,\}ZJ = = 'ﬂ 6 Haze“) (BJHD —| _ ~|+J = ___:______,____S (e ’%E) G - (6')?) _ﬁ +(I+ﬁ)l 44(1):: 070714-029an ._. 0.76546Josqmmd ' 1T YEN—1‘: aﬁcOs(Jir—n~%*%> 1— 3(0.7654)cos(311‘n +1T+g> gEn3= 2J3 605(1En + 3.2% cos( égn + 1m) 01.2 No'H. 'Hnod xx rs xEn] {:rom “Li (a) Y7: ConvCElllll all]! 1],“) V Thfs re Frescn-Is -Hne Pu'l-lzer coeFchienJrs') Polynoml'al coc¥?1'a‘en+\$_ Haas): 1+ 12" +1-z'2- + 1.2'3 + 1-2'4 + 1.2’5 5 “(iv-Dr- 2 2"“ = l—(JZ— a l_ I I<=o _ II N 3‘ n I S u srx Poin‘l” runm'ng Sum Rifer (b) To dag-Famine. 4m frequency response,we use maﬁabu The Code is “faker! dl'feaHY 'ﬁrom lab ‘3 poi—Lin 0m upda‘l’c +0 the Rl-ter COEPR. Normalized Radian Frequency q.2(c3 The recons£ruoHOW rock: is [7.00 Hz. (V2. 0? Prob.Q.l) Dish/19 the Prequencq vegPon-se From CLI (b)'. and = 243 (as ( 3}. Izoo-L 421:) + 2.2% (.05 331200er 1;) 24- 505): 2J2— LOS< BOOTY-t-E) + 2.2‘16 cos< QOOTH: +51%ﬁ) ____—______—————————____________—___—_—_‘—-—_ \$3 Cascade, Sys-tem (00 “3(1) = YCz) YCZD= (3.2" Y(z)+q,_Z'ZYCZ)+ Vaﬁz.) Q31”3Y{Z)1' lac V211) YCz) [ 1 — 61.2-l —az7_*7- -q32_'3] -= ED V2 (2) YCZ.) : Lo V162) l—q,z" ~azz'z—q3ri': (b) L¢+ a,=q2= 3': , a3=o bo= '7 =3 ﬁnd HCz) {4(2): chzh 141(23- 143(2) 143(2) -= ’7 z 7 1324 Jar"- —o |-e?_'—Z“—.2’_z-Z Haﬁz.) = '7 Hi ZOO—2“) +4.(z)= \$ 2-1-2—3 ha): 9+Z—:=2(l+ﬂ,z_,) = 2"(1-2‘2—D 2 H H”): 2"'(|-Z‘Z)°£(H'7':Z"‘)- 7 (WWI-z") (a) Gofnc? back +0 the. general Case, consider-We I"m?" 9"- re‘SP'DY‘S‘i 1013‘] of —the casmdtd sys+em being “Ln-J: ggfn'nol]. and £149. -PiH-ex coe~FF1'cr‘en+S i0”, 92,05’b.1 and Hoe. H‘me—sku’P-E no]. = SZ—nd = '5 2"(I—zz)°A(l+iZ" - b 2 ° :- I—Q.2“‘ —q22"Z—q32'3 7.” (l+-'—Z"~Z’Z 4.1—) 110 2. Z 9 z l—Q.Z—' *qu-‘Z “(132—3 . _ l 1-? QI=_iZ ’92: I’ Q3"? 14k¢nwe have,: 2" sLb‘, = 82'“ {.9 100:4 and nd=l , Jrhen we chain the, desired impulse resPonse. 