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IEEE TRANSACTIONS ON COMMUNICATIONS, V OL . COM-28, NO. 8 , AUGUST 1980 1269Short Term Frequency Instability Effects in Networks

of Coupled OscillatorsWALTER R.-BRAUN, M E M B E R , IEEE

Absrrucr-A set of N identical oscillators (clocks) are connected toform amutuallysynchronousnetwork.Generalmatrixexpressionsare derived for the phase error spectrum at each oscillator when thenetworkoperates in the inear egime.For ertain opologies ofpractical interest hephaseerrorvarianceand hecorrelationbe -tween the errors at different oscillators is computed and compared intheir behavior for large N . The results presented assume no thermalnoise and no elay between the oscillators. However, the theory can eeasily extended to include these two effects.

I. INTRODUCTIONT E problem of synchronizing many oscillators occurs innume rous distributed systems, such as digital communica -tion netwo rks, distributed logic systems, and electrical powergeneration [ 1 - [8 ] . Depending on the application, sucha net-work synchro nization may have a dual purpos e: besides pro-viding system coherence itmay enhance the frequency stabili ty.A system where both benefits are highly desirable is the SolarPower Satellite under stud y by NASA. Here, a large num ber ofpower amplifiers have to be synchronized to produce a high-power microwave beam from a large aperture phased array.Good frequency stability is imperative in order to minimizeinterference in adjacent freque ncy bands.This paper tudieshe effect of networkopology nsystem coherenceand requency stability. t shows that hemaximum increase in stabilityattainable is proportion al tothe number of oscillators and that most of the improvem entcan be attained with very low connectivity. This is particularlyimportantor systems distr ibut ed over wide geographicalareas.Section I1 defines the problem and presents the linearizedsystem equation in matrix form for an arbitrary network. Sec-tion 111 applies the theo ry to various topologies of special in-terest. It begins with a simple chain of oscillators where eachnode is slaved to the previous one (this could be one branchin a tree network ) and proceeds through systems of increasingcomplexity o he fully connectednetwork . t provides ex-plicit equations for the differential phase error and the phasestability of the system. It also dem onstrates the effect of thesynchro nization oop transfer functionon hestability. Sec-tion N summarizes the results and presents the conclusions.

11. NETWORK MODELConsider a network ofN nodes, where each node representsan oscillator (PLL) phase locked t o a w eighted average of thephases of all other oscillators. Fig. 1 shows the kth oscillatorof uch a ystem. Here, e l , ..., ON are the ando m phase-processes of the I'LL'S, $ k is the oscillator phase noise at nodeManuscript received June 15, 1979; revised April 3, 1 9 8 0 .The au tho r i s wi th the LinCom Corpora t ion , Pasadena, A 91 1 05 .

*kFig. 1. Linearized model for a yp ica l node.

k, andak l , I = 1, .-,N are the weights used in no de k to averageth e N phase processes. We will assumeak k = 0 (1)

The n, if all PLL's are identicaland the hase variations are smallenough to allow a linearization of the loop equa tion we findfor the steady-state behavior of the kth scillator

where @k an d j[lk are the Fourier transforms of the time functions e k an d $ k , respectively, and H ( o ) is the PLL transferfunction. Since the oscillator noise appears only in its filteredform in the above equation, it is convenient to define the newprocessG k ( 0 )= [ 1 -H(0)]k ( o ) . (5

The system operation can be described m ore concisely in vec-tor notatio n by defining the weighting matrix A(w) and thephase vectors 0 an d +:

where t stands for transpose. Eq uation (4) then takes the form@(a) A(w)@(w) G(w) (9)

0090-6778/80/0800-1269$00.75 0 1980 IEEE

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or, withI as the N X N identi ty matr ix@(o) [ I - A ( w ) ] -%(w). (10 )

Th e above description of hekthnod e excluded "master"oscillators described by akl = 0 , 1 = 1, -., N , an d e k = $ k .Such nodes can be included in (IO) by setting i i k = \ kk fo rmas ter oscillators. The spec trum and cross-spectrum (viz., theFourier ransformof he cross-correlation func tion)of heprocesses 0 can now be easily foun d:[ Se k , ( w) l = E { ~ ~ * ' } = [ I - A I - ' [ s ~ I [ I - A I - ' * ~11)where [S $] is a diagonal matrix with the spectrum of the kthoscillator (filtered by 1 - H for all but the master oscillators)in its kth positio n. f there are no mas ter scillators, all diagonalelements of [S$ ] are identical and,(11) reduces t o

[So,,] = S ~ ( w ) [ I - A ] - ' [ I - A ] - ' * f . (12)111. SPECIAL CASES

The results of Section I1 will now be applied to several net-work configurations of practical interest. The following topo-logies will be considered.Single Chain: A linear array where the first oscillator is amaster. The kth clock is controlled b y the (k- ) th, k = 2 , ...,N [Fig. 2(a)] .Double Chain: A ineararraywhere the kth oscillator islocke d to the verage phase of its twoneighbors. T he oscillatorsat the two end s of the hain are slaved to thei r neighbors [Fig.

