Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers


Science and Technology, 7 (2002) 157-167  
© 2002 Sultan Qaboos University 
 

Joint Symbol and Frame Synchronization for 
Direct- Detection Optical Communication Systems 

Eesa M. Bastaki* and Harry H. Tan**   

*Department of Electrical Engineering, Faculty of Engineering, United Arab Emirates 
University, P.O.Box 17555, Al- Ain, U.A.E, Email: eesa@uaeu.ac.ae, **Department of 
Electrical Engineering, School of Engineering, University of California, Irvine, Ca, 92717, 
U.S.A.  

تزامن الرموز واإلطارات في أنظمة االتصاالت الضوئية ذات الكشف المباشر      

 عيسى بستاكي و هاري تان

, يتعرض هذا البحث لمشكلة تزامن الرموز واإلطارات في أنظمة االتصاالت الضوئية التي تستخدم الكشف المباشر :  خالصة
اصة بتزامن الرموز واإلطارات لحاالت االحتماالت القصوى       ويستنبط البحث العالقات الخ   . بافـتراض معـرفة توقيت الشقوق     

 .كما يناقش البحث أسباب الخطأ في اشتقاق جورج هايدس. المثلى وللحاالت دون المثلى
 

ABSTRACT: The problem of joint symbol and frame synchronization in direct-detection optical 
PPM communication systems under the assumption of known slot timing is considered here. The 
optimum maximum-likelihood (ML) and sub-optimum rules for this joint symbol and frame 
synchronization problem are derived. The reason of Georghiades's (1985) incorrect ML rule is 
discussed in this paper. 
 
KEYWORDS: Frame Synchronization, Direct-Detection, Optical PPM Communication, Optimum 
Maximum-Likelihood. 

1. Introduction 

U nfortunately, as will be discussed in this paper, Georghiades' derivation and his reported ML rule (Georghiades, 1985) is not correct. This is because that derivation did not consider the 
end effects of each frame properly and also invoked an invalid symmetry assumption to simplify 
the structure of the reported decision rule. In this paper, the correct optimum ML rule is derived.  
Simulation results presented here shows a significant improvement in correct synchronization 
probability performance of the correct optimum ML rule over Georghiades' incorrect ML rule.  In 
particular, for high signal-to-noise ratios, the synchronization probability performance of the 
correct ML rule tends to the random data-limited upper bound while the performance of 
Georghiades' incorrect ML rule pre-saturates at a significantly lower level.  We shall also consider 
a sub-optimum ML rule that accounts for the end effects of each frame, but also assumes the 
invalid symmetry assumption.  This sub-optimum rule has a performance intermediate between that 
of the optimum ML rule and Georghiades' incorrect ML rule. 

2.   Joint Symbol and Frame Synchronization Problem 

We consider PPM modulation over the direct-detection optical Poisson channel in which each 
M-ary symbol duration is divided into M time-slot divisions, and a rectangular light pulse is sent in 
the time slot associated with the transmitted symbol.  The channel output is a Poisson process with 
intensity rate ns λλ +  when a light pulse is transmitted and nλ  otherwise.  Here sλ  is the photo-

157 



EESA M. BASTAKI and HARRY H. TAN 

detector count rate due to the light pulse and nλ is the count rate due to dark current and 
background noise.  Data transmission is formatted in successive frames with periodically inserted 
fixed synchronization patterns.  Each frame is assumed to consist of N data symbols composed of a 
fixed L-symbol sync pattern and N-L random data symbols.  No assumption is made to preclude 
the presence of the sync pattern among the random data symbols.  We shall represent each M-ary 
symbol as a M-dimensional vector ),...,( 10 −= Mddd

                    
slot,th -i 

 where  

,...,1

,...,1i

,..., K

−

1−

S

,...,L

,...,L

0
id

ˆ 0
id

 





=
          otherwise.;0

 theis pulselight   theif;s
id

λ
 

 
Let the sync pattern be given by 

),,( 10 −= LSSSS                                    (1) 
where for ,  10 −≤≤ Li

).,( 10 −= Miii SSSS             (2) 
 
