Modeling of Jitter Characteristics for the Second Order Bang-Bang CDR

Transcription

1 Modelng of Jer hrcerscs for he Second Order Bng-Bng D H Adrng nd Hossen Mr Nm eceved Aug ; receved n revsed 8 Se ; cceed 3 Oc ABSA Bng-Bng clock nd d recovery BBD crcus re hrd nonlner sysems due o he nonlnery nroduced y he nry hse deecor BPD. he secfcon of he D frequency resonse s deermned y jer olernce nd jer rnsfer. n hs er jer rnsfer nd jer olernce of he second-order BBD re chrcerzed y formulng he me domn wveforms. As resul new equon s resened o on corner frequency. Also he jer olernce s exressed n closed form s funcon of loo rmeers. he roosed mehod s generl enough o e used for desgnng BBD. he nlyss s verfed usng ehvorl smulons n MALAB. Smulon resuls demonsre he vldy of he resul oned y nlycl equons. KEYWODS lock nd D ecovery D Bng-Bng Phse Deecor BPD Jer rnsfer nd Jer olernce. NODUON Nowdys he volume of he d rnsored n elecommuncon sysems s nocely growng whch mens h he ndwdh requred for d rnsmsson s lso ncresng. herefore ecuse of hvng hgh ndwdh ocl fer s used for d rnsmsson. However fer ssng hrough he fer he d re dsored nd her mlude s decresed due o her non-del effecs. hus he d rnsored y he fer re requred o e regenered n he recever. However due o hgh rnsmsson seed of he d hose crcus re needed whch cn roerly c hgh seed frequency. lock nd d recovery D crcu usng ng-ng nry hse deecor BPD re wdely used n communcon sysems mnly ecuse of her hghfrequency cles []. Exmles nclude mulggherz clock mullers [] nd ocl recevers SM SONE [3]. he dvnge of BPD over lner hse deecors s h BPD exhs very hgh gn n he vcny of Δφ= nd s le o oere hgher seeds []. Snce n lner deecors nd n cse f hse dfference exss ulse rooronl o he hse dfference s genered; however f frequency s hgh he wdh of he roduced ulse would e very nrrow. n oher words he verge ouu vlue of he PD would e very smll nd s resul he volge requred for djusng frequency would no e rovded nd D would no roerly work. Bu f here s lrge or smll hse dfference due o hvng very hgh gn BPD roduces consn nd lrge volges n he ouu o move he D loo owrd he lock. n oher words he BPD ouu hs wo dsnc nd deermnsc levels deendng on he sgn of he nu hse-dfference. hs hs led o s lcon hgher seed frequency. he rnsfer chrcersc of BPD s shown n Fg.. he ouu of BPD s + nd - for osve nd negve hse errors resecvely. hus D sed on BPD BBD s nonlner sysem due o he nheren nonlner hse o volge rnsfer funcon of he BPD. hus unlke lner D he nlyss of ng ng loo s comlced. Ouu olge Δ Phse Error Fg.:rnsfer chrcersc of he BPD. As s well known he jer he nu leds o errors he ouu of D; herefore jer nlyss s very morn for evlung he erformnce of BBD. n rccl dgl communcon sndrds he secfcon of he D frequency resonse s deermned y jer olernce nd rnsfer. he frequency orresondng AuhorH. Adrng s wh he Dermen of Elecrcl Engneerng Nour Brnch slmc Azd Unversy Nour Mzndrn rn. eml: hdrng@su.n.c.r H. Mr Nm s wh he Bol Unversy of echnology Bol Mzndrn rn. h_mre@n.c.r Amrkr MS ol. No. Fll 37

