メrs,The authors would like to acknowledge H K.
Иοレο″た′g̀″Chiou of the Chung―Shan Science and Technology lnstitute for hisvaluablc suggcstions and help in thc tcsting,andこHuang
K.ヽ ofthcg。
Nano Device Laboratory for help on high iequcncy phase ineasu血⊂)IEE 2002

,θ Ocわ ら′″2θθ′レ′A/ο
′2θθ2θ′ff
cs Z̀′″/sO″ ′Eたòό″′「2θθ2θf′′
DOI:′ θ fθイ9々′

ο″イ′
Hsiung Linばο
Wen‐
Chich Wu and Hao‐λ D"α″″ ′げ̀″ツfν シ ZilDこ 2′ルク″,″α′物J″α″じ″rrたα′E″gレ ″&,屹E々ο
̀/sゴ̀̀′̀Jο
R響 ろ′
Jc crC力′″α)E‐
mail:wcw@eplCenteree nm.edu.情ReFerencesl BUCHWAL2 A W,and MARTNL K W: ̀High‐

speed voltage‐

controlled

り 二′ , 1991, 27,
oscillator with quadrature outputS', Elec力"
pp 309‑310),

2

HAЛ MIRI,A,and LEE,TH:̀A general theory ofPhase nOiSe in elecucal
rs,1998,33,(2),pp.179194
oscillators',EEE J Sa″ ´Sraた α κ ′H:̀Jitter and phase noise in nng
3 HAJIMIRI,A,LIMOTYRAKIS,S,and LEE,■oscillators',EEE J SariJ̲sraκ cira J● 1999,34,(6),pp.790‑804

Modelock‐ a,oiding synchronisation method
H.¨A.Tanaka and A.Hasegawa

running
where θ, and ωi represent thc oscillation phasc and frec‐eし △ω″ iS interpretOd as
麟 quency of the ih oscillatoL respect市
the locking rangc ofcach oscillator which is assumed to bc small andthc samc for al oscillators. The phasc lagΦ rcnects signal delay9
which cannot be neglected fOr the case of radiative coupling.
However,ifthc coupling is made by one‐wavclength wavcguides,Φ
is assumed to be O and we focus on this casc.The abovc discussion
valid for any ncs、 rk topolo働L For the 2D oscillator array withswitchcd coupling as shown in Fig. 1,(1)can be mOdined to thc
following phasc cquationθθnら(一Φ+ Δω″= 島5 7 S i十:=ωl)′(2)

ノwhere the addi●
onal paralneter tt takes 1/O dependng on the on/Off■h oscillator and the couplingstatcs of the switch between thellatOstS(dハne絆o r k . T h c s u m mΣ
d oおtn a k e nrぉa t t a C e )n 。

to the ilh oscillato■Hcrc,wc assumc cach switch independently takes
t h e o n / o f f s t a t c s a l t c m a t i v e l y f o r . /a しt ,i rm e S ps ep ca in v■e l y l t
is clear that ifall switches are on,(2)reduCed tO(1).COnversel"ifa=ω ぉ which implies that all oscilla‐
switches are ot(2)becomes θ′tors are n∝c ̲running.Between these two extremes,wc have inter‐re connected bym e d i a t e s t a t e s w h e r eo aonsyc i l l a t o r s w=ilt ha島
t h e n e″●o r k a n d a c e r t a i n a t n o u n t o f o s c iiltlha島
t =o Or s a■r e″ork, foming a percolation‐
like, global
disconnected from the ne●ne,"Юrk of intcracting oscillators.

A simp19 synchronisatton method avoidmg the modelock phcnomcnon
has been realised ushg dynanic ncmork cOupling Two‐dhlcnsional
amys of millheter‐ wavc power‐combining and beatn‐scannmg
control systFnS are considcrcd as an applicaJon of the method Thc
cfFect limitations and robustness of thc proposed method are mvcs‐
tigated numcncaly″″f Networks of interconnected oscillators cmerge in aI″rra′̀riοwide range of engineering issucs. Examples are known in milll‐
metre―wave power‐ combining and beam‐ scanning control systems
111,a nOvcl VLSI clocking[2],and JOSephson junction arrays,/o‐dimensional(2D)synchrOnOus arrays
where implcmcntation of卸
has been a technical chanenge since a number of oscillators arc
packed in a limited space and are required to oscillate in unison As
opposed to one‐ dimensional(lD)linear arrays of oscillators,planar
arrays of oscillators can somctimes exhibit 2D phase‑lagged stablesynchronous patterns called modclock or travelling(splral)WaVes
This modelock has been a notorious hazard sincc it hampers thephase synchronisation of the oscilator arrays.
desired in‐
h a 2D array of voltage‐controlled oscillators cCOs)fOr vLSI

