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2228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO . 10, O C T O B E R 1982Cycle Slips in Phase-Locked Loops: A Tutorial Survey

G E RD A SCH E I DA N D H E I N RICH M E Y R,M EM B ER ,IEEE

Abstract-Cycle slips in phase-locked loops are statistical, nonlinearphenomena. This makes a mathematical analysis extremely difficult.As a consequence, the results of such an analysis are not easily ac-cessible to the practicing engineer.

I t is the purpose of this survey paper to present a self-containeddiscussion of cycle slipsin phase-locked loop avoiding advanced math-ematical tools. Basedon the results of an extensive experimentalstudywe explain theunderlyingprinciple of thecomplexinteraction he-tween nonlinear ity and noise. The results are complemented by sim-ple,approximateanalysiswhich agreeswellwiththeexperimentalfindings.

In addi tion, we present a new and complete set of diagrams on cycleslip statist ics not presently available in the literature.

I . INTRODUCTIONMOST digital communication systems employcoherentdem odulation techniqu es. This requires a receiver capableof accu rately estimating t he phase of t he received signal.The circu its for generating a carrier reference are all basedon the same fundam ental principle: a locally generated refer-ence an d the received signal are non linearly processed to gen-erate an error signal which is s ubs equ ent ly use d.to adjust thephase of the VCO toth at of the incoming signal. Conse-que ntly, this suggests tha t these synchron izers have the samemathe matically equivalent baseband m odel shown in Fig. 1 .Depending onth eparticularmodulation,differentnon-linearities g(4) and noise processes n ( t ) will be obtained[2, p.1161.The f a c t t h a t t h e r e e x i s t sa comm on equivalent model is ofgreat importance since it allows one to transpose the resultsobtainedfo rthe phase-locked loop (PPL) tomorecomplexcircuits suc h as Costas loo ps, etc.Due to the noise , theVCO phase is a rand om process. Whenthe VCO phase variance becomes large, a pheno men on occurswhich is inherent to the nonlinear i ty in the loop. The VCOphase is increased to such an ext ent tha t the VCO slips one orseveral cycies w ith respect to the inp ut phase. The occurrenceof a slip is an event with very low p robability for weak noise,butthepro ba bili ty increases steeply with increasing noisepower. As an exam ple, the cycle slipping rate of a first-orderloop is plotted in Fig.2 versus the loop signal-to-noise ratio

p (which is defined in Sec tion-11). The sam e figure shows thephase variance a i2 obtained when thenonlinea rity is takenintoaccountandthe phase variance q i n 2 when the linearManuscriptreceivedNovember 1 , 1981; revisedMarch 22, 1982.Thisworkw assupportedbyth eDeutscheForschungsgemeinschaft( D F G ) u n d e r C o n t r a c tMe 651 /3 -3 .T h e a u t h o r s a r e w i t h A a c h e n T e c h n i c a l U n i v e r s i ty , D - 5 1 0 0 A a c h e n ,Wes t Germany .

+Loop fi l ter -

Fig. 1. Equivalent baseband mo del of synchronizers .theory is used. While the departure of the variance u i 2 fromthe linear theo ry is small, the slip rate varies by orders of manitude within a small interval.A great deal of effort has beinvested in the past in theoretical as weli as experim ental computersim ulatio n studies of cycle slips. Lindseyan dCharles [ l o ] presented results o n exp erim ental cycle slip dis-tributions. Mean timebetween slips ha s been obtained bymeans of computer simulation in[113 and [12 1 . This shortlist is on ly a sampling of note worth y papers.A comprehen-sive list on the sub ject can be foun d in th e books on PL[1],-[5] and in a bibliography [13 ]. Lately, there has been first commercial produ ct announced which measures ope rathreshold based on sliprate [14] .The purpose of this paperis twofold :

1) to present a set of new experim ental data on cy2) to sup port these results with simple analyses in order torates andshow the influenceof the various loop param eters.The main difference betw een the past work and this pais that we examin e the actual physical pheno meno n of aslip while the previous authorsreported overall statistics(which is very impo rtant inform ation).In SectionI1 a brief survey of w ell-known eq uations and re-sults is given. We feel that such a survey is necessary becauseth edefinit ions and notations in thePLLliterature are no tuniform. We have employedstate variables sincethis is th e

only mathem atical description that allows on e to analyze theloopund er nonlinear operatingcondition s. Normalized anddimensionless variables have beenused. The mainadvantageof such a normalization is that the num berof parameters isreduced to a few parameters having physical meaning.Section I11 of the paper is entitled Understanding CycleSlips. We dem onstra te that the statevariable x1 (proportionalto the cap acit or voltage in a second-order l oop ) is of utimportance in understanding slips. Forexample, x1 deter-mines wh ether or not the cycle slips occur in bursts.Of great

0090-6778/82/1000-2228$00.75 0 1982 IEEE


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ASCHEID AND MEYR: CYCLE SLIPS IN PHASE-LOCKED LOOPS 2229

Fig. 2. Phase variance and normalized sliprate n as a function of th eloop SN R p .

assistance, if no t indispensable, in explaining the complicatedinteractio n of noise and nonlinearity in aPLL, are trajectoriesfo rthe phase error @(t)andtheotherstate variable x l ( t )taken during a typical experiment. These trajectories, togetherwith simple, appro xima te com putatio ns, allow one to obtain agoodunderstandingof the influence th e various parametershave on cycle slips.In Secti on IV-A we give an overview of the expe rime ntalconfiguration.Theexpe rimen tal results are discussed in thenext subsection. Finally, we briefly discuss the optimizationof theloop parameterswhen the mean timebetween cycleslips is to be maximized.11. LOOP EQUATIONS

A . Basic Equations andDefinitionsWe begin byrecapitulating some well-known equationstogether with the notation to be used in the sequel. The mater-ial is covered in detail i n any book onPLL's.The input to thePL L equals the sum of signal s ( t ) and noise

n(t)Y ( t )= + n(t> (1)

where the signal s ( t ) is given b ys(t) = f i A sin (wot+ e ) ( 2 )

an d n(t) is a narrow-band Gaussian noise processn(t) = f i n , ( t ) co s wot + a n S ( t )sin wet. (3 )The features of this process together with the definitionofequivalent noise bandwidth B , , of an IF-filter preceding the

PL L and the signal-to-noise ratio at the outp ut of the IF-filterare summarized in T able I(a).

