Aquisition time parameter for first order phase lock loop with sine wave input in additive Gaussian nois

Schuchman, L.

English

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ELLCQMM, 1 N C .  
NORM, S.W. W A ~ ~ G T ~ ,  D.C 2 
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COWER S N DU 
TITLE- Acqu i s i t i on  T ime  i n  a F i r s t  
Order Phase Lock Loop 
TM- 70-2034-6 
DATE- J u l y  1 0 ,  1 9 7 0  
F l L l H G  SUBJECT(S)- Communication Theory Phase 
( A S S I G H E D  BY AUTHOR( s)- Lock Loops Nonlinear Tracking 
Devices Dif fus ion  Processes  
ABSTRACT 
I n  t h i s  paper  t h e  moments of  t h e  a c q u i s i t i o n  t i m e  
parameter  TA are de r ived  f o r  a f i r s t  order phase lock loop 
when the  i n p u t  t o  t h e  loop i s  a s i n e  wave i n  a d d i t i v e  Gaus- 
s i a n  no i se .  I t  i s  shown t h a t  except  f o r  d i f f e r e n c e s  i n  boun- 
dary va lues  t h e  moments of t h e  a c q u i s i t i o n  t i m e  and t i m e  t o  
cyc le  s l i p  (T)  v a r i a b l e s  s a t i s f y  i d e n t i c a l  r e c u r s i v e  second 
o r d e r  d i f f e r e n t i a l  equat ions .  
h 
It i s  demonstrated t h a t  as t h e  s i g n a l  t o  n o i s e  
r a t i o  ( p )  i n  t h e  loop  i n c r e a s e s ,  bo th  t h e  mean a c q u i s i t i o n  
t i m e  E[TA] and mean t i m e  t o  c y c l e  s l i p  E [ T ]  i n c r e a s e .  
n 
Thus, as the s i g n a l  t o  n o i s e  r a t i o  i n c r e a s e s  it 
takes longe r  f o r  t h e  phase error t o  reach a s tab le  p o i n t .  
But it i s  a l so  demonstrated t h a t  once such a p o i n t  i s  reached,  
it takes a much longe r  t i m e  f o r  t h e  phase error t o  change by 
a 2.rr increment.  I n  a d d i t i o n ,  it i s  shown t h a t  t h e  r a t i o  
E [T J /E [TA] i n c r e a s e s  as  p i n c r e a s e s .  
A 
The r e c u r s i v e  second o r d e r  d i f f e r e n t i a l  equa t ion ,  
which i s  s a t i s f i e d  by t h e  a c q u i s i t i o n  t i m e  moments of  a 
second o r d e r  loop,  is  g iven ,  Unfor tuna te ly ,  computer memory 
requirements  f o r  t h e  s o l u t i o n  of  t h i s  equa t ion  i s  so g r e a t  
t h a t  no computed r e s u l t s  w e r e  obtained.  
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SUBJECT: Acqu i s i t i on  T i m e  i n  a F i r s t  O r d e r  DATE: J u l v  10, 1 9 7 0  
Phase  Lock Loop 
i C a s e  9 0 0  FROM: L.  Schuchrnan 
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r. /. 
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TM - 70-2034-6 
TECHNICAL MEMORANDUM 
INTRODUCTION 
I n  previous memorandum I s t u d i e d  t h e  problem of 
cyc le  s l i p p i n g  i n  a phase lock loop. 1'2 
t h e  Nth moment of t h e  t ime-to-cycle s l i p  parameter ( T )  
s a t i s f i e d  a r e c u r s i v e  d i f f e r e n t i a l  equat ion  f o r  t h e  n t h  
o r d e r  phase lock loop. The phenomena of  c y c l e  s l i p p i n g  i s  
c l o s e l y  r e l a t e d  t o  t h e  problem of a c q u i s i t i o n .  I t  i s  shown 
i n  t h i s  memorandum t h a t  t h e  Nth moment of t h e  a c q u i s i t i o n  t i m e  
parameter  (T ) f o r  a f i r s t  o rde r  loop s a t i s f i e s  the  s a m e  
r e c u r s i v e  d i f f e r e n t i a l  equat ion  t h a t  T s a t i s f i e s .  The d i f -  
f e r ence  i n  determining t h e  moments of Ta from those  of 
T l i es  i n  t h e  boundary va lues  t h a t  t h e  moments of t h e  two 
parameters  must s a t i s f y  . 
