Mechanism of Jitter Amplification in Clock Channels

Transcription

1 Mchanism of Jir Amplificaion Cloc Channls Fangyi Rao Agiln Tchnologis, Inc. Sammy i Junipr Nwors

2 Oul Inroucion Thory of Jir Amplificaion Comparison wih Simulaions Summary

3 Oul Inroucion Thory of Jir Amplificaion Comparison wih Simulaions Summary

4 Inroucion v v ou channl SJ + DCD + RJ SJ ou + DCD ou + RJ ou In cloc channls, ISI ir is complly lima by h prioic 11 cloc parn Oupu ir is nirly uc by pu ir Jir is amplifi lossy channls vn whn h channl is lar, passiv an noislss A SJ,ou > A SJ, DCD ou > DCD RMS RJ,ou > RMS RJ,

5 Masurmn on DCD an RJ Inl 6 Two 15-ch passiv cloc channls a 6.4Gbps an 9.6Gbps On wih 1 ohm mach bacplan, h ohr wih 81 ohm mismach bacplan Amplificaions ar foun o crass wih aa ra S. Chauhuri, W. Anrson, J. McCall, an S. Dabrai, Jir amplificaion characrizaion of passiv cloc channls a 6.4 an 9.6 Gb/s, Proc. IEEE 15 h Topical Mg on Elcric Prformanc of Elcronic Pacagg, Scosal, AZ, Oc. 6, pp. 1-4.

6 Simulaion on DCD an RJ Rambus 7 8, 16, an 4-ch PCB cloc channls a aa ras bwn an 6G DCD an RJ amplificaions scal uniquly wih loss Inicas a l bwn h ffc an channl loss C. Man, S. Chang, D. Oh an C. Yuan, Jir Amplificaion Consiraions for PCB Cloc Channl Dsign, IEEE 16 h Topical Mg on Elcr. Prformanc Elcron. Pacag., Alana, GA, pp , Oc. 7.

7 Rsuls of SJ, DCD an RJ Agiln 1 SJ amplificaion is foun o crass xponnially wih SJ frquncy DCD an RJ scal uniquly wih loss F. Rao an S. i, Frquncy oma analysis of ir amplificaion cloc channls, Proc. IEEE 1 h Topical Mg on Elcric Prformanc of Elcronic Pacagg, Tmp, AZ, Oc. 1, pp hp://cp.liraur.agiln.com/liwb/pf/ en.pf

8 Oul Inroucion Thory of Jir Amplificaion Comparison wih Simulaions Summary

9 Susoial Wav Rprsnaion of Cloc Signal igh orr harmonics ar havily anua lossy channls Rpa 11 cloc signal can b approxima by a susoial wav Jir can b rprsn by phas moulaion v v v Acos[ ] A [ ] complx conuga : funamnal frquncy, half of aa ra : phas moulaion ha rprsn ir

10 Susoial Jir Amplificaion * c c A v. ] [ * c c A c c A v ou. ] [. ] [ * * cays xponnially wih lossy channl LSB is anua lss han h carrir is by a facor of -/ A ga of PM is uc a h channl oupu, lag o ir amplificaion Inpu signal wih SJ a Oupu signal USB LSB USB LSB carrir carrir frquncy carrir LSB USB

11 Susoial Jir Amplificaion con AM an PM ar uc a oupu by pu SJ, causg impairmns im an volag Oupu SJ Oupu signal R ]cos Im xp[ A v ou * ] cos[ R ou SJ ransfr funcion/amplificaion facor: ampliu raio bwn oupu an pu SJ 1 F SJ Jir amplificaion is h rsul of smallr loss LSB han funamnal Whn rflcion is small, can b rplac by forwar S-paramrs AM PM