94 (0) {e3 (0..) ﬁancee-fat impulse, rasponse’ ﬁequtnq/ reaponse’, Z-‘I'rans-Form, LTI sgs’mms, dn'aqrerence/ eq wan-Eran . Fumdﬂ‘qn roachfsi Wa're given thc sys+am e—q—bea—Jhe-H and mus-r defermme, ihe Impulse response, hEn'] ; —an Frequency response) HCcJQ) ; and em. du'ﬁFerence equal-Hon, S+ar+rn9 ﬁ’or‘n ﬁne sys+am ﬁtncth'on HLL) we; mm cab-ham arah‘o 01° Polynoml'als wf-Hn iLK Giver! that-IL H62) == Ycz) we, XCZ) collc¢+ X623 +erms and Y’CZJ farms , We. than Par-Form -the inverse warcmsform +0 ob-l-m‘h 4725/16 Folf-FFerence, eﬂua-tl'on, terms. To «hm ob+afn -bhe ['mBulge resgonse)we. apply SD] (M ihe l'npwl' and hEn] as the, 0v+pu+, , A To ob'Jr-m‘n the. Eequenglg response, we LenL 2:6“) and eva Wad-C H(Z:e:]a’) Answer HCz)= 721—821”? = 7a"-8a"Z+Qz-3 23 YCZ)= 72""! —¢?Z'Z+qZ—3 XCZ) YCL)= Xéz).7-z*' HXéz) 3.2—2-+ )(zz).92-3 yin? = 7x£n~!]—5*x[n—z]+9x£n-3] W hm: 795%] — Xﬂn-z] + 935n- 3] -_-.__—-—-—_—-_-—__—-—-._-_—-_'-—I——h—-—F——'—‘—'—I—____-_ - H" __ -‘h _-" .n H(2'51“)" 7‘3”) "3.3sz —+ Cle'Jw3 (53 QM: - (I) S 9+6m-Rmd'1‘oﬂ to du'PFerence equah‘Oﬂ‘. uih'phf ou+ +0 obfam ilne sys+em Punch‘OVI In 0\ Form +ha1' you Com ﬂake, #16, Inverse, 24722113- Form to Obtain -!:,|ne dig-“arenas eﬂuah'on. (Z) Sys-tem Hmah'on -to :‘mFulse. responsc'. Sl’mflom +0 above, exceF'l‘ (3) SYeren: Rmch‘on ‘f'D wcrequency response: use. Z-‘xejW and SI’mPh'Fy. Soluh'on S '0-7 ' . -' (3) [4(1): metJ ﬁz“l)({ _ 3!] 2-1) = 31.311 -1 _ej0.71r .: jZTr “2 Ha): :1. +1.176 z +z‘z‘ him-1':- SEn] 1' 1.176 SEW-17+ (QB—2] Since. YEn]-‘-’ Lin—1* xtn] win—.1 = Min—J *' Li'lé x Envij+xﬂ1~zj (23 I"DI—J ’- SDI—J 'F' M76 SEn—I] 1-9131-1] (3) Ha): I + I.I'Iez"1— z-Z Hm 2:;33: = 1+ Ins 65‘; + 63”” 3 Haiti?) q~"+(C3 ljfn]: xEn—ﬂj—xtn-H] 3 Egroaches (I) mmermce c‘31”‘"‘°"""‘m +0 I“'mf’t’lsae. response: 8‘! inspec’n‘on we. Com ob-Lm'n ache. -Pf|-|:er L0€¥ﬁ‘cx’¢n+8 and Wr\'+e ouf Hne \‘m?ulse fesPons€, (13 d1'¥?&re,nce, equa’n'oﬁ Jco ‘Frequanaj response: we’. +akc, +Jne Zﬁans’form 04‘— Jshe impulse. rewonse, {'D obiﬂih “the 6‘4st eqmﬁoﬂ. We subst'hue. Z: 631“) and Sim-Plx'ﬁj. (3.) d;-P—Fer£nce_ eﬁbmh‘ow f0 sqsklm hnah‘on: We: +aKe, the: z-aerns-Rmrm 01° Jaime impulse Faggnsa —or‘- +ake the szsms-form 01"— £146. 0‘4'We/renoe. ﬂaw-U0“ and ﬁnd iii} =-H(z) XLL) SON-Wong m hm: SEA-'1] - \$13-11] (7.) 