2228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO.

10, OCTOBER 1982

Cycle Slips in Phase-Locked Loops: A Tutorial Survey

GERD ASCHEID AND HEINRICH MEYR, MEMBER,IEEE

Abstract-Cycle slips in phase-locked loops are statistical, nonlinear

phenomena. 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.
Loop filter
-
It is the purpose of this survey paper to present a self-contained
discussion of cycle slipsin phase-locked loop avoiding advanced math-
ematical tools. Based on the results of an extensive experimentalstudy
we explain theunderlyingprinciple of thecomplexinteraction he-
tween nonlinearity and noise. The results are complemented by sim-
ple,approximateanalysiswhich agreeswellwiththeexperimental
findings.
In addition, we present a new and complete set of diagrams on cycle Fig. 1. Equivalent baseband model of synchronizers.
slip statistics not presently available in the literature.
theory is used. While the de‘parture of the variance ui2 from
I . INTRODUCTION the linear theory is small, the slip rate varies by orders of mag-
nitude within a small interval. A great deal of effort has been
M OST digital communication systems employcoherent
demodulation techniques. This requires a receiver capable
of accurately estimating the phase of the received signal.
invested in the past in theoretical as weli as experimental and
computersimulation studies of cycle slips. Lindsey and
Charles [ l o ] presented results on experimental cycle slip dis-
The circuits for generating a carrier reference are all based tributions. Mean timebetween slips has’ been obtained by
on the same fundamental principle: a locally generated refer- means of computer simulation in [ 113 and [ 121 . This short
ence and the received signal are nonlinearly processed to gen- list is only a sampling of noteworthy papers. A comprehen-
erate an error signal which is subsequently used.to adjust the sive list on the subject can be found in the book’s on PLL‘s
phase of the VCO tothat of the incoming signal. Conse- [1],-[5] and in a bibliography [13] . Lately, there has been a
quently, this suggests that these synchronizers have the same first commercial product announced which measures operating
mathematically equivalent baseband model shown in Fig. 1. threshold based on sliprate [14].
Depending ontheparticularmodulation,differentnon- The purpose of this paper is twofold:
linearities g(4) and noise processes n(t) will be obtained
[2, p. 1161. 1) to present a set of new experimental data on cycle slip
The f a c t t h a t t h e r e e x i s t s a common equivalent model is of rates and
great importance since it allows one to transpose the results 2) to support these results with simple analyses in order to
obtainedforthe phase-locked loop (PPL) tomorecomplex show the influenceof the various loop parameters.
circuits such as Costas loops, etc. The main difference between the past work and this paper
Due to the noise, theVCO phase is a random process. When is that we examine the actual physical phenomenon of a cycle
the VCO phase variance becomes large, a phenomenon occurs slip while the previous authorsreported overall statistics
which is inherent to the nonlinearity in the loop. The VCO (which is very important information).
phase is increased to such an extent that the VCO slips one or In Section I1 a brief survey of well-known equations and re-
several cycies with respect to the input phase. The occurrence sults is given. We feel that such a survey is necessary because
of a slip is an event with very low probability for weak noise, thedefinitions and notations in thePLLliterature are not
buttheprobability increases steeply with increasing noise
uniform. We have employed’state variables sincethis is the
power. As an example, the cycle slipping rate of a first-order only mathematical description that allows one to analyze the
loop is plotted in Fig. 2 versus the loop signal-to-noise ratio loopunder nonlinear operatingconditions. Normalized and
p (which is defined in Section-11). The same figure shows the dimensionless variables have beenused. The mainadvantage
phase variance a i 2 obtained when thenonlinearity is taken of such a normalization is that the number of parameters is
intoaccountandthe phase variance q i n 2 when the linear
reduced to a few parameters having physical meaning.
Section I11 of the paper is entitled “Understanding Cycle
ManuscriptreceivedNovember 1 , 1981; revisedMarch 22, 1982. Slips.” We demonstrate that the statevariable x1 (proportional
ThisworkwassupportedbytheDeutscheForschungsgemeinschaft
(DFG) under ContractMe 651/3-3.
to the capacitor voltage in a second-order loop) is of utmost
The authors are with Aachen Technical University, D-5100 Aachen, importance in understanding slips. Forexample, x1 deter-
West Germany. mines whether or not the cycle slips occur in bursts. Of great

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

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

Fig. 2. Phase variance and normalized sliprate n as a function of the

loop SNR p.

assistance, if not indispensable, in explaining the complicated We modelthe phase detector as an ideal multiplier. The
interaction of noise and nonlinearity in a PLL, are trajectories output of the multiplier in Fig. 3(a) equals (double frequency
forthe phase error @(t)andtheotherstate variable x l ( t ) terms are neglected)
taken during a typical experiment. These trajectories, together
with simple, approximate computations, allow one to obtain a
goodunderstandingof the influence the various parameters
have on cycle slips. with
In Section IV-A we give an overview of the experimental
configuration.Theexperimental results are discussed in the KD =KmKIA
next subsection. Finally, we briefly discuss the optimization
fiKl = amplitude of VCO-signal
of the loop parameterswhen the mean timebetween cycle
slips is t o be maximized.
Km = multiplier gain
11. LOOP EQUATIONS e = phase of input signal
A . Basic Equations andDefinitions
e' = phase of VCO
We begin byrecapitulating some well-known equations
together with the notation to be used in the sequel. The mater- @ = t9 - 6' = phase error
ial is covered in detail in any book onPLL's.
and
The input to thePLL equals the sum of signal s(t) and noise
n(t)
Y(t)= + n(t> (1)
where the signal s(t) is given by An exact equivalent circuit of the phase detector is shown
in Fig. 3(b) andthe power spectrum of n'(t) is shown in
s(t) = f i A sin (wot + e ) (2) Table I(b).
Under the assumption of small phase error the PLL has the
and n(t) is a narrow-band Gaussian noise process mathematically equivalentmodel shown in Fig. 4. The vari-
ance o i 2 of theoutput phase duetothe noise disturbance
n(t) = f i n , ( t ) cos wot + a n S ( t )sin wet. (3) n'(t) is
1 r+-
The features of this process together with the definition of
equivalent noise bandwidth B , , of an IF-filter preceding the
PLL and the signal-to-noise ratio at the output of the IF-filter
are summarized in Table I(a). wlth the loop transfer function
2230 TRANSACTIONS
IEEE ON COMMUNICATIONS, VOL. COM-30, NO. 10, OCTOBER 1982