I t  was shown t h a t  
n 
a A 
n 
The phase lock loop i s  schemat ica l ly  dep ic t ed  
i n  F igure  1. 
- .  ,, 
+ n(t)' 
FIGURE 1 - PHASE LOCK LOOP 
BELLCOMM, INC. - 2 -  
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The loop consists of three components; a multiplier, 
a time-invariant linear filter g(t) whose Laplace transform is 
G ( s ) ,  and a voltage controlled oscillator (VCO). If the loop 
is operating on a noiseless sine wave, then the steady state 
output of the VCO is a quadrature replica of the input signal 
which may differ only in amplitude. The multiplier o u t p u t  x(t), 
under these conditions, will have only a double frequency term 
which the linear filter and VCO configuration will not pass. 
The linear filter has a second and equally important function 
which is to reduce the effects of noise that normally gets into 
the loop. 
FIRST ORDER PHASE LOCK LOOP ( G ( s )  = 1) 
Viterbi3 has shown that the behavior of a first order 
phase lock loop in a noiseless environment can be described 
by the following differential equation 
Equation (1) describes how the frequency error changes 
with phase error. This can be depicted graphically as in Figure 2.  
FIGURE 2 - SYSTEM TRAJECTORY FOR FIRST-ORDER LOOP 
(SAME AS FIG. 3.2 OF REFERENCE 3) 
B E L L C O M M ,  I N C .  - 3 -  
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L ..i 
The phase e r r o r  w i l l  move u n t i l  t h e  frequency error 
i s  reduced t o  zero .  If t h e  frequency e r r o r  i s  p o s i t i v e ,  t h e  
phase e r ror  w i l l  i n c r e a s e  wi th  t i m e .  O n  t h e  o t h e r  hand, i f  
t h e  frequency error i s  nega t ive ,  t h e  phase e r r o r  w i l l  d ec rease  
wi th  t i m e .  Using t h i s  f a c t ,  t h e  p o s s i b l e  t ra jec tor ies  of t h e  
phase e r ro r  are dep ic t ed  i n  f i g u r e  2 .  Thus, t h e  va lues  of  4 
f o r  which '' - 0 ,  a r e  given by 
w - w  
( 2  1 -1 0 AK n = o , t 1 , + _ 2 ,  . - . = 2nn + s i n  'n 
w - w  -1 0 
A K  = (2n- l )n  - s i n  'L 
where w e  no te  t h a t  t h e  <n p o i n t s  are u n s t a b l e  and w i l l  l e a d  t o  
f u r t h e r  migra t ion  of t h e  phase e r r o r  t o  t h e  n e a r e s t  s t a b l e  $n 
p o i n t .  
I t  i s  t o  be noted t h a t  equat ion  ( 2 )  i s  v a l i d  only  
when w-wodAK. 
lock. The t i m e  f o r  t h e  phase error t o  move from an i n i t i a l  
s e t t i n g  ' ( 0 )  t o  a p o i n t  4' can be determined from equat ion  (1). 
Thus , 
Thus, i f  w - w  >AK t h e  loop w i l l  never a t t a i n  
0- 
'I r 
' J d' ' ( @  = (w-w,) - AKsin' 
Equation (3), f i g u r e  2 ,  and symmetry cons ide ra t ions  
T.3 as allows u s  t o  w r i t e  t h e  expected a c q u i s i t i o n  t i m e  
- d' (0) 
2 n  i ' o  io ( w - w o )  d' - AKsin' 
' 0  ' (0 )  
+ 
( 4 )  
BELLCOMM, INC. - 4 -  
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I n  d e r i v i n q  equat ion  ( 4 )  w e  assumed @(o) t o  be uni- 
formly d i s t r i b u t e d  about some $n  va lue .  