We modelthe phase det ecto r as an ideal multiplier. Th eout put of the multiplier in Fig. 3(a) equals (double frequencyterms are neglected)

withK D =K m K I Af i K l = amplitude of VCO-signalKm = multiplier gaine = phase of inpu t signale' = phase of VCO@ = t9 - 6'= phase error

an d

An exact equivalent circuit of the phase dete ctor is shownin Fig. 3(b) andthe power spectrum of n'(t) is shown inTable I(b).Under the assumptio n of small phase error thePL L has themathematically equivalentmodel sho wn in Fig. 4. The vari-ance o i 2 of theoutpu t phase dueto the noise disturbancen'(t) is

1 r+ -

wlth the loop transfer function


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2230 IEEETRANSACTIONS ON COMMUNICATIONS, V O L . COM-30, NO. 10 , OCTOBER 1982TABLE I(a)

POWER SPECTRUMOF THE NOISE PROCESSN ( T )Sn,,I 0 ) ) It :

F I F ( w ) : I F f i l t e r s p e c t r u m ,I F I F ( * W ~ ) /= 1

m

E q u i v a l e n tb a n d w i d t h : BI F = & ~ I F I F ( ~ ) 1 7 d ~

N o i s ep o w e r : P N = & j S n n ( w ) d o = N 60w

_ mo I F

S i g n a lp o w e r : P s = A

(b)Fig. 3 . (a ) Multiplier type phase detector. (b) Mathematically equiv-alent model of the phase detector.

F(s ) : loop filter.The equivalent loop bandwidthBL is defined as ~

If the equivalent noise bandwidth BL is much smallerthan the bandwidthBIF,the loop "sees" a wh ite noise processwith constan t power spectral density (TableI):

N o i s e p o w e rs p e c t r u m S n , J w )

N o i s e v a r i a n c e : o n . =-~N o B I F2 A ' 2SNRiI

K O / s -=Fig. 4. Equivalent linearized baseband model of the phase-locked loop.

NOS,*,,(m) = -2'2'4Inserting (8) and (9) into (6) yields

The inverse variance is often called signal-to-noise ratio inthe loop andgiven the symbo lp .1 A 2p = - = - -u,ij NOBL

Using signal and noise pow er, (1 1) can be rewritten

This form is particularly well suited for measu rement. Thdefinition of p is not unique in thePLL literature. In the f(1 1) it is used by Lindsey [2], Viterbi [3] ,and Blanchard [4].Gardner [5, p . 321 has a factor of 0.5 in his definition. Hisdefinition is equally arbitrary andis as valid asours.B.State Variable Descriptionof the Second-OrderPLL

To describe and understand the nonlinear behavior of phase-locked lo op we need a state variable d escription.For thesecond-order PLL with loop filter (Fig.5)


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ASCHEID AND M EYR: CYCLE SLIPSIN PHASE-LOCKED LOOPS 2231

L

1 + S T ,1 + S T ,F(s) =--

Fig. 5. L o o pfi l te r of a second-orderPL Lwithimperfectintegrator(lag-lead filter).

I - 1 - 2 3 P + P 2 t- - I I1

T2T1- +- _

x

-Fig. 6 . Equivalent model ofa second-orderPL L withimperfectinte-grator using normalized variables.1- T 2 P l

1 + S T ,the non linear differential equa tions are

(1 3) we arrive, after some algebraic manipulations, at

Thestate variable y l ( t ) is proportionalto th ecapacitorvoltage u c ( t )of the loopfilter

and has the physical dimension of angular frequency.The state equati ons as they stand are of little value. Theparameters can vary over orders of mag nitud e; their influenceonloop performance is hard to oversee. Intro duci ngappro-priate normalized and dimensionless variables yields a set ofequations with parameters more easily interpreted. We intro-duce the normalized timer and replace y , by the dimension-less quant ity x1 as follows:

+ [ 1 - 2 { p+p2 ] I Z :A mathematically equivalent model of thePL L using thenormalized variables is show n in Fig. 6.In (18) only the two parameters, loop damping{ and thefrequencyratio /3, appear (aside fromth enatural frequency

a, absorbed in 7). The reader will noti ce the use ofw, andin a nonlinear sta te equa tion. These qu antiti es are formallydefined in the same way as for a linear sy stem.The values of practical interest of 5 lie in the range of0.5 < < 5. We can also delimit values for 0.The ratioT2 /T ,is given by

T2-= p(2{ - 0)>0.TI (1 9)From (19) we find

Using the definitions of the natural frequen cya, and the loopdamping f ac to rIn the case of the comm only used high-gain loop we have0< 1 , typically 0.001< p < 0.1. If we formally set /3 = 0 weobtain the equationsfor the second-order loopwith uerfectloop integrator. For reasonably small0 the dynamic behavior

0, = rF { = 3 0, [T 2 +1 1 (17) forthe imperfect second-orderloop is practicallyindistinguish-KOKD able fromthe perfect integratorloopwith /3 = 0. In most of


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2232 IEEETRANSACTIONSONCOMMUNICATIONS, VOL. COM-30,NO. 10 , O C T O B E R 1982o ur discussions o n cycle slips we set 0 = 0 for simplicity. Thereis one imp ortan t differen ce, however, in the steady state. Inthe noise free case th e stat e variables assume th e values

x1 = -- (1 - 2 tp + 02) .AUThe loop with perfect integratorcompensates a(normal-ized) frequency difference Aw/w, with a value of the inte-grator outputx1 of exactly Aw/ o , . The steady state phaseerror 4 is zero. With an im perfect integrator therealways existsa static phase error whichsubstantially increases th e cycleslip rate as will be seen in alater section ofthis paper.

