Nonlinear phase noise in coherent optical OFDM transmission systems.
ABSTRACT We derive an analytical formula to estimate the variance of nonlinear phase noise caused by the interaction of amplified spontaneous emission (ASE) noise with fiber nonlinearity such as selfphase modulation (SPM), crossphase modulation (XPM), and fourwave mixing (FWM) in coherent orthogonal frequency division multiplexing (OFDM) systems. The analytical results agree very well with numerical simulations, enabling the study of the nonlinear penalties in longhaul coherent OFDM systems without extensive numerical simulation. Our results show that the nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM, which is in contrast to traditional WDM systems where ASEFWM interaction is negligible in quasilinear systems. We also found that fiber chromatic dispersion can reduce the nonlinear phase noise. The variance of the total phase noise increases linearly with the bit rate, and does not depend significantly on the number of subcarriers for systems with moderate fiber chromatic dispersion.

Article: Fiber nonlinearity mitigation by improved zerointerpolation method in coherent optical OFDM systems
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ABSTRACT: An improved zerointerpolation method with signal clipping to improve fiber nonlinearity tolerance in the longhaul coherent optical orthogonal frequency division multiplexing system (COOFDM) is presented. The new technique is implemented by interpolating zero subcarriers and selecting the odd subcarriers to carry data and clipping conventional COOFDM signal at zero. With such a scheme, the effect of fiber nonlinearity can be mitigated, and the wanted signal carried on odd subcarriers is orthogonal to clipping noise, which falls on even frequencies. Simulation shows that the system Q value is improved by more than 2 dB at the length of 960 km Key wordscoherent optical communication–fiber nonlinearity–signal clipping–orthogonal frequency division multiplexing (OFDM)Wuhan University Journal of Natural Sciences 01/2011; 16(1):3842.
Page 1
Nonlinear phase noise in coherent optical OFDM
transmission systems
Xianming Zhu1,* and Shiva Kumar1,2
1Corning Incorporated, SPTD011, Corning, NY 14831 USA
2On leave from Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada
*zhux@corning.com
Abstract: We derive an analytical formula to estimate the variance of
nonlinear phase noise caused by the interaction of amplified spontaneous
emission (ASE) noise with fiber nonlinearity such as selfphase modulation
(SPM), crossphase modulation (XPM), and fourwave mixing (FWM) in
coherent orthogonal frequency division multiplexing (OFDM) systems. The
analytical results agree very well with numerical simulations, enabling the
study of the nonlinear penalties in longhaul coherent OFDM systems
without extensive numerical simulation. Our results show that the nonlinear
phase noise induced by FWM is significantly larger than that induced by
SPM and XPM, which is in contrast to traditional WDM systems where
ASEFWM interaction is negligible in quasilinear systems. We also found
that fiber chromatic dispersion can reduce the nonlinear phase noise. The
variance of the total phase noise increases linearly with the bit rate, and
does not depend significantly on the number of subcarriers for systems with
moderate fiber chromatic dispersion.
©2010 Optical Society of America
OCIS codes: (060.2330) Fiber optics communications; (190.4380) Nonlinear optics, fourwave
mixing
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1. Introduction
Coherent optical orthogonal frequency division multiplexing (OFDM) has drawn significant
attention in optical communications due to its high spectral efficiency using hundreds of
subcarriers with higherorder modulation formats and its robustness to fiber chromatic
dispersion and polarization mode dispersion [1–5]. However, due to the large number of
subcarriers, OFDM is believed to suffer from high peaktoaverage power ratio, which makes
it less suitable for legacy optical communication systems with periodic inline chromatic
dispersion compensation fibers [6]. In Ref [7], a simple formula for estimating the
deterministic distortions caused by fourwave mixing (FWM) is developed, and it is found
that the nonlinear limit in OFDM systems is independent on the number of OFDM subcarriers
in the absence of dispersion. Ref [8]. analytically studied the combined effect of dispersion
and FWM in OFDM multispan systems and concluded that dispersion could significantly
reduce the amount of FWM. Recently, significant research effort has been put in nonlinear
compensation for coherent OFDM systems [9–16]. Of particular interest is the digital
backward propagation [14–16], a technique in which the signal is propagated backwards in
distance using digital signal processing (DSP) so that the deterministic linear and nonlinear
impairments can be compensated. However, the nonlinear phase noise caused by the
interaction between amplified spontaneous emissions (ASE) noise and fiber Kerr nonlinearity,
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also known as GordonMollenauer effects [17], cannot be compensated using digital
backward propagation. Nonlinear phase noise has been studied extensively for singlecarrier
systems [17–32]; however, to the best of our knowledge, nonlinear phase noise effects have
not been investigated for OFDM systems.