T h e  e v a l u a t i o n  of equat ion  ( 4 )  leads t o  t h e  fo l lowing  
conclusion.  I n  t h e  n o i s e l e s s  case, it t a k e s  an i n f i n i t e  amount 
of t i m e  f o r  t h e  loop t o  acqu i r e  lock. Since t h i s  r e s u l t  does 
no t  ag ree  w i t h  o u r  exper ience  w i t h  phase lock loops,  i t  i s  
necessary  t o  look a t  t h e  noisy  model i n  o r d e r  t o  o b t a i n  r e s u l t s  
which are more r e a l i s t i c  and more meaningful. 
I n  t h e  presence of a d d i t i v e  Gaussian n o i s e  equat ion  
(1) becomes 
= (w-w,) - AKsin' (t) - n ( t ) s i n O , ( t )  d t  1 
where n , ( t )  and n , ( t )  are independent Gaussian p rocesses  of 
zero mean and i d e n t i c a l  low pass  s p e c t r a l  d e n s i t i e s .  The 
s p e c t r a l  d e n s i t i e s  of n l ( t )  and n , ( t )  are assumed f l a t  over  
a s u f f i c i e n t l y  wide range so t h a t  w i t h  respect t o  t h e  c losed  
loop bandwidth, they  can be approximated a s  w h i t e  n o i s e  w i t h  
t h e  same magnitude of No/, w a t t s / H z .  
one s i d e d  n o i s e  s p e c t r a l  d e n s i t y .  
No i s  t h e  h e i g h t  of t h e  
I t  has been shown' t h a t  t h e  v a r i a t i o n  of @( t )  des- 
cribes a homogeneous f i rs t  order Markov p rocess  so t h a t  w e  may 
apply t h e  fo l lowing  r e s u l t .  
Define $ ( t = o )  = $(o) w i t h  a > $ ( t ) > b ,  then t h e  f i r s t  pas- 
sage t i m e  T a b ( @ ( o ) )  i s  t h e  random v a r i a b l e  de f ined  as t h e  t i m e  
t o  reach either boundary a or  b t h e  f irst  t i m e ,  g iven t h a t  ' ( t )  
s tar ts  a t  $o. 
Mathematically w e  say 
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ELLCOMM, I N C .  - 5 -  
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Let $(t) be a homogeneous Markov Process and if 
T = T ($(o)) is a proper random variable whose moments of 
n order nin exist, then E [Tab ($,)I satisfies 
ab 
- 0  
with 
E [T(a) n 3  = E[T(b)n] = o 
We define the acquisition time random variable 
given $(o), as T 
T ( $ 1  satisfies equation (6). To evaluate equation (6) we 
use the boundary values to determine the initial conditions 
on EFA($)n]and 
( $  ( 0 ) )  = TA($) (where a = $o and b = 2 ~ - $ ~ )  arb 
A 
To do this we note that 
J 
a 
where 
u(t) = p (x)dx i 
and 
p(x) = - p sin x + p- 
4BL 
0 
AK and Qo=w-w 
I t  
f BL = A2 
NoBL 
p = -  
g ( z )  = - - pn EpA ( z  1 n-l] 
4BL 
BELLCOMM. INC. 
while 
with 
- 6 -  
t X 
u(z)g(z)] dz+C1 e - u ( t ) ]  dt+C2 
b a 
we have 
c2 = 0 
We are thus in a position to solve equation (6) 
for the nth moment of the acquisition time, random variable, 
given $(o). 
However, one may ask at this time whether TA($) as 
defined, is the random variable which determines statistically 
the acquisition time of an actual first order phase lock loop? 
The first possihle objection to the definition is 
that it appears to restrict the possible unknown phase to a 
2.rr interval about $o. 
only substitute $n for $o in the definition and note that since 
= $o+2.rrN, equation (8) will give identical results for On @n 
and 4,. 
to a 2.rr interval about 4,. 