111. UNDERSTANDING CYCLE SLIPSWe already kn ow from Fig.2 that the cycle slip rate is ex-

tremelydependentonthe signal-to-noise rati o. As will beshow n presently, the damping factor1 also hasa strong in -fluence on the slip rate as well as on the mann er of occu rrenceof the slips.In Fig. 7 the probabil i ty that the time between two slipsTs is short er than t is plotted fo r two values of t . For t =0.24 we observe a steep increasein P(T,< t ) for small t ,i.e., man y slips have a very short durati on com pared to themean time between slips. This means that the slips occu r inbursts: if viewed o n an o scilloscope we see clusters of slipsofvery short duration separated from each other by long t imeintervals.

For large damping, the distribution P(T, < t ) is virtuallyexponential . From probabil ity theory we know that exponen -tially d istribu ted times betw een events imply statistical inde -pendence of these events. We therefo re conclu de that thecycleslips are independent, isolated events for large damping.Having recognized the burst-likeappearance of slips forsmall dampingfactorsan dthe isolatedevent structurefo rlarge damping, we presentlywant to explainthis differentbehavior.A . Loops withSmall Damping Factors

In Fig. 8 we have plo tted an exam ple of phase error$Iandstate variable x1 as a functio n of normalized timer fo r 5 =0.24 and a very small SN R. We clearly recognize t he burst-likenature of the cycle slips. B efore we turnour a t tent ion to thebursts of cycle slips we exam ine the behavior of the loop be-tween twobu rsts , Le., in itstrackingmode. A damping of( = 0.24 means that in th e tracking m ode (where linearizationapplies), we expect to observe aweakly damp ed oscillationof $(7) an d x1(7)sustained by th enoise process n'(r).Th e average period of this oscillation is

2n

in normalized time. From Fig.8 we find an average period of7, which is n ot far from the predic ted num ber. We would also

___----_.__--.-__._-----

, 1 , , , , 1 , , , ,0. 50 .100.150.200. 250. w,tto

Fig. 7. Probab il i ty distr ibutionP ( T , < t ) (solid lines) of the time tween slips and condit ion al distr ib utionP (T , < t I T , > t o ) (dashedl ine ) , condi t ioned on the fac t tha tT, exceeds a given t imeto . Sig-nal- to-noise ratiop = 2 (numeric ratio) .

L Y I-2n I 10. 20. 40. 60. 80. 100. TFig. 8. Trajectories of phase error@(T ) and sta te var iab lex l (T) versnormalized time forp = 0.03, f = 0.24, an d p = 2 (numeric ratio) .

expect an oscillation ofx l ( r ) with the same per iod as @ 'b ushifted by1 ~ 1 2 ,and this is indeed found in Fig.8. T o go onestepfurther , x1 an d $I must have approximatelyth esameamplitude, because of the normalizations that have beenperformed. Forx1 an d $I we read o ff from Fig.8 a value o f 1;converting the average amplitude of $I into degrees gives ap-proximately 60".The weaker the noise, th e smaller the am plitud e ofx1(7)and @(T) since th e weak noise can push the loop only slaway from its stable equilibrium. Fro m linear the ory we uo2= l / p ; hence,th eamplitude of th eperturbed sinusoidwould bem.For strong noise, as in Fig.8, it is very prob-able that the amplitude of the fluctuations exceeds t90"phase error (ma xim um restoring force), in which case a cyslip is v ery likelyto occur . Furtherm ore, in this case thex1(7)variable follows t o a wrong value. After comp letion of thcycle slip, x1(T), which is responsible for frequency correctiohas awrong initial value (loop stress) and,consequently, ismu ch mo re susceptibleto anoth er cycle slip.The exp erimentally determinedaverage value of x1 taken atthe com pletion of a cycle slip as a func tion ofp is shown inFig. 9(a). Indic ated in this figure are the normalized pull- ofrequency given by [SI


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ASCHEID AN D M E Y R :C Y C L ESLIPS IN PHASE-LOCKE D LOOPS 2233

p+ 0.70(b )

Fig. 9. (a) Condit ionalexperimenta lm e a n Fl of th estatevariablex1(7), t a k e n a t t h e i n s t a n tO(T) = +2n ( c o m p l e t i o n of a cycle slip).(b ) Cond it ional experimenta l varianceoX12 = (x1 - F1)2 t a k e n a tt h e c o m p l e t i o nof a cycle slip.

= 1.8(1 -I- 0, the phase error rapidly passes this region ofpositive feedback to reach @ = -2n, which, of course, isequivalent to zero phase error. At completionofthe firstslip, x1(7) assumes a rando m value of slightly more tha n thepull-o ut freque ncy. During th e following thre e cycle slips thevalue of x1 increases to amaximumbefore it is slowly de-creased to its corr ect average value ofx1 = 0. Another burst isshown in Fig. 12. In con trast to the previous burst we do noobserve a pumping up of thex1 variable during the first cycleslips. Rath er, the value ofxl,taken at the comp letionof acycle slip, fluctuates around the experimental mean valuex1before it takes on a valuex1


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2234 IEEETRANSACTIONSO NCOMMUNICATIONS, VOL. COM-30,NO. 10 , O C T O B E R 1982


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ASCHEID AN D MEY R: CYCLE SLIPS IN PHASE-LOCKED LOOPS 2235

Fig. 1 3 . Phase error@(T) and phase detec to r outpu tsin @(T) during abea tno te .

