A technique for maintaining synchronization between optical PPM (pulse-position modulation) pulses and a receiver clock by means of a delay-tracking loop is described and analyzed. The tracking loop is driven by a doubly stochastic Poisson process that contains information about the location of the desired slot boundaries. The slot boundaries are subject to slowly varying random delays that are ultimately tracked by the loop. The concept of fractional rms delay error is introduced to quantify the effects of signal and background induced shot noise on the performance of the delay-tracking loop

Rodemich, E. R.

Tan, H. H.

Vilnrotter, V. A.

English

NASA Technical Reports Server

TDA Progress Report 42-87 July-September 1986 
A Synchronization Technique for Optical PPM Signals 
V. A. Vilnrotter, E. R.  Rodemich and H. H. Tan 
Communications Systems Research Section 
A technique for maintaining synchronization between optical PPM pulses and a 
receiver clock by means of a delay-tracking loop is described and analyzed. The tracking 
loop is driven by a doubly stochastic Poisson process that contains information about the 
location of the desired slot boundaries. The slot boundaries are subject to slowly varying 
random delays that are ultimately tracked by the loop. The concept of  fractional rms 
delay error is introduced to quantify the effects of signal and background induced shot 
noise on the performance of the delay-tracking loop. 
1. Introduction 
Optical pulse-position modulation (PPM) is an accepted 
signaling format for the transmission of information over 
optical channels. With this coding technique. L bits of infor- 
mation are encoded onto one of M = 2L PPM words by estab- 
lishing a one-to-one correspondence between the possible 
states of L binary digits and the location of an optical pulse 
among M possible slots. If the slot duration is T seconds, then 
each PPM word requires M T  seconds for transmission. For a 
direct-detection system, maximum likelihood decoding con- 
sists of counting the number of photons in each slot and 
selecting the PPM word corresponding to  the greatest count. 
To accomplish this, however, requires accurate synchroniza- 
tion between the received PPM slot boundaries and the count- 
ing boundaries defined by the receiver clock. Synchroniza- 
tion for binary PPM optical signals by means of “early-late” 
gates and decision-directed feedback has been considered 
previously in the literature (Refs. 1, 2). Here we consider a 
novel technique for synchronizing M-ary PPM symbols without 
the use of decision-directed feedback, thus effectively remov- 
ing the dependence of the synchronization subsystem on 
decoder performance. The description and analysis of this 
tracking system is the subject of the following sections. 
II. Tracking System Description and 
Analysis 
The slot synchronization technique described here relies 
on the addition of a small “dead time” before and after each 
optical pulse, in order to derive an error signal proportional 
to the delay error that may exist between the received slot 
boundaries and the receiver clock. (Decoder performance is 
not affected if the pulse intensity at the transmitter can be 
increased to maintain constant average pulse count at the 
receiver.) Thus, T = T~ + 2Td, where T~ is the duration of 
the optical pulse and T~ the duration of each “dead-time’’ 
interval. 
The detector output can be modeled as a doubly stochastic 
Poisson process, represented by a sequence of randomly occur- 
ring unit impulses whose average rate at any given time depends 
on the total signal and background power collected by the 
24 
https://ntrs.nasa.gov/search.jsp?R=19870005899 2020-03-20T12:24:34+00:00Z
receiver. Consider the case where the receiver clock is syn- 
chronized with the received optical data (we do not address 
the acquisition problem here). Doppler effects and trans- 
mitter or receiver clock instabilities tend to  introduce random 
delay components between the received data and the receiver 
clock which, if uncorrected, may eventually cause loss of 
synchronization. Assuming that the tracking system main- 
tains synchronism between the received data and the receiver 
clock, transmitter clock instabilities and Doppler-induced rate 
changes can be modeled in terms of the delay process AT = 
Arc, + AT,, whose components are transmitter clock drift 
and Doppler-induced delay, respectively. Typically, AT is a 
slowly varying function of time that can be treated as a 
constant over a great many PPM words. 
A block diagram of the delay-tracking loop is shown in 
Fig. 1. The detector output process x(t;Ar), which contains 
