Optics and Photonics Journal, 2013, 3, 171-174
doi:10.4236/opj.2013.32B041 Published Online June 2013 (http://www.scirp.org/journal/opj)
A Novel Method with Martingale Theory for Phase
Noise Analysis in Coherent Optical Communication
Chengle Sui1, Qiangmin Wang2, Shilin Xiao1, Pingqing Li1
1State Key Lab of Advanced Optical Communication System and Networks, Shanghai Jiao Tong University, Shanghai, China
2School of Information Security Engineering Shanghai Jiao Tong University, Shanghai, China
Received 2013
ABSTRACT
Phase noise has a great influence on the performance of coherent optical communication. In this paper, martingale the-
ory is introduced to analyze the phase noise effect for the first time as far as we know. Through Fubini’s Theory and
martingale representation th eory, we proved that 0
1exp()( )
t
s
jWlsds
T
, which denotes the phase noise effect, is a pre-
dictable martingale. Then Ito’s formula for solution to stochastic differential equation is utilized for the analysis of
phase noise effect. Using our method, a nonrecursive formula for the moments of phase noise is derived and signal-
noise-ratio (SNR) degradation in coherent optical OFDM due to phase noise is calculated with our method.
Keywords: Martingale; Brownian Motion; Ito Itegral; Derivation of Moments; Phase Noise
1. Introduction
Phase noise greatly limits the performance of coherent
optical communication [1], especially on the condition
that recent advances in modulation formats and multi-
symbol detection are more and more widely introduced
in optical communication system nowadays[2-3]. There-
fore, the impact of phase noise has been investigated in a
lot of great works. Generally, evaluating the impact of
phase noises is difficult, because it’s too hard to fully
assess the stochastic property of laser phase noise (LPN)
Wiener process and nonlinear effect induced phase noise
accurately.
There are mainly two methods to evaluate phase noise
effect: alternative moment method [4] and perturbation
method [5]. Both of these two methods try to find a cer-
tain kind of expansion to approach the random process.
Thus, the two methods will either be ineffective in some
cases [6] or not accurate enough [7].
In this paper, we introduce martingale theory to ana-
lyze phase noise. Random process 0
1exp()( )
t
s
jWlsds
T

is proved to be a predictable martingale, which is meas-
urable with respect to proper nature filtration. Thus, the
process can be substituted to Ito’s formula for solution to
stochastic differential equation. Then we take advantage
of Ito’s formula to derive a non-recursive formula for the
moments of phase noise. And the SNR degradation caused
by LPN is approximated th rough Ito isometry. These two
works are examples of the application of martingale the-
ory for the analysis of phase noise in coherent optical
communication.
2. Model and Theory
Phase noise, frequency offset and nonlinear noise are the
significant impairments of coherent optical communica-
tion. The signal before detection can be described as




Reexp (2
tktNeff
EAjtW n  
L
(1)
where k

is the phase of modulation,

represents the
frequency offset and t
W

is a Wiener-Levy process which
is related to LPN. The stochastic characteristics of Wie-
ner-Levy process in a laser are known:
12
212 12
2
tt
WW tt tt  

  (2)
Here ()
H
z

denotes full 3dB line width of laser.
NL

and eff n corresponds to nonlinear noise and addi-
tive Gaussian noise respectively. According to [8], for
the frequency offset C


, random process 2t tW

 

t
tW    is a Brownian motion with drift and
denotes a geometric Brownian motion. As it is stated in
[4-5], if we simply look into the impact of frequency
offset and LPN, phase noise effect can be denoted as
exp 2

0
1exp 2
T
s
s
Wds
T
  
.
Our idea starts from the equation stated in [7]

C. L. SUI ET AL.
172
0
11
exp()( )exp( )
T
s
jWlsdsjWlsds
TT
0
T
s


 





(3)
00
1
exp()( )exp( )
n
T
ss
jWlsdsjnWls ds
TT

 



1
T



(4)
In coherent optical communication systems, the sym-
bol period is very small because of high-speed transmis-
sion. Thus, (3-4) can be quite accurate according to integral
property. If the time window is a rectangle over
the whole symbol period, which is most frequently used
in reality, we could simply to investig ate the integral of a
()ls
Brownian motion 0
1t
ts
YW
T
ds.
Let 0
1
exp 2t
t
SjtWd
T
 



 


s
s



. Now we will
prove that the real rand process 0
1t
s
Wds
T is a predict-
able martingale, and so that t, the geometric random
process, can be adapted to Ito Formula.
S
Stochastic Fubini’s Theorem: The stochastic process
s
W is a martingale with respect to a filtration


0
tt



and probability measure P. Then the underlying filtration
probability sp ace can be repre sented as




0, . t 
0
,, t  

tt
If is a bounded random
variable, and is predictable for filtration
(, ,):trw RRR






, then
for 0T
[0, ][0, ]
00
(,,) ()(,,) ()
tt
T
RR
Ts
s
rw lrdrdWsrw lrdWdr
 (5)
Utilizing (10), yielding
000
00
11
1
11
1()
tts
ts s
tt t
s
s
s
YW dsdW ds
TT
dsdWts dW
TT



 
(6)
Now the equation above can be utilized to prove
0
1t
s
Wds
T is a predictable martingale.
Martingale Representation Theorem: Let Ws be a
Brownian motion on a standard filtered probability space