The variance of this incremen t is easily fou nd t obe

~ b , ,= 2.8 = 0.24 U A ~ ,= 0.81. (33)The random fluctuation is mu ch larger than the systematicdriving force El.The different appearance of the two burstsis now easily understood . The systematic forceaw l is coveredby the random fluctuations; the values ofx1 taken at the com-pletion of a cycle slip are within abandof20Axlwidth,

centered aroundX 1 . What looks like a systematic pumping upeffect in Fig. 11 is no thing bu t anormal statistical fluctu ation .Our simple analysis is only approximatebutpredicts, re-markably well, the duration ofB slip within a burst as well asthe statistical fluctuations of th e increm enta x l .Th e distinctive featur es in region (c) are tha t th e cycle slip-ping stops andx1 rapidly converges toward zero. Immediatelyafte r the last cycle slip, the phase error rapidly increases. Therestoring force sin@ is large and has the correc t polarity durin gthe convergenceinterval. Integ ration of the restoringforceproceeds rapidlyso tha t x, quickly moves toward zero.In passing, formulas (30) and (32) give a hin t as to wh y theloopperm anen tly loses lock for decreasing p . The averagepull-in effect El is inversely proportional to (x1)?,while thevariance is inversely pro po rtio nal top and x l . From Fig. 9(a)we know that the averageE l increases for decreasing p . There-fore, the bursts tend to bec om e longer until the loop eventu-ally loses lock com pletely.B. Overdamped Loops (5> 1)

Having understoodth erathercomplexstructureofth ecycle slips of an underdam ped loop we turn our attentio n toloops having damping factors 5 > 1. As already noted, cycleslips occur in this case as isolated events, no t in bu rsts. Whythis d ifference?

The answer is again foun d by inspection o fE l in Fig. 9(a).We first n ot e th atXl is, for reasonably large p , less than 1 /3 ofthe pull-o'k:frequency. Seco ndly , the funct ionXl ( p ) is essenti-ally flat for p > 4. It remains for all p much smaller than thepull-out frequencyw p o / o n(no t to me ntio n the pull-in rangeup /wn) .From thisand the variance u X l 2 in Fig. 9(b) it isclear th at bursts of cycle slips are extrem ely unlikely events.The mea n time between cycle slips converges for lowp towardthe values of a first-order loop whi ch always performs be tterthan a second-order loo p, provided there is no frequency dif-ference betw een VC O and signal.We still owe an explanation as to why the variable x1 takeson such small values. As always, linear theo ry is instrum entalin getting a good u nderstanding of the no nlinear behavior.

A large damping implies a large time cons tant for thex1-integrator; the response of such an integra tor to a noise eveis very sluggish and small in am plitud e. The refore, the prop or-tional path of the filterdetermines the sho rt-time transientswhile the integral path compensates for an eventual frequencydifference between VCO and signal. Thus, for the compu tationof theshor t-time transients of @(r )we may approximatelyassume x1 to be constant. But the input to the integrator ofthe loop filter equals the input to the 6(r)-integrato r,if multi-plied by 1/21. Hen ce, 'the increm entsofx, an d 0 are approxi-mately equal:

Since 8= - we may write(34)

for the short-time transients.A typicalcycle slip is displayed in Fig. 14 . We observe avery similar shape of x1(T)an d @(r)up to the end of thecycleslip. Subse quently, the value ofx1(7) slowly decreases to itsinitial value of x1(7) = 0 with a time con stant of 25, corre-sponding to t he neglected pole of the second-order loop. No te,however, that @(T) afterthe slip is muc h less affected byHaving recognized the similarity of X ~ ( T )an d @(T) [as pre-dicted by (35)] we are in a position to comp ute an estimate of

x1(7) at the completion of a cycle slip. Let us denote byroan d r Z n , the beginning and end of a cycle slip, respectively,and define a cycle slip as part of t he trajecto ry@(r)such that@ ( T ~ )= 0 an d @(T< T ~ ~ )f 0. Furthermore, we assumex1( T ~ )= 0. Under these assumptions we obtai n forx, (r2*)

X1 (7).

r2,,- ro : random variable.Neglecting the integral path of the loop filter for the shotime interval( T ~ ~- r0), th e PLL is governed by a first-orderdifferential equation


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2236 IEEET R A N S A C T I O N SO NC O M M U N I C A T I O N S , VOL. COM-30, NO. 10, O C T O B E R 1982

Fig. 14. Cycle slipof an overdamped loop( 5 = ,1 .5 ) for p = 2.5 (nu-mer ic ra t io ) . Note tha t-x1(7) is displayed.

