Statistical behavior steady state phase error and time to cycle slip for phase lock loop

Schuchman, L.

English

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BELLCOMM, INC, 
955 L'ENFANT PLAZA NORTH, S . W . ,  WASHINGTON, D.C. 20024  
COVER SHEET FOR TECHNICAL MEMORANDUM 
TITLE- Character iz ing the Behavior of 
Phase Lock Loops 
F I L I N G  CASE NO(S)- 900 
TM- 70-2034-2 
DATE- February 2 0 ,  19'70 
AUTHORE+- L. Schuchman 
F I L I N G  SUBJECTfS) Communication Theory 
( A S S I G N E D  BY A U T H O R ( S ) ) -  phase Lock Loops 
Nonlinear Tracking Devices 
Diffusion Processes 
ABSTRACT 
The order  of a phase lQck loop i s  determined by t h e  
l i n e a r  f i l t e r  i n  the loop. Thus w e  say a phase lock loop i s  
of n + l  o rder  when t h e r e  are n f i n i t e  poles  i n  t h e  Laplace 
t r a n s f e r  funct ion of i t s  l i n e a r  f i l t e r  under t h e  r e s t r i c t i o n  
t h a t  t h e  number of zeros of t h i s  Laplace transform i s  no 
g r e a t e r  than n. 
I n  a previous memorandum (TM-69-2034-8) t h e  behavior 
of a f i r s t  and a p a r t i c u l a r  second order  phase lock loop w a s  
described i n  t e r m s  of t h e  s t a t i s t i ca l  behavior of two parameters,  
namely t h e  s teady s t a t e  phase e r r o r  $e and t h e  t i m e  t o  cyc le  
s l i p  T. This memorandum genera l izes  t h e  wogk j u s t  descr ibed 
i n  t h a t  t h e  s t a t i s t i c a l  behavior of 9, and T i s  charac te r ized  
f o r  a very l a r g e  c l a s s  of n- o rde r  phase lock loops. 
A 
t h  
It i s  t o  be noted t h a t  t h e  procedure developed i n  
t h i s  memo can be used t o  cha rac t e r i ze  t h e  behavior of s e v e r a l  
o the r  nonl inear  devices  a s ,  f o r  example, t h e  delay lock loop. 
m " - ,  
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R. L .  Se lden  
J. L. S t r a n d  
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B .  P .  T u n s t a l l  
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BELLCOMM, I N C .  
955 L’ENFANT PLAZA NORTH, S.W. WASHINGTON, D. C. 20024 
SUBJECT: Charac te r i z ing  t h e  Behavior of 
Phase Lock Loops - C a s e  900  
~ 
DATE: February 2 0 ,  1970 
FROM: L. Schuchman 
TM-70-2034- 2 
TECHNICAL MEMORANDUM 
The phase lock loop  t o  be d i scussed  i s  shown i n  
Figure 1 
Figure 1 Phase Lock Loop 
The loop c o n s i s t s  of  three components, a m u l t i p l i e r ,  
a t ime- inva r i an t  l i n e a r  f i l t e r  g ( t )  whose Laplace t ransform i s  
G ( s )  , and a vo l t age  c o n t r o l l e d  o sc i l l a to r  (VCO) . I f  t h e  loop 
i s  o p e r a t i n g  on a n o i s e l e s s  s i n e  wave, then  t h e  s t eady  s ta te  
ou tpu t  of t h e  VCO i s  a quadra tu re  r e p l i c a  of t h e  i n p u t  s i g n a l  
which may d i f f e r  on ly  i n  amplitude.  The m u l t i p l i e r  ou tpu t  
x ( t ) ,  under t h e s e  cond i t ions ,  w i l l  have only a double frequency 
t e r m  which t h e  l i n e a r  f i l t e r  and VCO c o n f i g u r a t i o n  w i l l  n o t  
pass .  The l i n e a r  f i l t e r  has a second and e q u a l l y  important  
func t ion  which i s  t o  reduce t h e  e f f e c t s  of n o i s e  t h a t  normally 
g e t s  i n t o  the  loop. 