Single Loop: Circular array obta ined from the single chainDouble Loop: Circular array obtained from the double hainCompletely Coupled System: A system where every oscilla-

2(b) l .by slaving t he f i r s t oscillator to the last one [Fig. 2( c ) ] .by coupling the two end oscillators [Fig. 2(d)].to r has the average of all other clock phases as its inpu t.A . Single Chain

Here the matrix A takes the form

A = H ( o )[! ;- 0 01. (13)The inverse of [ I - A] is then (with w suppressed for concisenotation) 0 0' . 1 0[ I - A ] - '

- 1Hr(2

HN - 1

01 -H

HN-

01

. . . H

(e )Fig. 2. Network configurations analyzed. (a) Single chain, (b)dochain, (c) single loop, (d) do uble loop, (e) comp letely coupled sSubstituting this ma trix nto (12) the phase spectracancomputed for any given oscillator spectrum. The largest vtion may be expecte d at the Nth oscillator so i ts spectruc o m p u t e d n e x t :

I N - 2SeN(w)= IH(w) 12N-2 + I 1 2 kk=bI 1 -H(w) l 2 SJ, w)i

= { I H(w) + 1- H (w )1 - H(w) IIt can easily be show n that the term between braces aboidentical t o on e for a first-order loo p, Le., f H(w) = 2( ~ W L jw). Figs. 3 and 4 how he frequency responstwo second-order loops. In Fig. 3 the loop damping t is to on e which eads to a high gain near the loop natura lquency. Increasing th e damping to t = 2 reduces the peaksiderably (Fig. 4).We n ote the following limiting cases which hold for alltransfer functions.

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BRAUN: NETWORKS OF COUPLED OSCILLATORS 127

an i - . .L . .- - ..L . . . ...j-.-lL...I . ..---A,R E 0. 5 1 E I. 5 2 8 25 1 0W W ,

Fig. 3. Frequency response for single chain (second-order PLL, < = 1) .

0.01 I I I I I I I I

w/w,0. 6 0. 5 1.E 1.5 2.0 2.5 1 0 1 5 4.E 4.5Fig. 4. Frequency response for single chain (second-order PLL, = 2).

i.e. , in b ot h cases the stabili ty has not been improved or degraded over the perfo rman ce o f single oscillator. Th e variancof he first phase noise is clearly = uIL and he correlation betw een the first and last phase noise is given by1 "-- H N - (w)SQ(w) w. (17O e l N - 2 n -00

Hence the phase difference betw een the first and last elemehas th e variance

B. Double ChainFor this case the following matrix represents the systemtopology

1A = -2

0 w 0 'H 0 H .0 H 0 .. . .. . .. . . . .: I

Using thecofactormet hod recursively anyelement of [ IA ] -' can be found. If we are interested itl the variance ancorrelation of the phase noise in th e two end-oscillators onlit suffices to com pute the first an d last row of the invertematrix. The following expressions result then for the spect ra

where

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1 - -H*)Ll - d m * ]

+4 ($JN-i [( + ( 2 ) 3+di=P N - l 1 - 4 m N - l

= 1 - P N = 2 .Figs. 5 to 7 show the system response Se ( a ) / S $ (w ) for first-an dsecond-order PLL's. Note hat he peaks in thesecond- C. Single Looporder oop esponses areconsiderably m aller han or hesingle chain; Also, the stability is improv ed for large N at lowfrequencies,

we have to invert the matrixFor the limiting case H = 1 we find

1 2 N - 3S,g;(a)=-- 2 (N - )2 SJ, and we find for the phase error spectrum

i.e., this part of the phase noise is com mo n to all oscillators.For H(o) = 0 on the other hand the results as expec ted:The ratio Si&, is shown in Figs. 8 to 10 as a functionfor different loops. The limit forH ( o ) = 1 is

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BRAUN: NETWORKS OF COUPL E D OSCILLATORS 127

Fig. 5. Frequency response for double chain (first-order PLL). Fig. 8 . Frequency response for single loop (first-order PLL).

l a O t

Fig. 6 . Frequency response for double chain (second-order PLL, w/w,5 = 1) . Fig. 9. Frequency response for single loop (second-order PLL, 5 = 1

I I I I I I I I8 0 1.0 20 3.0 4 .0 9 0 68 7.0 & B 9Fig. 7. Frequency response for double ch+n (second-order PLL, w/w,

5 = 2) . Fig. 10. Frequency response for single loop (second-order PLL, 5 = 2

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1274 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1It seems then that the stab ility can be increased in his con-figuration by choosing large loopandwidth. 1.0 1 I I I I I ID. Double Loop

This case is characterized by the following matr ix:ro H o .-- HI

The results for arbitrary values of H are somewhat involved sowe restrict ourselves to th e imiting cases. For H = 1 we Qnd1

=Nwhile for H = 0

so =s,.Hence in the se limiting cases the com plicatio n of phase lock-ing to tw o oscillatorshas not improved the stability of hesystem over th e single loop performance.E. Completely Coupled System

The completely coupled system is characterized by the ma-trix

Inverting [I- A ] we findN - 2I N - 1 I-- H + IH 12/(N- 1)l2 I 1 - H I2S,(w).