(3 7)

No te that the right-hand side of (37) multiplied by- 1/2{equals theintegrand in (36).Integration of thestochasticdifferential eq uatio n (37) from T~ t o 7 2 n yields

bu t @ ( T ~ )= 0 and I $ ( T ~ ~ )I = 2 n , by definition. R eplacing theright-hand side of (36) by- $ ( ~ ~ ~ ) / 2 {yields

As in the case of small damping factors {, x1(72.)appearsas a frequency detuning . However, th e detunin g is too smallcompared to the pul l -outrange to produ ce a burst. The inversedependence of ~ ~ ( 7 ~ ~ )is experimentally well confirmed; seeFig. 9 . Du.e to th e coupling ofxl(T) an d $(T), the result (39)is, of course, o nly an app roximation.So far we have d iscussed two examp les of loop s of{ = 0.24and { = 1.5 damping and have found a tendency that weaklydam ped loops burst while overdamped loop s do not. It wouldbe interesting to identifyaboundarybetween burstingan dnonburst ing. Of course, such a bound ary cannot be arigid one,bu t would merelyprovide informa tion as to wheth er a loopis more l ikely to bur st or no t,On the average, a loop will burst only if the mean value o fx1 taken at the com pletio n of the slip is larger than the pull-out freque ncy . Using (39) and (23) yields the inequality

n/{> 1.8(1 + {) condit ionfo rburst . (40)Solving (40) we find thatloopswith damping factorsof

{ 0 we obtain

or slightly rearranged


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A S C H E J DD M E Y R : C Y C L ESLIPS IN PHASE-LOCKEDLOOPS 2237

Fig. 16. Trajectories o (7 ) a nd Xl(7) fo rth esameparameters asinFig. 15, b u t f o r Aw/w , = 16. The loop sta r t s with correc t in i t ia lcond i t ions at7 = 0 and drops lock af te rf i s t slip.

In general, fo r a givenA w / o , an d {lo,there exists an in-terval of x1 for which the inequality is not true. If the loopslips a cycle and x1 accidentally assumes a value inthis in-terval, resuming lock wouldbepurely bychance.Such be-havior is clearly unacceptable in a practical appl icatio n. Thequestion arises whetherthere are values of Aw/w, and {/Pfor which thequadraticform (44) is positive for all valuesof xl. Then,th eloo p could always reduce the difference(Ao/w,) - x1 to zero:Indeed,thequadraticform is strictly positive if the dis-crim inan t is negative.

or

A o / w , < 22.But the right-hand side of (46) is noth ing but the pull-in

frequencywp/on; see (24).Inconclusion,th eloopmus t be designed suchthatl Aw /w , I is sufficiently sm aller than the pull-in freq uen cyu p ;this is particularly im po rtan t for lowSNR's. From (46) oneconcludes that a perfect integrator@ = 0) realized by meansof an active loop filter is preferable. In practice, however, dueto ever present drift curren ts, there will always be a limit onthe maximum permissible frequency difference.For the ratio betw een positive and negative cycle slips, thefollowing formula valid fo r a first-order loop has been derived:

N +N--= exp (4.?) (47)In any case, the s um

Fig. 1 7. Simplified block diagram of experimental configuration.

IV. EXPERIMENTA . Experimental Configuration

The configuration is divided into two parts, the experimenitself,' in analog ha rdw are, and a microprocessor (pP) systemfor con tro l of parameters and recording of measured dat a. Inthis section,a survey of t he hardware and a functional des-cription of thepP system, based on aZ80 CPU, will be given.For a mo re detailed discussion see[6 ] .A block diagram of the analog hardware is shown in Fig.17. An unm odulated carrier is provided by a crystal (Xtal)os-cillator. T he signal po wer can be set by a variable a tten uat or.Wide-band Gaussian noise from a rand om noise generator isadded. The noise power can be varied by a second atten uator.Both signal and noise may also be switchedoff.

Filtered by theIF quartz filter, thenoise becomes a narrow-band Gaussian process as d escribed in TableI. At the outputof the filter, the signal an d the noise power are measured. Thfilter ou tpu t is also the inp ut to thephase detector of the PLL .The phase detecto r is of the multiplier type, the only oneusable at low SNR. The passive loop filters are exchangeable.The VCO output is notonlyconnectedtoth eloop phasedetecto r, but alsoto a reference phase d etecto r.Th e un disturbed carrier is directed along a reference pathto the other inputof this lineark180 phase detector to deter-mine the actual phase error@(t).A second quartz filterha sbeen inserted into the reference p ath, adjusted to com pensa tefor the phase s hift of the IF filter.The connections between the analog hardware and the psystem are marked by double lines in Fig. 17. Thep,P sets thevariable attenuators in steps of approximately 0.05 dB/bit .The center frequency of theVCO is adjusted by a digital-to-analog con verter.The digital power meter is conn ected to thepP system byan IEC-bus. Thu s, thepP can not only read off values from thepower meter,but also send comm ands to the power mete r,such as zero calibration and mod e com man ds. If the power ismeasured indBrn, a fo ur digit value results, with, a least signifi-cant digit of 1 /1 00 dB,.

Three analog-to-digitalconverters, all buff eredby sampleand hold amplifiers, allow the pP system t o record values ofthe phase detectoroutput of thePLL (ug ) , th ecapacitorvoltage (u,) of the loopfilter of th e second-order PL L, and theactual phase erro r @.


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2238 IEEETRANSACTIONS O N COMMUNICATIONS, V O L . COM-30, NO. 10 , OCTOBER 1982In add ition, the outp ut of the reference phase detector isconne cted to a cycle slip d etecto r, which provides a signal toth e pP whenever the phase error exceedsa value of 2r. Bymeans of a second signal the direction of the cycle slips to thef l system is recorded.T he pP system and the analog hardw are are strictly sepa-rated to avoid signal interferen ce. The analog hardware is built

into a shielded box with sepa rate voltage supply .In addit ion,the analog hardware itself is again builtin five subblocksshielded individually to avoid internal interference. These aresignal path , noise p ath, IF path ,reference path , and thePLL.The funct ion of theE.Lp system in themeasurem ent are1) zero calibration of the power m eter,2) signal and noise pow er setting,3) VCO center frequency adjustment,4) samplingand storing of the measured data during5) frequent recalibration in measurements of long duration.The actual parameters and the measured data are stored to-gether in a random access memo ry(RAM). The stored para-meters are t he measured signal and noise power, a filter identi-fier, the digital-to-analog conv erter values, the offset of phasedetector output(uo) ,and capacitor voltage(uJ.Th e microprocessor system is connectedto a larger machine(PDP 11/60) for two purposes.