In this paper, we derive an analytical formula to calculate the nonlinear phase noise
induced by the interaction of ASE with SPM, XPM and FWM in coherent OFDM
transmission systems. The analytical model is verified with numerical simulation results,
enabling the study of the nonlinear phase noise in coherent OFDM systems without lengthy
simulations. With the analytical model, we quantitatively compare the nonlinear phase noise
induced by SPM, XPM and FWM, separately, and find that the nonlinear phase noise induced
by FWM is dominant compared to that induced by SPM and XPM. This is in contrast to the
results of Ref [27]. for WDM systems, in which it is found that ASEFWM interaction is
negligible in quasilinear systems. This difference is likely due to the subcarriers of OFDM
systems interacting coherently, since they are derived from the same laser source. We also
study the effects of fiber chromatic dispersion and the bit rate on the total phase noise in
coherent OFDM systems and find that, the total phase noise decreases as fiber chromatic
dispersion increases, achieving the limit of linear phase noise. The total phase noise scales up
as bit rate increases.
The remainder of the paper is organized as follows. Section 2 describes the mathematical
analysis of the nonlinear phase noise. In section 3 we show the validation of the mathematical
model with numerical simulation of coherent OFDM systems, and study the impact of fiber
chromatic dispersion, number of subcarriers, and bit rate on the variance of the total phase
noise. Section 4 gives the conclusion.
2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems
The nonlinear Schrödinger equation governing light propagation in optical fiber is [33]
2
2
2
2
( )
z
( , )
u t
∂
∂
( , )
u t
∂
exp[
γ
( )] ( , )
w
z
( , )
u t
0,
2
j u t
t
β
∂
−+−=
zz
zz
z
(1)
where
2( )
β z is the dispersion profile, γ is the nonlinear coefficient,
α z is the fiber loss/amplifier gain profile.
There are large numbers of subcarriers in OFDM systems, making each subcarrier a quasi
cw wave due to low bit rate. The OFDM signal can be described as [8]
0
( )
z
( )
s ds
α
w
=∫
z
,
and ( )
()
/2 1
−
/2
( , )
u t
( , )exp
u t
z
,
N
ll
lN
jt
ω
=−
= ∑
z
(2)
where N is the total number of subcarriers,
ωπ
=
is the frequency offset from a reference, with
time. First we derive the analytical formula for the variance of nonlinear phase noise
including the interaction of ASE noise with SPM and XPM. Second, we include the nonlinear
phase noise variance induced by FWM.
( , )
lu t z is the slowly varying field envelope, and
T
2
lblock
l T
block
as the OFDM symbol
2.1 SPM and XPM induced nonlinear phase noise
Inserting Eq. (2) into (1) and considering the effects of SPM and XPM only, we have
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2
2
l
22
2
2
22
22
exp[
γ
( )]
w z
2.
lll
lll
lml
m l
≠
uu
∂
u
ju
tt
uuu
ββ
β ωωω
∂
∂
∂∂
∂
−−+=
−−+
∑
z
(3)
Within each OFDM block,
lu is constant; therefore, the first and second order derivative
of
(3) can be written as
lu with respect of time, appearing in Eq. (3), can be ignored. Now the exact solution of Eq.
22
2
l
2
( , )
u t
( ,0)exp
u t
( )
z
2,
2
llefflm
m l
≠
jj L
γ
uu
β
ω
=++
∑
zz
(4)
where
1 exp(
−
)
( )
z
.
eff
L
α
−
α
=
z
(5)
We define
sL as the fiber span length, and the signal is periodically amplified by
amplifiers located at
κ
, κ = 1, 2, …, M, where M is the total number of fiber spans.
sL
Consider the noise added by the amplifier located at
sL
κ
. Let us expand the noise field as a
discrete Fourier transform
/2 1
−
/2
( )exp( ).
N
ll
lN
n tnjt
ω
=−
= ∑
(6)
Strictly speaking, the noise field should be expressed as a Fourier transform instead of a
discrete Fourier transform. In other words, we have approximated the noise as a field with 2N
degrees of freedom (DOF) instead of infinite degrees of freedom. For a linear system that
employs matched filter at the receiver, 2N DOFs accurately describe the noise process [32]. In
Ref [17]. it is argued that 2 DOFs per carrier (or total 2N DOFs) is sufficient to describe the
noise field even in a nonlinear system. The total field immediately after the amplifier located
at
κ
is
sL
()
/2 1
−
∑
/2
( ,) ( ,)exp().