Osci:lator
Fig.1 2D s9j″ sw′た力ι′α″ α、ぃ οSο″ねraぉ ″clochng[2],mode10Ck is avoided by adding a special phase detectoreD)thc reSponse of wich decrcases monotonically beyond a.phasestate crcuits such as
difFerence of π/2 to each VCO.However・ h solid‐MESFET oscillators for millimetrc‐ wave generation, 饉e interac■onbe柳′
een oscillators is due to thèittecJOn̲10chng'mechanism andtrp″″gs̀ο:=12000(cyC:e),7..=350:貶
T o n = 5 0 0 :多隊

rresponding to the FD。
the rcsulthg synchronisation charactcHsucs●responsc)cOme from the inttmsic nonlheaHty ofthe oscillato■ Thus,insuch cases avoiding modelock by ttg the syhchronising charactcr―
istics may not bc straightfomard(if nOt impossible),as Opposed to thecasc of thc VLSI clockhg circuit.
h this Letter we propose an altematlve synchronishg method thatavoids modelock by htroducing dynamic couplhg vnth only on or
m me
switches to the array(sCe Fig.1).The basic idea comeslikc)2D nctworks of oscilla‐
observation that nonrcgular ocrcolatiOn‐
tors amm thein̲phase synchronisation state by desLucion ofthècore'of the splral wave pattem(centre Of the modelock).SyStematic
coherency σ,0:1
eness of
numencal simulations are carned out to consider the enbct市―scfrrarゎ
″swave power‐combin‐
Fig.2 Cο力a″″fb ″ο″げ σq′ ′2θθθcたο
the method for possible applications to millimetre‐ν Zts″cッ̀′
ing and bcam‐scanning control systems The liinitations and robustncssof the method are also investigated。 cases ofWe perfonned nurne五cal simuladons of 2)fOr鯖(■″
where,ワ =(350,150)and(Z",■∂=(500,0),respeclveゝ

.

あ′動 a″ ″α″お J″2D οsι′′ο″α′ηッSI We assume here a weakorks with random
loo mals arc madc for 30×30 oscillator array nci″
coupling between attacCnt Oscillators,and sufnciently uniforln oscil‐ωjinitial oscillationphascrs and switching states The natural ttqucncics
lator characte五stics.Under such conditions,a systematic dcHvation ofa r c c h o s e n u n i f o m lmy[お095,1.05]atrandOm.To measure ule dcgrethe phasc cquation for oscilla,ors Can be madc(e g see[1]),WhiChofphase synchronisation,the phase coherency
σ isintroduced as fbllows:
Crゴ iblly takes the fonowing fom:

一の& ン の十￨ ￨ ￨ : l■ a = け Δら
186①
EL[ε ttON′ CS LETTfRS=書eXズσ)1膳ら嗚ら

74:わ FebrtJar/2θ θ2 1/o138 Arο

0.4

It is noted that
σ=l cOrrcsponds tO the in‐phase synchronised statc andthe lower coherency σ indicates loss Ofin‐phasc synchrow̲Fig.2 shows
the result of this simulation(aner 12 000‐cycle oscillations)A clcar

1潔
躙: ぶ毀
器蹴税辮畿

̲ I針鶴ど諄躍

dfFerencc is observed
be館een mc
case ″of(■e,7砂=(500,0)and■
1809
case of(■me fomer has a widesprcad coherency
た,■∂=(350,150).月
2 ctlrNK,V,and cHANDRAKASAN,A:LActive GHz clock nemOJ、/｀.