12 DCD Amplificaion DCD is a spcial cas of SJ a = DCD amplificaion facor F DCD 1 A =, LSB bcoms a DC shif Oupu = DC + Funamnal + ighr orrs harmonics ngligibl unr loss Signal is shif vrically by DC, causg imbalanc bwn 1-bi an -bi cycls, i.. DCD DC shif is proporional o, which is always aroun B rgarlss h loss Funamnal is proporional o Th largr h loss a, h smallr h funamnal ampliu, an h largr h DCD

13 Ranom Jir Amplificaion Inpu RJ: assum whi nois wih fla powr spcral nsiy C C Oupu RJ ou F SJ C RJ amplificaion facor: RMS raio bwn oupu an pu RJ 4 1 F ou RJ

14 Comparison wih Squar Wav Formulaion Inpu cloc signal is mol by squar wav. Jir is appli o ransiions. v nt n Oupu signal is compu wih suprposiion of sp rspons R lvn v R lt R mt ou l mo Oupu ir is masur from crossg im shif o ou n m m 1 h nt mt m 1 m h nt mt For pu SJ m = cosmt, DTFT yils ou n c. c m m : channl lay h: R/ or impuls rspons. FT[h]= o nt o o cos nt

15 Comparison wih Squar Wav Formulaion con SJ amplificaion facor Convrg o susoial wav rsul afr nglcg highr orr harmonics an usg phas lay of a for o o SJ F 1 F SJ Equivalnc of F SJ bwn susoial an squar wav formulaions auomaically las o quivalnc of F DCD an F RJ

16 Scalg of DCD an RJ Amplificaion wih Loss Channl loss mol SJ amplificaion facor for < xp F SJ 1 Jir is amplifi a all frquncis blow. F SJ grows xponnially wih frquncy. DCD an RJ amplificaions F DCD an F RJ grow xponnially wih aa ra DCD an RJ scalg wih loss F DCD F RJ F DCD F RJ cosh ln1 D / 8 5 ln1 sh ln1 D D D log1

17 Oul Inroucion Thory of Jir Amplificaion Comparison wih Simulaions Summary

18 Comparison wih Simulaions 4 sgl-n channls wih iffrn losss Inpu cloc signal is mol by squar wav wih ir appli a ransiions Oupu signal is compu wih sp rspons suprposiion 1 million bis pr simulaion

19 Susoial Jir Channl a 1Gbps wih pu SJ amp = 5ps an frquncy =.5, an 3Gz Simula oupu ys Simula oupu ir isribuions

20 Susoial Jir con F SJ channl 1 a 1Gbps an channl a 1 an Gbps Thory 1 F SJ 1 S1 S1 S1 S1 Thory loss approx. F SJ Two horical pricions wih S,1 an wih loss mol is xrac from srion loss slop agr wih simulaions Loss mol is shown o b a goo approximaion hs channls F SJ grows xponnially wih ir frquncy F SJ is snsiiv o aa ra, as pric by h loss mol

21 DCD Oupu ys of channls 1, an 3 a 1Gbps wih 5%UI pu DCD Th highr h loss, h largr h oupu DCD

22 DCD con F DCD channls 1 an Thory 1 F DCD 1 S1 S1 S1 S1 Thory loss approx. F DCD

23 DCD con Oupu DC shif p 4 DC shif a channl 1 oupu V DC A DCDpp, UI VDCDpp, UI V =pa/4 is h pu squar wav ampliu

24 Ranom Jir Channl 1 wih 1ps pu RJ a 8, 1 an 16 Gbps aa ras Simula oupu ys Simula oupu ir isribuions

25 Ranom Jir con F RJ channls 1 an Thory 1 F RJ S1 S1 41 S1 S1 Thory loss approx. F RJ 8 1

26 DCD an RJ Amplificaion Scalg wih Loss All 4 channls, ach a iffrnc aa ras up o 16Gbps Insrion loss is masur a funamnal Scalg Thory F DCD F RJ cosh ln1 D / 5 ln1 sh ln1 D D 1 1