2 domn chrcersc of jer rnsfer mus mee some requremens. Frs he BBD loo ndwdh should e very smll. Second he ek of he jer rnsfer mus no e greer hn secfc vlue. he moun of ekng n he jer rnsfer mus e less hn. db for he SONE sndrd. herefore he nlyss of ekng s requred. Moreover he jer olernce secfes he moun of olerle nu jer whou ncresng he error re gven jer frequency. Mny effors hve een mde o nlyze he nonlner loo dynmc of BBD [5]-[]. he effec of jer cn e ccurely nlyzed usng Mrkov models u her lcly s lmed o frs-order loo whou loo fler ccor [5] [7]. n [8] he sedy-se nlyss of BBD s erformed for he cses h here s no ny jer n nu nd he vlue of ouu jer mlude s derved. Moreover [9] nd [] mnly focus on he chrcerzon of he jer rnsfer nd olernce of BBD n resonse o lrge snusodl nu jers. n [9] he jer rnsfer roeres re nvesged for frs order BBD whou loo ccor nd n [] usng n roxmon lner model of second-order ng ng loo s nroduced nd he slewng effec s ulzed o derve exressons for jer rnsfer jer olernce nd jer generon. However n [] he nlyss mehod s sed on lrge off-ch vlue ssumon for he ccor []. he off-ch ccor ncreses he numer of exernl comonens nd n coun; lso coules nose from off-ch o he conrol volge of he volge-conrolled oscllor O n he D lock. Anoher ssue s h he ond wre nducor drsclly ncreses he hgh-frequency mednce of he loo fler nd mkes he D more sensve o hgh-frequency nose []. hus desgners re sll lookng for he wys for desgnng loo rmeers nd esme he frequency resonse chrcersc. herefore he frs se s ccure nlyss of he D loo. n hs er jer rnsfer nd jer olernce for BBD wh frs order loo fler re chrcerzed y formulng he me domn wveforms. As resul new closed form equons re resened for he jer rnsfer nd he jer olernce. he roosed mehod s generl enough o e used for desgnng BBD. he roosed nlyss does no furher need ny ssumons on sysem rmeers eseclly loo ccnce. Behvorl smulons wll e used o vlde he nlycl resuls wh rculr emhss on he jer rnsfer nd he jer olernce chrcerscs. hs er s orgnzed s follows. Secon refly revews he BBD rchecures n he lerure. Secons 3 nd nroduce he nlyss of me domn wveforms o derve he jer rnsfer nd he jer olernce secfcons resecvely. he ccurcy of he roosed nlyss s evlued n secon 5. Fnlly he er wll e concluded n secon 6.. BANG-BANG D AHEUE Fg. shows lock dgrm of ycl second order BBD whch s comosed of d smler PD chrge-um frs order loo fler nd volgeconrolled oscllor O. he PD deecs he hse dfference eween he d-smlng clock nd he cener of he ncomng d nd generes le or erly sgnls for he chrge-um. he chrge-um sules he loo fler h ncludes ccordng o he sgnls genered y PD. he volge over he loo fler s he O conrol volge nd deermnes he frequency nd hse of he smlng clock. Wh hs feedck mechnsm he d-smlng clock rcks he cener of he ncomng d s nd he clock o whch he d s synchronzed wll e recovered n he recever. n some rccl sysems second ccor s usully dded n rllel wh nd nd ncreses he loo fler order o wo. hs ccor s used o suress he sudden jum on he O conrol volge roduced y chrge njecon nd clock feed hrough of he wo swches nd mroves he rnsen chrcerscs. f hs ccor s much smller hn for exmle ou one-ffh o one-enh of he me domn wveforms nd frequency resonse