information about the location of the transmitted data- 
pulses, serves as input to the traclung loop. This is a doubly 
stochastic Poisson process with random rate h(t;Ar) = h,(t;Ar) 
+ nb,  where hs(t;Ar) is a random rate due to  the signal while 
nb is a constant average count rate due to multimode back- 
ground radiation. The receiver clock generates a squarewave 
g(r ;q)  of nominal period r and noisy delay estimate = 
*+ Arcr. The com onents of 6 are the loop’s estimate of 
Arcr due to instabilities within the receiver clock. The input 
to the loop filter is the product of the detector output process 
and clock waveform. Defining the delay error as AT, = AT - ‘c.9 r this process can be represented as 
the received delay A AT, and an independent drift component 
(1) 
A y(t;Ar,) = x(t;Ar)g(t;Arn) 
where the notation implies that the product contains informa- 
tion about the delay error Are. Indeed, foi M-ary PPM symbols 
with dead-time, the time averaged mean value given AT and 
Q, clearly depends on AT,: 
T 
 AT,) 4 & IT E {x(t;Ar)g(t;G} d t  
where 26 is the extent of the linear range, E is the expectation 
operator, ns is the Poisson rate due to signal when a pulse is 
present, and a = 2nJMr. Note that 7 (Arc) does not depend 
on the back round induced count rate since the time average 
Figure 2 illustrates the behavior of 7(ArE) as a function of 
AT, over the interval IAr, I < 712, assuming that rd < 714. 
of nb g(t; a8 7,) over an integral number of periods is zero. 
It is convenient to  decompose y(t;Ar,) as 
where n(t;Are) 4 y(t;ArE) -  AT,) is viewed as an additive 
noise process, containing the conditioning parameter Are. 
The power spectral density of this noise process conditioned 
on Arc, Sn(o;Ar,), is derived in Appendix A. There it is shown 
that for w = 0 the noise power spectral density does not 
depend on AT,, but takes the value (% + nb), where 
is the average Poisson count rate generated by the signal. 
Thus, in a suitably narrow band of frequencies around the 
origin. the power spectral density of the equivalent noise 
process may be approximated as 
2n S n ( w ) 5 < + n b ;  Iwl<< - M r  (4) 
If AT, varies slowly enough to be considered constant over a 
great many PPM symbols, then  AT,) and Sn(o;ArE) may be 
approximated by replacing the infinite limits in their defining 
equations by finite values that encompass regions of con- 
stant Are. In that case, we conclude that for frequencies 
near the origin, the process y(t;Ar,) behaves as though it 
were composed of a slowly varying term proportional to the 
delay error: plus an independent additive white noise process 
n( t )  characterized by the spectral level (T + nb),  as defined in 
equation (4). 
Assuming that lArcI << 6 ,  the operation of the tracking 
loop can be described by means of the linear loop model 
shown in Fig. 3 (Ref. 3). Using the differential operator 
p 4 dldt, 100p upeiaiioii Cali be ejiijiesjed in ieriiis of the 
equation 
A? = Arc + AT, - Are = p F @ )  G [&Are + n ]  
(5) 
Arc A arc, -ATcr 
where G is the gain of the voltage-controlled clock and n = 
n(t;Ar,) as in Eq. (3) (the time index has been deleted for 
the sake of notational simplicity). Solving for Are yields 
25 
Are = [ l  - H @ ) ]  (Arc + AT,) - H @ )  A ff (6a) 
The tracking system should be designed to track Arc and AT, 
accurately, in which case their contributions to Are can be 
ignored. Under this condition the steady-state delay-error is 
due entirely to the equivalent noise process n(t) .  Assuming 
that IH('jw)I2 takes on significant values only in an interval 
I w 1 << 2n/Mr, the variance of the delay-error becomes 
where BL is the one-sided loop bandwidth defined by 
Substituting for cy and Sfl(0) in equation (7) yields 
(9) 
From equation (9) it is clear that the fractional rms delay error 
(uc,/rp) depends only on BI,, ns and nb.  and is independent of 
rp for any fixed value of S : 
- 
Note that even in the absence of background radiation (uE/rp) 
does not become arbitrarily small, but approaches its quan- 
tum-limited value d m .  
It is apparent from equations (6b) and (8) that BL depends 
only on the loop filter transfer function F @ )  and the product 
aG .  Since G is independent of the other system parameters, 
constant loop bandwidth can be maintained by adjusting G 
so as to keep CUG constant. Thus, BL can be held fixed even 
as % = w P / 2  varies. The behavior of (u / T  ) as a function of 
% for various loop bandwidths and background rates is shown 
in Fig. 4. 
E P  
Other meaningful fractional rms delay errors may also be 
defined in terms of (ue/r ). namely 
P 
(>)= ( 1 -  7) (4) ; O < T , < ~  7 ( l l b )  
and using Eq. (2b), 