■bingＦｏ
腰ｏＦω
鳳ｏ

Ｄ ｂｏ
日一
■一︒
Ｈ一Ｈ一︒
︒

distribution,which is due to the gcneration of modclock(sOmcimes
dist五buted PLLs' ISSCC Dig Tech Papers, San Francisco, USA,s e v e r a l s p i r a l c o e x i s t h t h e a r r a y ) C O n v e r s e L t h e h t t e r h a s Fcbmarya s h a2000,pprp
174‑175
3 sTEPHAN,KD,and MoRCAN,WA: ̀Analysis of intennJection‑lockcd
distribu」
on belowσ=1,where no modelock remains and all(cO:mectedl
oscillators for integrated phased arrays',EEE rra″ s ittc′o″ Zた′οッoscillators cxhibitphasein‐ synchronisation.
セch, 1987,35,pp 771‑781
Such avoidance of mOdelock in s、ntched array neヽ″orks can beexplained as f01lows.Figs.3αand b shows typical examples for cases
without control oscillatOrs,and wlth control oscillators attachcd to theoutside of the array9 respect市
ely.cFor the latter case,driving f"quen‐ciesω:ofthe control oscillatos are gradually set to be low to high from
Channei capacity of b:t‐ inter:eaved coded
the tOp to the bo■
Om ofthe array shown in Fig 3b.)In bOth Cases ofmodu:ation schemes using 8‐
ary s:gna:
Figs.3a andら ,it is Observed that(1)inhiall光(multiple)stable sphl
conste‖ ations
waves e対 st(at this stage all switches are on),(li)ShOnly aner theswitching ls initiatct SOmc cores of the splral waves are destroycd orS.Y Lc GoffmOVCt(ili)■ e spiral waves conlnue to bc destroy軋(1りfhally they
are lost and a coherent pattem is stabilisec and(V)at thiS Stage,weThe。切CCt Ofthe work dcschbed was to detemme tte 8 ary signal sc億
obtain compttely coherent pattems by making di swiches on agJn.
that maximise the cOding gains achieved by povc「cmcicnt bit‐inに■caved coded modulttion(BICM)SChCmcs ovcr an addit〜ewhite Gaussiah noise challnel isTo■ena the charmel capacity

linit of BICM for scvcra1 8‑ary corlstellations has beOn cvaluatcd lt
is shclwn■
latぬe most suitablc constcllation for designhg a BICM
scheme depcnds on the desired spcctal emciency of the system
ルrrOグ′criο
″I Bit―interleaved coded modulation(BICM)iS a band̲
width‐emcient coding technique made up of serial concatenation ofbinary error‐
correcting cOding,bit―by‐bit interleaving,and high‐ordermodulation using Gray or quasi‐ Gray labelling[1].BICM has provento be a very power‐ emcient approach provided that state‐ of‐the―artcodes, such as turbo codes[2]or loW‐
density panty―check codes[3,4],are emp10yed[5].It iS pOsSible to design BICM schemes by
cmploying any h″ o―dlmensional signal constellation.Howevet somerecent studies have indicated thatthe choice ofthe signal set rnay haveFig。3 助′″ Ю″おα′わ″ ′raraSS rο ω力′″″ ′arra s レ 2D
αttηッa strong innuence on the error perfomance ofthe system[6]In thiS
οsciJJàοrs
Lettet wc address the problem of nnding the mOst suitable 8‐aryαWithout control oscillators
constellationsfordesigning
BICM
schemes,onadditive
whiteGaus―b Wtth control oscillators
sian noise(AWGN)channels,using the conccpt ofcapacity limit The
idea of evaluating capacity limits to■n d the best signal set comes
iom the factthat powe卜 emciOnt BIcMs usually employ state‐ of―the‐D おε s s ゴ
οだ I n t h e a b o v e s i m u l a t i o n s o f ( 2 ) , t h e C a sΦe = O f i s
art codes, and are thus capable of achieving near‐ capacity perfor‐
considered We also considered cases of Φ 二=0 01π and O.lπ,model―
mance As a result,the actual error perfomance ofthese BICMs canling mismatches to one― wavelength interconnections For theΦ =0 01π case with 30 x30 arrays,several per cent Of the mals did be predicted accurately by evaluating their capacity liinits.

