HW08s09_soln - E65 209 5m 1000 ms sozwons 3(cg The...

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Unformatted text preview: E65 209;} 5m 1000: ) ms; sozwons 3;: (cg The alafferlgnre equa‘ti‘om represcnfi. o; 5190””— runm‘ng average R‘I‘Eer', 4. c = .L -- Yn? 2 SXEV‘. K] 4. = _L ”Jfik 5. Z 8 uglna the GEOmele'C, games {4:0 (see gabn 6-23 {in—(1,5?) _.__ e _ . . A 5 Sm (66/2,) " 5'” 565/2") 6 ”J2“ SsmCfi/z) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 (at) Magnitude as a function of frequency .3 —2 I -1 Normalized Radian Frequency KEN]? 7+|3 (05(Qon)) fi‘nol \ffn]:c - I"! HécJGo) = O = HCC‘J") sm( 5 lgee/z.) : Sin 6 $42.) - n O SW0(Z:nW Wa:2-_’Tf',4fi’ (011'. S '55—» ’ *5 1 “We Suhceocao'éfi n A Luci—2.1? fl: 5 5 _—|—_‘—-fi—|——.—'-_ IF a DC {V‘lPU'f ls OkPPh'ed +0 (1 L‘Poi'n‘l’ averaging R’l-Jcer; Jane. ow‘p‘uf = i‘nPui ._°.' °) 3D1=7=c J ‘the sinusol'ol is blocked! er fine Px'H-er: Conc€E+S Impulse responserl, Fz'Her coem'cienfifi'equencg Response AEEFOO‘CJ‘: The frEquench response—afar} FIK Sy5+em rs relm‘ed "f0 4H6: Impulse re spanse as: Hafiz”: NZ! 1: e‘fik K ’ whare [3L {:0 represen‘ls “he #Her Ceeffl'aénf-‘S. There-Fore, we. R'rs’r deiermx‘ne, WNW:- imfvlse responga mnd+hen (an ole+ermm6 the Frequency response_ (b) Defermr'na 4he frequency resPonse 010 {he overall Cascade. €1/‘43‘h‘3m.r Hc’le") FIIZ Fl:/‘ffir§ MI Frequency Response) (OSCGCJECIW) MMHPh’cafion o-P Frequency Responses. I aggroaok , Fn'r‘er, we use {he Suva-n Imlobtls-e resPDnse 04‘ Sys'ffimi +0 de-iermm‘ne. HS Rrefiufncu/ VPSPOflS‘Q. Nem‘, we mulh’PlY {he +w0 Frequency W-’5PW’5€’S Since Convolu‘bmn ofimpu‘sfi responses Lmresponds 4:0 muHiPh'cmfiOI’l 0P fireo‘uehayl regponses ‘POY LTT 9149M.m3. Finally, we, mann'Pmiafe +148, preqmency resfamnga exlaressl'on +0 gePaere OH+ Jche amPliJrude porh'on. 4 Afiswef: _ .n 'r» .n ,3.“— Mn] - 23En—k] ) Maj-:3): Mame-away“ {=0 + egjiblf— ’3 ~" 3‘2 4"“3 :304 (E'Jm-veJ >’(|+33”+eJ +3.3 +61" - .fi r" 6-3502. -—€_th7 H C 6311:) ‘~ __ e _‘ To S-EPara‘fE 00+ —Hr1e. amPlufude porh'on, _J$4.s @502: 5 H 6"J”Z";> _ 9“”? 2‘ a swarms) e 3 J .p- .r- f'Pq.‘ ' ‘ _ h Mei“) : eJ‘” SCJM‘TO 33.n(w-35) (c) Pia-f 4kg, innaflmfuole 0p 45kt; LIN/era“ ‘Preqmencxf regponsa 010 Hoe €613.01de Syd-em“- M} AmplfluolejMajm’fiode) Frequency RESPOHSB flip-Breach: The magma-fuck of H46 ‘Preoiugnm/ response is 4cm. Megabit: value. 