The performance of a carrier phase locked loop (PLL) driven by a periodic Doppler input is studied. By expanding the Doppler input into a Fourier series and applying the linearized PLL approximations, it is easy to show that, for periodic frequency disturbances, the resulting steady state phase error is also periodic. Compared to the method of expanding frequency excursion into a power series, the Fourier expansion method can be used to predict the maximum phase error excursion for a periodic Doppler input. For systems with a large Doppler rate fluctuation, such as an optical transponder aboard an Earth orbiting spacecraft, the method can be applied to test whether a lower order tracking loop can provide satisfactory tracking and thereby save the effect of a higher order loop design

Chen, C.-C.

Win, M. Z.

English

NASA Technical Reports Server

/TDA Progress Report 42-104 February 15, 1991
N91-18314
J
Ii _.
L
Steady-State Phase Error for a Phase-Locked Loop
Subjected to Periodic Doppler Inputs
C.-C. Chen and M. Z. Win
Communications Systems Research Section
The performance of a carrier phase-locked loop (PLL) driven by a periodic
Doppler input is investigated. By expanding the Doppler input into a Fourier
series and applying the linearized PLL approximations, it is easy to show that, for
periodic frequency disturbances, the resulting steady-state phase error is also pe-
riodic. Compared to the method of expanding frequency excursion into a power
series, the Fourier expansion method can be used to predict tile maximun_ phase-
error excursion for a periodic Doppler input. For systems with a large Doppler-rate
fluctuation, such as an optical transponder aboard an Earth-orbiting spacecraft,
the method can be applied to test whether a lower order tracking loop can provide
satisfactory tracking and thereby save the effort of a higher order loop design.
I. Introduction
Coherent carrier phase recovery using a phase-locked
loop (PLL) has become an integral part of digital comnm-
nication systems [1-3]. By performing coherent demodu-
lation using the recovered signal carrier, the receiver can
achieve 3 dB of performance gain over systems using non-
coherent demodulation techniques. The ability to recover
and track the incoming carrier phase can also lead to a
significant performance gain in related applications such
as coherent ranging and spacecraft, navigation [4].
The performance of a phase-locked receiver depends
critically on the ability to accurately recover the carrier
phase. Synchronization errors between the incoming sig-
nal and the local reference can quickly lead to a degraded
signal-to-noise ratio (SNR) and a large power penalty. The
design of the loop, therefore, must ensure proper phase
tracking under the operating conditions. In general, the
perforlnance of the PLL is influenced by the additive cir-
cuit noise, the oscillator frequency noise, and the frequency
characteristics of the signal it is designed to track. The ef-
fect of channel noises on the performance of the PLL has
been studied extensively [1-4]. It is shown that, in gen-
eral, the residual phase tracking error due to the additive
noise increases with PLL bandwidth, whereas the tracking
error due to the oscillator frequency noise decreases with
increasing loop bandwidth. With a given SNR, therefore,
68
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there is an optimal choice of receiver bandwidth and PLL
design that minimizes tile root-mean-square (rms) phase
tracking error.
In addition to the channel and oscillator noises, the
performance of the PLL is also affected by the frequency
characteristics of the source. A simple first-order loop can
provide adequate tracking only when the free-running fre-
quency of the voltage-controlled oscillator (VCO) is equal
to the frequency of the incoming signal. A second-order
loop is needed to track a constant frequency offset, whereas
a third-order loop is needed to track a signal with lin-
early varying frequency. In general, higher order loops are
needed to compensate for higher order frequency distur-
bances. Ilowever, higher order loops present extra design
complications since control loops higher than second order
are not unconditionally stable. Furthermore, for systems
with large dynamic frequency fluctuations, the frequency
perturbations with orders higher than tile control loop can-
not be completely compensated by tile loop. As a result,
some residual phase tracking error always remains.
Since the performance of the PLL-based receiver de-
pends on the ability to accurately track the carrier phase,