27 Summary Jir is amplifi by channl loss I occurs vn whn h channl is lar, passiv an noislss Amplificaion aris from smallr anuaion ir LSB han h funamnal In lossy cloc channls, ir is amplifi a any ir frquncy blow Nyquis Th ffc grows xponnially wih ir frquncy an aa ra DCD an RJ amplificaions scal uniquly wih loss cloc channl Tx ir mus b appli pu simulus whn simulag l prformanc orr o capur h amplificaion ffc

28 DsignCon 14 Mchanism of Jir Amplificaion Cloc Channls Fangyi Rao, Agiln Tchnologis, Inc Sammy i, Junipr Nwors

29 Absrac Jir amplificaion cloc channls is analyz analyically rms of signal ransfr funcion or channl S-paramrs. Th prioiciy of h cloc parn limas h r-symbol-rfrnc ir so ir a h channl oupu is nirly uc by pu ir. A phas moulaion PM approach is mploy o riv h ir ransfr funcion an amplificaion facors for susoial ir SJ, uy-cycl-isorion DCD an ranom ir RJ. Rsuls monsra ha ir amplificaion is h consqunc of smallr anuaion a h ir lowr siban LSB han a h funamnal, which is a a highr frquncy han h LSB. Scalg quaions of DCD an RJ amplificaions wih channl loss is oba by mployg an xponnial loss mol. I is shown ha ir is amplifi by lossy channls a any frquncy blow Nyquis an h ffc grows xponnially wih ir frquncy an aa ra. Amplificaion facors of SJ, DCD an RJ ar also riv wih h squar wav rprsnaion of cloc signals, an h rsuls ar shown o rcovr hos usg h PM approach whn high orr harmonics ar nglc. Th hory is vrifi by simulaions. Auhors Biography Fangyi Rao is a masr ngr a Agiln Tchnologis. rciv his Ph.D. gr horical physics from Norhwsrn Univrsiy. o Agiln EEsof 6 an wors on Analog/RF an SI simulaion chnologis ADS an RFDE. From 3 o 6 h was wih Canc Dsign Sysms, whr h vlop h company's armonic Balanc chnology an prurbaion analysis of nonlar circuis. Prior o 3 h wor h aras of EM simulaion, nonlar vic molg, an mical imagg. Sammy i is a snior lcrical ngr a Junipr Nwors. Prior o Junipr h was a chnical lar a Cisco Sysm for mor han 11 yars an a prcipal ngr a Rambus Inc. for six yars. Prior o Rambus h was a sign ngr a iffrn firms clug Tanm Compur an Philips. rciv his BS gr Elcrical Engrg from Univrsiy of Bagha.

30 1. Inroucion igh sp rconnc prformanc is crasgly flunc by ir as aa ra avancs. Th amoun of ir is moula by channl isprsion as signals propaga h sysm. I is obsrv boh masurmns an simulaions ha ir can b amplifi by a lossy channl vn whn h channl is lar, passiv an noislss [1]- [5]. Th ffc happns o iffrn ir yps clug susoial ir SJ, uy-cyclisorion DCD an ranom ir RJ. In paricular, DCD an RJ amplificaions cloc signals ar foun o scal uniquly wih channl loss [], icag ha loss is rsponsibl for h ffc. Th mchanism of ir amplificaion cloc channls is xpla horically [5]. I is monsra ha ir amplificaion is h consqunc of smallr anuaion a h ir lowr siban LSB han a h signal carrir, which is a a highr frquncy compar o h ir LSB. Such anuaion iffrnc amplifis h phas moulaion PM, which is quivaln o ir, h channl oupu signal, lag o ir amplificaion. Th scalg of DCD an RJ amplificaions wih channl loss is riv usg an xponnial loss mol. Jir is foun o b amplifi by lossy channls a any frquncy blow Nyquis, an h ffc grows xponnially wih ir frquncy an aa ra. In his papr ir amplificaion cloc channls is analyz analyically usg h chniqus vlop [5]. Th avanag of usg cloc signals is ha h prioiciy of h 11 cloc parn limas h r-symbol-rfrnc ISI ir so ir a h channl oupu is nirly uc by pu ir. Two approachs ar mploy h suy. In h firs approach, h rpa 11 cloc parn is approxima by a susoial wav wih frquncy a half of h cloc aa ra an wih phas moulaion ha rprsns ir. Jir ransfr funcions an amplificaion facors of SJ, DCD an RJ ar riv rms of signal ransfr funcion or channl S-paramrs. Scalg quaions of DCD an RJ amplificaions wih channl loss ar oba. In h scon approach, a mor ralisic squar wav rprsnaion is us o mol h cloc signal wih ris an fall gs bg shif by ir. I is shown ha h squar wav formulaion yils h sam rsuls as h susoial formulaion os whn high orr harmonics ar ignor. Thorical pricions ar confirm by numrical Mon Carlo channl simulaions runng on million bis.. Jir Transfr Funcion an Amplificaion.1 Susoial Jir