010 Jc'he ampl:'{ude,. There OWL) we P|o+ Jaime, abgolufie valu—e 04‘ {he armpit-{viola "Ponti’on over a fan/23.9, oPa’n'ng‘) surr. ’ . A ;\ Answef- \ H(¢J Ml : \ a Sm ( W35)! Vadues xix/hare. \Siln(v’\\3‘2.§)\ owe on maximum} if} .25 : if 3:11;) Say 2M0 values occur :2. ’ "'2‘ T . 0” 65pm: om 211* l‘ _ vo* EJE'IIJW “3:022Wq‘? S S ‘5‘ ’ 3’” BY symme‘h‘x’ ‘, "Hm: mega-hue, vcalufg (AVG/i ‘1)'éfl , r” Gmaximm ‘- 2.11“ " 4—77 < Minl'mm 5 5 5 J 5‘ , v 3.2 (d) an3= 7605 (0.3110 —O-}1T‘)——3 Ea Y”? W: FMQHWGV RBS'DDnge’ Supuposr'fion) Cogugmla symmC/fiuy W1 The wsme can be expanded' mm +W mewy wnJ'LLgcfie ayponenh'alg M9109 Euler’s Formula. Tim: 1969qu respm'rse, May be deiermfndd “For one of the cornme ConJuflaQS . Since. the FH-er weffi'cfcn‘fs are. recu'j we. can appfy conjugonlc QYmme‘h‘y f0 Obtain {he Fragmenoy resForIse. 01f- Hag Mnjugafc's. Frefiumgj. And fl'nafly, we use Hue Pri‘na'PLL r210 Superposrfi'on To Sum +106. m‘ aufipufs +0 obflain 4mg 5:me odpuf quJ- I+'§ I'mpor‘fanf +0 no“: Hm? +113 1929:4ch resFonse mug+ be muifi'Ph'fd by 47178 orflgfnbd {mpg/“f campmx exponanh’al 4:0 obml’n flue and-Poi . Answer": The procedure. 1‘9 deflated below: 'o.3rm ..’O.I1-r 7 J 1 —e e. H 2 "—3 'o-Brrn‘a ('5. - oufpuf= chJ0'3")u:e,J em 4- - Z _‘o.3m ‘o. . . L 13" e-fJ HT -—-) -__3 HCe‘Jo'a’")= H+CeJo.3n) 7cosfa3mq 70.1w): 2. Ou'l-PU'I‘ = #{330-3100ézé-J0-3lgjomn- ... JD3TYn—ol1'r __ 031Tn 0] 3m_ Zia 9’1 - “(103“)“ 96’ «5.:J WHWcPBn’) ‘ * .S'fi‘“ —.J(0-311)"I-.5 J0 _, . _ _ “mpgfl : a 6, 33(0 31:“ 2.9) ._" H1799?" .; 60””- . 0.707»; = LLH e, J ’0311' J'O‘Ss‘fl' H*(e.J ) : 1. Lu 6: 5m: ' cm coscosm - 0.9510 In me you nee-.5] 47%“; mHy~flni+yi ‘031111 gmrr -‘ 3511' r‘a-B-rm 'o.|1't' tim— 3:” 6“ . l-LHeJO +-22‘-°'J c’ «:Arleflo'w" , gage-m {so-31W jofls-n‘ .13.] = _§ 01.9 cJo‘aTmEJ -1- 9.9 C, C ‘5“ Q— 9:34:01) PIO‘I‘ +hc frequencxf regPonge magnl4'uole For“ HCeJ‘fi) over the Precluenmi row/193 —’n'é$£-1T’ Concegfs'l Pregnancy Response) Majnflruc’e, Amgln‘i-uo‘e, APPr0-a_cl'1f WC -Foc;+or the eXpreSSI'on «for Hm: ‘FYECI menu] response. +0 €epara‘ie 00+ 4W3. amplitude ?orh'on_ We fines/1 Jmke, Jche QbSOIU‘lE value 0-? the (lymph-Mole, +0 \li‘eu fine ma w-lude. To Fla}J we, dejrefmine. {he MAKIMA (And mm'mma. I+ r'3a150 aoseible +0 use MM Lola to genercflrb 5'— Plot” Answer ‘3 - Hfiefi‘fi): 7 +76th 1?“ (€JZW+eTJZ“°> 2 A _rn’ 34v w~__ .w 4’ 4— a2‘fil‘fif ”T ‘r (5) Draw ‘the SP€C~+VUm, O? “31] “hint”. inPQ-f if) K Approach: __________._._.