1 -- H - F 2 / ( N - 1)

Fig. 11 . Frequency response for ompletely oupled ystemfirst-order PL L).

a0 I 1 I I I I I I0.0 1.0 20 3.0 4.0 5.0 E'0 7.0 E0

Theatio Se/Sb is shown in Figs. 1 1 to 13 fo rifferent loops. l.ev1 I I IFor the cross-spectrum between two arbitrary node s e find2 N - 2

N - 1 (N - 1)2e(H)-- IH l2' 8 k (w) = N -2

11 -N--l . I2- P (N )I 1 -H I2S,(0)and hence for the spectrum of the hase difference' 8 i- e k (W )

N - 2 3N--42 1--H +- 21 N - 1 1 (N-lf IH12-- WH)N - 1I 2 I1N - 2 H-H2(N- 1)a el I I I I I I I Iae 20 + 0 5 0 ~0 lae 128 14.0 1E 0

Fig. 13 . Frequency response for completelycoupledsystem secondorder PLL, 5 = 2).w/w,

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BRAUN : NETWORKS OF COUPLEDSCILLATORS 1275

Fig. 14. Frequency response for phase d i fference in comple te ly coupledsystem (first-order PLL).

IBB 2 0 4.0 E8 E0 1U 0 128 14.0 16.0 1EBw/w,

Fig. 16. Frequency response for phase d i fference in com ple te ly couplesystem (second-order PLL, r = 2).

FuzLu>YzY0

118 2 0 2 8 4 5. 8 6.8 1. 8 E0----LA/wFig . 15 . Frequency response for phase d i fference in complete ly coupledsystem (second-order PLL, r = 1) .

The erm in braces is plotted in Figs. 14 o16 or hreetypical loops. Note hat he low frequency response is thesame as in Figs. 11 t o 13 while the limit for H = 0 is at 2 sincethe oscillators are decoupled in this case.IV.SUMMARY AND CONCLUSIONS

The results presented in the last section show that the net-work requency stability can be reatly increased throughmutu al coupling, articularly in the low frequency regionwhere it is needed mo st due to the f type spectrum typicalfor oscillators. This improvement can be realized to a largeextent even ina networkwith low connectivity. Since thephase error is only reduced inside the oo pbandw idth t isbest to widen the andw idth as far as the hermal noisepermits.

161

171

[SI

REFERENCESH . Inose , H. Fujisaki,and T.Sai to ,The ory of mutually yn-chronized ystems, Electron.Commun. apan. vol. 9,Ap r .1966.A .Ge rsh o a n d B . J.Karafin,Mutual ynchronization of geo-graphically separated oscil lators, Bell Sysr. Tech. J . . vol. 45. pp.1689-1704. Dec.1966 .J . Yamoto ,M .On o ,an d S. Usuda,Synchronization of a PCMintegrated elephonenetwork, IEEE Trans.C o m m un.Technol . .v o l . C OM -1 6 , p p . 1-1 1, Feb.1968.M. R. Mi l ler ,Feasibil i tytudies of synchronized-oscil latosy s t e ms for PCM telephone networks, Proc . Inst. Elec . E ng . , vol.116, pp . 1135-1 142. J u l y 1969.H . Otto, Synchronisierverfahren von integriertenPCM-Netzennach dem Phasenmittelungsprinzip (Synchronization procedures forintegratedPC M Networksusing hephase-averagingprinciple),Nachrichtentech. Z . . vol. 23. pp. 402-41 1, Aug. 1970.J. Yamato . S. Nakaj ima,an d K. Saito.Dynamicbehavior of asynchronization ontrol ystem or n ntegrated elephone net-wo rk , IE EE T ra ns . C o m m un. , vol. COM-22, pp. 839-844, June1974.W . C. Lindsey and A. V . Kantak, Mathematical models for hetimeransferetworks,resentedtnt. onf. omm un.,Toron to , Ont . , Canada, June 1978 .issue, pp. 1260-1266.- Network synchronization of random signals in noise, this

*Walter R. Braun (S71-M76) was born iZurich,Switzerland,onApri l 6 , 1947. He received he D iplom i n electrical ngineeringfrom the Swiss Federal Insti tute of TechnologyZurich , in 1972 andhe M . S . E . E . an dPh.D.degrees rom heUniversityofSouthernCali-fornia, osAngeles, in 1973nd976.e-spectively.He joined LinCom Corporation in 1976 wherhe hasbeenworkingon ynchronizationprob-lems of digital communication systems andn thsimula t ion of satell i te communication l inks.