14(1): 2-7 _2._u r- r~ .4" r r ’f- _ _' I ._ 00“? IMZ -—-'LOZ. “(1:63”) = 63°07 ~€Jw 1-- 93} (aw —-e_J ) 'h -_- ' r . _ A HCeJm) e. . ad smuo Z) ___ Zlew/Z w‘l)smcﬁj L) ._________________________—-— 3 _ +7 .. (3 H623 — Z "Z N as about 0L4- (01) Lin] = LAD—'7] - uLn- H] W'- (0 Impulse resFonse, +o olf-PPexence equq'h'on.’ Reer'nLc, Mae: impulse. response. as a serves 0; impulses. Use Hoe eoeFchn'en-ls +0 express {ﬁne dyiﬁgrgnce, aciua'b‘on, (’33 Impulse, response, +0 eye-I'm Rknah'oln'. Take. {the Zﬂ-hransform o-F Mae. impulse resPoYISQ. (3) IMPUlse resFonst +0 ﬁ‘eﬁluanaj reePonse'. use, we. 939+£m ﬁne-Hon and Subs-U'bd'ﬁ -/~ L=er Sow—H ons'. hm __________..— (0 kg] = min-7] -u[n—n] ' 7 'n w1= Sin-11+81"‘ﬂ*5[“""]*SEH'ﬂ gin'lr- xEn—1]+ gin—9] + x134] +‘I~[n-|0] M— '2_ : ..'r () 2 + 2—9 fz—q if Z__l0 :- 3 m A ' (D s H'CZDl25ef-‘lu = a = l—e'Ja . 1‘ .A f“- _.\$7 -sz Jeo2_ —Jm2 eJ . e (e e" ) .n = a”? e”; — a”; ./~ " -Jloq p a 3m ( “’2- _ @— San (/52, HE). C 3 :HWGD a Sm (“/z) 5““(w/z) GL4 (a) A roaches '. (I) Syg-I—em ﬁnah'on +0 Frequency regPon§€'_ Subeh'h’rc 2=6j\$ [n+0 -l:|ne svgi-em Puma-tom ﬂqmahoﬁ and smeltl'FY. (75) eye-fem Fundrioﬂ +13 imp/Ase resPonse', IFS. easier +0 ﬁnd —l:he diﬁaence, eqmﬁon R’rs‘f and then obtain the inn ulse resgonse diva/Hg b3 le-H’inrj Win14: kin and x nj—QSEn] (3) Sqeﬁm Marion +0 th—Ference {qua-Hon: Sl‘nce. ﬁne. sys+em «Pane/“Hon is equal -to im V‘QHO 0'9 W ZMnS‘FDTm. 0‘? the owl—PW:— Toﬁ ﬁne 'IBVDT’ we cross-mul‘h'ply +1) oben an expression wi-th YLL) «arms on 'Hne laf—‘t and M32.) {ermg 0n “H118. n‘g‘n‘t‘. VW. Unevx hke. Jane. 2vfraﬁ\$¥o~rm a? eadn (bide. l0. ahw'oh'ﬂg by ‘8 will”? " “:3” “J D A} H. 95* (00 YD‘] '-'- '12: \jtn—l] +XEn-1-l —x[n—z] xEn'l = u [n] Conceﬂs'. Z—+ran s-Form, 3y s+cm fwna'l'I‘OYI, (lavage, Z-Jnr‘am sForm . Aggroaohi In the Problm gin-hmelm-1'J approach 2 is “the bed” 0 ﬂown) 2- domain. Firstwe. -F\'no| the syg em Moi-Ion. Mead", we and ﬁne. zv-h'cms-Form o-F Hm: Mpu'i‘ and mul-HPlxd H— 103 -E/\ne SY\$+£m%ch‘on. The. reguH- I"; the OM‘I’PO‘I' i'n {ﬁnal-domain. Finally, We take Hm; vans-Form o? the. DU+PU+ +0 Obwih 'Mne OUL‘I‘pU‘l' {Y1 dx’gcre—fe. ‘bl‘me domain. Sow-tron '. Ya.) .~.