BELLCOMM, I N C .  - 2 -  
1 I n  my previous  memorandum , t h e  behavior  of a f i r s t  
and a p a r t i c u l a r  second order loop  w a s  desc r ibed  i n  terms o f  
the  s t a t i s t i c s  of t w o  parameters ;  namely, t h e  loop s t a t i o n a r y  
phase error cp (t)= e l  ( t ) - e 2  (t) I and the  t i m e  t o  c y c l e  s l i p  
T. S ince  both  these parameters are random, t h e i r  behavior  w a s  
c h a r a c t e r i z e d  s t a t i s t i c a l l y .  That i s  cp (t) w a s  desc r ibed  i n  
t e r m s  of i t s  p r o b a b i l i t y  d e n s i t y  func t ion  while  T w a s  g iven i n  
terms of i t s  f i r s t  f o u r  moments which were shown t o  f i t  t hose  
of a Pearson Type I11 d e n s i t y  func t ion .  
A 
A 
A t  t h i s  p o i n t ,  the  work of two o t h e r  r e s e a r c h e r s  i s  
b r i e f l y  descr ibed .  
he obta ined  s o l u t i o n s  f o r  t h e  t i m e  t o  Cycle s l i p  and t h e  s t eady  
Lindsey2 r e c e n t l y  publ i shed  a paper i n  which 
t h  s t a t e  d e n s i t y  of t h e  phase error f o r  a N- o rde r  gene ra l i zed  
t r a c k e r .  
which an n- o r d e r  phase lock loop  i s  c h a r a c t e r i z e d  by N f irst  
o rde r  Markov d i f f e r e n t i a l  equat ions .  T h i s  l eads  t o  t h e  
s t a t i s t i c a l  c h a r a c t e r i z a t i o n  of  the  t r a c k e r  by a N dimensional 
Fokker-planck equat ion  (forward Kolmogorov equat ion)  . I n  
a d d i t i o n ,  Lindsey e l e g a n t l y  demonstrates  t h e  analogy between 
the theo ry  of continuous Markov processes  and Maxwell's wave 
equat ions .  
Tausworthel has ob ta ined  r e su l t s  f o r  t h e  mean t i m e  
H i s  approach i s  an ex tens ion  of V i t e r b i ' s '  work i n  
t h  
t h  t o  c y c l e  s l i p  f o r  t h e  N- o rde r  loop. I n  a foo tno te  t o  h i s  
paper ,  he describes a d i f f e r e n t i a l  equat ion  t h a t  t h e  
character is t ic  func t ion  of t h e  t i m e  t o  c y c l e  s l i p  parameter 
satisfies.  U s e  of t h i s  r e s u l t  could a lso l e a d  t o  my r e s u l t s  
f o r  t h e  n-moment of t h e  t i m e  t o  cyc le  s l i p  parameter for  an 
t h  a r b i t r a r y  N- o r d e r  loop. 
t h  
In  t h i s  t e c h n i c a l  memorandum, t h e  behavior  of t h e  
phase lock loop is  c h a r a c t e r i z e d  by a Markov l i k e  
process  which  l e a d s  quick ly  and simply t o  t h e  s t a t i s t i c a l  
c h a r a c t e r i z a t i o n  of t h e  loop by both  t h e  forward and backward 
Kolmogorov d i f f e r e n t i a l  equat ions .  W e  then  make use of t h e  
t h  work of Dar l ing  and Sieger t '  t o  d e r i v e  r e s u l t s  f o r  the  n- 
moment of the t i m e  t o  cyc le  s l i p  parameter f o r  an 
N- o r d e r  loop. T h i s  r e s u l t  i s  i n  t h e  form of a d i f f e r e n t i a l  
equat ion.  I t  has  been found by t h e  au tho r  t h a t  a l though a 
s o l u t i o n  t o  t h i s  equat ion  can be obta ined ,  t h e  eva lua t ion  of 
t h e  s o l u t i o n  i n  most cases  i s  a s i g n i f i c a n t  t a s k ,  w h i l e  direct 
eva lua t ion  of t h e  d i f f e r e n t i a l  equat ion  by 
s imula t ion  on a d i g i t a l  computer i s  r e l a t i v e l y  simple.  
t h  
B E L L C O M M ,  I N C .  - 3 -  
The class of l i n e a r  f i l t e r s  fo r  which ou r  r e s u l t s  w i l l  
be v a l i d  i s  de f ined  as 
G ( s )  = C -F H ( s )  
where C i s  a c o n s t a n t  and 
m 
i - a . s  
1 . .  
t h e  sets of [ai] and [bi] are c o n s t a n t s  chosen so  Chat t h e  phase 
lock loop  i s  s t a b l e  and p h y s i c a l l y  r e a l i z a b l e .  