TABLE I(a)
POWER SPECTRUM OF THE NOISE PROCESS N ( T )

Sn,,I 0 ) )
I

t :

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

FIF(w) : I Ff i l t e rs p e c t r u m ,
Noise variance:on. = -
NoBIF
~

I F I F ( * W ~ ) /= 1 2A' 2SNRi
m

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

0

Npooi w
s ee r : PN
w

= &jSnn(w)do= N 6
I
o IF
_m

pSoi w
g ne ar :l Ps = A

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

NO
S,*,,(m) = -2'
2'4

Inserting (8) and (9) into (6) yields

The inverse variance is often called signal-to-noise ratio in

the loop andgiven the symbol p .

(b) 1 A2
p=-=--
Fig. 3. (a) Multiplier type phase detector. (b) Mathematically equiv- u,ij NOBL
alent model of the phase detector.
Using signal and noise power, (1 1) can be rewritten

F(s): loop filter. This form is particularly well suited for measurement. The
definition of p is not unique in the PLL literature. In the form
The equivalent loop bandwidth BL is defined as ~

(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. His
definition is equally arbitrary and is as valid as ours.

B. State Variable Descriptionof the Second-Order PLL

If the equivalent noise bandwidth BL is much smaller To describe and understand the nonlinear behavior of the
than the bandwidth BIF,the loop "sees" a white noise process phase-locked loop we need a state variable description. For the
with constant power spectral density (Table I): second-order PLL with loop filter (Fig. 5)
ASCHEID AND MEYR: CYCLE SLIPS IN PHASE-LOCKED LOOPS 2231

Fig. 5. Loopfilter of a second-orderPLLwithimperfectintegrator

(lag-lead filter).

I -1-23P+P2 t- -
1
I I

-
Fig. 6 . Equivalent model of a second-order PLL with imperfect inte-
grator using normalized variables.

1 + S T , T2 we arrive, after some algebraic manipulations, at

F(s) = --
1 +ST, T1
- +
-_ 1-T2Pl
1 +ST,
(1 3)

the nonlinear differential equations are

+ [ 1 -2{p+p2]IZ:

A mathematically equivalent model of the PLL using the

Thestate variable y l ( t ) is proportionaltothecapacitor normalized variables is shown in Fig. 6.
voltage uc(t)of the loopfilter In (18) only the two parameters, loop damping { and the
frequencyratio /3, appear (aside fromthenatural frequency
a, absorbed in 7). The reader will notice the use of w, and
in a nonlinear state equation. These quantities are formally
and has the physical dimension of angular frequency. defined in the same way as for a linear system.
The state equations as they stand are of little value. The The values of practical interest of 5 lie in the range of
parameters can vary over orders of magnitude; their influence 0.5 < < 5. We can also delimit values for 0.The ratio T 2 / T ,
onloop performance is hard to oversee. Introducingappro- is given by
priate normalized and dimensionless variables yields a set of
equations with parameters more easily interpreted. We intro- T2
-= p(2{ - 0)> 0. (1 9)
duce the normalized time r and replace y , by the dimension- TI
less quantity x1 as follows:
From (19) we find

In the case of the commonly used high-gain loop we have

Using the definitions of the natural frequencya, and the loop
0 < 1, typically 0.001 < p < 0.1. If we formally set /3 = 0 we

rF
damping factor
obtain the equations for the second-order loop with uerfect

0, = {= 3 0, [ T2 + 1 1
KOKD
(1 7)
loop integrator. For reasonably small 0 the dynamic behavior
forthe imperfect second-orderloop is practicallyindistinguish-
able fromthe perfect integrator
loop
with /3 = 0. In most of
2232 TRANSACTIONS
COMMUNICATIONS,
IEEE ON VOL. COM-30,
NO. 10, OCTOBER 1982

our discussions on cycle slips we set 0 = 0 for simplicity. There _.__--.---

__._-----___----
is one important difference, however, in the steady state. In
the noise free case the state variables assume the values

AU
x1 = -- (1 - 2 t p + 02).

The loop with perfect integratorcompensates a(normal-

ized) frequency difference Aw/w, with a value of the inte- , 1 , , , , 1 , , , ,

grator output x1 of exactly Aw/o,. The steady state phase 0. 200.

150.
100.
50. 250. wt,
to
error 4 is zero. With an imperfect integrator therealways exists
Fig. 7. Probability distribution P(T, < t ) (solid lines) of the time be-
a static phase error whichsubstantially increases the cycle tween slips and conditional distribution P(T, < t I T , > t o ) (dashed
slip rate as will be seen in alater section ofthis paper. line), conditioned on the fact that T , exceeds a given time to. Sig-
nal-to-noise ratio p = 2 (numeric ratio).
111. UNDERSTANDING CYCLE SLIPS
We already know from Fig. 2 that the cycle slip rate is ex-
tremelydependentonthe signal-to-noise ratio. As will be
shown presently, the damping factor 1 also hasa strong in-
fluence on the slip rate as well as on the manner of occurrence
of the slips.
In Fig. 7 the probability that the time between two slips
Ts is shorter than t is plotted for two values of t. For t =
0.24 we observe a steep increasein P(T,< t ) for small t ,
i.e., many slips have a very short duration compared to the
mean time between slips. This means that the slips occur in
bursts: if viewed on an oscilloscope we see clusters of slips of
very short duration separated from each other by long time L Y I