In the joint symbol and frame synchronization problem, the channel output corresponding to N 
transmitted symbols are observed.  Since the pulse slot timing is known, the sufficient statistics are 
the photon counts in the  NM time slots corresponding to the selected  N  transmitted symbols.  Let 
 

),...,,...,,,...,( 1)1(1210 −−−−= NMMNMMM KKKKKK    (3) 
 
denote this vector of NM photon counts.  There are NM possible starting positions for the sync 
pattern S.  The joint symbol and frame synchronization problem is to estimate this starting position.  
We consider the maximum likelihood approach here.  The optimum ML rule estimates the sync 
pattern starting position as , where 0m̂ 1ˆ ≤≤ NMm  is chosen to maximize the likelihood that 

 are the ML photon counts corresponding to the transmitted frame sync pattern ),...,( 1ˆˆ −+LMmm KK S . 

2.1     ML Rule 

Consider a candidate position m, 0 ≤≤ NMm . The starting position of S corresponds then 
to the count .  So  are the counts corresponding to random data symbols preceding mK ),...,( 10 −mKK
S ,  are the counts corresponding to ),...,( 1−LMmK +mK S , and  are the counts 
corresponding to random data symbols following 

),...,( 1−+ NMLMm KK
.  In order to consider all NM-candidate 

starting positions, we need to consider the (N-L) random data symbols preceding and following S .   
 
Hence denote  

)( 1−= Nddd      (4) 
 
to be the N-L random data symbols following S  and 
 

)ˆˆ(ˆ 1−= Nddd      (5) 
 
to be the N-L random data symbols preceding S  where 
 

),,...,( 1−= Mii dd      (6) 
 

),ˆ,...,(ˆ 1−= Mii dd      (7) 

 158



JOINT SYMBOL AND FRAME SYNCHRONIZATION  

and 







=
                                                        ;0

.;ˆ,
otherwise

slotthj in the is symbolthi for the pulse  theifsj
i

j
i dd

λ     (8) 

 
Table 1 illustrates the relation between ,,, dSK  and d̂  for N=4, L=2 and M=2 for each of the 

NM candidate starting positions m. 
The ML rule chooses its estimate to be the value of m that maximizes 
 

m.given  K ofon distributiy Probabilit)|Pr( =mK     (9) 
 
Similar to the approach taken in [1], )|Pr( mK  can be derived by averaging over the random data 

,d̂ and d  

.d̂ and d m,given  K ofon distributiy Probabilit)ˆ,,|Pr( =ddmK    (10) 
 

For the direct-detection optical channel the components of K  are all conditionally independent 
Poisson random variables given m, d̂ and d .  In examining Table 1 it can be seen that there are three 
separate cases to consider 

 
a) Case I 0  M) (mod m and )(0 =−≤≤ MLNm  
Suppose m=qM, where 0 .  Then LNq −≤≤ )ˆ,,|Pr( ddmK  depends on (N-L-q) kd ’s and q kd̂ ’s for a 
total of (N-L) i.i.d. random data symbols. 
 
b) Case II   ( 11) −≤≤+− NMmMLN  

)ˆ,|Pr()ˆ,,|Pr( dmKddmK =  depends on (N-L) i.i.d. kd̂ ’s. 
 
c) Case III 0 0)(mod  and  1)( ≠=−−≤≤ kMmMLNm   

 
Table 1: Relationship between KddS ,ˆ,,  for M=2,L=2,N=4. 