0
,, ,
ttP


 

if Mt is a square integrable random
variable measurable with respect to 

0
tt

, then there
exists a predictable process ϑt which is adapted to


0
tt

 such that
00
t
ts
M
M  
ds
(7)
Therefore, the stochastic process 0
1t
s
Wds
T (i.e.
0
1t
s
Wds
T according to (16)) can be represented by Ito
integral, which is called as predictable representation
property[13]. It means the geometric random process St
can also be a solution of Ito’s formula.
Ito’s Formula: let partial derivative of function exits
and is continuous, then
(, )'(, )(, )
1
''(,)
2
ttt
t
dft WftWdWftWdt
ftWdt


t
(8)
By substituting f with , yielding
t
S
2
2
(2 )
2
ttt
tt
dSjS dWjS dt
TT
 
t
 (9)
3. Applications
In the section, the analysis above will be used to intro-
duce some meaningful utilization in optical coherent
communication, including moments of phase noise, SNR
degrade due to LPN. These discussions here will reveal
the way to apply martingale theory for analyzing phase
noise effect.
3.1. Non-recursive Formula for the Moments of
Phase Noise
Reference [9] proved a recursive formula for the mo-
ments of phase noise. Method of moments is very im-
portant in performance evaluation of digital communica-
tion systems [6]. Here a different approach is revealed to
get a non-recursive formula for the moments of phase
noise. By substituting (4) into (6), a new stochastic dif-
ferential equation can be denoted as
22
2
2(1)
2
nn
ttt
nt t
dSjSdWjnnnSdt
TT
 

 


n
t
(10)
Integrating and then taking expectation on both side of
the equation, yielding
22
02
0
1
2(1)
2
t
nn n
t s
E
SESjnnnsESd
T
 

s

 
 


 


(11
Since the expectation of Wiener process is zero
so
)
, by
lving the integral equation (11), we can get
22
02
2
2
1
exp2( 1)
2
(1)
exp 2
nn
t
ESSjnnnt t
T
nn t
jn t
T
 







 
 







 
 
(12)
(Note: 000
1
exp 21
t
s
t
SjtWds
T
 





 


)
The result is much simpler than that of [9], and it’s
easy for calculation. Figure 1 shows the first three mo

C. L. SUI ET AL. 173
Figure 1. The first three moments as function of laser lin
ents as function of laser line width. We can observe
3.2. SNR Degradation Due to Phase Noise in
Cohe systems are very sensitive to the
t
e
width.
m
that the expectation value of both recursive and non-re-
cursive formula is almost the same except that the third
moments of recursive solution suffers fierce shock when
laser line width is below 400 kHz. It’s quite unreasonable
because phase noise effect should be smaller if laser line
width degrades. Therefore, the non-recursive formula is
more effective than the recursive formula given by [9].
Optical OFDM
rent optical OFDM
phase noise of laser. The impact of phase noise in wire-
less OFDM has been widely evaluated, including [10-11].
Here, we show a totally different method to give an
evaluation on phase noise effect in optical OFDM system.
As it’s widely known, if we simply investigate the effect
of phase noise, the signal before deciding could be writ-
ten as
0,kk k
x
aI n (13)
where 00
1exp( )
T
s
I
jW ds
T
, and the SNR degradation
se noise can be denduo to phaoted by [10]
2
0
10 (1 )(1)
ln10
DESNR (14)
Thus, the key point is to calculate 2
00
()EEI. Ex-
pand (3) by Taylor series, yielding
2
000
1
exp( )12
T
sss
jWdsW dsWds
TTT


(15)
Therefore, we can firstly get
1
TT
j
22
22
1
T
000
2
2
00
1
() 1
2
11
1
T
ss
TT
ss
EI EWdsWds
TT
EWdsEW ds
TT
 
 

















(16)
Now Ito isometry can be applied to calculate (15). Ito
isometry: if f is an elementary function, then




00
,,
s
EfswdWEf swds






22
tt
 (17)
Therefore, the derivation for the mean variance of is

2
0
23
22
0
1T
[] ( )
11
3
ts
t
EYEts dW
T
tsds t
TT













(18)
So (16) can be simplified as

22
01
16 EIT

 .
Similarly, we can get

22
01
60
I
T

 . And from
get that Ito’s Formula, it’s easy to2 s
2
s
EW

. So the
parameter will be



I.
2
22
00
EEI  
0
into (14), the SNR deadation due
to Substituting 2
0
Egr
phase noise in coherent optical OFDM can be ex-
pressed by

11 2
6ln10
DNT   SNR (19)
The result fits well with [10], yet the method is much
simpler. Figure 2 shows the relationship between laser
width and SNR degradation when the tran smission rate is
set at 40Gbps with 64 carrier frequencies. If BER is set
to be 7 10

, the value of SNR for M-QAM and M-PSK
are appmated by 10( 1)M roxi

and 2 15/ sin(/)
M
 [10].
Figure 2. SNR degradation as function of laser line width.

C. L. SUI ET AL.
174
It
4. Conclusions
ngale theory was introduced to ana-
’s easy to observe that MPSK modulated signal suffers
more SNR degradation due to phase noise. And with the
increasing of signal constellation’s size, the signal be-
comes more and more sensitive to the phase noise. If the
laser line width is 1MHz and the modulation format is
64QAM, extra 2.5dB SNR needs to be compensated for
the phase noise effect.
In the paper, Marti
lyze the effect of phase noise for the first time. Through
stochastic Fubini’s theorem and martingale representa-
tion theorem, we proved the process 0
1tWds is a pre-
s
T
dictable martingale which can be applied to Ito Form
5. Acknowledgements
orted by the Nationa
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