intervals. -2n I 1
0. 20. 40. 60. 80. 100. T
For large damping, the distribution P(T, < t ) is virtually Fig. 8. Trajectories of phase error @ ( T ) and state variablexl(T) versus
exponential. From probability theory we know that exponen- normalized time for p = 0.03, f = 0.24, and p = 2 (numeric ratio).
tially distributed times between events imply statistical inde-
pendence of these events. We therefore conclude that thecycle
expect an oscillation of x l ( r ) with the same period as @ ' b u t
slips are independent, isolated events for large damping.
shifted by 1 ~ 1 2and
, this is indeed found in Fig. 8. To go one
Having recognized the burst-likeappearance of slips for
stepfurther, x1 and $I must have approximatelythesame
small dampingfactorsandthe isolatedevent structurefor
amplitude, because of the normalizations that have been
large damping, we presentlywant to explainthis different
performed. For x1 and $I we read off from Fig. 8 a value of 1 ;
behavior.
converting the average amplitude of $I into degrees gives ap-
A . Loops withSmall Damping Factors proximately 60".
The weaker the noise, the smaller the amplitude of x1(7)
In Fig. 8 we have plotted an example of phase error $I and
and @(T) since the weak noise can push the loop only slightly
state variable x1 as a function of normalized time r for 5 =
away from its stable equilibrium. From linear theory we know
0.24 and a very small SNR. We clearly recognize the burst-like
nature of the cycle slips. Before we turn our attention to the uo2 = l / p ; hence,theamplitude of theperturbed sinusoid
bursts of cycle slips we examine the behavior of the loop be- would be m. For strong noise, as in Fig. 8, it is very prob-
tween twobursts, Le., in itstrackingmode. A damping of able that the amplitude of the fluctuations exceeds the 90"
( = 0.24 means that in the tracking mode (where linearization
phase error (maximum restoring force), in which case a cycle
applies), we expect to observe aweakly damped oscillation slip is very likely to occur. Furthermore, in this case the x1(7)
variable follows to a wrong value. After completion of the first
of $(7) and x1(7)sustained by the noise process n'(r).
cycle slip, x1(T), which is responsible for frequency correction,
The average period of this oscillation is
has awrong initial value (loop stress) and,consequently, is
2n much more susceptible to another cycle slip.
The experimentally determinedaverage value of x1 taken at
the completion of a cycle slip as a function of p is shown in
in normalized time. From Fig. 8 we find an average period of Fig. 9(a). Indicated in this figure are the normalized pull-out
7, which is not far from the predicted number. We would also frequency given by [SI
SLIPS
ASCHEID
CYCLE
AND MEYR: IN PHASE-LOCKED LOOPS 2233

The average of x1 as a function of p increases rapidly with

decreasing p , see Fig. 9(a). For p = 1 (OdB) x1 has come very
close to the pull-in frequency of the loop. Since the experi-
mental variance ox [Fig. 9(b)] taken at the completion of a
cycle slip also increases with decreasing p , it is very likely that
x1 assumes a value larger than the pull-in frequency once the
loo$$lfas started slipping cycles. Therefore, once the loop has
lo& lock it stays out of lock with high probability. Control of
the VCO is lost and its frequency wanders off from the signal
frequency. This phenomenon has been called drop-lock in the
literature.Actually,there is no well-defined noise threshold
below which the PLL falls out and stays out of lock; there is
nofundamental differencebetweenrepeated cycle slips and
drop of lock (but see Section 111-C on loop detuning).
The wandering off of the VCO at extremely low SNR is

p+ 0.70
easily explained if we recognize thatthe restoringforcein
the equivalentmodel of Fig. 6 can be neglected, if x1 is
larger than the pull-in frequency. But neglecting the restoring
force sin @ is equivalent to opening the feedback system (see
Fig. 6). In this case, the input to the integrator representing
the VCO consists of the noise process plus xl(T). Asis well
known, the variance of an integrated noise process increases
with T ; hence, it is unbounded and the VCO wanders off.
It is very instructive to examine loop behaviorduringa
burst of slips. We want to examine in detail the first, the last,
and a cycle in the middle of a burst. Examples of bursts hav-
ing the same parameters are shown in Fig. 10 and in expanded
(b) scale in Fig. 11 and Fig. 12.
Fig. 9. (a) Conditionalexperimentalmean Fl of thestatevariable Due to noise, the magnitude of both the phase error @(T)
x1(7), taken at the instant O ( T ) = +2n (completion of a cycle slip). and of X ~ ( T ) increase at the beginning of the first cycle slip
(b) Conditional experimental variance oX12 = (x1 - F1)2 taken at
the completion of a cycle slip. [Fig. 11, region (a)] . (To correctthe negative going phase
error x1(7) should become negative.) The phase error passes
through -n/2, the point of maximum restoring force toward
= 1.8(1 -I-<) -T. In the interval of --71 < @ < -2n, the restoring force has
o n
thewrongpolarity;there is positive feedback in theloop.
and the pull-in frequency [SI of the second-order loop with Since x1 > 0, the phase error rapidly passes this region of
imperfect integrator positive feedback to reach @ = -2n, which, of course, is
equivalent to zero phase error. At completionofthe first
slip, x1(7) assumes a random value of slightly more than the
pull-out frequency. During the following three cycle slips the
value of x1 increases to amaximumbefore it is slowly de-
creased to its correct average value of x1 = 0. Another burst is
If theloopstartswith zero phase error @(O) = 0, then shown in Fig. 12. In contrast to the previous burst we do not
xl(0) = op0/on
is the maximum initial frequency deviation observe a pumping up of the x1 variable during the first cycle
for which the loop does not skip one or more cycles but re- slips. Rather, the value of xl,taken at the completion of a
mains in lock.The pull-in frequency is themaximum fre- cycle slip, fluctuates around the experimental mean value x1
quencydifference for which theloop will lock, eventually, before it takes on a value x1 < wpo/w,, such that the loop
after a long acquisition period. Both frequencies hold for the can pull in.
noise-free case only,but serve as a good indicatorforthe Looking at the two (entirely different) bursts, the question
noisy case. arises whether we have observed examples of twodifferent
For a small damping factor the average value o f x l is-even phenomena or whether there exists a common mechanism in
for large SNR-only slightly smaller thanthepull-out fre- both cases. We will see thatthereindeed exists a common
quency. This means that a burst of cycle slips rather than a mechanism in both cases consisting of a systematic force driv-
single cycle slip is to be expected. For p < 5 (7.0 dB) and = < ing the PLL towards its stable equilibrium and a random per-
0.24 the average of x1 is larger than the pull-out frequency turbation. We will first analyze the force and later on compute
and a single cycle slip is very unlikely. In addition, the larger the variance of the random perturbation.
the difference Ixl - opo/on 1, the longer the mean duration A typical cycle in the middle of a burst is shown in region
of the burst will be. (b) of Fig. 11. It is typical, in the sense that a value x1( 7 )
2234 TRANSACTIONS
COMMUNICATIONS,
IEEEON VOL. NO.
COM-30, 10, OCTOBER 1982
ASCHEID AND MEYR: CYCLE SLIPS IN PHASE-LOCKED LOOPS 2235