——- Ahéw Pf f , QSSumt‘ng ithc XE) = 3 4r Scos (321001? "Q for .— 00411: 40C: SPeC‘Hum, Sampling , Ail‘asinfl ‘To de+arm{ne, KEN] we, Sample XCJc‘) 0H: {ha gw‘em SamPIMS Pa+8103, Since 9; 4 39mm: 0-? Mt), all'asmg occurs. We then apply EMICrg-Formulcx 1'0 the COSl‘nfi +0 o‘c-i-evmine, the spec/Ml c0n+en+ and 90* ocmrdtnahl, The 'DC Jcerm 1‘8 OCCOan‘bfi-Ryra‘go, Atty:— 3+ 5cos( 330cm t) -{:= (1:73 =__f:}_ F3 3300? = KB]: 3“” 5505( $300 n E: + ScoS (jig—Tm) =5cos(§ln +145“) : . 31in 3+ 5502(7) _' 3 XEn1=3+é€Ji§£n +2€J751fl 2 “* Also n§4e 4mm (57 For Hoe Same. KHz), defiermlne a St’mP|e grmulfik ”Bar 4AM: 0M+Po+ yU-J ‘For‘ ——Oo<t<;0:: Concce'fs '. FIR Q'Har, Reconsi‘rucfi'On, Time‘pomaghj SuperPomfion) C0r1ju3a+e1.€ymme,+r~y. APPWO‘GH given fined we. lcnow 4918 Praguehck/ resPonse 019 the Q'H—arJ we Page eac-Jn 31990479“ Componen+ +hromfik? To Since. Had {TH-er - 0“. 9+ Loe-FG‘cfen'ks are. de+ermine "the W +0 X130], we read, we have- use QuPer'POSI’HOW and add qu @596ch [WJW—‘e Symmflh‘y ou—hau‘l’s. Ngfie -Hno~+ +0 de+ermine 44/16, -For '. “’59 .7 Y i?) HUB-3) H (c3 ) om+Po+ , We. mué+ Mulfiplq Hne #equeficy e by dune fnPqu comPLaX eyponenfio. YES?OHS . . truerr 44m: 919nm 910‘] To final V0.4) we, recon; using Irv-"i“; ‘ .h :7." m Ari-swell" 1 From Par+(5~); HCer) = {$6. J we. '4- CosCZw Applyx‘n3.+lfie Dc: componen+ _3', HCe'o): 8-0 ny—COSCO): 14' ‘5 The. ou+pu1 :‘S‘. 3°W= ‘rZ APP'Y the, poSf-HVC Comphzx axloonenhalh ._.'8"rr . . ”a J c 2:3)?— zgngnf. H (6') g) :8 IF“ [1+ OS( ‘1‘ :0 ’31—“ 0 IO. (a) (on+'o|‘. We Could have. also easily Seen this Rom £Vl€ Heguenovl respo'nge, 19lo+ in Pad ((1.) gEn1= 42 +0 50;): 42 (OD SULPPDSE— tho-t the inPLfi' Sl'gfia' 1's x05): tacos‘fimowg axPlam why {he ou’rpof grand t9 Ya) as will be. Zero For -Oot;'(:<r90 Mi Sampling, Specfium) $PerP09u'41'0fl) Frequency Response. AEEroaoh ; The ian‘f xii-t) mus-1' be Sampled 0d“ ‘PS. SifiCC. £9 >3‘Pn'10k'x, alt'astag doean‘ ocean, To o‘e‘l’erml'ne. 44315 specyjrmvi Con+an+ O"? @ X(+>), Mia use, «’m‘g. I'olenHHcs {:0 express Mb) me. 0\ sump? cogineg, We then Sam U. +108 shamed +0 défiermine, )(En , We, Mae-n use. Hoe. Freque/nc‘j Hon. +0 Pfcjmre, ow+ Ho (on X(t)= \66033(20Q1=t) In gmera‘l (033(14):- CosfiA)‘ C082CM)=CUS(UL)‘[LEOJS% 2. 