measurement, and

1) different types of measu remen t programs can be loadedfrom the PDP to the pP s ystem ,which gives high flexibility.2) after completion of a measurement the contents of therandom access memory are sent tothe PDP for processing,such as evaluation of statistics orgraphical display.The main advantages of the m icroprocessor c ontro l may bementio ned here. As the actual parameters are stored toge therwith the measured data, confusio n of different measurementsor erroneou s readings of the parameters are avoided. The factthat the attenu ators can only adjust discrete power levels hasno inf luence on the accuracy of the measu remen t, as only theactu al parameters are used for the evaluation o f the statistics.Furthermore,it is advantageous tha t long-timemeasure-me nts are also possible. So me num erical examples o f the dura -t ionof a completemeasurementare given in Table 11. Anequivalent loop bandw idth ofBL = 750 Hz is assumed for thecom puta tions . In the described configuratio n, this value can

not be exceeded very much, since it still has to be muchsmaller th an the IF f il ter bandw idthB I F = 14.96 kHz. Dis-played in Table I1 are the mean time betwee n cycle slipsE(Ts)and the mean duration of a com plete measurementE(Tm) fo ra few values of e(BL Ts) .E(TM)is comp uted for1600 events to be recorded, whchis the maximum number that can be stored in theRAM, andan arbitrary chosen minimum of300 events to be recorded. Ascan be seen, the mean duration of the experimen t increasesrapidly and forE(BLTs) = l o 8 an unacceptable duration re-sults. Besides, the results will not be reliable for such low rates.If higher rates ofE(BLTs) are supposed to be measured,the durationo f the experimen t can only be shortened by in-creasing the equivalent loop bandwidth BL. On theotherhand , this decreases the m inimum distance betwee n cycle slips

TABLE I1MEAN DURATIONO F AN EXPERIMENTE(T,)

BL = 750HzE[BLT,] E[T,] E[T,]/1.600 events E[T,]/300 events101 13.3 ms105 21.3 s133.3 s 60 h 1 1 h1os 37 h 463 days

at lower signal-to-noise ratios. But th e m inimum distance th acan be recorded with the described configurationi s limited bythe speed of bo th the analog-to-digital converters and the system.Ano ther aspect has to be taken into consideratio n at hsignal-to-noise ratios. The me an time betw een slips increaexponential ly withp. Assume the bandw idthBIF is measuredy i t h x percent uncertainty.Thecomp uted value of p (seeTable I) will have the same un certa inty. The slip rate will havan uncertainty proportional to exp( P X / ~ O O ) .An exponential dependence ofp will lead to an increasingasymmetry in the distr ibution of the direction of theslips if the phase detector is not symmetrical to phase error4 = 0 or if the VCO gain is not sym metr ical for positive annegative detuning .With th e descrlbed configu ration, measurements of the sliprate of a first-order loop , wh ere analytical results exist, wmade . In the range l o 0 . t o ] . (50)

Using Bayes rule it yields

The condit ional distr ibutiondescribes th e statistics betwbursts, rather than slips. As exp ected ,(51) is p ractically indis-tinguishable from an exp onential distribution(Fig. 7).


12/14

ASCHEID CYCLE SLIPS IN PHASE-LOCKED LOOPS 22396.00 0 . 50I o f ~ r s torder loop p ( x 1 l Q = - 2 T L 1

Fig. 18. Normalized meantime between slipsof asecond-order PLLas a func t ion ofp with f as parameter. Zero loop de tuning. Solidl ine shows analy t ica l resu l t sfor first-order PLL.6 . 00 ,

Fig. 19. Normalized meantimebetween slips of a second-order PLL(t = 1.0; p = 0.03) with loop detun ingA w normalized to the pull-in f requency wp .

The effect of loop detuning onE(T,) is shown in Fig.19.The frequency differenceA o is normalized to th e pull-in fre-quency u p .The importance of the state variablex1 has been discussedat length in this pape r.A key finding of this paper has been the behav ior ofx1 im -mediately following aslip;aconsistentdeparturefromth ecorrect value has been identified. The statistics of the cond i-tional means E [ x l I$J = ?27~] and variance E [ ( x l -A typical distributionof x1 immediatelyfollowing a slipis displayed in Fig.20 fo r { = 0.7. As a consequence of theoccu rrence of bursts a significant skewness is visible.If one

excludes the very sh ort time interval between slips from con-sideration, the skewness disappears and both sides o f the dis-tributio n assume a symmetrical sh ape.

114= +2n] can be found in Fig. 9.

0. 40

0 . 3 0

0 . 21?1

0 . 1 0

0 . 0 0-2 K -K 0.(b )Fig. 20. (a )Experimentallyderivedconditionalprobabili tydensityfunc t ion p ( x l I6)fo r a second-order loop withS = 0.7 an d p = 2.1.(b ) Condi t iona l probab i l i ty dens i ty func t ionp ( x 1 I@ ,T , > to ) ob -tained if slips of dura t ion T , shor te r thanto Q E(TJ are excluded(E(T,)ltO = 43.5) .