 
m m(mod M) 76543210 KKKKKKKK  

0 0 1303120211011000 ddddSSSS  
1 1 0

3
1
2

0
2

1
1

0
1

1
0

0
0

1
3

ˆ dddSSSSd  
2 0 1

2
0
2

1
1

0
1

1
0

0
0

1
3

0
3

ˆˆ ddSSSSdd  
3 1 0

2
1
1

0
1

1
0

0
0

1
3

0
3

1
2

ˆˆˆ dSSSSddd  
4=(N-L)M 0 1

1
0
1

1
0

0
0

1
3

0
3

1
2

0
2

ˆˆˆˆ SSSSdddd  
5 1 0

1
1
0

0
0

1
3

0
3

1
2

0
2

1
1

ˆˆˆˆ SSSddddS  
6 0 1

0
0
0

1
3

0
3

1
2

0
2

1
1

0
1

ˆˆˆˆ SSddddSS  
7 1 0

0
1
3

0
3

1
2

0
2

1
1

0
1

1
0

ˆˆˆˆ SddddSSS  
 

Here )ˆ,,|Pr( ddmK  is a function of N-L-q-1 entire kd  vectors and q entire kd̂  vectors as well 
as a function of part of another kd  vector and part of another kd̂  vector.  The partial vectors are at 
the two ends of the NM-vector K.  So )ˆ,,|Pr( ddmK  depends on N-L-q  kd 's and q+1 kd̂ 's for a 
total of N-L+1 i.i.d. random data symbols. This case then differs significantly from the first two 
cases above. 

 159



EESA M. BASTAKI and HARRY H. TAN 

Georghiades' incorrect derivation (Georghiades, 1985) of the ML rule makes the mistake of 
assuming that only cases I and II hold and does not consider case III.  In order to derive )|Pr( mK , 
let us first consider )ˆ,,| ddmKPr( for each of the above three cases. 

 
a) Case I   0  M) (mod m and )(0 =−≤≤ MLNm
Assume that 

,qMm =      (11) 
where .  Here LNq −≤≤0

)ˆ,,|Pr( ddmK ')(
1

0

1

0
)!(

]')[( TS
L

i

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

TS λλ +−
−

=

−

= ++
∏∏

+++
=  

')(
1 1

0
)!(

]')[( Td
qN

Li

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

Td λλ +−
−−

=

−

= ++
∏ ∏

+++
∗ ')

ˆ(
1 1

0
)!(

]')ˆ[( Td
N

qNi

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

Td λλ +−
−

−=

−

= ++
∏ ∏

+++
∗   (12) 

where T' is the pulse slot duration.  We also adopt the convention here ∏  
=

=
n

mi

if 1)(

whenever n<m.  Moreover indices are interpreted modulo NM. 
 
b) Case II   ( 11) −≤≤+− NMmMLN  
 
Here 
 

)ˆ,|Pr()ˆ,,|Pr( dmKddmK =  

')(
1

0

1

0
)!(

]')[( TS
L

i

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

TS λλ +−
−

=

−

= ++
∏∏

+++
= .

)!(
]')ˆ[( ')ˆ(

1 1

0

Td
N

Li

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

Td λλ +−
−

=

−

= ++
∏∏

+++
∗   (13) 

 
c) Case III 0 0)(mod  and  1)( ≠=−−≤≤ kMmMLNm  
Assume that 
 

,kqMm +=      (14) 
where . .10 and 10 −≤<−−≤≤ MkLNq
 
Here 

)ˆ,,|Pr( ddmK ')(
1

0

1

0
)!(

]')[( TS
L

i

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

TS λλ +−
−

=

−

= ++
∏∏

+++
= ')(

1)1( 1

0
)!(

]')[( Td
qN

Li

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

Td λλ +−
−−−

=

−

= ++
∏ ∏

+++
∗  

')(
1

0 )1(

1 1
)1(

)!(

]')[( TdkM

j jkMN

K
n

j
qN n

j
qN

jkMN

e
K

Td λλ +−−−

= ++−

−− −−
++−

∏
+

∗
')ˆ(

1

)1(

1 1
)1(

)!(

]')ˆ[( TdM

kMj jkMN

K
n

j
qN n

j
qN

jkMN

e
K

Td λλ +−−

−= ++−

−− −−
++−

∏
+

∗  

')ˆ(
1 1

0
)!(

]')ˆ[( Td
N

qNi

M

j mjiM

K
n

j
i n

j
i

mjiM

e
K

Td λλ +−
−

−=

−

= ++
∏ ∏

+++
∗                (15) 

 
Note that 

∏∏
−

=

−

=

+−
1

0

1

0

')(
L

i

M

j

TML nse λλ      (16) 

and that 

∏∏
−

=

−

=
++

++1

0

1

0
!