31 In lossy channls high orr harmonics ar havily anua an h 11 cloc parn can b approxima by a susoial wav wih frquncy a on half of h aa ra. Jir h pu cloc signal, v, can b rprsn by phas moulaion as ] cos[ A v 1 whr is h funamnal frquncy of h cloc signal, a consan phas offs, an h phas moulaion ha rprsns ir. Whn is small, Eq. 1 can b lariz as ] xp xp xp [xp A v Consir a susoial ir a frquncy. xp xp * 3 Subsiuion of Eq. 3 o Eq. yils ]} xp[ ] xp[ xp ] xp[ ] xp[ {xp * * A v 4 Eq. 4 shows ha h PM spcrum is shif by h carrir an spli o h lowr siban a -, h uppr siban a +, an hir complx conugas. Assum h signal ransfr funcion of h channl is. Th oupu signal, v ou, is givn by

32 ] [1 xp ] [1 xp ]} xp[ ] xp[ xp ] xp[ ] xp[ xp { * * A A A v ou 5 whr + an - ar f as * * 6 Noic ha + = - *. For small Eq. 5 can b rwrin as R cos ] Im xp[ A v ou 7 whr R + an Im + no ral an imagary pars of +, rspcivly. Th phas moulaion h oupu signal is givn by h R + rm Eq. 7 as ] cos[ R ou 8 whr is h phas of [+ / +- /- ]. Equaion 8 shows ha a SJ is uc h oupu by h pu SJ. Th ir ransfr funcion, f as h ampliu raio bwn oupu an pu SJ, is oba as 1 F SJ 9 Equaion 9 scribs h rlaion bwn ir amplificaion an channl isprsion. In a lossy channl, as illusra Fig. 1, cays wih frquncy xponnially. Th

33 lowr siban of PM a - is anua lss han h carrir is, proucg a ga h oupu PM ha las o ir amplificaion. Equaion 9 monsras ha h amplificaion, oma by h - rm, ariss primarily from h anuaion iffrnc bwn h LSB an h funamnal. Equalizaions ha compnsa high frquncy loss ruc h amplificaion ffc. B Frquncy LSB Carrir USB Figur 1. Mchanism of ir amplificaion lossy channls. I shoul b po ou ha h pu SJ also ucs an ampliu moulaion h oupu signal, which is givn by h Im + rm Eq. 7 as A ou A xp[ Im ] A A Im s[ ] 1 whr is h phas of [+ / -- /- ]. This ampliu moulaion is a susoial a frquncy an causs y high impairmn a h channl oupu.. Duy-cycl-isorion Whn pu ir is absn, h ial pu ransiion im by zro-crossg of v xprss Eq. 1 wih = as cos s n n 1 n1 of h n-h bi is rm 11