it is important to estimate the amount of residual track-
ing error due to tile higher order frequency perturbations.
The analysis of PLL performance given a time-dependent
Doppler input is generally accomplished by expanding the
Doppler signal into a power series and then analyzing the
effects of different power terms separately. The power-
series expansion provides a simple and intuitive method of
expanding the Doppler frequency term. Ilowever, it is dif-
ficult to justify dropping the higher order terms since the
steady-state phase error due to higher order terms does
not converge.
For a very special class of system, the Doppler signal
is periodic. In this case, the phase-locked loop equation
can be examined by performing a Fourier decomposition
rather than a power-series expansion. In this article, the
procedure of analyzing the residual phase tracking error
using a Fourier expansion of the frequency perturbation is
outlined.
II. Carrier Phase Tracking Loop
The essentials of a carrier PLL include a phase detec-
tor, a loop filter, and a VCO. Shown in Fig. 1 is a typical
implementation of a radio-frequency (RF) PLL. The l)hase
detector detects the phase difference between the incom-
ing signal and the output of the VCO. For carrier phase
tracking applications, an RF mixer is generally used as
the phase detector. The mixer output is filtered by the
loop filter with transfer function F(s). The sum frequency
term at the mixer output is filtered by the loop filter such
that the loop effectively responds only to the difference fre-
quency term. When the frequency of the signal is equal to
that of the VCO, the difference frequency term is simply
proportional to sin ¢(t), where ¢(t) is the phase difference
between the signal and VCO output.
The filtered phase-difference signal is subsequently in-
jected into the receiver VCO. The output frequency of the
VCO is linearly dependent on the input voltage signal.
When the loop eventually reaches a locked condition, the
phase-error signal, ¢(t), will be such that it is governed by
the following loop equation:
de(t)
dt-- + AKf(t) ® [n(t) + sin ¢(t)] = ¢3(t) + fN(t) (1)
where n(t) is the additive noise, fg(t) is the oscillator fre-
quency noise, and/3(t) is tile frequency error between the
signal and the local oscillator. Tile loop mechanization
is represented by the signal amplitude, A; the VCO gain
constant, K; and the impulse response of the loop filter,
f(t). For systems operating with an ample signal-to-noise
ratio, the effect of additive noise is usually very small. At
the same time, a relatively high SNR allows oscillator fre-
quency noise to be tracked out. For this analysis, therefore,
the focus is on the frequency detuning term.
If the phase error is small (the loop is in lock), the sine
function can be approximated by its argument, and tile
phase error of the PLL can be adequately described using
the following linearized form:
de(i)
d---i--+ AK f(t) O ¢(t) = fl(t) (2)
Tile integral-differential equation in Eq. (2) can be simpli-
fied into a linear differential equation of the form
d'_¢(t) de(t)
dr-----7--+ ... + aa ---d-l- + a0¢(t)
dn-a_(t) b d_(t)
b,_-i dt,__-----_ +...+ _+bo_(t) (3)
where the coefficients {aj } and {bj } are related to the loop
transfer function H(s) by
69
bn_l sn-1 + ... + bls + bo 1
s '_+...+als+ao s + AKF(s)
= 111 - H(s)] (4)
Equivalently, it can also be simplified into a set of first-
order differential equations [5]. Given the frequency de-
tuning process,/3(t), Eq. (3) can be solved for the steady-
state phase error. For a time-varying frequency detuning
process, one method of simplifying the analysis is to ex-
pand f3(t) into a Taylor series and then retain only terms
sufficient for the analysis. Since the system is linear, the
solution obeys ttle superposition principle and is equal to
the suln of solutions of individual expansion terms.
In general, the solution to Eq. (3) includes the transient
response and the steady-state response terms. The tran-
sient response, which depends on the initial conditions,
dies out after a time period that is on the order of the
inverse loop bandwidth. If the phase error converges to
a constant, the steady-state solution can be easily solved
by Laplace transforming Eq. (3) and using the final value
theorem, i.e.,
,B(s)
lira ¢(t) = lira s.*(s) = lira (5)
t-o_ ,-o ,-o s + AKF(s)
where F(s) and B(s) are the Laplace transforms of I(t)
and fl(t), respectively. Equation (5) is applicable only
when the steady-state solution exists as a constant value.
For higher order perturbations, the final value in Eq. (4)