)'(N

i

M

j
K

K
n

mjiM

mjiMTλ           (17) 

 160



JOINT SYMBOL AND FRAME SYNCHRONIZATION  

are independent of m.  Also note for the following cases that 
 
a) Case I 
 

∏ ∏∏ ∏
−

−=

−

=

+−
−−

=

−

=

+−


























 1 1

0

')ˆ(
1 1

0

')(
N

qNi

M

j

Td
qN

Li

M

j

Td n
j

in
j

i ee λλ '))(( TMLN nse λλ +−−= ,       (18) 

b) Case II 

∏∏
−

=

+−−
−

=

+− =
1

'))((
1

0

')ˆ( ,
N

Li

TMLN
M

j

Td nsn
j

i ee λλλ     (19) 

c) Case III 

∏ ∏∏ ∏
−

−=

−

=

+−
−−−

=

−

=

+−































 1 1

0

')ˆ(
1)1( 1

0

')(
N

qNi

M

j

Td
qN

Li

M

j

Td n
j

in
j

i ee λλ ,'))(1( TMLN nse λλ +−−−=       (20) 

Define 

∏∏
−

=

−

=

+−

++

++

=
1

0

1

0
!

')(
1

)'(N

i

M

j
K

K
nTMN

mjiM

mjiM
ns

T
eC

λλλ      (21) 

 
(C1 is independent of m).  Then using (16) - (20) in (12), (13) and (15) we obtain for the three cases 
the following expressions for )ˆ,,|Pr( ddmK . 
 
a) Case I   . LNqqMm −≤≤= 0,

)ˆ,,|Pr( ddmK ∏∏
−

=

−

=

+++=
1

0

1

0
1 )/1(

L

i

M

j

K
n

j
i

mjiMSC λ ∏ ∏
−−

=

−

=

+++∗
qN

Li

M

j

K
n

j
i

mjiMd
1 1

0

)/1( λ ∏ ∏
−

−=

−

=

+++∗
1 1

0

)/ˆ1(
N

qNi

M

j

K
n

j
i

mjiMd λ    (22) 

 
b) Case II  ( .11) −≤≤+− NMmMLN  

)ˆ,,|Pr( ddmK ∏∏
−

=

−

=

+++=
1

0

1

0
1 )/1(

L

i

M

j

K
n

j
i

mjiMSC λ ∏∏
−

=

−

=

+++∗
1 1

0

)/ˆ1(
N

Li

M

j

K
n

j
i

mjiMd λ       (23) 

 
c) Case III  .10,10, −≤<−−≤≤+= MkLNqkqMm  

)ˆ,,|Pr( ddmK

∏
−−

=

− −−∗
kM

j

Td qNe
1

0

'1 (

∏∏
−

=

−

=

+++=
1

0

1

0
1

' )/1(
L

i

M

j

K
n

j
i

T mjiMs SCe λλ

−−
++−+

K
n

j
qN

jkMNd 1
)1()/1 λ

1 ˆ
1∏

−

−=

− −−∗
M

kMj

d Ne

∏ ∏
−−

=

−

=

+++∗
qN

Li

M

j

K
n

j
i

mjiMd
1 1

0

)/1( λ

.)/ˆ1( 1
' )1(

−−
++−+

K
n

j
qN

T jkMNq d λ

∏ ∏
−

−=

−

=

+++∗
1 1

0

)/ˆ1(
N

qNi

M

j

K
n

j
i

mjiMd λ

         (24) 

 
We next use (22) - (24) to obtain 

 
          (25) ∑∑ −−=

d d

LN ddmKMmK
ˆ

)(2 ).ˆ,,|Pr()|Pr(

In order to do this, define 
,)(12

LNMCC −−=        (26) 
 
which is independent of m.  Also define for each i, 10 −≤≤ Li , 
 

,pattern  sync  theofsymbolth -i for theposition slot  pulse )(ˆ Sij =   (27) 
and 

)./1( nsx λλ+=                     (28) 

 161



EESA M. BASTAKI and HARRY H. TAN 

It then follows that 

,)/1( )(ˆ
1

0

mijiMmjiM KK
n

M

j

j
i xS

++++ =+∏
−

=

λ          (29) 

 

∑∏ ++
−

=

+
i

mjiM

d

K
n

M

j

j
id )/1(

1

0

λ ,
1

0
∑
−

=

++=
M

j

K mjiMx              (30) 

 