remn relvely unchnged nd re smlr o hose of he frs order loo fler [3]. As resul n hs er he dded ccor s chosen much smller hn nd BBD wh frs order loo fler s nlyzed. he hse-domn model of BBD s shown n Fg. 3. hs model ws mlemened for ehvorl smulon o verfy he ccurcy of he derved equons. n Fg. 3 K O s he O gn nd s he chrge um curren. serl d srem lock Alexnder PD le erly DD cnl O ecovered lock Fg.: Srucure of he second order D wh ng-ng hse deecor. n Nonlner Block sgn PD Frs - order Loo Fler s cnl O K O s Fg.3: Phse domn model of he second order BBD. ou 38 Amrkr MS ol. No. Fll

3 3. JE ANSFE ANALYSS he jer rnsfer funcon s defned s he ro of he ouu sgnl jer o he jer led on he nu versus frequency. Jer rnsfer cn e consdered s gn y whch he D enues or mlfes he nu jer. Jer rnsfer mus mee some secfcons. As n exmle n SONE O-8 he cuoff frequency f s MHz nd he ekng s. db [3]. n nry D he jer rnsfer funcon srongly deends on he nu jer mgnude. hus he lner models cnno e led o he nry D nlyss. For neglgle jer ekng nd jer frequency close o f he wveforms of BBD hve ycl rooyes shown n Fg.. Also s he nu jer frequency exceeds f he ouu jer mgnude egns o fll s shown n Fg. 5. cn e resonly ssumed h ll of he wveforms re symmerc round he orgn nd hve he sme erod of P. Where P s he erod of he led nu snusodl jer. Accordng o endx A he ouu hse P s oned s follows ou where.5 KO K O 3 he nl hse of φ ou occurs = whch s oned from s follows [9] KO cn e seen from h φ ou hs rolc she. As shown n Fg. he nu s snusodl wveform nd s resonle o ssume h φ n= φ snω P + θ. hus from Fg. snθ =φ φ. Also noe h f he jer ekng of he jer rnsfer funcon s roxmely zero jer frequences close o f φ ou sll rcks φ n closely so s Δφ s mnmzed nd he jer rnsfer roches uny. n oher words φ n φ ou for f f. follows h ou 5 sn Equon 5 holds for ny n he nervl [ ] lke he me menoned n 6 h leds o n eser soluon. Becuse of he erm sn. consderng oher mes n 5 mkes he soluon comlex nd fndng he unknown rmeer wll e mossle. 6 Fnlly he corner frequency cn e derved y susung 6 no 5 s cos 7 elcng no 7 yelds Amrkr MS ol. No. Fll 8 And fnlly he equon for he corner frequency s s 9. 9 Equon 9 cn e used o derve n roxme vlue for corner frequency f = P of he jer rnsfer. s herefore ossle o roxme he enre jer rnsfer wh frs order low ss fler s ou n s s n o P cnl PD P n mx ou Fg.: Wveforms of he BBD jer frequency close o f. n n ou Fg.5: Wveforms of he BBD s jer frequency exceeds f. As execed cn e seen from 9 h n nry D he ndwdh of he jer rnsfer deends on he nu jer mgnude φ. Also o ncrese f one of loo rmeers or or K O mus e rsed. Bu s ncreses he jer ndwdh s decresed. Noe h he ove nlyss s vld when he vlue of he jer ekng s roxmely zero. hs s equvlen o lrge dmng fcor n he vcny of ω []. hus hs condon should e me. he jer ekng does no occur when he mxmum ouu hse sys smller hn he mxmum nu hse. n order words he exresson should e hold. 39