The answer is again found by inspection ofEl in Fig. 9(a).

We first note that Xl is, for reasonably large p , less than 1/3 of
the pull-o'k:frequency. Secondly, the functionXl ( p ) is essenti-
ally flat for p > 4. It remains for all p much smaller than the
pull-out frequency w p o / o n(not to mention the pull-in range
u p / w n ) .From thisand the variance u X l 2 in Fig. 9(b) it is
clear that bursts of cycle slips are extremely unlikely events.
The mean time between cycle slips converges for low p toward
the values of a first-order loop which always performs better
than a second-order loop, provided there is no frequency dif-
ference between VCO and signal.
We still owe a n explanation as to why the variable x1 takes
on such small values. As always, linear theory is instrumental
Fig. 1 3 . Phase error @(T) and phase detector output sin @(T) during a in getting a good understanding of the nonlinear behavior.
beatnote.
A large damping implies a large time constant for the x1-
integrator; the response of such an integrator to a noise event
The variance of this increment is easily found to be is very sluggish and small in amplitude. Therefore, the propor-
tional path of the filterdetermines the short-time transients
while the integral path compensates for an eventual frequency
difference between VCO and signal. Thus, for the computation
of theshort-time transients of @(r)we may approximately
assume x1 to be constant. But the input to the integrator of
the loop filter equals the input to the 6(r)-integrator,if multi-
plied by 1/21. Hence,'the increments ofx, and 0 are approxi-
mately equal:

, 2.8
~ b ,= = 0.24 , 0.81.
U A ~ = 3)(3 (34)
The random fluctuation is much larger than the systematic
driving force El.The different appearance of the two bursts Since 8<r>= - we may write
is now easily understood. The systematic force awl is covered
by the random fluctuations; the values of x1 taken at the com-
pletion of a cycle slip are within abandof20Axlwidth,
centered around X 1 . What looks like a systematic pumping up
effect in Fig. 11 is nothing but a normal statistical fluctuation.
Our simple analysis is only approximate but predicts, re- for the short-time transients.
markably well, the duration of B slip within a burst as well as A typicalcycle slip is displayed in Fig. 14. We observe a
the statistical fluctuations of the increment a x l . very similar shape of x1(T)and @(r)up to the end of thecycle
The distinctive features in region (c) are that the cycle slip- slip. Subsequently, the value of x1(7) slowly decreases to its
ping stops and x1 rapidly converges toward zero. Immediately initial value of x1(7) = 0 with a time constant of 25, corre-
after the last cycle slip, the phase error rapidly increases. The sponding to the neglected pole of the second-order loop. Note,
restoring force sin @ is large and has the correct polarity during however, that @(T) afterthe slip is much less affected by
the convergenceinterval. Integration of the restoringforce X1 (7).
Having recognized the similarity of X ~ ( Tand
) @(T) [as pre-
proceeds rapidly so that x, quickly moves toward zero.
In passing, formulas (30) and (32) give a hint as to why the dicted by (35)] we are in a position to compute an estimate of
looppermanently loses lock for decreasing p. The average x1(7) at the completion of a cycle slip. Let us denote by ro
pull-in effect El is inversely proportional to (x1)?,while the and r Z n , the beginning and end of a cycle slip, respectively,
variance is inversely proportional to p and x l . From Fig. 9(a) and define a cycle slip as part of the trajectory @(r)such that
we know that the average El increases for decreasing p . There- @ ( T ~=) 0 and @(T< T ~f ~0. Furthermore,
) we assume
fore, the bursts tend to become longer until the loop eventu- x1( T ~ =
) 0. Under these assumptions we obtain for x, (r2*)

ally loses lock completely.

B. Overdamped Loops (5 > 1)

Having understoodtherathercomplexstructureofthe r2,,- ro : random variable.
cycle slips of an underdamped loop we turn our attention to
loops having damping factors 5 > 1. As already noted, cycle Neglecting the integral path of the loop filter for the short-
slips occur in this case as isolated events, not in bursts. Why time interval ( T -~r0),~ the PLL is governed by a first-order
this difference? differential equation
 2236 TRANSACTIONS
IEEE
COMMUNICATIONS,
ON VOL. COM-30, NO. 10, OCTOBER 1982

Fig. 14. Cycle slip of an overdamped loop ( 5 = ,1.5) for p = 2.5 (nu- Fig. 15. Trajectories @(T) and x1(7) of a second-order loop with 5 =
meric ratio). Note that -x1(7) is displayed. 1.0 and p = 2.8 and a loop stress of A w / w , = 8.

ference as shown by examplein Fig. 15. Because the differ-

(3 7) ence between maximum restoring force and the stable equilib-
rium point is reduced to 1 - sin @Ao/o,), theloop slips
Note that the right-hand side of (37) multiplied by - 1/2{ much more often to this side than would be the case for zero
equals theintegrand in (36).Integration of thestochastic detuning.
differential equation (37) from T~ to 7 2 n yields In the experiment, it was observed that if Aw/w, was in-
creased above the pull-in range, theloop,with very high
probability, completely lost lock after once slipping a cycle, a
behavior not observed for the same signal-to-noise ratio with
zero detuning.
To discuss this phenomenon, a few preliminary remarks are
but @ ( T ~ )= 0 and I $ ( T ~ ~ I) = 2n, by definition. Replacing the
helpful. In the noise-free case the steady-state value of x1 re-
right-hand side of (36) by - $ ( ~ ~ ~ ) /yields
2{
mains constant, that is, x l = 0. This is only possible if the
signal (- oxl) at the input of the x1 integrator is exactly com-
pensated by a staticphase error such that