∑ ∏
−−

++−−−
−−

=
−−

−
+

qN

jkMN
j

qN

d

kM

j

K
n

j
qN

Td de
1

)1(1
1

0
1

' )/1( λ kxe
kM

j

KT jkMNs += ∑
−−

=

− ++−
1

0

' )1(λ  (31) 

and 

∑ ∏
−−

++−−−

−−

−

−=

− +
qN

jkMNqN

d

K
nqN

M

kMj

Td de
1

)1(1

ˆ
1

1
'ˆ )/ˆ1( λ ).(

1
' )1( kMxe

M

kMj

KT jkMNs −+= ∑
−

−=

− ++−λ   (32) 

 
Using (29) - (32) in (25) then yields a relatively simpler expression for )|Pr( mK .  We express the 
results for the three above cases below in terms of  
 

2ln)|Pr(ln)( CmKmL −=      (33) 
 
For the ML rule, it is equivalent to choose m to maximize L(m) since C2 is independent of m.  The 
expressions for L(m) are identical in cases I and II, but different in case III. 
 
a) Case I and Case II Either m(mod M) = 0 or 1)( −≤≤− NMmMLN  
 

∑
−

=
++=

1

0
)(ˆ)ln()(

L

i
mijiMKxmL ∑ ∑

−

=

−

=

+++
1 1

).ln(
N

Li

M

oj

K mjiMx           (34) 

 
b) Case III 1)(0 and 0M) mod( −−≤≤≠= MLNmkm . 
 

∑ ∑∑
−

=

−

=

−

=
++ 















+= ++

1 1

0

1

0
)(ˆ ln)ln()(

N

Li

M

j

K
L

i
mijiM

mjiMxKxmL

ln(')(ln
1

' )1( TkMxe s
M

kMj

KT jkMNs −+















−+
















+ ∑

−

==

− ++− λλ












+








+








− ∑∑

−−

=

−
−

=

++−++− kxex
kM

j

KT
M

j

K jkMNsjkMN
1

0

'
1

0

)1()1( lnln λ

)M               (35) 

 
The ML rule is to choose m, 10 −≤≤ NMm  to maximize L(m), where L(m) is given by (34) for 

cases I and II, and by (35) for case III.  The expression for L(m) differs in case III from cases I and 
II because of the two partial symbols at the ends of the frame.  Note from (34) and (35) that 
computation of L(m) for each m uses the counts corresponding to all N symbols in the frame.  
Georghiades' incorrect ML rule is based on a likelihood formula which for each m uses only the 
counts corresponding to the L symbols of the sync pattern.  Since a low sync pattern overhead is 
desired for high data rate throughputs, the frame length N is often significantly larger than L in 
practice.  Moreover, reasonably large values of L are usually employed to reduce the probability of 
replication of the sync pattern in the random data portion of the frame.  Hence it is meaningful to 
consider the case when N becomes large relative to both L and M.  In particular, the complexity of 
implementing the decision rule should be examined. Table 2 gives the total number of 
computations in terms of the number of addition, multiplications, integer power, logarithm and 
exponential function evaluations to compute L(m) for all m. 

 162



JOINT SYMBOL AND FRAME SYNCHRONIZATION  

In this table, there are a total of N+L(M-1) m's for cases I and II and N(M-1)-L(M-1) m's for case 
III.  The total number of computations is evaluated in cases I and II for all N+L(M-1) m's and also 
in case III for all N(M-1)-L(M-1) m's. 

It can be seen from this table that, asymptotically for large N and fixed L and M, the 
complexity of implementing the ML rule based on computing L(m) is of order .  Georghiades' 
incorrect ML instead has an asymptotic complexity of order N.  In order to obtain this reduced 
complexity Georghiades makes the incorrect assumption that A(m) is independent of m for all 

, where 

2N

10 −≤≤ NMm

.ln)(
1

0

1

0
∑ ∑
−

=

−

=








= ++

N

i

M

j

K mjiMxmA              (36) 

 
Table 2: Total number of computations for ML rule. 

