A Novel Method with Martingale Theory for Phase Noise Analysis in Coherent Optical Communication

doi:10.4236/opj.2013.32B041

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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

Email: [email protected], [email protected]

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()( )

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()( )

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





(1)

where k



is the phase of modulation,



represents the

frequency offset and t



is a Wiener-Levy process which

is related to LPN. The stochastic characteristics of Wie-

ner-Levy process in a laser are known:

212 12

WW tt tt

 



 

(2)

Here ()



denotes full 3dB line width of laser.



and eff

n corresponds to nonlinear noise and addi-

tive Gaussian noise respectively. According to [8], for

the frequency offset C





, random process 2t









 



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



1exp 2

Wds

 



.

Our idea starts from the equation stated in [7]

C. L. SUI ET AL.

172

exp()( )exp( )

jWlsdsjWlsds





 











(3)

exp()( )exp( )

jWlsdsjnWls ds



 













(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

ds.

Let 0

exp 2t

SjtWd

 















s







. Now we will

prove that the real rand process 0

Wds

T is a predict-

able martingale, and so that t, the geometric random

process, can be adapted to Ito Formula.

Stochastic Fubini’s Theorem: The stochastic process

W is a martingale with respect to a filtration











and probability measure P. Then the underlying filtration

probability sp ace can be repre sented as









0, .

t



,, t

 



If is a bounded random

variable, and is predictable for filtration

(, ,):trw RRR













, then

for

0T

[0, ][0, ]

(,,) ()(,,) ()

rw lrdrdWsrw lrdWdr

 (5)

Utilizing (10), yielding

000

1()

tts

ts s

tt t

YW dsdW ds

dsdWts dW







 

(6)

Now the equation above can be utilized to prove

Wds

T is a predictable martingale.

Martingale Representation Theorem: Let Ws be a

Brownian motion on a standard filtered probability space







,, ,

ttP





 



if Mt is a square integrable random

variable measurable with respect to





tt



, then there

exists a predictable process ϑt which is adapted to





tt



 such that





ds

(7)

Therefore, the stochastic process 0

Wds

T (i.e.

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

(, )'(, )(, )

''(,)

ttt

dft WftWdWftWdt

ftWdt





t

(8)

By substituting f with , yielding

(2 )

ttt

dSjS dWjS dt





 (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

2(1)

ttt

nt t

dSjSdWjnnnSdt

 



 





(10)

Integrating and then taking expectation on both side of

the equation, yielding

2(1)

nn n

t s

SESjnnnsESd

 





 

 





 





(11

Since the expectation of Wiener process is zero

)

, by

lving the integral equation (11), we can get

exp2( 1)

(1)

exp 2

ESSjnnnt t

nn t

jn t

 















 

 























(12)

(Note: 000

exp 21

SjtWds

 



















)

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

width.

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

aI n (13)

where 00

1exp( )

jW ds

, and the SNR degradation

se noise can be denduo to phaoted by [10]

10 (1 )(1)

ln10

DESNR

(14)

Thus, the key point is to calculate 2

()EEI. Ex-

pand (3) by Taylor series, yielding

000

exp( )12

sss

jWdsW dsWds

TTT





(15)

Therefore, we can firstly get

000

() 1

EI EWdsWds

EWdsEW ds

 

 

































(16)

Now Ito isometry can be applied to calculate (15). Ito

isometry: if f is an elementary function, then









EfswdWEf swds











 (17)

Therefore, the derivation for the mean variance of is



[] ( )

EYEts dW

tsds t

























(18)

So (16) can be simplified as



EIT



 .

Similarly, we can get





. And from

get that Ito’s Formula, it’s easy to2



. So the

parameter will be







EEI





into (14), the SNR deadation due

to Substituting 2

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



, the value of SNR for M-QAM and M-PSK

are appmated by 10( 1)M

roxi



and 2

15/ sin(/)



[10].

Figure 2. SNR degradation as function of laser line width.

C. L. SUI ET AL.

174

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-

T

dictable martingale which can be applied to Ito Form

5. Acknowledgements

orted by the Nationa

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