cycle slips. Such an optim izatio n is a formidable task tha t canbe carried out on ly numerically o n a digital com puter or in th eform of an experiment.In a second-order lo op there are essentially tw o loo p para-meters, namely bandwidth BL and loop damping{, to be op ti -mized in a two-dimensional search. Th e loo p parameters haveto be clearly distinguished from the signal parameters{Ps,N o ,Af} which are fixed q uantities. In a first step we want to op -timize the loop bandwidthBL for a given dam ping{. For thispurpose we seek a suitable normalized representation of themean time between cycle slipsE(T,) as a function of the ba ndwidth BL. It is natu ral to mod ify the familiar plot of normal-ized mean time between slipsE[BLTs]versus p as depicted inFig. 19.The signal-to-noise ratio is afunctionofth etwo signalparameters Ps,N o , and the loop parameterBL :

C Optimum LoopParameters ps 1N o BLThe usual approach in designing a PLL uses linear theoryto p =-- (52)determine th e loop parameters for a given setof specifications.For the next step, if necessary, th e slip rate for the resultingparameters can be ob tained from Fig.18 and checked againsta specified maximum permissible numberfo rthe particularapplication.

loop parameters to achieve maximummean timebetween NoAf BL

If we multiply both numerator and denominator byAf weobtain instead o f (52)For certain applications, it is of interest to optim ize the =__ps _.Af


13/14

2240 IEEE TRANSACTIONSO NCOMMUNICATIONS, V O L . COM-30, NO. 10, O C T O B E R 1982The three signal parameters can be grouped into a single quan-t i ty b :

NoAfPS

b = -4-0 0 /

b = 0,)

, a

w hc h i s re la ted to th e loopparameters y an d p as follows:

withAfBLy = -: normalized loop stress.

From (55) we learn thatfo rplott ing E[BLT, ] versus peither the signal parame terb can be kep t constant andy variedor vice versa. The,two possibilities lead to differentsetsofcurves: the case whereafixed relative offset Af/BL is main-tained is dep icted in Fig. 1 9 (with a different normalization),while in Fig. 21 we have plotted the curves for a constantb ona double-logarithmic scale.The dashed curves in Fig. 21 display E [ B L T , ] for afirst-orde r loop where an analytical formula exists[ 2 ]:

Th e ma xim um in these curves is qualitatively easily u nde r-s tood. A small p implies a large bandwidth BL :

resulting in a large num ber of slips dueto poo r noise suppres-s ion. Forincreasing p , the band width is decreased a nd the loopslips less cycles. If, however, the ban dwid th is decreased be-yond the optimum , the static phase error caused byAf startsto interfere and the mean time between slips starts to increaseagain. The smaller b is , the smaller the optimum bandw idth;in the limiting case for b = 0 we obtain a strictly increasingcurve. The curves labeled with 0 an d + are experimentalresults for a second -order loop with{ = 1 .O and /.? = 0.03.Because the normalized quantity log ( E [ B L T , ] ) is dis-playedin Fig.21[andnot E(T,)] th eopt imum BL is notfound at the loca t ion of the maximum ofthese curves. We willshow below h ow the optimization can be carried out bymeansof a graphic procedu re.Multiplying n umerator and deno minator ofp by E(T,) (1 1)and taking the logarithm yields

or rearranged

Fig. 21 . Analytical lyderivedmeantimebetweenslipsfo rfirst-orderloop (dashed curves) and experimental results for second-oloop( 5 = 1.0; p = 0.03) with b = A f / ( P $ N o ) as parameter .For any givenE(T,), (60) represents a straight line in 21 ;with increasing E(T,) the linemoves upwards.Le t us assume that su cha lineintersects a curvelog(EIBL T,] )as illustrated for b = 0.5, and let us label the poin t of isection P,,and the correspondingp by p i l . Solving (58) fo rBL yields

which determines the bandw idth of the loop fora given E(T,)and parameters p , an d b . Jn our example the line also inter-sects the same curve at the po intP2.This means that the sammean timebetw een slips is obt ained fo rtw odifferentloopbandwidths.If we now increase E(T,), then according to(60)th estraightline moves upwards until the two poin tsPI and P zf inally converge to o ne po int de fiedas the tangen t of thcurvewith the slope given by (60). Since for values ofE(T,) abovethat point no intersection exists,we have found the maxim uachievable E(T,) fo ranyloopbandwidth.Therefore,th eopt imumbandwidthfor a given signal para me ter b can befound by constructing the tangent to th e particular cInFig. 22 we have plotted log ( E [ B L T , ] )as a function ofp fo rthetwo damping factors = 0.7 and = 1.0. We observe aslow increase up to the max imu m and a steep descent the opt imum.As expected, the loop with the larger dam

performsbetter.For b = 0 thefunction log [E(BLT, ) ] in -creases monotonically. In theory , any desiredE(T,) can thusbeobtainedwith a sufficientlynarrow loopbandwidth. Wealso see that the first-ord er loop always performs betterthe second-order loo p forb = 0 (but compare the two lofo r b = 0.5). This is not surprising. For zero freque ncy deting, the integ rator in the loop filter is superfluou s, only cing additional slips by feigning a loo p stress.Our discussion on optim al loop parameters is brief and icomplete. A more detailed discussion can be found in [ 7 ] .I twas foun d in this stud y that the loop parameters for m um E(T,) were distinctly different from those obtained linear Wiener filtering theory. In part icular ,the favorite= 0.707 was foun d to be too small ;a better choice is =1.0-1.2.