 
Total Number of Computations in Cases I and II 

Computation Total Number 

Addition [NM-LM+L-1][N+L(M-1)] 

Multiplication N+L(M-1)   

Integer Power [NM-LM][N+L(M-1)] 

Logarithm [N-L+1][N+L(M-1)] 

Total Number of Computations in Case III 

Computation Total Number 

Addition [(N-L)M+L+5][N(M-1)-L(M-1)]   

Multiplication 3[N(M-1)-L(M-1)] 

Integer Power [NM-LM][N(M-1)-L(M-1)] 

Logarithm [N-L+3][N(M-1 (- L(M-1)] 

Exponential 2[N(M-1)-L(M-1)] 

 
 
Suppose this assumption is true.  Then the ML rule can be based on choosing m to maximize L(m) 
- A(m).  The expressions for L(m)-A(m) in cases I, II and III are as follows. 
 
a) Case I and Case II Either m(mod M) = 0 or 1)( −≤≤− NMmMLN . 
 

)()( mAmL − .ln)ln(
1

0

1

0

1

0
)(ˆ ∑ ∑∑

−

=

−

=

−

=
++ 













−= ++

L

i

M

j

K
L

i
mijiM

mjiMxKx            (37) 

 
b) Case III  1)(0 and 0M) m(mod −−≤≤≠= MLNmk . 

)()( mAmL −

ln 'e
j

Ts













+ −λ

∑ ∑∑
−

=

−

=

−

=
++ 













−= ++

1

0

1

0

1

0
)(ˆ ln)ln(

L

i

M

j

K
L

i
mijiM

mjiMxKx

)ln(')(
1

)1( MTkMx s
M

kM

K jkMN −+






−+







∑
−

−=

++− λ














+














+














− ∑∑

−−

=

−
−

=

++−++− kxex
kM

j

KT
M

j

K jkMNsjkMN
1

0

'
1

0

)1()1( lnln λ

                      (38) 

 163



EESA M. BASTAKI and HARRY H. TAN 

The asymptotic computational complexity of computing all of the L(m)-A(m) is of order N.  
Unfortunately, A(m) is not independent of m for all 0 1−≤≤ NMm .  Hence the ML rule cannot be 
implemented by choosing m directly to maximize L(m)-A(m). 

2.2   Simplified ML Rule 

Although A(m) is not independent of m for all 0 1−≤≤ NMm
0

, it is independent of m for 
restricted sets of integers m.  Specifically, define for each k, 1−≤≤ Mk , 

 
}.) (mod and 10:{ kMmNMmmBk =−≤≤=    (39) 

 
It can then be seen from (36) that for each k, 0 1−≤≤ Mk , A(m) is independent of m for all 
. This property suggests the following approach towards reducing the number of 

computations to implement the ML rule.  In order to describe this approach define 
kBm ∈

 
{ })(max 1 xf

Ax

−

∈
         (40) 

 
to be the value of x achieving the following maximum: 
 

{ })(max xf
Ax∈

        (41) 

 
 
First note that since A(m) is independent of m for kBm ∈ , 
 

{ })(max 1 mLm
kBm

k
−

∈
= { })()(max     1 mAmL

kBm
−= −

∈
          (42) 

But 

{ } { }






=

∈

−

−≤≤

−

−≤≤
)(maxmax)(max 1

10

1

10
mLmL

kBmMkNMm
{ })(max  1

10
k

Mk
mL−

−≤≤
=   (43) 

 
 
Equations (42) and (43) suggest the following two-step implementation of the ML rule. 
 
Step 1 
    For each k, , determine 10 −≤≤ Mk

{ })()(max m    1k mAmL
kBm

−= −
∈

.         (44)  

 
Step 2 
    The ML decision rule is the value  given by m̂
 

{ }.)(maxˆ 1
10

k
Mk

mLm −
−≤≤

=                   (45) 

 
The advantage of this two step approach is because the asymptotic complexity (fixed M, L 

asymptotic in N) of computing the [L(m)-A(m)]'s for all kBm ∈  is of order N for each k.  So the 
asymptotic complexity of step 1 is of order N.  The asymptotic complexity of computing each of 
the  in step 2 is also of order N.  So the total number of computations involved in 
implementing steps 1 and 2 is asymptotically of order N.  Table 3 gives the exact number of 
additions multiplications, integer powers, exponential and logarithm function evaluations.  We 
shall call this rule the simplified ML rule. 

smL k )'(

 164



JOINT SYMBOL AND FRAME SYNCHRONIZATION  

Table 3: Total number of computations for simplified ML rule. 
 

