Phase acquisition times of type II and III loops typical of the Advanced Receiver are studied by computer simulations when the loops are disturbed by gaussian noise. Reliable estimates are obtained by running 5000 trials for each combination of loop signal-to-noise ratio (SNR) and frequency offset. The probabilities of acquisition are shown versus time from start of acquisition for various loop SNRs and frequency offsets. For frequency offsets smaller than one-fourth of the loop bandwidth and for loop SNRs of 10 dB and higher, the loops acquire with probability 0.99 within 2.5 B sub L for type II loops and within 7/B sub L for type III loops

Aguirre, S.

English

NASA Technical Reports Server

TDA Progress Report 42-84 October-December 1985
N 8 6 - 2 2 7 9 3
Acquisition Times of Carrier Tracking Sampled Data
Phase-Locked Loops
S. Aguirre
Communications Systems Research Section
Phase acquisition times of type II and III loops typical of the Advanced Receiver are
studied by computer simulations when the loops are disturbed by gaussian noise. Reliable
estimates are obtained by running 5000 trials for each combination of hop signal-to-
noise ratio (SNR) and frequency offset. The probabilities of acquisition are shown
versus time from start of acquisition for various loop SNRs and frequency offsets. For
frequency offsets smaller than one-fourth of the loop bandwidth and for loop SNRs of
10 dB and higher, the loops acquire with probability 0.99 within 2.5/BL for type II loops
and within 7/BL for type HI loops.
I. Introduction
During the acquisition mode, the nature of the phase-locked
loop is highly nonlinear. This is further complicated by the
presence of noise, thus precluding a detailed analysis of the
behavior of the loop during this mode of operation.
Very little analytical work has been published in the open
literature on the subject of acquisition. Viterbi (Ref. 1) and
Lindsey (Ref. 2), among others, developed approximate expres-
sions for the time necessary to acquire in a noiseless environ-
ment. Their expressions are useful when the frequency offset
is large compared to the bandwidth of the loop. Hurd and
Anderson (Ref 3) and Holmes (Ref. 4) present results of
acquisition experiments in a noisy environment for a digital
transition tracking loop (DTTL) symbol synchronizer and
continuous time carrier tracking loops, respectively. Although
all previous sources serve as a general guidance, more specific
results are needed for the proposed carrier tracking loops of
the DSN Advanced Receiver (Ref. 5).
In this article, we show results based on Monte Carlo simu-
lations of phase acquisition for types II and III sampled data
loops typical of the Advanced Receiver, where the type of
loop indicates the number of perfect integrators present,
including that contributed by the NCO. Plots of probability
of acquisition versus normalized time are presented with
initial frquency detuning (frequency offset) as a parameter,
for different loop signal-to-noise ratios.
II. Description of the Sampled Data
Loop Model
A block diagram of the nonlinear baseband model employed
in the simulation is shown in Fig. 1. The symbols contained in
the diagram are defined as follows:
A = rms value of carrier signal (V)
88
https://ntrs.nasa.gov/search.jsp?R=19860013322 2020-03-20T15:04:30+00:00Z
= input phase at time tn (rad)
<t>0 = initial phase offset (rad)
J2Q = initial frequency detuning (rad/s)
<t>n = phase estimation error at time tn (rad)
AK = loop gain
W = zero mean white gaussian noise sample with variance
NJ2T
T = update time (s)
F(z) = G, +
1-z-1 (1-z-1)2
= loop filer
N(z) = r(z + l ) /2z 2 (z- l )
= NCO transfer function including the
transport lag of the loop
0)
(2)
The loop modeled in Fig. 1 is the proposed implementation
for the carrier tracking loop of the Advanced Receiver (Ref. 5).
Operationally, the behavior of the loop is characterized by
0 = 0 - 0Yn n n
= 6n -AKF(z~) {sm(0n)+
(3)
(4)
The PLLs considered are sampled data analogies of continuous