4 o As shown n endx B he followng morn resul cn e derved s. KO 3 Equon s suffcen condon o gurnee he jer ekng s neglgle. f nd K O re held consn nd s decresed he sly would e defnely desroyed. However K vco s he O rmeer nd cnno e djused y he desgner. hus f he ccor s seleced o e smll oher rmeers cn e ncresed so h he sly s no desroyed. he ccor cnno e seleced very smll snce rronl vlues would e resuled for nd. However f he roduc of s excessvely ncresed he condon s voled nd ekng would defnely occur n he ouu. hus he D rmeers should e desgned so h he roduc of s he mxmum o mxmze sly. Addonlly condon should no e voled n order no o hve ny ekng n he ouu. herefore desgnng should e n he oundry se of. nvesgng end mes of nervls from he unknown rmeer cn e exressed s omnng 3 nd 9 gves. 8 3 n he SONE sndrd P =f s gven. Also φ s known. hus he negve vlue of he unknown rmeer '' s derved y solvng. As resul s generlly used for desgnng he BBD loo rmeers o mee SONE jer rnsfer requremens loo ndwdh nd jer ekng. he desgn mehodology for he D s descred s follows he rmeers P =f nd nu jer mlude φ re gven. herefore usng he negve vlue of he rmeer s clculed. Hvng he O gn nd he on-ch loo ccor he chrge um curren cn e esly oned from. 3 n order o mnmze jer ekng mus e ssfed. hus he negve vlue of cn e deermned nd he loo ressor cn e clculed usng 3.. JE OLEANE ANALYSS n he D snce he ncomng d my exh susnl nose nd exerence consderle enuon he d s mus delly e smled y he recovered clock her mdons so s rovde mxmum dsnce from he decson hreshold nd d rnson ons. hs conce hs een shown n Fg. 6. For slowly vryng jer he ncomng d he recovered clock rcks he hse vrons n order o ccurely smlng he d o vod ncresng he BE. For rdly vryng jer he clock cn no fully rck he nu hse vrons flng o smle he d omlly nd creng greer BE. he jer olernce deermnes he ek-o-ek mlude of he nu jer for gven jer frequency h cn e led o he D nu whou worsenng he error re BE of -. Jer olernce reresens he ly of D o recover n ncomng serl d correcly dese he led jer. As llusred n Fg. 7 he secfcon s yclly descred y jer olernce msk s funcon of he jer frequency. As n exmle n SONE O-8 he D mus whsnd ek-o-ek jer of 5 U [U=Un nervl] f he jer frequency vres re elow Hz []. n order o cheve good jer olernce hgh ndwdh s requred so h D cn rck he jery nu sgnl nd e le o recover he ncomng serl d. On he conrry jer rnsfer comlnce s cheved y lmng he ndwdh of D o ensure h hgher frequency jer s reduced. However n nry D he loo ndwdh cnno e defned n gven vlue ecuse he loo ndwdh srongly vres wh he nu jer mgnude. del D Noseless D lock U.5U Fg.6: Noseless d smled y clock. Jer olernce U dbdec Fg.7: Jer olernce Msk. f f f 3 f - dbdec Jer Frequency Decson hreshold As nu jer frequency exceeds he loo ndwdh frequency he ouu jer mgnude egns o fll s shown n Fg. 8. Smlr o he nlyss of secon 3 he ouu hse P cn e oned from. As shown n Fg.8 he nu sgnl s snusodl wveform nd cn e exressed s follows n sn 5 where snθ =φ φ. Amrkr MS ol. No. Fll