does not converge, and the Laplace transform cannot be
used to solve for the steady-state response. Fortunately,
it is known from the linear differential equation theory
that the genera[ solution to Eq. (3) for a driving force of
the form fl(t) = ant '_ is a polynomial of order n. The
steady-state phase error can therefore be solved by substi-
tuting the polynomial of order n into the right-hand side
of Eq. (3) and then matching tile coefficients.
Although the steady-state phase error can be solved
by assuming a polynomial general solution, the resulting
polynomial is diverging at t --. _. Since most physical
systems do not have unbounded frequency variation, the
higher order perturbation eventually (lies down. The anal-
ysis of the PLL performance using higher order perturba-
tion is therefore limited to the time period within which
the perturbation is present. The loop design is said to be
adequate if the effects of the higher order perturbation are
small. Because of the coml)lexily of designing higher order
tracking loops, the analysis of the time-varying Doppler
term is generally limited to third order or less. Justifica-
tions for dropping the higher order terms, however, can be
very difficult since the solution is not bounded.
III. Fourier Expansion of the Time-
Dependent Doppler Signal
Since the frequency fluctuation at the input is generally
bounded, polynomial approximation to the Doppler signal
will eventually become greater than the input. If the time
period of interest is longer than the time for the polynomial
approximation to deviate from the signal, a better (higher
order) approximation is needed to analyze the PLL behav-
ior. For some class of missions such as Earth-orbiting satel-
lites, however, the periodic orbit will result in a periodic
Doppler input that should intuitively result in a periodic
phase variation. For such a system, the steady-state solu-
tion can be more easily derived by expanding the Doppler
signal into a Fourier series. The resulting linearized PLL
equation can be written as
d¢(t___)+AKf(t)®¢(t) = /3(t) = _ c_e/k''°' (6)
dt
where w0 is the fundamental frequency (reciprocal of the
period) of the perturbation. From linear differential equa-
tion theory, it is known that a linear differential equation
responding to a sinusoidai driving term with frequency w0
will exhibit a general solution with an identical frequency.
Again, the particular solution (transient response) is ex-
pected to die down with a time constant that corresponds
to the eigenvalues of the characteristic equation. Further-
more, by using the superposition principle, solutions to dif-
ferent harmonics can be solved individually. Consequently,
if the Doppler stimulus can be expanded into a Fourier
series, the solution can be found using the superposition
principle.
Two examples can now illustrate the power of this tech-
nique.
Example 1: First-Order Loop. It is well known
from linearized PLL theory that the first-order loop can
be used to track a constant frequency detuning (Doppler)
with a constant phase offset, lligher order perturbation
can result in a loss of lock. If such a loop is used to track a
periodic frequency variation of frequency w0, conventional
analysis cannot adequately predict the resulting loop per-
formance, llowever, by performing the spectral expansion
of the Doppler signal
7O
3(t) = cosin_ot (7)
the general solution to the first-order loop can be written
as
aoc 0 ¢,doC 0
¢(t) - a°2_w02sinw0t+ a°__aa02cosw0t (8 /
where a0 = AK = 4BL is related to tile bandwidth of the
loop. It is seen from Eq. (7) that a first-order loop can be
used to track a periodic Doppler input, provided that the
loop bandwidth, frequency variation, and Doppler period
satisfy the condition for linearizing the loop equation, i.e.,
¢(t) << 1 for all t. Furthermore, the resulting steady-state
phase error is periodic with the same period as the driving
term, but falls slightly out of phase from the driving input.
Example 2: Perfect Second-Order Loop. It is
known from the linearized loop theory that a perfect
second-order loop can be used to track out a constant
Doppler rate with a steady-state phase error. Tile transfer
function for the loop filter is
l +r2sF(,) - (9)
7" 1S
By substituting Eq. (9) into tile PLL equation and con-
verting the resulting equation back to tile time domain,
d2¢(t) de(t) d/3(t) (10)dt-------T- + aa T + a0¢(/) - at
where al = AKr2/7"1, and a0 = AK/rl. The general
solution to this equation can be reached by substituting
the solution of the form
¢(t) = Psinw0t+Qcosw0t (11)