time loops. For the type II and III loops with update time T,
sampled data PLLs analogous to continuous loops of one-
sided bandwidth b/T Hz have parameters (see Appendix for
details):
AKTGl = rd
AKTG2 = rd2
AKTG3 = krd3
C? =
 7^U"3FTT
(5a)
(5b)
(5c)
(6)
where //(z) is the closed loop transfer function. These band-
widths are approximately b IT for b « 1, and are somewhat
wider for larger b. The analogy is used so that the typical
continuous loop parameters (Ref. 6 and TR 900-4501 ) can
be specified as inputs to perform the simulations (note that BL
and b here correspond to BL* and BLTof Ref. 5).
III. Description of Simulation
Introducing the state variables un, vn, defined as the out-
puts of the first and second integrators in the loop filter F(z),
it can be shown (see Appendix) that the difference equations
describing the behavior of the PLL are given by:
= V n- l
+ k d u n
(11)
(12)
The simulations were conducted by solving Eq. (4) with help
from Eqs. (8)-(12) with zero initial conditions for the state
variables.
In order to obtain accurate estimates, 5000 independent
runs, produced by 5000 nonoverlapping sequences of pseudo-
random noise, were employed to generate a single probability
of acquisition curve for a given frequency offset and loop
signal -to-noise ratio SNR = A2INQBL. The phase offset for
each run was randomly selected between (-TT, w).
To assure phase locking, each simulation was run for a
maximum of 5Q/BL . The loop was declared to be in lock when
the magnitude of the phase error dropped below 90 deg for
at least 10/BL. This threshold is arbitrarily chosen so that
PLLs operating with small loop SNRs (large variance of the
tracking phase error) may be studied.
The simulations were performed on a VAX 11/750 com-
puter. To generate the noise samples, a gaussian noise genera-
tor was synthesized using the direct method (Ref . 7) based on
where r is the damping parameter, and fc is a type III loop
gain component (k = 0 for type II loop). The actual loop
noise bandwidths, BL , of the sampled data loops are given by:
'Tausworthe, R. C., and Crow, R B , Practical Design of Third Order
Phase-Locked Loops, Technical Report 900-450, Jet Propulsion
Laboratory, Pasadena, Calif, Apr 27, 1971.
89
the uniform random number generator provided by the VAX
11/750. The simulations were run for BLT = 0.02, which is
representative of the numbers presently imposed by both
hardware speed (which influences 7) and bandwidth require-
ments to track input dynamics.
IV. Summary of Results
Figures 2-5 present the distribution functions resulting
from simulations of phase acquisitions. The ordinates of
these figures are the probability of phase acquisitions occur-
ring within time t. Time t, as indicated on the abscissa, is
normalized by the bandwidth of the loop. For example,
consider a type II loop with bandwidth BL = 50 Hz, a loop
SNR of 10 dB, and a frequency offset of 25 Hz, or BL/2.
From Fig. 3(b), phase acquisition with probability of 0.95 is
achieved at BLt = 3 or t = 3/BL = 0.06 s. The figures show
that for the normal range of loop SNRs, greater than 10 dB,
and for frequency offset values smaller than one-half the loop
bandwidth, the loops attain phase lock with high probability
(0.99) within 5/BL or 15/BL, for type II and type III loops,
respectively. Acquisition times are approximately reduced by
one-half if the offset is reduced to BL/4.
Larger frequency offsets may require an excessive time to
lock. Aiding of acquisition by sweeping the local oscillator or
by estimating the frequency offset by a fast Fourier trans-
form (FFT) is indicated when the frequency offset is large.
Since type II loops lock faster than type III loops, it is often
useful to acquire with a type II loop and then change to a
type III. This can be done without loss of phase lock.
Our results are in close agreement with the earlier results
for symbol synchronizers and for continuous time PLLs.
This is not surprising for the continuous time PLL since for