5 n cnl ou mxmum hse error s clculed s mx cos n ou ou PD P mx P Fg.9: Aroxmon of φou dshed lnes y 8 sold lnes. Fg.8: Wveforms of he BBD s jer frequency exceeds f. s nsrucve o qunfy he jer olernce of ycl D loo nd comre he resul wh he msk shown n Fg. 6. A gven jer frequency we mus ncrese he mgnude of he nu excess hse φ n unl he error re egns o rse. n oher words he hse error Δφ roches π [= hlf un nervl U] rngng he smlng edge of he clock close o he zerocrossng ons of d. hus s shown n Fg. 6 n roxme condon o vod ncresng BE s s follows []: mx n ou mx U 6 Equvlenly form nd 5 we hve: n ou 7 sn he me whch Δφ mx occurs cn e clculed y equng he frs derve of 7 o zero. As cn e seen he clculon of hs me s dffcul. o resolve hs rolem we roxme φ ou y 8. hs roxmon s shown n Fg. 9 nd cn e exlned n he followng. he me when Δφ mx occurs s roxmely equl o he me whch Δφ mx= φ nφ ou reches he mxmum vlue. cos ou Amrkr MS ol. No. Fll 8 where φ cn e oned ccordng o Fg. 8 nd y 5 for = s sn 9 herefore from 5 8 nd 9 we ge mx n ou sn. cos As resul he me when Δφ mx=φ n-φ ou s mxmzed cn esly e clculed. Equon revels h Δφ mx occurs ω = π or equvlenly =. omng ck o 7 nd susung = he elcng no yelds mx where =πω. Equng Δφ mx o π yelds he mxmum olerle nu jer φ =G J: G J 3 he ove equon conns wo oles he orgn nd wo zeros. onsequenly s deced n Fg. G J fll re of dbdec for ω <ω nd dbdec for ω <ω <ω rochng U [=π] for ω >ω. he SONE msk s lso gven n hs lo. G J U.5 SONE - dbdec Fg.: Jer olernce Msk. - dbdec ω ω ω he corner frequences cn e clculed y he followng relonsh. Equon 3 ndces h he jer olernce s rooronl o he O gn P curren nd fler ressor whle nversely rooronl o he smller ccor n he fler. Usng 3 gven nu jer mlude φ he frequency whch Δφ mx wll e equl o π s oned y solvng he followng equon. A comrson of he resened nlyss wh smulon resuls wll e demonsred n Secon SMULAON ESULS AND DSUSSON n order o vlde he roosed nlyss nd

6 modelng roch wo desgn exmles re crefully erformed nd smuled usng MALAB smulor. Sysem level smulon s erformed y he hsedomn model of BBD whch hs een shown n Fg. 3. he corresondng loo rmeers re clculed usng he desgn mehodology descred n Secon 3. n hese sysem level smulons Ds re desgned for π nu jer mlude lhough he loo rmeers vry from desgn o desgn. As n exmle he desgnng mehod ccordng o he SONE-O-8 sndrd ws gven n whch f =MHz. hus 6 f MHz.5 f By susung hs vlue of n nd y ssumng φ =π he followng cn e oned onsderng wh ws menoned revously he vlue of = < s ccele. n such cse 3 s used n order o hve less ekng: Snce he vlue of s negve hus = he vlue of K O s usully defne nd s seleced equl o. 9 Hz. Moreover vlue of ccor s on-ch nd s seleced equl o f. herefore nd should e clculed. he vlues of nd resul n 9 3 KO. 6.7 nd K O 7A..8 K Equons nd demonsre h wh he ncrese n he vlue of ccor nd ncreses nd decreses resecvely. he ccor cnno e seleced very smll snce rronl vlues would e resuled for nd. le shows he rmeer vlues of he desgned Ds. n order o vlde he nlyss he vlues desgned for loo rmeers re he resonle vlues for rccl mlemenon s well whch hs een confrmed n [9] nd [3]. As cn e seen from 9 nd n O-9 he D ndwdh s very smll nd crees smll nd lrge ressor from nd 3. hus he loo ccor s ncresed nd K O s reduced o on resonle vlues of nd P. ABLE DS LOOP PAAMEES SONE O-8 O-9 f MHz KHz f 5 f =.5 5 f 5 f K O MHz 5MHz 7μA μa P.8KΩ 5.7KΩ 6 A. Jer rnsfer Smulons Sysem level smulon s erformed y he hsedomn model of he BBD whch hs een shown n Fg. 3. n order o cheve he jer rnsfer chrcersc hrough smulon snusodl jer wh vryng mlude ws led frequences from 5 khz o 3 MHz nd ech jer frequency he ouu jer mlude s mesured. For exmle he jer wh n mlude of.5u nd frequency of 5MHz s led s φ n=πsnπ 5 6. hen he mgnude of φ ou s mesured whch equls φ o=.rd. Afer h he φ oφ ro s exressed n db h s log.