Modeling of the CML FD noise-to-jitter conversion as an LPTV process

Transcription

1 Modlig of h CML FD ois-o-ir covrsio as a LPV procss Marko Alksic. Rvisio hisory Vrsio Da Comms. //4 Firs vrsio mrgd wo docums. Cyclosaioary Nois ad Applicaio o CML Frqucy Dividr Jir/Phas Nois Aalysis fil CycloNoisAdFDir_v..doc. Frqucy Aalysis of LPV ysms ad Applicaio o h FD ois-o-ir graio fil LPVadFDir_v..doc. hory of LPV sysms ad cyclosaioary ois procsss. Formulaio of h sysm fucio of a LPV sysm Assum a LPV sysm wih a impuls rspos fucio hτ whr rprss h curr im im a which h sysm is obsrvd ad τ rprss h arrival im of h ipu impuls. Obviously τ i is assumd ha h sysm is causal ad sic h sysm is priodic hτ hττ. A sysm fucio ca b dfid which rprss h im-varyig rasfr fucio of h sysm [Zadh 95] τ τ h τ dτ h τ dτ ic h sysm is priodically-im varyig ca b rprsd by is Fourir sris whr h Fourir cofficis ar im-idpd ad ar calld harmoic rasfr fucios d 3 I 3 π/. ubsiuig from io 3 3 bcoms

2 ad fially τ h τ dτ d 4 τ h τ dτ d 5 Lowr limi i h ir igral i 5 is acually / sic oly o priod of h impuls rspos fucio is obsrvd. Aohr way of fidig h harmoic rasfr fucio follows [Gardr 99]. If hτ is h impuls rspos of a LPV sysm h h is priodic wih. r rprss h im diffrc bw h occurrc of h impuls a h ipu of h sysm ad h im a which h impuls rspos of h sysm is obsrvd. ic h is priodic i ca b rprsd by is Fourir sris h h 6 whr h is h h d 7 h sysm ca agai b rprsd by h sysm fucio h d h d 8 h dfiiio 8 of h sysm fucio is idical o h o giv i wh τ i is subsiud by. Fucio h- is acually h dlayd by. hrfor is Fourir sris is h h 9 h sysm fucio ow bcoms

3 d d h Afr rarragig bcoms d d h I is agai brok io is Fourir sris ad h cofficis of h sris ar h harmoic rasfr fucios d d h h uppr limi i h our igral i should b / sic oly o priod of h impuls rspos fucio is obsrvd. Compac form of h xprssio for h harmoic rasfr fucio is d h 3. Characrisics of cyclosaioary ois Cyclosaioary radom procsss ar dscribd i [Gardr 99]. A radom procss is said o b cyclosaioary i h wid ss if is ma valu ad auocorrlaio fucio ar priodic. For h aalysis of propagaio of cyclosaioary ois hrough a liar im varia or ivaria sysm auocorrlaio fucio ad is Fourir rasform i.. powr spcral dsiy ar of gra irs. h auocorrlaio fucio of a cyclosaioary procss R saisfis R R 4 whr ad ar h im isacs a which h procss is obsrvd ad is h priod. 4 ca also b wri as R R τ τ τ τ 5

4 whr τ is h diffrc bw ad subscrip is omid i 5. ic auocorrlaio fucio is priodic i ca b rprsd as a Fourir sris τ τ R R τ 6 whr π/ ad R τ τ τ R d 7 A im-varia powr spcral dsiy ca b dfid as h Fourir rasform of h auocorrlaio fucio τ τ R τ dτ 8 I cas of a cyclosaioary ois procss ca b rprsd by h Fourir sris 9 whr d Cofficis ar calld cyclic spcra ad i ca b show ha hy ar Fourir rasforms of h cofficis R τ from 3 τ τ R dτ.3 Filrig of cyclosaioary ois hrough a LPV sysm Wh cyclosaioary ois wih a kow powr spcral dsiy i.. cyclic spcra is passd hrough a LPV sysm whos harmoic rasfr fucios ar also kow h rsulig oupu ois powr spcral dsiy ca b foud by applyig h marix quaio from [Roychowdhury 998]

5 whr ad sigifis h rmiia marix rasposd marix of complx cougas. As a xampl assum h LPV sysm ad h ipu cyclosaioary ois procss hav oro harmoic rasfr fucios ad cyclic spcra rspcivly for ad. h h oupu ois cyclic spcrum for h saioary compo is giv by 3 whr sigifis h complx couga. As ca b s i h quaio abov h LPV sysm iroducs dow ad up covrsio of h ipu ois procss. o s his v mor clarly assum ha h ipu ois procss is us a si wav a frqucy i.. -δ- ad - for. h ipu procss is saioary i his cas. h h saioary compo of h oupu ois is

6 δ 4 As ca b s from 4 h oupu ois spcrum ow coais compos a ad ±. hrfor wh a ois procss saioary or cyclosaioary is passd hrough a LPV sysm h oupu ois is cyclosaioary.. Applicaio of h hory o ir graio of h CML frqucy dividr. FD ir graio as a LPV sysm h ois-o-ir impuls rspos fucio of a frqucy dividr or ay ohr swichig circui ca b drmid ihr hrough simulaio or aalyically. r w will assum ha ois-o-ir impuls rspos fucio is alrady kow ad ca b rprsd as follows [ ] u u x h τ τ τ 5 whr τ is h arrival im of h curr impuls is h mom a which h sysm is obsrvd ad u is h uiy sp fucio. h im isa τ corrspods o h omial arrival of h clock sigal which rims h sysm. xτ rprss h crossig im variaio causd by h ois impuls a τ corrspods o h impuls ssiiviy fucio IF from [aimiri 998]. xτ ca b obaid hrough simulaio or drmid aalyically. h plo of xτ for a FD circui is giv i Fig..