into Eq. (10) and equating tile coefficients. It is found
that the general (steady-state) solution due to tile periodic
Doppler input is given by
= c0 0'a (ao-- wo)2 + a caosinWot
c0 0(a0 -
+ - ----_,,_----¢- ,, coswo/ (12)
(a0 - _'5)_ + ai_
Again, the constants must satisfy tile constraints that
¢(t) << 1. Note that the loop bandwidth can be related to
the constants a0 and al by
ao al
BL = --+-- (13)
4al 4
The difference between the Fourier solution and the
power-series solution can be seen in Fig. 2 where the
steady-state phase response of the linear loop to a sinu-
soidal frequency excursion has been plotted. The power-
series solution was calculated by expanding the sinusoid
into a power series and retaining the first, two terms
(Doppler rate and second derivative of Doppler). It is
seen from the figure tl,at the power-series solution is a
close approximation to the actual solution during the ini-
tial 1/4 period. IIowever, as soon as the approximation
to the sinusoid breaks down, the power-series estimate di-
verges, whereas the actual solution remains bounded.
IV. Discussion
Expanding the Doppler signal into a Fourier series offers
a different perspective in predicting the PLL performance.
Unlike the power-series expansion method that, although
intuitive, cannot adequately predict the performance un-
der higher order perturbations, the periodic expansion of
the Doppler results naturally in a periodic phase solution.
As a result, the theory can predict a bounded solution even
when the driving force (Doppler) ha.s a higher order con>
ponent. For applications where a good phase synchroniza-
tion is essential, such as coherent data conmmnications,
the Fourier solution can provide an adequate estimate of
the maximum phase-error excursion.
An example for the problem occurs in tile design of the
optical phase tracking loop between a low Earth-orbiting
satellite and a ground station. At. the operating wave-
length of 1 pro, the relative Doppler rate of the two ter-
minals can change from +300 MIIz/sec to -300 Mltz/sec
within 30 seconds. Given a PLL of 20 kHz bandwidth,
it is difficult to predict whether tile loop can remain ade-
quately in lock during the period. By approximating the
Doppler signal near the l)ortion of the orbit with maxi-
nmm frequency change as a sinusoid, however, a simple
estimate of the PLL performance can be obtained. For a
frequency excursion with 3-Gtlz amplitude and 60-second
period, it can be shown that the loop can adequately track
the Doppler with less than 0.2 radian of residual error.
Although tile Fourier expansion provides a bounded so-
lntion for a periodic varying Doppler signal, there are some
71
practical limitations in applying the Fourier analysis tech-
nique. First, the solution presented above ignores tile tran-
sient solution. This is true only when the period of the dy-
namic signal is long compared to the loop-response time.
Furthermore, the results are derived only for a linearized
equation. For a nonlinear PLL equation, a periodic driv-
ing force can excite higher harmonic terms. Finally, for
a predictable periodic driving force, it is a usual practice
to apply a periodic estimator correction term at the VCO
input to compensate for the periodic driving force. In this
case, the detuning is small and the loop will essentially
respond only to the noise inputs.
References
[1] W. C. Lindsey and M. K. Simon, Telecommunications Systems Engineering, En-
glewood Cliffs, New Jersey: Prentice-ttall, 1973.
[2] A. J. Viterbi, Principles of Coherent Communications, New York: McGraw-Itill,
1966.
[3] J. J. Spilker, Digital Communication by Satellite, Englewood Cliffs, New Jersey:
Prentice-IIall, 1977.
[4] W. C. Lindsey, Synchronization Systems in Communication and ControIj
Englewood Cliffs, New Jersey: Prentice-Itall, 1972.
[5] C. T. Chen, Introduction to Linear System Theory, New York: IIolt, Rinehart
and Winston, 1970.
72
n(tl
fN + _(t_
¢LO
Fig. 1. Equivalent-noise block diagram of a carrier phase-locked
loop.
1.0
0.5
=o
Lu
=< 0
_3
rr
-0.5
-1.0
I I
FOURIER SOLUTION
--_-- POWER SERIES SOLUTION (TWO TERMS)
%.
\
LOOP BANDWIDTH = 50 Hz \
MAXIMUM FREQUENCY
EXCURSION = 5 kHz
\
PERIOD OF EXCURSION = 300 sec \
I Z \
0 1oo 200 300
TIME, sec
Fig. 2. Residual phase tracking error for a second-order phase-
locked loop subjected to a periodic Doppler Input. The PLL was
analyzed by using the Fourier decomposition method and a power-
series solution.
73


Steady-state phase error for a phase-locked loop subjected to periodic Doppler inputs

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