.)
As in the case of small damping factors {, x1(72 appears
as a frequency detuning. However, the detuning is too small
As long as x1 < Aw/w, < 1/p (hold-in range) the loop
compared to the pull-outrange to produce a burst. The inverse
remains in lock.
dependence of ~ ~ ( is 7experimentally
~ ~ ) well confirmed; see
If, due to noise, the loop slips a cycle, thereexists a random
Fig. 9. Du.e to the coupling of xl(T) and $(T), the result (39)
difference (Ao/w,) -x1 after completion of the slip. A neces-
is, of course, only an approximation. sary condition for the PLL to resume lock is that it is capable
So far we have discussed two examples of loops of { = 0.24 of reducing this difference to zero, on the average. This might
and { = 1.5 damping and have found a tendency that weakly
require a long pull-in period involving many cycle slips.
damped loops burst while overdamped loops do not. It would
Mathematically,
__
this condition requires thatthe average
be interesting to identifyaboundarybetween burstingand
pull-in voltage sin C#J for a given difference (Ao/w,) - x1 must
nonbursting. Of course, such a boundary cannot be rigida one,
be larger in amplitude than the decay oxl of the leaky inte-
but wouldmerelyprovide information as to whether a loop grator.
is more likely to burst or not,
On the average, a loop will burst only if the mean value of
x1 taken at the completion of the slip is larger than the pull-
out frequency. Using (39) and (23) yields the inequality
Otherwise, the value of x1 decays to zero and thePLL faus
n/{> 1.8(1 + {) condition
burst.
for (40) out of lock forever,as has been the case in Fig. 16.
The worst case occurs for large frequency differences
Solving (40) we find thatloopswith damping factorsof (Aw/w,) - xl. In this case we may use the result of (29) if we
{ < 0, 9 will burst. replace x1 b y x l - Ao/w,. Then for Aw > 0 we obtain

C. Frequency Detuning
A frequency difference Aw betweensenderand receiver
causes a static phase error; the x1 variable assumes a value of
x1 = A o / o , in order to compensate for the frequency dif- or slightly rearranged
AND ASCHEJD MEYR: CYCLE
SLIPS IN PHASE-LOCKED
LOOPS 2237

Fig. 17. Simplified block diagram of experimental configuration.

Fig. 16. Trajectories o(7) and Xl(7) forthesameparameters asin
Fig. 15, but for A w / w , = 16. The loop starts with correct initial
conditions at 7 = 0 and drops lock afterf i s t slip.
IV. EXPERIMENT

A . Experimental Configuration
The configuration is divided into two parts, the experiment
In general, for a given Aw/o, and { lo,there exists an in- itself,' in analog hardware, and a microprocessor (pP) system
terval of x1 for which the inequality is not true. If the loop for control of parameters and recording of measured data. In
slips a cycle and x1 accidentally assumes a value inthis in- this section, a survey of the hardware and a functional des-
terval, resuming lock wouldbepurely bychance.Such be- cription of the pP system, based on a Z80 CPU, will be given.
havior is clearly unacceptablein a practical application. The For a more detailed discussion see [6] .
question arises whetherthere are values of Aw/w, and {/P A block diagram of the analog hardware is shown in Fig.
for which thequadraticform (44) is positive for all values 17. An unmodulated carrier is provided by a crystal (Xtal) os-
of xl. Then,theloop could always reduce the difference cillator. The signal power can be set by a variable attenuator.
(Ao/w,) - x1 to zero: Wide-band Gaussian noise from a random noise generator is
Indeed,thequadraticform is strictly positive if the dis- added. The noise power can be varied by a second attenuator.
criminant is negative. Both signal and noise may also be switched off.
Filtered by theIF quartz filter, the noise becomes a narrow-
band Gaussian process as described in Table I. At the output
of the filter, the signal and the noise power are measured. The
or filter output is also the input to thephase detector of the PLL.

Ao/w, < 2 2. The phase detector is of the multiplier type, the only one
usable at low SNR. The passive loop filters are exchangeable.
The VCO output is notonlyconnectedtotheloop
detector, but also to a reference phase detector.
phase

But the right-hand side of (46) is nothing but the pull-in The undisturbed carrier is directed along a reference path
frequency wp/on; see (24). to the other input of this linear k180° phase detector to deter-
Inconclusion,theloopmust be designed suchthat mine the actual phase error @(t). A second quartz filterhas
lAw/w, I is sufficiently smaller than the pull-in frequency u p ; been inserted into the reference path, adjusted to compensate
this is particularly important for low SNR's. From (46) one for the phase shift of the IF filter.
concludes that a perfect integrator @ = 0) realized by means The connections between the analog hardware and the pP
of an active loop filter is preferable. In practice, however, due system are marked by double lines in Fig. 17. The p,P sets the
to ever present drift currents, there will always be a limit on variable attenuators in steps of approximately 0.05 dB/bit.
the maximum permissible frequency difference. The center frequency of the VCO is adjusted by a digital-to-
For the ratio between positive and negative cycle slips, the analog converter.
following formula valid for a first-order loop has been derived: The digital power meter is connected to the pP system by
an IEC-bus. Thus, the pP can not only read off values from the
N+
-=
N-
exp (4. ?) power meter, but also send commands to the power meter,
(47) such as zero calibration and mode commands. If the power is
measured in dBrn, a four digit value results, with, a least signifi-
cant digit of 1/100 dB,.
In any case, the sum Three analog-to-digitalconverters, all bufferedby sample
and hold amplifiers, allow the pP system to record values of
the phase detectoroutput of thePLL ( u g ) , thecapacitor
voltage (u,) of the loopfilter of the second-order PLL, and the
actual phase error @.
2238 TRANSACTIONS
IEEE ON COMMUNICATIONS, VOL. COM-30, NO. 1 0 , OCTOBER 1982