It  i s  assumed t h a t  the  loop i n p u t  i s  a s i n e  wave p l u s  
a s t a t i o n a r y  narrow band Gaussian process  n ( t )  of zero  mean. 
The n o i s e  n ( t )  may be r ep resen ted  a s  
. 
where n l ( t )  and n2 (t) a r e  independent Gaussian processes  of zero 
mean and i d e n t i c a l  l o w  pas s  s p e c t r a l  dens i t i e s .The  parameter wo rad i s  
taken as t h e  qu ie scen t  frequency ou tpu t  a t  the  VCO. 
d e n s i t i e s  of n ( t )  , n l ( t )  and n2 ( 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  wi th  r e s p e c t  t o  t h e  c losed  loop 
bandwidth they can be approximated as w h i t e  n o i s e  wi th  t h e  same 
magnitude of N /2 watts/Hz. ** 
0 
The spectraTec 
* 
T h e  r eade r  i s  r e f e r r e d  t o  section 2 , 7  of Reference 3 for a 
N o  = KTo wat t s /Hz  and i s  t h e  one s ided  no i se  s p e c t r a l  
more d e t a i l e d  development of t h e  no i se  model. 
** 
d e n s i t y  where K = Boltzmann's c o n s t a n t  and To = system n o i s e  
temperature ,  degrees Kelvin.  
B E L L C O M M ,  I N C .  - 4 -  
I f  w e  d e f i n e  t h e  i n p u t  t o  t h e  phase lock loop a s  j z  A s i n [ u o t  + e 2 ( t ) l  
then w e  may w r i t e  the m u l t i p l i e r  ou tpu t  x ( t )  8s 
Since  t h e  VCO w i l l  n o t  pas s  t h e  double  frequency 
t e r m s ,  they  can be ignored. When t h e  cont ro l  s i g n a l  t o  t h e  VCO 
i s  app l i ed  a t  t = O f  t h e  VCO frequency becomes mQ 3- k 2 e ( k )  where 
K2 i s  a p r o p o r t i o n a l i t y  c o n s t a n t  a t  t h e  VCO whose dimensions 
are r ad ians  per second p e r  v a l t .  Thus t h e  t i m e  derivative of 
the loop phase e r r o r  i s  
where 
w i t h  
n '  ( t) = -nl (t) s i n e 2  (t) + n2 (t) cose2 (t) 
Using equat ion  (1) w e  can w r i t e  equa t ion  ( 5 )  as 
BELLCOMM, I N C .  - 5 -  
where 
f t  
j o  
f t=  K h ( t - u )  [Asin+ (u)  + n '  (u) ]du 
K = K1K2 def ined  as the loop aga in  
h ( t )  i s  the i n v e r s e  Laplace t ransform of H ( s ) .  
Th i s  work p a r a l l e l s  t h a t  given i n  Reference 1, and 
much of the d e t a i l  i s  omitted from t h e  d e r i v a t i o n s  t h a t  fo l low 
s i n c e  such d e t a i l  can be obta ined  from the re ferenced  work. 
and w r i t e  t h e i r  j o i n t  d e n s i t y  d i s t r i b u t i o n  as , 
we d e f i n e  f o u r  variables d e s c r i p t i v e l y  i n  F igure  2 
A s  i s  shown i n  Appendix C of Reference 1, i n t e g r a t i n g  b o t h  s i d e s  
wi th  r e s p e c t  t o  $1 and then f a c t o r i n g  o u t  P ($O, f t )  l e a d s  t o  
* 
f t  d i f f e r s  from i t s  d e f i n i t i o n  given i n  TM-69-2034-8 
only by a cons t an t  m u l t i p l i e r  f o r  t h e  second o rde r  loop. 