34 which h scon quaion nsurs ha vn bis ar logic 1. Wih pu DCD h ransiion im is shif from by n1 n 1 s n 1 whr is half h pa-o-pa DCD. As a rsul, all vn bis ar longr whn > han all o bis. Equaion 1 icas ha DCD is quivaln o a SJ a frquncy, an h quivaln PM is cos 13 or xp 14 Th ir amplificaion facor for DCD is hus givn by Eq. 9 a = as 1 F DCD 15 Th mchanism of DCD amplificaion can b unrsoo uiivly rms of h DC shif rouc by pu DCD [1]. No ha a = h LSB bcoms a DC componn. Subsiug Eq. 13 o Eq. yils v Acos A Acos 16 Th pu DC shif prouc by DCD is A /. Th oupu signal is v ou Acos[ ] A Acos[ ] 17 Th oupu DC shif is A /. Eq. 17 shows ha h oupu signal is compos of funamnal an DC componns if h scon harmonic is ignor lossy channls. As illusra Fig., h DC shif causs all logic 1 bis h 11 parn o b longr whn > han all logic bis a v ou =, lag o DCD h oupu. Th zrocrossg im shif from h ial crossg im givn by

35 can b calcula from h DC rm an h funamnal slw ra a, as, which is ou n1 A / 1 n1 n A Equaion 18 givs h sam DCD amplificaion facor as Eq. 15 os whn h rm is ignor. No ha mos channls is aroun B. Equaion 18 an Fig. show ha h highr h loss a h funamnal, h largr h oupu DCD..3 Ranom Jir Figur. Mchanism of DCD amplificaion. RJ h pu signal is assum o b whi nois, an is avrag powr is givn by h graion of h powr spcral nsiy PSD wih h ir Nyquis frquncy, which quals. C C 19 whr C is h consan pu RJ PSD. Th oupu RJ powr is givn by h ir ransfr funcion an C as ou C F SJ Th RJ amplificaion facor, f as h RMS raio bwn oupu an pu RJ, is

36 F RJ 1 4 ou 1 Whn impanc mismach h channl is ngligibl, Eqs. 9, 15 an 1 can b rplac by channl forwar S-paramrs. 3. Equivalnc bwn Susoial an Squar Wav Rprsnaions Whil all iscussions so far ar bas on h susoial wav rprsnaion of h cloc signal, i can b shown ha sam rsuls can b oba usg h squar wav rprsnaion. 3.1 Susoial Jir In Fig.3 h pu cloc signal is rprsn by a 11 squar wav whos n-h ransiion im is a, whr T is h uni rval an h pu ir a h n-h bi. No ha T= sc is half of h aa ra. As iscuss [3], [4] an [5], h oupu signal of a lar channl can b calcula by lar suprposiion as lvn v R lt R mt ou whr R is h channl sp rspons. l mo m v nt τ n

37 Figur 3. Squar wav rprsnaion of cloc signal Whn h pu ir is zro, hr is no ir h oupu u o h prioiciy of h cloc parn. For a givn lay, v ou crosss h sam valu a = nt+ for any gr n. Wih h prsnc of pu ir, ir uc v ou can b masur by h crossg im shif, which is rm by v ou nt ou n lvn lvn R nt R nt ou n lt lt mo l R nt mo R nt mt ou n mt m 3 whr ou n is h shif of h n-h crossg. For small ir, larizaion of Eq. 3 yils m 1 h nt mt m ou m n 4 m 1 h nt mt m whr h = R/ is h channl impuls rspons, an is Fourir ransform FT is h ransfr funcion us prvious scions. Consir a susoial pu ir a frquncy, m cos mt 5 Subsiuion of Eq. 5 o Eq. 4 yils ou n ~ m h nt nt mt{xp[ mt] xp[ mt]} m h nt ~ nt ~ nt mtxp mt 6 which iniy -1 m ~ = xp- mt is us. is h iscr-im Fourir ransform DTFT of sris hmt+ f as Th rlaion bwn DTFT an FT is [6] ~ ~ h mt xp mt 7 m xp[ ] 8