In addition, the output of the reference phase detector is TABLE I1

MEAN DURATION O F AN EXPERIMENTE(T,)
connected to a cycle slip detector, which provides a signal to
the pP whenever the phase error exceedsa value of 2r. By
means of a second signal the direction of the cycle slips to the BL = 750 Hz
E[BLT,] E[T,] E[T,]/1.600 events E[T,]/300 events
f l system is recorded.
The pP system and the analog hardware are strictly sepa- 101 13.3 ms 21.3 s
rated to avoid signal interference. The analog hardware is built 105 133.3 s 60 h 11 h
1 os 37 h 463 days
into a shielded box with separate voltage supply. In addition,
the analog hardware itself is again builtin five subblocks
shielded individually to avoid internal interference. These are at lower signal-to-noise ratios. But the minimum distance that
signal path, noise path, IF path,reference path, and thePLL. can be recorded with the described configuration i s limited by
The function of theE.Lp system in themeasurement are the speed of both the analog-to-digital converters and the pP
system.
1) zero calibration of the power meter,
Another aspect has to be taken into consideration at higher
2) signal and noise power setting,
signal-to-noise ratios. The mean time between slips increases
3) VCO center frequency adjustment,
exponentially with p. Assume the bandwidth BIF is measured
4) samplingand storing of the measured data during
yith x percent uncertainty.Thecomputed value of p (see
measurement, and
Table I) will have the same uncertainty. The slip rate will have
5) frequent recalibration in measurements of long duration.
an uncertainty proportional to exp (PX/~OO).
The actual parameters and the measured data are stored to- An exponential dependence of p will lead to an increasing
gether in a random access memory (RAM). The stored para- asymmetry in the distribution of the direction of the cycle
meters are the measured signal and noise power, a filter identi- slips if the phase detector is not symmetrical to phase error
fier, the digital-to-analog converter values, the offset of phase 4 = 0 or if the VCO gain is not symmetrical for positive and
detector output (uo), and capacitor voltage (uJ. negative detuning.
The microprocessor system is connected to a larger machine With the descrlbed configuration, measurements of the slip
(PDP 11/60) for two purposes. rate of a first-order loop, where analytical results exist, were
made. In the range lo0.’ < E(BLTs) < lo5 excellent agree-
1) different types of measurement programs can be loaded
ment occurred (see Fig. 18). At low rates, a degradation ap-
from the PDP to the pP system,which gives high flexibility.
pears due to the limiting of the noisewithin the phase de-
2) after completion of a measurement the contents of the
tector. At higher rates the duration of the measurement is too
random access memory are sent to the PDP for processing,
long.
such as evaluation of statistics orgraphical display.
The main advantages of the microprocessor control may be B. Results
mentioned here. As the actual parameters are stored together Thedistribution P(T, < t ) was found t o be very nearly
with the measured data, confusion of different measurements exponential for sufficiently large damping factors and signal-
or erroneous readings of the parameters are avoided. The fact to-noise ratios:
that the attenuators can only adjust discrete power levels has
no influence on the accuracy of the measurement, as only the P(T, < t ) = 1 - exp (-t/E(T,)). (49)
actual parameters are used for the evaluation of the statistics.
Furthermore,it is advantageous that long-timemeasure- The mean time E(T,) between slips versus p is displayed in
ments are also possible. Some numerical examples of the dura- Fig. 18.
tionof a completemeasurementare given in Table 11. An For small damping factors, a significant departure fram the
equivalent loop bandwidth of BL = 750 Hz is assumed for the exponential distribution is visible for very small t ; see Fig. 7
computations. In the described configuration, this value can for atypical example. This featuremust be attributed to
not be exceeded very much, since it still has to be much bursting of the slips.
smaller than the IF filter bandwidth B I F = 14.96 kHz. Dis- If we exclude all slips occurring very shortly after a pred-
played in Table I1 are the mean time between cycle slips E(Ts) ecessor, the conditional probability that T, < t under the con-
and the mean duration of a complete measurement E(Tm) for dition that T, exceeds to results in
a few values of e ( B L T s ) .
E ( T M )is computed for 1600 events to be recorded, w h c h P[T,<tlT~>to]. (50)
is the maximum number that can be stored in the RAM, and
an arbitrary chosen minimum of300 events to be recorded. As Using Bayes’ rule it yields
can be seen, the mean duration of the experiment increases
rapidly and for E(BLTs) = lo8 an unacceptable duration re-
sults. Besides, the results will not be reliable for such low rates.
If higher rates of E(BLTs) are supposed to be measured,
the duration o f the experiment can only be shortened by in- The conditional distributiondescribes the statistics between
creasing the equivalent loop bandwidth B L . On theother bursts, rather than slips. As expected, (51) is practically indis-
hand, this decreases the minimum distance between cycle slips tinguishable from an exponential distribution(Fig. 7).
AND ASCHEID CYCLE SLIPS IN PHASE-LOCKED LOOPS 2239

6.00 0. 50
I o f ~ r s torder loop p(x1lQ=-2TL1

Fig. 18. Normalized mean time between slips of a second-order PLL

as a function of p with f as parameter. Zero loop detuning. Solid
for first-order PLL.
line shows analytical results

6. 00, 0. 40

0. 30

0 . 21?1

0 . 10

0. 00
-2K -K 0.
(b)
Fig. 20. (a)Experimentallyderivedconditionalprobabilitydensity
function p ( x l I 6)for a second-order loop with S = 0.7 and p = 2.1.
Fig. 19. Normalizedmeantimebetween slips of a second-order PLL (b) Conditional probability density function p ( x 1 I@, T , > to) ob-
(t = 1.0; p =
0.03) with loop detuning A w normalized to the pull- tained if slips of duration T , shorter than to Q E(TJ are excluded
in frequency w p . (E(T,)ltO = 43.5).