Total Number of Computations for Step 1 

Computation Total Number 

Addition (N-L)(M-1)[M(L+2)+L+4]+[N+(M-1)L](ML+L-1) 

Multiplication [3(N-L)+L](M-1)+N 

Integer Power (N-L)(M-1)M(L+2)+[N+L(M-1)]ML 

Logarithm (N-L)(M-1)(L+5)+[N+(M-1)L](L+1) 

Exponential 2(N-L)(M-1) 

Total Number of Computations for Step 2 

Computation Total Number 

Addition [(N-L)M+L+5](M-1)+(N-L)M+L-1   

Multiplication 3M-2 

Integer Power [(N-L)M](M-1)+(N-L)M 

Logarithm (N-L+3)(M-1)+N-L+1 

Exponential 2 (M-1) 

 
2.3    Sub-optimum Rule 

The incorrect Georghiades ML rule is invalid because it does not consider the case III 
expression for L(m) and also because it assumes the constancy of A(m) for all m. We next consider 
sub-optimum rule which assumes just the constancy of A(m).  This rule then chooses m to achieve 
the following maximum, 

{ }.)()(max
10

mAmL
NMm

−
−≤≤

          (46) 

 
Similar to the above discussion on the simplified ML rule, the asymptotic complexity of this 

sub-optimum rule is of order N.  It is interesting to consider this rule because it only assumes the 
constancy of A(m) for all m and does not also ignore the case III expression for L(m) as does 
Georghiades' incorrect ML rule.  Hence, in comparing the performance between the ML rule, the 
sub-optimum rule and Georghiades' incorrect ML rule, it is possible to access the performance 
deteriorations due to these two incorrect assumptions.  The section below discusses the relevant 
performance results. 

3. Simulation Results 

Computer simulation was used to evaluate the correct synchronization probability 
performance.  Figures 1 to 6 give the simulation results on the correct synchronization probability 
of the ML, sub-optimal and incorrect ML rules for (N,L,M)=(4,2,2), (8,2,4), (12,2,8), (12,4,8), 

 165



EESA M. BASTAKI and HARRY H. TAN 

(12,6,8) and (20,4,4) respectively.  These Figures give the correct synchronization probability as a 
function of the average signal count 'Tsλ  per slot for an average noise count per slot 4'=Tnλ  
photons.  It can be seen that the true ML rule performs from 3 to 5 db better than the incorrect ML 
rule.  The performance of the sub-optimal rule is intermediate between that of the ML rule and the 
incorrect ML rule.  In fact the performance of the sub-optimal rule is closer to that of the ML rule 
than to that of the incorrect ML rule.  This suggests that the constancy of the A(m) is a more robust 
assumption than ignoring the case III  expression for L(m). 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 166



JOINT SYMBOL AND FRAME SYNCHRONIZATION  

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 

References 

BASTAKI, E.M., EESA, M. 1989. “Frame Synchronization in Communication Systems,” Ph.D. 
Dissertation, University of California, Irvine, U.M.I., Ann Arbor, MI. 

GEORGHIADES, C.N. 1985. "Joint Baud and Frame Synchronization in Direct-Detection Optical 
Communication," IEEE Transactions on Communications, vol. COM-33, pp. 357-360, April 
1985. 

 
 
 
Received 28 June 2001 
Accepted 16 January 2002   
 

 
 
 
 
 
 
 
 

 167


	Joint Symbol and Frame Synchronization for Direct- Detection Optical Communication Systems
	Eesa M. Bastaki* and Harry H. Tan**
	
	
	
	
	*Department of Electrical Engineering, Faculty of Engineering, United Arab Emirates University, P.O.Box 17555, Al- Ain, U.A.E, Email: eesa@uaeu.ac.ae, **Department of Electrical Engineering, School of Engineering, University of California, Irvine, Ca, 92





	Joint Symbol and Frame Synchronization Problem
	2.1     ML Rule
	2.2   Simplified ML Rule
	2.3    Sub-optimum Rule

	Simulation Results
	References