力一＞

晩７

ヽ １ ︐ ノヽ １ ︐ ノ一

﹁fLfεTROrVノθS lfTTERS 741わ／ １ ヽ)７,洗ψ一,Gル４

H‐A.Tanaka(Dψα″
初 rげ]acrra″
たE″gレ ″g・ル′【F″ν
カメars′ヶ̀″′Ю‐6〃 ″ ″たα′ゴ
ο″s PEの ,あり,ユ ὴ′α″
げ Eセε)
E‐
mail:htan@synchr01.ee uec acjp
A.H,segawaの9α″
勧 rグ E′ rra″
icsα″′Ca″″″ガcariο
″,ct/z̀ευ′jν
ιぉゴα″Σ′5И gッsr 2θθ′／ １ ヽｐ

③ IEE 2002
E′ac″り″たsL̀′たrs O″″″θ Nοr 2θ
θ2θf′イ
DOI=′ θ fθ
イ9ノel:2θ
θ2θfノイon some general results g市en h[1],we Can demonstrate that,under theconstraint Of unifonn―hput distnhl● on and assuming ideal(h缶ite―depth)bit‐by‐bit htrlea宙 ng,the capacity C of a BICM system uslng
2″‐
ary constelladon is expressed over an AWGN channel asΠ日Cο″cルs,ο
″sI Although a theoretical study has not bcen completet
systematic numeHcal simulations supportthe efncacy ofthe presented
method fOr avoiding modelock The required switches are expected tobe realised using silnple circui、̲Cη αごク ″″Jrげ β力ι″ω trsレg″ ガοぉ ∂‐αッ slg″α′S′なr‐̀6/Sο
Consider a 2層
modulation
inodelledbyvo‐
adiinensiOnal signal set
町S ofsize lSI=2"Let c=(Cl,̲ ,C″ }∈{0,1}″denOte a set of″bits
a t t h e m o d u lⅢa tuotr, a n d ノ
the correspondlng channel output Basedに

not converge to the cOmpletely cOherent state(atleast for 12 000‑cycleoscillatiOns)10×10 and 20× 20 arrays are alsO sirnulated under thesalne conditiOns, and 100%cOnvergence is obtained for Φ =001πThis implies that smaller arrays have better phasein‐ synchrony9 andt h i s i s w h a t t h e s i m u l a t i o n i n [ 3 ] r e p o r t C d l n tΦh e= 0 c1 aπs e, o fnone of10x10,20×20,and 30× 30 arrays converged to the in‐phascstate, suggesting a certain upper limit OfΦ beyond which thepresented method is no longer egective.
For more ettct市 e synchronous nettvork topologles,we perfomed
sllnuladons of(2)on iangular la伍
ce networks under the satne
simuladon protOcol as that men● oned abovc As opposed to theabove square lattice cases, mangular lattices show better in― phascsynchrontt e g 10× 10, 20×20,and 30× 30 oscillator amys with aΦ =0 1π phase delay have shωll 1000/.in‐phase synchro,μwherc鳥 ジ denOtS expectation with respect to c andソ
ら is theEuclidean distance beiveenッ
and a signal s c S,and si,cl denotes the群
1器ゾぶ嘱f肌慶:l府鯖
鵠:驚』
T:l霧.雪mean Gaussian noisc.Capacity is here expressed in inforlnalon bits perchannel usc,where a channcl use corresponds to the transnllssion of asignal s c I覧From(1),itiS possble to evaluate the capacity lhit ofBICM for any
ミ、‐
dimcnsional constellation and thus dcterr― c thc signal sets thatma対 mise the coding gains achieved by power‐efncient BICM schcmcs.
In this Lettet hc BICM capacity is cvaluated for severa1 8‐ary signal
sets using Montc Carlo integration of(1).The 8‐ary constellations thatare considered in this work are 8‐
PSK,rcctangulat(4,4)with a Hngr a t i o o f l . 9 3 , o p t i m u m , ( 1 , 7 ) , m a n g l ec,Юs
a n ds [ 87 ‐,8].All except theFebrυar/2θθ2 1/ol 3θ A10.4

187