7 xτ Fig.. ypical xτ plo. τ [s] h impuls rspos fucio dfid i 5 is prsd i Fig. for wo valus of τ oghr wih xτ. I his xampl s. x 9

8 4 35 h τ 3 5 xτ 5 h τ τ [s] Fig.. Impuls rspos fucio for wo diffr valus of τ ubsiuig h impuls rspos fucio dfid hτ i 5 io 5 or w g h harmoic rasfr fucios. Fig. 3 shows sigl-sidd for..4 as fucios of frqucy fπ rahr ha. x 9

9 f x f [] Fig. 3. armoic rasfr fucios of h ois-o-ir covrsio sysm x 9. A mhod for obaiig h of a CML frqucy dividr Assum ha all h ois a h frqucy dividr oupu origias from h lvl shifr accordig o BDAIM simulaios dvics i h lvl shifr coribu o h FD oupu ois by mor ha ~6%. Also w will assum ha h ois is whi ad auocorrlaio fucio is giv by R σ δ 6 whr σ is h im-varyig powr of h ois procss. h of a swichig circui ca b drmid usig h followig mhod A ach im isa powr spcral dsiy of h swichig circui ca b foud by applyig h radiioal ois aalysis o a liar im-ivaria circui whr DC opraig poi of h w circui ar currs ad volags ha corrspod o hos of h swichig circui a im. I ohr words currs ad volags of h swichig circui ar fro a im ad ois aalysis is h prformd o ha im ivaria circui. As a xampl Fig. 4 shows of a CML lvl shifr ovr o priod of h oupu sigal s.

10 5.6 x [A/ ] x 9 [s] Fig. 4. of a CML lvl shifr Fig. 5 shows how ois powr spcral dsiy varis for diffr oupu volag lvls. o rla plos i Fig. 4 ad 5 oupu volag aks h valu of.9v a ims 56ps hrough 94ps ad valu.v from 6ps o 44ps i Fig. 4.

11 5.8 x f [A/ ] 5 V ou.v 4.8 V ou.9v x f [] Fig. 5. Powr spcral dsiy for diffr oupu volag lvls Now cyclic spcra ca b asily foud usig 7 ad hy ar show i Fig. 6. I ca b s from Fig. 6 ha h mos sigifica cyclic spcrum is h o for. his ca b xplaid by h fac ha sic h oupu volag swig of h CML lvl shifr is small oly 3mV i his xampl powr spcral dsiy dos o vary much wih im ad is mos sigifica harmoic is h o a DC.

12 6 x 5 4 f [A/ ] 3 highr harmoics > x f [] Fig. 6. Cyclic spcra of h CML lvl shifr im varyig powr spcral dsiy Fially h abov rsuls ar compard wih rsuls of h BDAIM priodic ois aalysis. Fig. 7 shows h cyclic spcra drmid accordig o h aalysis dscribd abov ad oupu ois spcrum drmid by BDAIM gr li. I ca b s from Fig. 7 ha h aalysis prsd hr machs h priodic ois simulaio rsuls.

13 6 x 5 BDAIM 4 f 3 [A/ ] highr harmoics > f [] Fig. 7. Compariso wih BDAIM h aalysis abov shows ha if all h ois sourcs i h circui ar whi ad oupu swig of h circui is sigificaly small quival cyclosaioary oupu ois is rprsd by oly o cyclic spcrum for ad ca b subsiud by a saioary oupu ois sourc whos powr spcral dsiy is qual o h powr spcral dsiy i h opraig poi wh o AC sourcs ar applid o h swichig circui. x.3 FD ir rsuls Jir of a CML frqucy dividr ca ow b simad by applyig wih harmoic rasfr fucios ad cyclic spcra calculad abov. As s h oly sigifica harmoic rasfr fucios ad cyclic spcra ar hos for. For ha raso rducs o ir 7 Jir powr ca h b foud as σ ir ir d d 8

14 Afr valuaig 8 for h xampl abov simad RM ir is foud o b σ ir 45.9fs. Wh his rsul is compard o h rsuls obaid usig h FD ir modl dscribd i [Lvaio 4] which yilds σ ir 5.8fs i ca b s ha h mhod prsd hr udrsimas ir assumig ha h modl prsd i [Lvaio 4] is valid. O oicabl drawback of h modl prsd hr is ha ir valu varis wih h oupu frqucy of h frqucy dividr. Whhr ir dos dpd o oupu frqucy should b cofirmd xprimally bu ha is o xpcd ad is i coradicio wih ohr publishd modls. owvr his mhod may b usful for fidig h phas ois of h frqucy dividr. Rfrcs [Zadh 95] L. Zadh Frqucy Aalysis of Variabl Nworks Proc. of IRE Vol [Gardr 99] W. A. Gardr Iroducio o Radom Procsss wih Applicaio o igals ad ysms cod Ediio McGraw-ill 99. [Roychowdhury 998] J. Roychowdhury al Cyclosaioary Nois Aalysis of Larg RF Circuis wih Mulio Exciaios IEEE Joural of olid-a Circuis Vol. 33 No. 3 March 998. [Lvaio 4]. Lvaio al Phas Nois i Digial Frqucy Dividrs IEEE Joural of olid-a Circuis Vol. 39 No. 5 May 4. [aimiri 998] A. aimiri ad.. L A Gral hory of Phas Nois i Elcrical Oscillaors IEEE Joural of olid-a Circuis Vol. 33 No. Fbruary 998.