38 By uilizg Eq. 8, Eq. 6 can b wrin rms of as xp xp ] xp[ xp xp ] xp[ nt nt o o o o ou n 9 r iniy xp nt = -1 n for o gr is us. Noic ha h firs rm Eq. 9 is h complx conuga of h scon. Thus, o o o o ou n nt xp ] xp[ cos xp ] xp[ 3 Equaion 3 shows ha a SJ is uc a h channl oupu by h pu SJ an h ir ransfr funcion is o o SJ F xp ] xp[ 31 In lossy channls, high orr harmonics can b nglc, an Eq. 31 can b approxima as xp xp ] xp[ ] xp[ SJ F 3 By choosg o b h phas lay of a, h phas of is cancll by xp. As a rsul, an ar boh ral an qual o ach ohr. Eq. 3 hn bcoms

39 F SJ 1 xp[ xp 1 ] xp[ xp ] 33 Equaion 33 is inical o Eq. 9. As xpc, h squar wav formulaion convrgs o h susoial wav formulaion whn high orr harmonics ar ignor. Equaion 33 also sablishs h quivalnc of DCD an RJ amplificaion rsuls bwn susoial an squar wav rprsnaions. Nvrhlss, rivaions for DCD an RJ ircly from h squar wav formulaion ar provi h followg wo scions. 3. Duy-cycl-isorion Subsiug Eq. 1 o Eq. 4 yils ou n 1 m ~ ~ n1 1 nt nt m h nt lvn m o h nt mt mt l xp l xp 34 whr is h phas lay of a as h SJ iscussion abov. Th DCD amplificaion facor is givn by Eq. 34 as F DCD l vn o l xp l xp 35 Afr nglcg high orr harmonics, boh Eq. 35 an Eq. 15, which is riv from h susoial wav rprsnaion, convrg o 3.3 Ranom Jir 1 F DCD 36

40 As po ou scion.3, RJ is uncorrla whi nois, an nm RJ m n 37 whr is h pu RJ RMS. Subsiug Eq. 37 o Eq. 4 las o xp ] xp[ 1 ~ ~ 1 1 o l RJ nt nt RJ m m m RJ ou n l l mt nt h mt nt h 38 whr h Parsval s horm [6] is appli o h numraor. No ha h graion rang Eq. 38 can b shif from [-, ] o [, ] u o h prioiciy of DTFT. As a rsul, h RJ amplificaion facor can b xprss, afr a variabl chang from o -, as xp ] xp[ 1 o o RJ ou n RJ F 39 whr is h phas lay of a as prvious iscussions. Afr nglcg high orr harmonics, Eq. 39 rcovrs Eq. 1 as shown blow. 4 1 xp xp ] xp[ ] xp[ 1 F RJ 4

41 4. Scalg of DCD an RJ Amplificaions wih Channl Loss Th scalg of DCD an RJ amplificaions wih channl loss obsrv [] can b riv usg an approxima loss mol scrib by xp 41 whr is h loss consan an h channl lay. Subsiuion of Eq. 41 o Eq. 9 yils h amplificaion facor for SJ blow h ir Nyquis frquncy as F SJ xp xp 4 I can b asily shown ha an ir is amplifi by lossy channls a any frquncy blow. Equaion 4 also icas ha F SJ grows xponnially wih ir frquncy. DCD an RJ amplificaions wih h loss mol ar givn by subsiug Eq. 41 o Eq. 15 an Eq. 1, rspcivly. F DCD xp xp 43 xp xp 1 F RJ 44 8 F DCD an F RJ ar shown o cras xponnially wih aa ra. Scalg of F DCD an F RJ is oba by rwrig Eq. 43 an Eq. 44 as F DCD cosh ln1 D / 45 5 ln1 1 F RJ sh ln1 D D 1 46 whr D = log 1 nos h channl loss B a h funamnal frquncy. 5. Comparison bwn Thory an Simulaion

42 A s of four sgl-n channls rma wih 5 Ohm ar us h suy. Thir S-paramrs ar gnra from EM simulaions. Th Svnsson-Drmr mol [7] is mploy o mol h subsra loss. Simula srion loss an rurn loss ar plo Fig. 4 an lis Tabl 1. Th cloc signal ransmi o h channl is rprsn by h 11 squar wav as shown Fig. 3. SJ, DCD an whi nois Gaussian RJ ar appli a h ransiions. Th channl oupu signal is calcula wih Eq. usg sp rsponss characriz by SPICE ransin simulaions. On million bis ar run ach simulaion. S,1 5Gz 1Gz channl B B channl B B channl B B channl B B Tabl 1. Channl srion loss a 5 an 1 Gz.