small values of BLT the sampled data loop behaves very
similarly.
If we use equations (3.21) and (10-18) of Refs. 1 and 2,
respectively, we can get an upper bound for the necessary time
to phase lock a noiseless type II continuous time PLL. These
equations show that when the frequency offset equals the loop
bandwidth, and r = 2, the loop locks in 1.9/BL. The simula-
tions indicate 9.2/BL with probability 0.99 when the loop
SNR = 16 dB.
Although the DTTL symbol synchronizer does not possess
a sinusoidal characteristic for its phase detector, in Ref. 3 it
was found that for low symbol SNR, acquisition is obtained
in about 1\BL for an optimized midphase integration window
of one-quarter of the symbol time and a frequency offset of
BL/2. Large symbol SNRs require roughly 0.6/fiL for the same
frequency offset. Interpolation of the data of Ref. 4 indicates
that a continuous loop locks in 3.8/BL when SNR = 10 dB
and the frequency detuning is BL\1. The close agreement in
acquisition times adds confidence in the results.
References
1. Viterbi, A. J., Principles of Coherent Communication, McGraw-Hill, New York, 1966.
2 Lmdsey, W. C., Synchronization Systems in Communication and Control, Prentice-
Hall, Englewood Cliffs, N J., 1972.
3. Hurd, W. J., and Anderson, T. 0., "Acquisition Time for Symbol Synchronizer for
Low SNR Coded Systems," Space Programs Summary 37-61, Vol II, pp. 57-63, Jet
Propulsion Laboratory, Pasadena, Calif, Jan. 31,1970.
4. Holmes, J. K , Coherent Spread Spectrum Systems, John Wiley and Sons, Inc., New
York, 1982.
5. Aguirre, S., and Hurd, W. J., "Design and Performance of Sampled Data Loops for
Subcamer and Carrier Tracking," IDA Progress Report 42-79, Jet Propulsion Labora-
tory, Pasadena, Calif., pp. 81-94, Nov. 15, 1984.
6. Tausworthe, R. C., Theory and Practical Design of Phase-Locked Receivers, Technical
Report 32-819, Jet Propulsion Laboratory, Pasadena, Calif., Feb. 1966.
7. Abramowitz, A., and Stegun, I., Handbook of Mathematical Functions, Section 26.8,
National Bureau of Standards, Washington, D.C., 1965.
90
Wn/A
< 1 00
< 095
D
O
090
085
080
- (b)
Fig. 1. Nonlinear baseband sampled data loop model
5 10
NORMALIZED ACQUISITION TIME, BLt
15
080
5 10
NORMALIZED ACQUISITION TIME, BLt
15
Fig. 2. Cumulative probability of phase acquisition as a function of Fig. 3. Cumulative probability of phase acquisition as a function of
normalized time for type II sampled data loop at 7-dB loop SNR: normalized time for type II sampled data loop at 10-dB loop SNR:
(a) Probability range 0 to 1, (b) Expansion of high probability region (a) Probability range 0 to 1, (b) Expansion of high probability region
91
080
5 10
NORMALIZED ACQUISITION TIME, BLt
15
Fig. 4. Cumulative probability of phase acquisition as a function of
normalized time for type II sampled data loop at 16-dB loop SNR:
(a) Probability range 0 to 1, (b) Expansion of high probability region
080
5 10
NORMALIZED ACQUISITION TIME, BLt
15
Fig. 5. Cumulative probability of phase acquisition as a function of
normalized time for type III sampled data loop at 10-dB loop SNR:
(a) Probability range 0 to 1, (b) Expansion of high probability region
92
Appendix
Setup of Equations for Simulation
From Ref. 4 and TR 900-450, a type II loop filter of a
continuous time PLL can be expressed as
1 +T.S 1 (A-l)
which is related to the commonly used parameters
r = AKT\lr l
= T2/T3
— = -— ( r—I one-sided loop bandwidth
y 4 T_ \ T~~ K I
(A-3)
(A4)
G2 = TVr, (A-6)
(A-7)
Substitution of Eqs. (A-2)-(A-7) into the transfer function of
the sampled data loop filter F(z) produces Eq. (5), Notice that
the term AKT was absorbed by the filter coefficients. With
(A.2) this in mind, define
Y'(z')
-
krd
(A-8)
Define
Gl =
where Y'(z) = TY(z) (see Fig. 1), from which Eqs. (8)-(12)
follow easily. These, in conjunction with Eq. (4), are the
(A-5) required equations for simulation.
93


Acquisition times of carrier tracking sampled data phase-locked loops

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