It should be emphasized that the above equations are meaning- 
ful only if the inequality (u,/S) << 1 is satisfied. Assuming 
this to  be the case, Eqs. ( l l a )  and ( l l c )  show that (uc/8) is 
a monotone decreasing function of T, in the interval 0 < r < 
r/4, while maintaining a constant minimum value of 2(uefr ) 
over the interval 7/4 < rd < r/2.  P 
26 
In typical applications, the loop bandwidth BL is chosen 
large enough to exceed the significant spectral components of 
Arc and AT,, thus allowing the loop to track these quantities 
accurately. The signal-set dimension M and slot-width r are 
generally determined on the basis of communications require- 
ments, while nb depends on the background environment and 
on the particular set of spatial and frequency predetection 
filters employed. In order to achieve the desired decoder error 
probability, some minimum average signal pulse count n 7 
must be maintained, even in the absence of synchronization 
errors. Under the condition (uc/8) << 1, random delay errors 
should have little effect on decoder performance if the decoder 
counts photons over the entire synchronous r-second time 
interval. Thus, a value of (ue/S) can generally be found to 
guarantee negligible decoder performance degradation while 
simultaneously assuring linear loop operation: let this value 
be denoted by y. Again using Eqs. (1 la) and (1 IC), a reason- 
able value for rd can be determined by calculating (uE/r ) 
from the given system parameters, and comparing it to  yf2. 
If (uc/rp) < y/2, then let 
S P  
Td = 7 [ 2 +  (y/(3)]-1 
However. if ( 4 1 7  ) > y/2, then the desired value nf (oe/6) 
cannot be realize$ with the initial set of system parameters, 
and hence at least one of the original parameters must be 
altered to accommodate tracking-loop requirements. Generally. 
the tracking loop will perform well, with dead-times substan- 
tially less than 7/4 seconds, when operating in high data-rate 
narrow band communication systems where a suitably small 
value of ( u ~ / T ~ )  can be achieved. 
111. Concluding Remarks 
A technique has been developed for maintaining synchroni- 
zation between a PPM-modulated optical data-stream and a 
receiver clock. This technique relies on the addition of a 
small “dead time” interval before and after each possible 
pulse location to derive an electrical signal proportional to 
the delay error between the received slot boundaries and the 
recelver clock. It was argued that in applications where only 
spectral components near the origin are of interest, and where 
the delay error remains essentially constant over a great many 
i 
PPM symbols, the driving process could De approximated by 
a slowly varying term proportional to the delay error, plus an 
effective additive noise process due to the inherent random- 
ness of the detected optical fields. Based on the above approxi- 
mations, the variance of the delay error due to both signal and 
background radiation was derived, assuming a linear loop that 
effectively tracked out the slowly varying delay error compo- 
nents. We observed that since linear loop operation typically 
implied rms delay errors on the order of a small fraction of the 
linear region, decoder performance should not be signifi- 
cantly affected if the decoder counts photons over the entire 
7-second slot time. It was shown that in applications where 
this is true, a simple algorithm can often be employed to  find 
“good“ design values for the dead-time interval in terms of 
the given system parameters, and a quantity the designer 
selects to  simultaneously ensure linear loop operation and 
acceptable decoder performance. However, a precise deter- 
mination of the dead-time interval (taking into account 
decoder performance degradation and possible non-linear 
effects within the loop) is beyond the scope and interest of 
this article. 
References 
1. Gagliardi, R. M., “Synchronization Using Pulse Edge Tracking in Optical Pulse-Position 
Modulated Communication Systems,” IEEE Transactions on Communications, Oct. 
1974. Vol. Com-22. No. 10. pp. 1693-1702. 
2. Gagliardi, R. M. and Karp, S . ,  Optical Communications, Wiley, N.Y., 1976. 
3. Lindsey, W. C., Synchronization Systems in Communications and Control, Prentice 
Hall, N.J., 1972, Chapt. 4. 
27 
A 
d t ; A r J  VOLTAGE 
TO 
DECODER CLOCK 
CO NTR 0 L LE D 
y = V + n  ,“‘d FUNCTION GENE RAT0 R 1 , I , 
VOLTAGE 
CONTROLLED 
CLOCK 
Fig. 1. Tracking loop block diagram Fig. 3. Linear loop model 
-I 
10-1 
Q 
- 
10-2 
b” 
I 
100 101 102 1 03 1 04 - 
“S Fig. 2. Time averaged mean value as a function of delay 
error ( T ~  < T / 4 )  
Fig. 4. Fractional rms delay error (u(T,,) as a function of average 