14/14

ASCHEID AN D MEYR: CYCLE SLIPSIN PHASE-LOCKED LOOPS6 . 00 ,

I ++ b - o 5

I t

Fig. 22 . Exper imen ta l mean time between cycle sl ipsfo r second-orderl o o p f o rf = 1.0 a n d f = 0.7 ( p = 0.03) w i t h b as paramete r .V. SUMMARY AND CONCLUSIONS

Cycle slips inphase-locked loop s are statis tical, nonlinearphenomen a. The complicated interaction of nonlinearity andnoise hasbeendescribed.Experimental results were com -plemented by simple analyses t o ob tain a quantitative under-standing of the influence of the various signal/loop parameterson cycle slip statistics.I t hasbeen shownthat a state variable representation isneeded to discuss cycle slips properly. Besides the phase error,which is defined in a +2n interval according to the definitio nof a slip (and not modu lo2n), the state variablex1 plays acentral role. This f act, which has not received pro per attentio nin the olde r literat ure, was first discussed in several theoreti calpapers; see [8],[9].From a theoretical point of view it is interestingto observethat the experimentally determined mean tim e between slipsfor a first-order loo p perfectly agrees with the results obt ainedvia Fokker-Planck techn ique (which assumes a wh ite noiseprocess).

ACKNOWLEDGMENTWe have greatly profitedfromthe expertise ofDr.F.Gardner who spentfreely of his t ime to review the man uscrip t.His detailed critique made the paper better than it would havebeen with out his help. We would also like to acknowledge theconstructive criticism mad e by Mrs. McKenzie, Dr.W. Braun,and Dr.C. Chie of LinCom and L. Po pken of RWTH Aachen.Thesupp ort of the DeutscheForschungsgemeinschaft(DFG) is greatly appreciated.

R EF ER ENC ES[ I ] W .C. L indseyan dM . K . Simon , Telecommunication Systems[2 ] W .C. L indsey , Synchronization Systems inCommunicationan d131 A.J.Vite rb i , Princ iples of Cohe rentCommunications. Ne w[4]A .Blanchard , Phase-LockedLoops.ApplicationtoCoherentRe -

Engineer ing. EnglewoodCliffs , NJ : Prentice-Hall.1973.Contr o l . EnglewoodCliffs, NJ: Prentice-H all,1972.York:McGraw-Hill , 1966.ceiverDesign. NewYork:Wiley,1978.

2241F. M. Gardner , PhaselockTechniques. New York:Wiley.1979.G . Asche idan d H . Meyr. "Microprocessor-controlled experimentto determinecycleslipstatistics: Hardw are and software."ERT-Rep .713/18 ,Sept .1981 .ERT,Rep. 713/19,Sept.1981 .W . C . Lindsey and H. Meyr. "Complete statistical descriptionofthe phase-erro r generated by correlative tracking systems ."I E E ETrans.Inform.Theory. vol.IT-23, pp. 794-802, M ar. 1977.D .Ryteran dH .Meyr ,"Theory of phasetrackingsystems ofarbitrary order: Statist ics of cycle slips and probabili ty distribof the state vector,"IEEE Trans. Inform. Theory,vol. IT-24, pp.1-7. Jan.1978 .F . J . Charlesand W . C. L indsey."Someanalytical and experl-P r o c . I E E E ,vol. 5 4 , pp . I 152-1 166, Sept.1966.mentalphaselocked loop

results for low signal-to-nowratios."R. W. Sannemann and J. R. Rowbotham. "Unlock characteristicsof the opt imum type11 phase-locked loop," IEEE Trans. Aerosp.Navi g . El ect r on . ,vol.ANE- I I , pp. 15-24, Mar. 1964.R. C. Tausworthe, "Cycle slipping in phase-locked loops. ' 'I E E ETr ans. Commun. Technol . , vol. COM-15. pp. 417- 421 ,June 1967.W .C.LindseyandR . C. Tausworthe, "A bibliography of thetheory and ap plication of the phase-lock principle," Jet PropulsionLab . ,Pasadena ,C A ,Tech .Rep . 32-1581, Apr . I , 1973."Statist icalloopanalyzer(SLA)," LinCom Tech.Rep. ,Mar .1982.

-, "Param eter optimiza tion in PLLs: An experim ental study,"

*GerdAscheid was born in Cologne,Germany,on June 14. 1951.Hereceived the Dipl.-Ing.degree from the Technical University of Aachen(RWTH Aachen), Germany, in 1978.He is now a Research Assistant at the Depart-ment of Electrical Engineering, RWTH Aachen,workin g toward the Ph .D. degree. His main inter-est is in synchronization,especially of band-width-efficient modulations.*HeinrichMeyr ("75) received the Dip l.-lng.an dPh.D.degrees from the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland,in 1967 and1973.respectively.From 1968 to 1970 he held researchpositionsat Brown BoveriCorporation,Zurich, and theSwissFederalInstitutefo rReactorResearch.Ftom 1970 to the summerof 1977, he was withHaslerResearchLaboratory,Bern.Switzerland,doingresearch in thefields of digitalfacsimileencoding and tracking system s. His last positionat Hasler was managerof the Resea rch Departm ent. During 1974 he was aVibitingAssistantProfessor with the Department of ElectricalEngi-neering.University of South ernCalifornia. Lo s Angeles. Since thes u m m e r of 1977 he has been a Professor of Electrical Engineering at theAachenTechnicalUniversity(RWTHAachen).Germ any. His researchinterestsincludesynchronization,estimation.and, i n particular.th einteractionofimplementationissues.and the design of controlan destimation/measurementsystems.Presently he is consu ltant to the 1BM

Research Laboratory, Zurich,i n the area of synchronization of local areacomputer networks andto Krohne Ltd. , Duisburg, Germany.i n the area ofdigitalsignalproce ssing . He has published work in variousfields an djournals and holds over a dozen patents.Dr. Meyr served as a Vice Chairman for the 1978 IEEE Zurich Seman d as an International Chairman for the 1980 National Telecomm unica-tions Conference, Houston. TX. He presently serves as Associate Editfo rth eIEEE TRANSACTIONS ON ACOUSTICS.SPEECH, A N D SIGNALPRO CE SSI N G .

pll_cycleslips

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