43 5.1 Susoial Jir Figur 4. Channl srion loss an rurn loss. A SJ wih 5 ps ampliu is a o h pu cloc signal. Oupu y iagrams of channl a 1 Gbps aa ra wih SJ frquncis of.5, an 3 Gz ar shown Fig. 5. Oupu ir probabiliy nsiy funcions masur a V ar plo Fig. 6. Thy xhibi h characrisic shap of h SJ isribuion scrib by 1 p x 47 x

44 whr is h SJ ampliu. Th oupu SJ ampliu can b masur from locaions of h wo pas h PDF. As shown Fig. 6, h oupu SJ ampliu a.5 Gz is h sam as h pu. A an 3 Gz, oupu amplius ar abou 1.4 an ims largr han h pu, rspcivly. As pric by Eq. 4, h oupu SJ amplificaion grows wih SJ frquncy. In Fig. 5, ampliu nois is foun o b uc by h pu SJ as pric by Eq. 1. Figur 5. Oupu y iagrams of channl a 1Gbps aa ra. Inpu SJ ampliu is 5ps. SJ frquncy is a.5gz, b Gz, an c 3Gz. Figur 6. Oupu ir isribuions of channl a 1Gbps aa ra. Inpu SJ ampliu is 5ps. SJ frquncis ar.5gz, Gz, an 3Gz. Simula SJ amplificaion facors as funcions of SJ frquncy channls 1 an a 1 an Gbps aa ras ar plo Fig. 7. Two ss of horical rsuls, calcula usg Eq. 9 bas on S,1 an usg Eq. 4 bas on h approxima loss mol scrib Eq. 41, ar also shown h plo. Loss consans h loss mols ar xrac from slops of srion loss. Figur 7 shows ha simulaion rsuls ar goo agrmn wih horical pricions. Th iscrpancy bwn rsuls givn by Eq. 9

45 an Eq. 4 is foun o b mor, icag ha h loss mol is a rasonabl approximaion hs channls. Comparison of rsuls channl bwn 1 an Gbps suggss ha F SJ is snsiiv o aa ra lossy channls, as pric by Eq. 4. Th amplificaion facor is foun o b grar han or qual o on a any SJ frquncy an grow xponnially wih i. Figur 7. SJ amplificaion facors oba from simulaions an horical calculaions wih Eq. 9 an Eq Duy-cycl-isorion Oupu y iagrams of channls 1, an 3 a 1 Gbps aa ra wih 5% UI pu pao-pa DCD ar plo Fig. 8. Th y cnr is shif upwar by DCD as pric by Eq. 17. Figur 8 shows ha as h loss crass from channl 1 o channl 3, h funamnal ampliu crass, an h oupu DCD crass. Simula DCD amplificaion facors as funcions of aa ra channls 1 an ar plo Fig. 9. Th rsuls ar agrmn wih boh ss of horical valus calcula usg Eq. 15 an Eq. 43 rspcivly. Amplificaion facors ar foun o b grar han or qual o on a all aa ras an grow xponnially wih aa ra, as pric by Eq. 43 lossy channls. Th oupu DC rm Eq. 17 can b rwrin rms of pu pa-o-pa DCD UI as

46 A 4 V DC A DCDpp, UI VDCDpp, UI 48 whr V =A/4 is h pu squar wav ampliu an quals 5V his cas. is B for all channls. In Fig.1, simula DC shifs channl 1 a 1 an Gbps aa ras as funcions of pu pa-o-pa DCD ar foun o agr wih Eq. 48. Figur 8. Oupu y iagrams of channls 1, an 3 a 1Gbps aa ra wih 5% UI pu pa-o-pa DCD.