6093“") 7‘ COS (LA) L- Cos£u):cos(au) 1 3. £09500 = ..LCOS(U~) i—J. i (C09 (“'2 u) + COSCU+QUJj1 2 Z Z (03(301) COSB(U~):‘ 3:. COS(U—) f [‘F CDSCUL) + 1‘“... (0‘9 xtt) : lg, (03% - 13 cosCQOOrrt) + 4cos€®001r0 (goowt ‘Fmoux —- $99.3 = 5OOHZ Pg > meaxj no altasur-Lg STI— KEN]: 1360S(—:&r—fl) + 4cos<$fifi) {Aging the Plof #90111 ParJr (a) +0 %'3 Arr-wire 4ha+ we. know" the. g eoW’I Compfim’mm 0 the cosince ‘me Ema‘s {mum _ The ou-HOOT For“ [ZCOS(E0) % 0 pure ou’r fine, Frequency” ‘0 response: {I} = 3:4 Mei-‘10: 0 , {3; : +31? “(9,355) =0 , The DOWN For 4:08 (@flfi’o Ii.— Msing SuPbrPDS-Hion; [JD—1:0 3., gtt): O ‘84 win] = mm +b5xfn—5] .n ‘.£3‘5 H (61)”) : be + bseJ (09m xffnjz'gcosCOZnnIma I I_.3 Y.[n]= O (2) >313]: B _____9 [j “#9 YZU‘KZS -5 (am-5) 2 O 50 _+ [0535?) ___ O (i _ g bo‘bS"O Mp'amfl bo+bs=$ (9.) 3b5:i Io b b _I b :J— : ° :5 o“- 5" 5 5 a, ___.___ (b) _ fl XEn'JW cosm’jm) we I: ——> cos(w,n) r _ "$.5- MM» ea- ! m. (HM =1 2LH(cj’~¢=)-o 'Q-S _;l;l_+_liefJ ' r 1 e'j-OS: i, _,i‘:)l~5 102-275-411" £31: (DJ—23;} fl 5 S “/3 303m “ya/2 3-5613 Kin]: -3 + 2(=:‘”J'"°‘3‘"mcs:—J + 26 e an]: :3 + 4—(08 (0.31m + 05173 00) I?) general YEn'j: Hej3°)o Aele’e'j‘g" Thus [IF WE know *the Frequency YesPomse QAHC't-IOVI; we can apply each ('omPowtn’F 01c- xfnj independenfly and {Age querposnhon +0 defirmine. Una overall Yé’SPOIflSCyEnl 'For' 0~ Q-P'oini' averafll'na Q‘H-dr”: mm) = SMM/z) e-JM (999 m mg- q sm(&iz) in fay-f) Lfl _. .-_. . n__#_1 'Dm'd’l lei' Furl die n I; we anon,- «tine DC. {warm -5#xttn]I Hm"): 1, Y.En]= (He15)(*3ei°)(eJ°") = '3 I? we. aflolfl 2a+J°3fifle+JWZ = X1131]: (6‘03" 2 3m (D'ZW'q/L “(exam 3 are (0.511;) = 0.2.18 ”J01?" YZD‘] : 23-4..)0‘5? CJU-B‘IW’! o 0.21? 6530.211 ~,-,-_ 0 Aggrej 0.31m e: 3033‘ '- -—. I 1 E‘TT! ' I; we. apply 25 Jogfl‘e J 2 a x3110] ., and snare. Ygtn]:kf[“] and 61066 {he ¥\\+er coe’E-F’iden‘ls cue, meal} we have. , {onjmgau Sqmme'h'fl For the heel rzsPonsel erjoan):w(€103fi) “(610.310 : OJISEJO'Z’W [paw 6-303er . 0‘le 630'?" ‘joafin {303114 it 93”] = 2 0.436: 6 ‘JW 3 WWWIW +9301]: ”3* 0.317. cos (0.31m +03%) 6.0; (D3. The, sYerem 1'8 HM L_TI,’ (ons:'a|e.(‘ 31,3111 Aha oMpu-r Wu'llnth be. 5141b] 0‘5 "m '“F‘” 4hEr3‘Ft'3re, i+ fmls we easr'eer hnearHY +695}. Consider“ )(IEn-l] as 0m inPU+ , “the cahoo‘} foggy be Shifted by i . Consider xEn—no] as animPU+ ) 'Féar' £516 sydam 4:0 be +1‘me—invarl‘mn‘i‘, XEn—fl01 1—6 yEn~Ho—_( ‘Por 0L“ imPo‘YS W x [n] 19~+3W= {I r? ”En—no] >05 0 1’? lXEfl-hfll <5 gin] = ginmo] LC) Consl'oharima Hoe analyeis “hm/6!; Jaime; Sydflm is not \‘nnecM‘1 100+ Hr {g tx'wwe mvarian‘i‘, ...
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