- *é-‘YCzD-z" + XML) 2" - X0052 YCz) [I ’f—z'-Z“I = X(1)[Z" ~24 Ycz‘) :chj = z-I-a-Z , z" (I 1“) X62) 1 + 3:2" 2-: (2+ é) The, zrkanS-fbrm 01° the. am} sitp'. I _. xtz) \-—z"' Mulﬁ'PLﬂinﬂ HCZ} X021)" YCL) Ya): (l~z_") . | = ' = 2*l (2+3; (1-2" 2*: |+%Z-: Takmg the mvexse. z~+mns-Porm 0‘1: YCZ) (+er pg 2:?) " (n—l) (Ji MIMI] = 3th 12. Ci.5(b) Kiln]: 2cm; (0.51Tn- E>+ COSCO.ZS?1n-TT) -Fbr -Do<n<oo Approach] Since we. de-fermrncd -Hne aye-Fem Wch‘on in par+ (an)J we, Subsb'm-l-e. Z=e~ia a+ each \$requencﬂ —From the: M 0+. We. {hen determine, +lne ou+pu+ Ear each inPU'I‘ 0081116 and Luge. +he Pm‘na'ple o-F Superposf-UOY‘J +0 ole+cxmrne ~Hne overall response. So/u-bl'onl £3= 0.371' 3 0.2510" HCz)= i-z“’ , 15+ z=eJ°\$"—- J 2+4. 2 maﬁa] = 1-1. 5 1, w=of L) "’1; ‘12:- 3"} Tr _T___ = __ 2 .. 2 =IZ_O4J U+—' -'-+’ _’.— 5.! 2 2. J 2 J g. 1" "rt: '0322. anal HCeJ )Jd‘oaogﬁ -— [zsseJ r g liege JOIDZS‘n’ 3;“: Hﬁc55)]&:olgw . Z= ﬂ " é' I “'15 HC Yw=02§q~r P = 0.436 *JOEBO 'I a"? +ia “ccjcg 'oé-eracl 6 \$=oasqr = 0547:“ =05+7eJouﬂu I3. “1.5 (b) con-FBI 3m: 1.15;. g¢05(O.STIn —12 —o.IOZS1-r) + 0.547 oos(O-Z\$W -—n- + 0.20677) ﬂLn-l = 1.53 (0503.511?! -' 0.602517) 1- 0.597605 (0-2577?) "' 0.791}- 1r) 01.5 (c) XEn3=IoS£n-5'j Com-egg '. lmPulse response, fl'mc- domain 3 Con Vol 1) tron, l'r‘l VEJ‘S a Z—‘h‘mn s ﬁbrm_ AEEroac/lﬂ: From Par+(a) we have H62), Take £he Inverse 2-ﬂansform +0 obi-m?! PIER]. Com/clue. hfn] wi—bh kin], SI‘YMC X110] J‘s a seal-gal and ghi'w‘l'teo‘ um’r impulse, [nil 1's easfl ob-ta'ned b A l ' Z sht'F-l: cmdva Scaring def-ﬁt 'Zt'DnQENY. l‘f. (1.5m) M33 [3 a sampled speech Si'ﬂnal- Concae-l's '. Aggroachi Since. +he. Inpu+ 1's a sequenéc o-F numbers, f+ makcs sense ~11: use, approach 1 J +ime—domm'n. Tl'mc- domam , ma+lab opera—hens, Answer Using ihe {Vii-er command,we womlol ape ('4'ij 1 t I, m E0, 13W] 4—— x~CO£PPrcren+\$ 00! Lab <-——- \/ — Coe~F-Fl'c.l'tn+3 .j = RHm (bb,ao, x) 15, ...
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## This note was uploaded on 01/27/2011 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Institute of Technology.

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HW09s09_soln - EOE 2025I HWﬁq Soluh’ons , SE'OQI (Ll...

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