π= db. he ouu mlude s lso mesured for oher nu frequences n he sme mehod nd he φ oφ ro s exressed n db. Jer rnsfer chrcersc s oned hrough connecng hese ons. Fg. los he heorecl nd smuled jer rnsfer of oh desgned Ds. hs lo revels -dbdec roll-off n erms of jer frequency. A comrson eween smuled nd clculed vlues shows he ccurcy of nlycl equons. As redced y he nlyss nd ccordng o Fg. he corner frequency of he jer rnsfer curve s decresed y ncresng he nu jer mgnude. hus f φ vres ω s oned from. he corner frequency vlues for dfferen nu jer mludes re summrzed n le. Also le 3 shows he jer ekng vlues for dfferen nu jer mludes for he wo desgned exmles. As redced he jer ekng s less hn.db. Jer rnsfer db Jer rnsfer db U Jer Frequency MHz.5 U.5 U.5 U -5.. Jer Frequency MHz Fg.: Smuled sold lnes nd clculed dshed lnes jer rnsfer of SONE O-8 nd SONE O-9 for dfferen nu jer mludes. Amrkr MS ol. No. Fll

7 ABLE HE ONE FEQUENY FO DFFEEN NPU JE AMPLUDES f φ O-8 O-9 U clculed smuled clculed smuled.5 MHz.97MHz KHz KHz.5.9MHz 3 MHz 75KHz 8KHz ABLE 3 HE JE PEAKNG FO DFFEEN NPU JE AMPLUDES Jer Pekng db φ O-8 O-9 U smuled smuled B. Jer olernce Smulon Jer olernce s mesure of he D cly n olerng he nu jer nd s usully cheved y lyng snusodl jer wh dfferen mludes gven frequency rnge. hus gven jer frequency we mus ncrese he mlude of he nu jer φ n nd oserve Δφ= φ n- φ ou unl he mxmum hse dfference ecome equl o π.herefore he mxmum jer mlude s funcon of jer frequency whch he mxmum hse dfference s equl o π s clled jer olernce. For exmle for O-8 we ly φ n=φ snπ 3 nd φ s grdully ncresed unl Δφ mx=π. cn e seen h n φ =.5U=5π Δφ mx=π wll ke lce. herefore he mxmum olerle nu jer mlude for jer wh frequency of KHz wll e.5u. hs mehod s led n dfferen frequences nd he mxmum olerle φ hs een oned for ech frequency. he jer olernce curve s loed y connecng hese ons. he smuled nd clculed vlues of he mxmum olerle nu jer mludes for dfferen jer frequences re summrzed n le. he clculed vlues of he jer frequency re gven y. Fg. shows he heorecl nd smuled jer olernce of oh Ds. As seen he good greemen eween he closed-form nlycl exresson 3 nd he smulon resuls confrms he roosed nlyss. As execed G J flls re of roxmely dbdec very low jer frequences nd dbdec for greer ω nd fnlly rochng π hgh ω. he SONE msk for O-8 nd O-9 re lso shown n Fg. whch shows h he SONE msk s ssfed. ABLE SUMMAY OF HE SMULAED AND ALULAED ESULS he frequency whch Δφ mx= π KHz φ O-8 O-9 U clculed smuled clculed smuled Jer olernce U. Jer olernce U. SONE O Jer Frequency KHz SONE O-9 Jer Frequency KHz Fg.: Smuled sold lnes nd clculed dshed lnes jer rnsfer of SONE O-8 nd SONE O-9 for dfferen nu jer mludes. 