The effect of loop detuning on E(T,) is shown in Fig. 19. cycle slips. Such an optimization is a formidable task that can
The frequency difference A o is normalized to the pull-in fre- be carried out only numerically on a digital computer or in the
quency u p . form of an experiment.
The importance of the state variable x1 has been discussed In a second-order loop there are essentially two loop para-
at length in this paper. meters, namely bandwidth BL and loop damping {, to be opti-
A key finding of this paper has been the behavior of x1 im- mized in a two-dimensional search. The loop parameters have
mediately following aslip;aconsistentdeparturefromthe to be clearly distinguished from the signal parameters {Ps,N o ,
correct value has been identified. The statistics of the condi- Af} which are fixed quantities. In a first step we want to op-
tional means E [ x l I$J = ?27~] and variance E [ ( x l - timize the loop bandwidth BL for a given damping {. For this
114 = +2n] can be found in Fig. 9. purpose we seek a suitable normalized representation of the
A typical distributionof x1 immediatelyfollowing a slip mean time between cycle slips E(T,) as a function of the band-
is displayed in Fig. 20 for { = 0.7. As a consequence of the width B L . It is natural to modify the familiar plot of normal-
occurrence of bursts a significant skewness is visible. If one ized mean time between slips E[BLTs]versus p as depicted in
excludes the very short time interval between slips from con- Fig. 19.
sideration, the skewness disappears and both sides of the dis- The signal-to-noise ratio is afunctionofthetwo signal
tribution assume a symmetrical shape. parameters Ps,N o , and the loop parameterBL :

C Optimum LoopParameters
ps 1
The usual approach in designing a PLL uses linear theory
to p =--’ (52)
No BL
determine the loop parameters for a given set of specifications.
For the next step, if necessary, the slip rate for the resulting
If we multiply both numerator and denominator by Af we
parameters can be obtained from Fig. 18 and checked against
obtain instead of (52)
a specified maximum permissible numberforthe particular
application.
For certain applications, it is of interest to optimize the = _ps
_ _.
Af
loop parameters to achieve maximum mean time
between NoAf BL
 2240 IEEE TRANSACTIONS
COMMUNICATIONS,
ON VOL. COM-30, NO. 10, OCTOBER 1 9 8 2

The three signal parameters can be grouped into a single quan-

tity b:

b = -No Af b = 0,)’

PS ,’ a
4-0 0 /

w h c h is related to the loopparameters y and p as follows:

with

Af
y = -: normalized loop stress.
BL Fig. 21. Analyticallyderivedmeantimebetweenslipsforfirst-order
loop (dashed curves) and experimental results for second-order loop
From (55) we learn thatforplotting E[BLT,] versus p (5 = 1.0; p = 0.03) with b = Af/(P$No) as parameter.
either the signal parameter b can be kept constant and y varied
or vice versa. The ,two possibilities lead to different sets of For any given E(T,), (60) represents a straight line in Fig.
curves: the case whereafixed relative offset Af/BL is main- 21 ;with increasing E(T,) the linemoves upwards.
tained is depicted in Fig. 19 (with a different normalization), Let us assume that sucha lineintersects a curvelog(EIBL T,] )
while in Fig. 21 we have plotted the curves for a constant b on as illustrated for b = 0.5, and let us label the point of inter-
a double-logarithmic scale. section P,, and the corresponding p by p i l . Solving (58) for
The dashed curves in Fig. 21 display E[BLT,] for afirst- BL yields
order loop where an analytical formula exists [ 2 ]:

which determines the bandwidth of the loop fora given E(T,)

The maximum in these curves is qualitatively easily under- and parameters p , and b . Jn our example the line also inter-
stood. A small p implies a large bandwidth BL : sects the same curve at the pointP2.This means that the same
mean timebetween slips is obtainedfortwodifferentloop
bandwidths.
If we now increase E(T,), then according to(60)the
straightline moves upwards until the two points PI and Pz
resulting in a large number of slips due to poor noise suppres- finally converge to one point defied as the tangent of the curve
sion. Forincreasing p , the bandwidth is decreased and the loop with the slope given by (60). Since for values of E(T,) above
slips less cycles. If, however, the bandwidth is decreased be- that point no intersection exists,we have found the maximum
yond the optimum, the static phase error caused by Af starts achievable E(T,) foranyloopbandwidth.Therefore,the
to interfere and the mean time between slips starts to increase optimumbandwidthfor a given signal parameter b can be
again. The smaller b is, the smaller the optimum bandwidth; found by constructing the tangent to the particular curve. In
in the limiting case for b = 0 we obtain a strictly increasing Fig. 22 we have plotted log (E[BLT,])as a function of p for
curve. The curves labeled with 0 and + are experimental thetwo damping factors = 0.7 and = 1.0. We observe a
results for a second-order loop with{ = 1.O and ?./ = 0.03. slow increase up to the maximum and a steep descent beyond
Because the normalized quantity log (E[BLT,]) is dis- the optimum. As expected, the loop with the larger damping
playedin Fig.21[andnot E(T,)] theoptimum BL is not performsbetter.For b = 0 thefunction log [E(BLT,)] in-
found at the location of the maximum of these curves. We will creases monotonically. In theory, any desired E(T,) can thus
show below how the optimization can be carried out bymeans be obtained with a sufficiently narrowloopbandwidth. We
of a graphic procedure. also see that the first-order loop always performs better than
Multiplying numerator and denominator ofp by E(T,) (1 1) the second-order loop for b = 0 (but compare the two loops
and taking the logarithm yields for b = 0.5). This is not surprising. For zero frequency detun-
ing, the integrator in the loop filter is superfluous, only caus-
ing additional slips by feigning a loop stress.
Our discussion on optimal loop parameters is brief and in-
complete. A more detailed discussion can be found in [ 7 ] .It
was found in this study that the loop parameters for maxi-
or rearranged mum E(T,) were distinctly different from those obtained via
linear Wiener filtering theory. In particular,the favorite
= 0.707 was found to be too small;a better choice is =
1.0-1.2.
ASCHEID AND MEYR: CYCLE SLIPS IN PHASE-LOCKED LOOPS 2241

6. 00, F. M. Gardner, PhaselockTechniques. New York:Wiley.1979.