To see that this is not the case one need 
This periodicity allows us to restrict our attention 
Secondly, we note that TA is a conditional random 
variable. Thus, if we were interested in the worst case sta- 
tistics, we could determine E 
2.rr interval and select that val 
for all possible values in a 
of $o for which 
PA 
is greatest, 
BELLCOMM. I N C .  - 7 -  
3 
Another approach i s  t o  average over  a l l  v a l u e s  of . Because of symmetry c o n s i d e r a t i o n s ,  assuming W - w  e i t h e r  
$0 0 
p o s i t i v e  o r  nega t ive  l e a d s  t o  i d e n t i c a l  r e s u l t s .  Thus, w e  
need average over only a   IT r a t h e r  than a  IT i n t e r v a l .  W e  - 
d e f i n e  t h e  moments o f - the  mean a c q u i s i t i o n  t i m e  a s  
21T 
r 
J 
0 
W e  assumed $o t o  be uniformly d i s t r i b u t e d  s i n c e  t h e  
receiver has l i t t l e  o r  no knowledge of t h e  t r a n s m i t t e r  phase 
o r  of t h e  s t a t i s t i c s  of t h e  t r a n s m i t t e r  phase. 
To compute t h e  moments E [ TA "1 w e  s imula ted  equa t ion  ( 6 )  
on a d i g i t a l  computer u s ing  a program called " M i m i c . "  This  
i s  t h e  same procedure t h a t  w a s  used t o  determine t h e  t i m e  t o  
c y c l e  s l i p  parameter  excep t  i n  t h i s  case C i s  no longe r  zero 
b u t  is  g iven  by equa t ion  (10). 
1 
Our r e s u l t s  are summarized i n  Tables I and 11. I t  
can be seen  from the data t h a t  as the  s i g n a l  t o  no i se  r a t i o  
i n c r e a s e s  and a s  t h e  i n i t i a l  phase o f f s e t  i s  inc reased ,  t h e  
mean a c q u i s i t i o n  t i m e  i n c r e a s e s .  
However, t h e  mean a c q u i s i t i o n  t i m e  f o r  a phase o f f -  
set of .875 and a s i g n a l  t o  n o i s e  r a t i o  of 2 0 ,  i s  only 1.82 
seconds- It  i s  a l so  i n t e r e s t i n g  t o  no te  t h a t  one would expec t  
the  mean a c q u i s i t i o n  t i m e  t o  be less than  t h e  t i m e  t o  c y c l e  
s l i p  s i n c e  on t h e  average the  phase e r r o r  has t o  t rave l  a 
shorter d i s t a n c e  t o  a boundary i n  t h e  former case. Table I11 
compares t h e  mean a c q u i s i t i o n  t i m e  w i t h  t h e  mean t i m e  t o  c y c l e  
s l i p .  It  can be seen t h a t  f o r  a g iven  s i g n a l  t o  no i se  r a t i o  
t h e  mean t i m e  t o  c y c l e  s l i p  i s  s i g n i f i c a n t l y  greater than  
Emax ['A] 
- 8 -  ELLCOMM. INC. 
TABLE! I 
M o m e n t s  of t he  A c q u i s i t i o n  T i m e  Parameter TA 
F M o m e n t  1 3 5 10 1/4 
7 
I 1 E m a x  P A ]  .962 1.11 1.31 .634 
.389 
.257 
.168 .548 .615 .733 
.194 .398 e 473 .491 .494 
1.53 
3.81 
1.45 
3.76 
1.17 2.104 
6.48 
1.810 
4.82 
S k e w n e s s  
Excess 
Excess 
Predicted* 
2.55 
6.6 4.86 3.51 3.15 2.05 
i TABLE I1 
M o m e n t s  of t h e  A c q u i s i t i o n  T i m e  P a r a m e t e r  TA 
= .875) 
($0 
I M o m e n t  20 10 I 12 I 15 
1 E m a x  PA] 1.48 11.56 11.68 1.82 
1.02 
*860 I I O g 5 1  
I .202 I .394 I .487 I ,611 .611 1 3:1:0 1 -654 
1.26 1.30 1.36 
3.00 3.84 
.683 
1.44 S k e w n e s s  
Excess 
2.077 1.56 1.118 1.16 
6.32 3.52 2.32 2.17 4.52 
I 6.47 1 3.65 1 1.87 I 2.02 12.38 12.54 12.77 Excess Predicted* 3.11 
*Excess predicted under the  a s s u m p t i o n  t h a t  TA i s  a P e a r s o n  
Type I11 r a n d o m  variable.  