BELLCOMM. INC. - 6 -  
Thus w e  see t h a t  t h e  c o n d i t i o n a l  t r a n s i t i o n a l  
p r o b a b i l i t i e s  s a t i s f y  t h e  Chapman Kolmogorov equat ions  given 
by equat ion  (11) 
PHASE 
ERROR 
7 
TIME 
where 
BELLCOMM,  I N C .  - 7 -  
But t h e  l e f t  s i d e  of equa t ion  (11) can be expanded 
with t h e  r e s u l t  t h a t  
Equating t h e  r i g h t  s i d e  of equa t ions  (11) and (13) we 
have 
I t  i s  easy t o  show t h a t  
- y l ( @ , - , f t )  = AKC s i n  G l t  + f t  - - d t  
2 2  C K No  
Y2(+1 , f t )  = 2 
f ) = O  f o r  n > 2 y n ( + l ,  t 
Equation ( 1 4 )  then  reduces t o  
2 
a@ 1 
4 
BELLCOMM.  I N C .  - 8 -  
W e  nex t  mu l t ip ly  both sides of equat ion  (15) by p ( f t )  
and then  i n t e g r a t e  over a l l  va lues  of  f t  w i th  t h e  r e s u l t  t h a t  
2 + -  4 
where E [ f t l $ l ]  i s  t h e  expec ta t ion  of f 
t rece ived  s i g n a l  a t  t i m e  t. The c o n d i t i o n a l  expec ta t ion  of f 
cannot be eva lua ted  e x a c t l y  s i n c e  it r e q u i r e s  a knowledge of 
t h e  p r o b a b i l i t y  d e n s i t y  d i s t r i b u t i o n  of $1 which w e  obviously 
do no t  know. 
given t h e  phase of t h e  t 
It has  been shown l f4  t h a t  f o r  a second o r d e r  phase 
lock loop t h e  approximation 
g ives  r e s u l t s  which agree  e x c e l l e n t l y  w i t h  experimental  f i nd ings .  
Tauseworthe has  shown t h a t  e can be obta ined  from a l i n e a r  
approximation of  t h e  phase lock  loop wi th  t h e  r e s u l t  t h a t  
4 
g 2  ( 0 )  
4BL 
e = AKC - (18) 
BELLCOMM, - 9 -  
The loop-noise bandwidth BL i s  de f ined  a s  
d s  - 1 BL - 4.rri 
V i t e r b i  computed BL and t a b u l a t e d  h i s  r e s u l t s  for s e v e r a l  f i l t e r s  
i n  R.eference 3. 
To i l l u s t r a t e  l e t  
G ( s )  = 1 + 2 , then S 
a - -  (AKJ 
AK+a 1 AK+a 
e = A K -  
4 (7
This i s  i n  agreement wi th  t h e  r e s u l t  p resented  i n  Reference 1. 
I f  t h e  l i n e a r  approximation can be made or,more 
generally,when E [ f t l $ l ]  can be shown t o  be independent of t then  
equat ion  (16) d e f i n e s  a s t a t i o n a r y  Markov Process so t h a t  w e  can 
use a r e s u l t  by Dar l ing  and S i e g e r t 5  t o  o b t a i n  t h e  fol lowing 
r e c u r s i v e  r e l a t i o n s h i p  f o r  t h e  moments of t h e  t i m e  t o  c y c l e  
s l i p  parameter T. 
h 
where E I T o ]  = 1 
/ /  
k / /  ,' ,' 
/ 
I 
L. SchucKman 2034-LS- j f 
Attachment 
References 
ELLCOMM, I N C .  
REFERENCES 
1. L. Schuchman, "Time to Cycle Slip in First and Second Order 
Phase Lock Loops", Bellcomm Technical Memorandum TM-69-2034-8, 
December 19, 1969. 
2. N. C. Lindsey, "Nonlinear Analysis of Generalized Tracking 
Systems", Proceedings of the I E E E ,  Vol. 57, No. 10, 
pp. 1705-17022, Oct. 1969. 
3 .  A. J. Viterbi, "Principles of Coherent Communications", 
Chapter 2,  McGraw-Hill, 1966. 
4. R. C. Tausworthe, "Cycle Slipping in Phase-Locked Loops", 
IEEE Transactions on Communication Technology, Vol. COM-15, 
No. 3 ,  pp. 417-421, June, 1967. 
5. D. A. Darling and A. J. E. Siegert,"The First Passage Time 
Problem for a Continuous Markov Process", Am. Math. Stat., 
Vole 24, pp. 624-639, 1953. 


Characterizing the behavior of phase lock loops

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