G . Ascheid and H . Meyr. "Microprocessor-controlled experiment
to determinecycleslipstatistics:Hardware and software."ERT-
Rep.713/18,Sept.1981.
I + -, "Parameter optimization in PLLs: An experimental study,"
ERT,Rep. 713/19,Sept.1981.
W . C . Lindsey and H. Meyr. "Complete statistical description of
+ b-o5
the phase-error generated by correlative tracking systems." IEEE
I t Trans.Inform.Theory. vol.IT-23, pp. 794-802, Mar. 1977.
D.RyterandH.Meyr,"Theory of phasetrackingsystems of
arbitrary order: Statistics of cycle slips and probability distribution
of the state vector," IEEE Trans. Inform. Theory, vol. IT-24, pp.
1-7. Jan.1978.
F. J . Charlesand W . C . Lindsey."Someanalytical and experl-
mentalphaselocked loop results for low signal-to-nowratios."
Proc. IEEE, vol. 54, pp. I 152-1 166, Sept.1966.
R. W. Sannemann and J. R. Rowbotham. "Unlock characteristics
Fig. 22. Experimental mean time between cycle slips for second-order of the optimum type 11 phase-locked loop," IEEE Trans. Aerosp.
loop forf = 1.0 and f = 0.7 (p = 0.03) with b as parameter. Navig. Electron., vol. ANE- I I , pp. 15-24, Mar. 1964.
R. C. Tausworthe, "Cycle slipping in phase-locked loops.'' IEEE
Trans. Commun. Technol., vol. COM-15. pp. 417-421, June 1967.
V. SUMMARY AND CONCLUSIONS W.C.LindseyandR. C. Tausworthe, "A bibliography of the
theory and application of the phase-lock principle," Jet Propulsion
Cycle slips inphase-locked loops are statistical, nonlinear Lab., Pasadena, CA, Tech. Rep. 32-1581, Apr. I , 1973.
phenomena. The complicated interaction of nonlinearity and "Statisticalloopanalyzer(SLA)," LinCom Tech.Rep.,Mar.
noise hasbeendescribed.Experimental results were com- 1982.
plemented by simple analyses to obtain a quantitative under-
standing of the influence of the various signal/loop parameters
*
Gerd Ascheid was born in Cologne,Germany,
on cycle slip statistics. on June 14. 1951.Hereceived the Dipl.-Ing.
It hasbeen shownthat a state variable representation is degree from the Technical University of Aachen
needed to discuss cycle slips properly. Besides the phase error, (RWTH Aachen), Germany, in 1978.
He is now a Research Assistant at the Depart-
which is defined in a +2n interval according to the definition ment of Electrical Engineering, RWTH Aachen,
of a slip (and not modulo 2n), the state variable x1 plays a working toward the Ph.D. degree. His main inter-
central role. This fact, which has not received proper attention est is in synchronization,especially of band-
in the older literature, was first discussed in several theoretical width-efficient modulations.

papers; see [8],[9].

From a theoretical point of view it is interesting to observe
that the experimentally determined mean time between slips
*
for a first-order loop perfectly agrees with the results obtained HeinrichMeyr ("75) received the Dipl.-lng.
and Ph.D. degrees from the Swiss Federal Insti
via Fokker-Planck technique (which assumes a white noise tute of Technology (ETH), Zurich, Switzerland,
process). in 1967 and1973.respectively.
From 1968 to 1970 he held research positions
ACKNOWLEDGMENT at Brown BoveriCorporation,Zurich, and the
SwissFederalInstituteforReactorResearch.
We have greatly profitedfromthe expertise ofDr.F. Ftom 1970 to the summer of 1977, he was with
Gardner who spentfreely of his time to review the manuscript. HaslerResearchLaboratory,Bern.Switzerland,
doingresearch in thefields of digitalfacsimile
His detailed critique made the paper better than it would have encoding and tracking systems. Hislast position
been without his help. We would also like to acknowledge the at Hasler was manager of the Research Department. During 1974 he was a
constructive criticism made by Mrs. McKenzie, Dr. W. Braun, VibitingAssistantProfessor with the Department of ElectricalEngi-
neering.University of SouthernCalifornia. Los Angeles. Since the
and Dr. C. Chie of LinCom and L. Popken of RWTH Aachen. summer of 1977 he has been a Professor of Electrical Engineering at the
The
support of the Deutsche
Forschungsgemeinschaft AachenTechnicalUniversity(RWTHAachen).Germany. His research
(DFG) is greatly appreciated. interestsincludesynchronization,estimation.and, in particular.the
interactionofimplementationissues.and the design of controland
estimation/measurementsystems.Presentlyhe is consultant to the 1BM
REFERENCES Research Laboratory, Zurich, i n the area of synchronization of local area
[I] W.C. LindseyandM. K . Simon, Telecommunication Systems computer networks andto Krohne Ltd., Duisburg, Germany.in the area of
Engineering. EnglewoodCliffs, NJ: Prentice-Hall.1973. digitalsignalprocessing. He has published work in variousfields and
[2] W.C. Lindsey, Synchronization Systems inCommunicationand journals and holds over a dozen patents.
Control. EnglewoodCliffs, NJ: Prentice-Hall,1972. Dr. Meyr served as a Vice Chairman for the 1978 IEEE Zurich Semmar
131 A.J.
Viterbi, Principles of Coherent
Communications. New and as an International Chairman for the 1980 National Telecommunica-
York:McGraw-Hill, 1966. tions Conference, Houston. TX. He presently serves as Associate Editor
[4]A.Blanchard, Phase-LockedLoops.ApplicationtoCoherentRe- fortheIEEE TRANSACTIONS ON ACOUSTICS.SPEECH, AND SIGNAL
ceiverDesign. NewYork:Wiley,1978. PROCESSING.