BELLCOMM, INC.  - 9 -  
TABLE I11 
h 
Comparison of Mean Time to Cycle Slip ( T )  and Maximum Mean Acqui- 
- '1 
i 
i 
i \.-. 
sition Time in a First Order Phase Lock Loop 
1/4 1.27 .268 
1 7.91 .648 
3 352.7 ,995 
5 18,308. 1.19 
In estimating the time to cycle slip probability den- 
sity distribution, the first three moments of the time to cycle 
slip parameter were fitted exactly to the first three moments 
of a Pearson Type 111 probability function. The Excess was then 
predicted assuming T to be a Pearson Type I11 random vari- 
able, and then this predicted excess was compared to that 
which was computed using the derived differential equation for 
the moments of T. It was noted then that the computed and pre- 
dicted values of Excess were extremely close in value. As can 
be seen in Tables I and I1 the computed and predicted values of 
the Excess of TA do not compare as well. The results indicate 
that as p increases, the Pearson Type I11 assumption becomes 
less valid. 
n 
h 
THE SECOND ORDER PHASE LOCK LOOP 
In a second order loop with a perfect integrator in 
c1 the loop [ G ( s )  = 1 + 
less case the pull-in range is theoretically infinite. In 
addition, he has shown that an approximate expression for the 
change of frequency per cycle for large values of cp is 
Viterbi3 has shown that in the noise- 
ELLCOMM, INC.  - 10 - 
‘1 
So t h a t  t h e  number of c y c l e s  t r a v e r s e d  by t h e  phase 
e r r o r  i n  o r d e r  t o  o b t a i n  lock f o r  an i n i t i a l  frequency o f f s e t  
of Ro i s  
where 
r = -  AK and i s  t h e  damping f a c t o r  of t h e  loop 
c1 
0 t h e  phase e r r o r  of the loop a t  t = o 
Assuming a p o s i t i v e  frequency o f f s e t  Ro w e  can w r i t e  
t h e  a c q u i s i t i o n  t i m e  random variable f o r  t h e  no i se  case as 
TA = T a , b ( 0 )  
where 
a = +m 
6 4  
T h i s  equa t ion  ho lds  f o r  va lues  of $(o) as l o w  as 3 and 
va lues  of r down t o  1. 
Using r e s u l t s  ob ta ined  i n  r e f e r e n c e  1 w e  f i n d  t h a t  
TA f o r  a second order loop sa t i s f ies  equa t ions  ( 6 )  and (7) w i t h  
Theref ore ,-. 
O X  
2 5 L  cos  x + - x + -  (l+r) 2 r  c1 u ( x )  = P 
w h i l e  
BELLCOMM, INC. - 11 - 
where AK+a . and B, = v 4BL Y =  L -1 2  4 
r P (1+-) 
p ( l + L )  r L 
Unfortunately, numerical results could not be ob- 
tained even for +(o) as low as 3 .  To see what the problem 
is, note that the a and b limits are very large and that u(x) 
for large values of x is proportional to x . Since u(x) is 
the argument of an exponential in equation 7 we see that the 
memory requirements f o r  computing E I T A 1  in a second order loop 
2 
1-1 
1 
2034-LS-ms Leonard. 'Schuchdan 
BELLCOMM, INC. 
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REFERENCES 
1. Schuchman, L., "Time to Cycle Slip in First and Second 
Order Lock Loops," International Communications Con- 
ference Record pp. 34-1 to 34-7, June, 1970. 
2. Schuchman, L., "Characterizing the Behavior of Phase Lock 
Loops,'' Bellcomm Technical Memorandum #70-2034-2- Febru- 
ary 20, 1970. 
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Acquisition time in a first order phase lock loop

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