47 Figur 9. DCD amplificaion facors oba from simulaions an horical calculaions wih Eq. 15 an Eq. 43. Figur 1. DCD uc DC shif channl 1 oupu signal oba from simulaions an horical calculaions wih Eq Ranom Jir

48 Figur 11 shows oupu y iagrams of channl 1 a 8, 1 an 16 Gbps aa ras wih 1ps pu Gaussian RJ. Oupu ir probabiliy nsiy funcions masur a V, plo Fig. 1, manifs Gaussian characrisics. Th oupu RJ RMS crass wih aa ra. Simula RJ amplificaion facors as funcions of aa ra channls 1 an ar plo Fig. 13. Rsuls ar consisn wih boh ss of horical valus givn by Eq. 1 an Eq. 44 rspcivly. Figur 11. Oupu y iagrams of channl 1 wih 1ps pu RJ a aa ras of a 8G, b 1G, an c 16G. Figur 1. Oupu ir isribuions of channl 1wih 1ps pu RJ a aa ras of 8, 1 an 16 Gbps.

49 Figur 13. RJ amplificaion facors oba from simulaions an horical calculaions wih Eq. 1 an Eq Scalg of DCD an RJ Amplificaions wih Channl Loss Figur 14 shows h scalg of F DCD an F RJ wih channl srion loss a h funamnal frquncy all channls a iffrn aa ras. Th horical scalg is givn by Eq. 45 for DCD an Eq. 46 for RJ. Agrmn is foun bwn simulaion an hory all cass. Th scalg curvs ar also consisn wih simulaion rsuls rpor [].

50 Figur 14. DCD an RJ amplificaion scalg wih srion loss oba from simulaions an horical calculaions wih Eq. 45 an Eq. 46. Th srion loss is masur a h funamnal frquncy. 6. Summary In his papr cloc channl ir amplificaion facors rms of ransfr funcion or S- paramrs ar riv. Amplificaion is shown o rsul from h smallr loss a h ir LSB han a h funamnal. Th amplificaion scalg wih channl loss is oba by usg an approxima loss mol. In his mol h amplificaion is foun o occur a any ir frquncy. Th hory is confirm by simulaion aa. Rfrncs [1] S. Chauhuri, W. Anrson, J. McCall, an S. Dabrai, Jir amplificaion characrizaion of passiv cloc channls a 6.4 an 9.6 Gb/s, Proc. IEEE 15 h Topical Mg on Elcric Prformanc of Elcronic Pacagg, Scosal, AZ, Oc. 6, pp [] C. Man, S. Chang, D. Oh an C. Yuan, Jir Amplificaion Consiraions for PCB Cloc Channl Dsign, IEEE 16 h Topical Mg on Elcr. Prformanc Elcron. Pacag., Alana, GA, pp , Oc. 7. [3] S. Chang, D. Oh an C. Man, Jir molg saisical l simulaion, Proc. IEEE Elcromagn. Compa. Symp., Droi, MI, Aug. 18-, 8. [4] F. Rao, V. Borich,. Abb an M. Yan, Rigorous molg of ransmi ir for accura an fficin saisical y simulaion, IEC DsignCon, Fb. 1.

51 [5] F. Rao an S. i, Frquncy oma analysis of ir amplificaion cloc channls, Proc. IEEE 1 h Topical Mg on Elcric Prformanc of Elcronic Pacagg, Tmp, AZ, Oc. 1, pp [6] A. Oppnhim an R. Schafr, Discr-im Signal Procssg, Prnic-all, Nw Jrsy, [7] C. Svnsson an G. Drmr, Tim oma molg of lossy rconncs, IEEE Trans. Av. Pacagg, vol. 4, no., pp , May 1.