signal count for various background rates and loop bandwidths 
28 
Appendix A 
The power spectral density of n(t;Ar,) can be derived 
using the results of Appendix B. Assume that AT, is held 
fixed. Since n(t;Ar,) =y ( t ;Ar , )  -u(Ar,) ,  it follows that 
where 
Consider the case where the detector observes signal plus 
multimode background noise fields. Thus, h(t;A7) = As(t;A7) t 
nb where hs(t;A7) is the random rate function associated with 
the signal while nb is a constant average count rate contrib- 
uted by the background fields. The square-wave generated by 
the receiver clock is denoted by g(t;<) as before. For 
M-ary PPM with dead-time the product h(t;Ar) g ( t ; c )  can 
be modeled over the interval [-T, T) as 
(A-2) 
where K = (T/Mr) - 1/2, 7 is the slot duration, andM7 the 
duration of a PPM word. Elements of the set {Qi} are inde- 
pendent random integers taking on values from 0 to (M - 1) 
with equal probability. Given AT,, p ( t )  and pb( t )  are defined 
as 
0; 0 G t < Td’ 7 - rd < t < 7 
ns; rd < t < + (A-3a) 
2 P S ( O  = 
( + Are < t < T - Td 
With Ps(w) a F b S ( t ) }  and P ( w )  a F @,(t)} ,  the Fourier 
transform of h(t;Ar) g ( t , @ )  restricted to  the interval 
[- T, 0, becomes 
where 
L J 
1 -jw(r-rd) -jw(r/2+Ar6) - 2e 
and 
Recognizing that 
29 
and making use of Eq. (A-4). it follows that 
+ IPs(w)12 (2K + 1) 1 - f i n i  M sin ’( - .‘.ilj wr (A-5) 
Note that 
To show this we first observe that y(w)  is a periodic function 
of w with period 2n/r.  Over any interval I that does not 
contain the points w, = 27rni/r, the denominator 
M sin (w7/2) 
is bounded away from zero, hence Iy(0)1~ is bounded, and 
Thus, the integral of ly(w)I2/(2K + 1) vanishes everywhere 
except within arbitrarily small neighborhoods around the 
points urn as K grows without bound. However. integration 
over any complete cycle yields 
K M-1 
&I2 (2K+ 1) i = - ~  Q = O  
1 
dwly(w)12/(2K+ 1) = 
27r 
Mr 
- _  - 
independent of K .  Therefore, this is the value of the integral 
over arbitrarily small neighborhoods around each point urn as 
K approaches infinity. 
We also make the useful observations that Pb(0) = 0. 
1Ps(0)12 = 4 n: AT:. and 
30 
T 
= lim IT k(r:Ar)dr  3T 
T- - 
= + l l b  
where 
Making use of the fact that 2T = (2K + 1)  M r .  substituting 
the above terms into equation (B-7) and using Eq. (A-1) 
finally yields 
Thus. the spectral density of the equivalent noise process 
consists of continuous and impulsive terms. The impulses 
occur at integer multiples of the radian frequency 27r/r; 
however, there is no impulse at the origin since the contribu- 
tion of 7 was subtracted out.  The value of the continuous part 
at the origin is 
Sfl(O;ArE) = % + nb (‘4-7) 
independent of 
unconditioned spectral density as well: 
Therefore, this must be the value of the 
Since for any \Arc\ < 6 ,  the continuous part of Eq. (A-6) 
begins to  deviate significantly from its zero-frequency value 
only as w approaches values on the order of 27r/Mr, it can be 
argued that in a suitably narrow band of frequencies around 
the origin (namely. 1 0  j << 2n/Mr) the spectral density of the 
equivalent noise process may be approximated by its zero- 
frequency value. 
Appendix B 
The power spectral density of the product of a doubly 
stochastic Poisson process and a deterministic function is 
derived. Let 
be a sample function of a doubly stochastic Poisson process 
with random rate A([). Let g(t) be a deterministic function 
and form the product .v(t) = g(t) x(t).  Define the power 
spectral density ofv( t ) ,  S,,(w), as in Ref. 2 (Chapter 4): 
03-2) 
1 
2T Sv(w) = lim - E  IYT(w)I2 
T+-  
where 
is the Fourier transform of g(t) restricted to the interval 
[- T, 7'). Using Eq. (B-1) yields 
P m  
(B-3b) 
First assume that h(t)  is known. For a given A(?) and con- 
ditioned on k arrivals in [- T, T), the joint probability density 
of the unordered arrival times tk = ( r l ,  r 2 ,  . . . , t k }  can be 
expressed as 
(B-4) 
where 
Thus, 
m rr 
j w ( t  --r ) h(tJ 
x e ' n (k) d t ,  d t , .  . . dtk 
I =  1 
J-T  
The first term accounts for those k elements for which II = m ,  
while the second term is due to the remaining ( k z  - k )  ele- 
ments of the double sum. Since for a Poisson process with rate 
m T ,  E ( k )  = mT and E ( k 2  - k )  = m;, it follows from Eq. (B-5) 
that 
In general h ( t )  is itself a sample function of a random pro- 
cess; hence an additional averaging is required over the sample 
space of A( t ) .  Thus, for doubly stochastic Poisson processes, 
the power spectrum is given by the expression 
31 


A synchronization technique for optical PPM signals

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