6. ONLUSON he jer rnsfer nd jer olernce nlyss of second-order ng-ng D hs een resened nd lso good nlycl equons were derved. he roosed roch offers wo dvnges comred o he convenonl desgnng mehods. Frs hs roch does no consder ny vlue resrcon o he ccor. Second new condon hs een resened o gurnee h he vlue of jer ekng s roxmely zero. he ccurcy of he resened exressons ws verfed hrough sysem level smulons. he resuls n hs work would hel desgners o omze he jer erformnce of BBD n sysem level desgn APPENDX A. he Ouu Phse Equon As deced n Fg. he wveform of O conrol volge cnl cn e wren s follows Amrkr MS ol. No. Fll 3

8 Amrkr MS ol. No. Fll cnl 5 As PD jums eween P nd + P due o ressor he oscllor volge conrol exerences jum of P P. hus 6 From Fg. snce for P he cnl reches - = P usng 5 nd 6 we cn wre 7 8 hus he wveform of O conrol volge cnl s oned s cnl 9 he ouu hse cn e derved s he negron of K O cnl O O ou K K 3 φ ou P cn e smlfed s ou 3 where he rmeers nd re oned from nd 3 resecvely. Accordng o Fg. snce φ ou = φ = - φ ou P usng 3 we hve 3 And O K 33 Susung 33 no 3 he ouu hse P cn e rewren s ou 3 B. New ondon o Avod Jer Pekng Fg. revels h he mxmum on of φ ou occurs = mx whch s oned from 3 s 35 mx 35 Susung 35 no 3 φ o cn e esly oned s 36 o 6 36 elcng 35 no we hve o 6 37 nd hence 38 Susung 38 no 9 we hve: 39 Usng nd 3 he followng morn resul cn e derved s. O K 3 8. EFEENES [] J. K. Km J. Km G. Km nd D. K. Jeong A fully negred.3-μm MOS -Gs serl lnk rnscever EEE J. Sold- Se rcus. vol. no My. 9. [] N. D Dl E. hller P. Gregorus nd L. Gzs. A comc rle-nd low-jer dgl L PLL wh rogrmmle col n 3-nm MOS EEE J. Sold-Se rcus. vol. no Jul. 5. [3] Y. S. Seo J. W. Lee H. J. Km h. Yoo J. J. Lee nd h. S. Jeong A 5-gs clock nd d recovery crcu wh 8-re lner hse deecor n.8-μm MOS echnology EEE rns. rcus Sys. Ex. Brefs vol. 56 no.. 6- Jn. 9. [] B. zv hllenges n he desgn of hgh-seed clock nd d recovery crcus EEE ommun. Mg.vol no Aug.. [5] N. D Dl Mrkov chns-sed dervon of he hse deecor gn n ng-ng PLLs EEE rns. rcus nd Sys. Ex. Brefs vol. 53 no Nov. 6. [6] B. hun nd M. P. Kennedy Sscl roeres of frs-order ng-ng PLL wh nonzero loo dely EEE rns. rcus nd Sys. Ex. Brefs vol. 55 no.. 6 Oc. 8.

9 [7] S. ernek J. P. Gleeson nd O. Feely Sscl nlyss of frs-order ng-ng hse-locked loos usng sgn-deenden rndom wlk heory EEE rns. rcus Sys. eg. Pers vol. 57 no Se.. [8] S. heng H. ong J. Slv-Mrnez nd A.. Krslyn Sedy-se nlyss of hse-locked loos usng nry hse deecor EEE rns. rcus nd Sys.. vol.5 no Jun. 7. [9].. Wlker Desgnng ng-ng PLLs for clock nd d recovery n serl d rnsmsson sysems n Phse-lockng n Hgh Performnce Sysems B. zv Ed. Pscwy NJ: EEE Press [] J. Lee K. S. Kunder nd B. zv Anlyss nd modelng of ng-ng clock nd d recovery crcus EEE J. Sold-Se rcus. vol.39 no Se.. [] J. L J. Slv-Mrnez B. Brunn Sh. okhsz nd M. E. onson A full on-ch MOS clock-nd-d recovery for O-9 lcons EEE rns. rcus Sys.. eg. Pers vol. 55 no.5.3- Jun. 8. [] B. zv Desgn of negred rcu for Ocl ommuncons Frs Ed. New York: McGrw Hll 3. [3] F. A. Mus nd A.. rusone Modelng nd desgn of mullevel ng-ng Ds n he resence of S nd nose EEE rns. rcus Sys. eg. Pers vol. 5 no Oc. 7. Amrkr MS ol. No. Fll 5