N
slsll
lN
u tLu tLnj t
ωκκ
+
=−
=+
(7)
Let
[]
22
2
l
2
( ,
t
) ( ,)
( ,0)
u t
' exp
⋅
/ 2()2.
lslsl
llseffslm
m l
≠
uL u tLn
njL j L
γ
Luu
κκ
β ω κκ
+
=+
=+++
∑
(8)
We assume that the ASE noise is a white noise process with power spectral density
from which it follows that,
ASE
ρ
,
*
'
*
,
',
'' 0,
ASE
lklk l k
block
T
lk
nnn n
nn
ρ
δ
⋅=⋅=
⋅=
(9)
where
, l k
δ
is the Kronecker delta function,
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,
1if
.
0 otherwise.
l k
lk
δ
=
=
(10)
Now treating
( ,
t
)
ls
uL
κ
+
as the initial field, Eq. (3) is solved to obtain the field at the end
=
z
, as
{
' exp()(
lDeff
njjML
of the optical system, located at
s
ML
[]
()
}
**
( ,
t ML
))' ' ,
lslsllll
uuLu nu n
γκ
+
=+ Φ +⋅−⋅+
(11)
where
D
Φ is the deterministic phase shift caused by dispersion, SPM and XPM, which has no
impact on the nonlinear phase noise, and is expressed as
22
2
l
2
/ 2()2.
Ds effslm
m l
≠
MLML
γ
Luu
β ω
Φ =++
∑
(12)
The linear phase noise is embedded in the term
'
ll
un
+
κ
(
u n
, and the nonlinear phase noise
caused by SPM and XPM by the amplifier located at
sL
=
z
is:
()
)
*
'
*
,
() ' .
SPMXPMeffsllll
MLL u n
κ
δγ
κ
+
Φ=⋅−⋅+
(13)
Squaring Eq. (13) and making use of Eq. (9), we obtain the variance of the nonlinear
phase noise caused by SPM and XPM
22
2
SPM
222
eff
,
2()()2.
ASE
XPMslm
m l
≠
block
T
N (equivalently the
MLLuu
κ
ρ
δ
γκ
+
Φ=⋅−+
∑
(14)
Assuming that the number of subcarriers carrying data is
e
oversampling factor is /
e
N N ) and each subcarrier has equal power, summing Eq. (14) over
all amplifiers, we obtain the nonlinear phase noise variance caused by SPM and XPM
()
2
SPM
22
eff
1
(1)(21)() 21,
3
ASE
XPMsesc
block
T
M MMLLNP
ρ
δγ
⋅
+
Φ=−−−
(15)
where
sc
P is the power per subcarrier. Equation (15) is our final equation for the nonlinear
phase noise variance taking into account the interaction of ASE with SPM and XPM. The
analytical model will be validated in the next section.
2.2 FWM induced nonlinear phase noise
Substituting Eq. (2) in Eq. (1), and considering only the FWM effect, we obtain the following
equation with the quasicw assumption
[]
()
2
l
*
r
2
p
2
q
2
r
22
,
exp ( )
z
exp.
22
l
lpq
p q r l
p l q r
≠
u
jujwu u uj
ββ
ωγωωω
+ − =
≠
∂
∂
−=−⋅−+−
∑
z
z
(16)
The solution to Eq. (16) for the field
lu is as follows
0000
0
0000
'*
r
,,,,, , ,
p q r l
β
*
r
,,,,, , ,
p q r l
0
()'exp( ')
z
'
'( ,
z
),
s
ML
lslpp
p q r l
+ − =
∑
lpqs
p q r l
+ − =
u MLujeuuujd
ujuuuY ML
α
−
γ
=+− ∆
=+
∑ ∫
z
zzzz
z
zzzz
z
(17)
where
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00
2
l
2
,,0
' exp(),
2
ll
uujβ
ω
=−
zz
z
(18)
with
0
,0
( )
z
ll
uu
=
z
.
, , ,( )
p q r l
β
∆
z is the phasemismatch factor given by
2
p
2
q
2
r
2
l
2
, , ,( )
p q r l
β
∆
().
2
β
ωωωω
=+−−
z
z
(19)
and
0
'
, , ,
p q r l
0, , ,
p q r l
β
( ,
z
)exp ( ')
z
'.
s
ML
∫
s
z
YMLejd
α
−
γ
= − ∆
z
z (20)
To obtain Eq. (17), we have ignored the depletion of FWM pumps appearing on the right
hand side (RHS) of Eq. (16), which is known as the undepleted pump approximation [34].
We have also assumed that the chromatic dispersion has been completely compensated using
digital backward propagation [14–16] at the end of the system.
Now consider the noise added by the amplifier located at
immediately after the amplifier is shown in Eq. (7). Equation (17) is solved using the initial
condition of Eq. (7). Replacing
0
, l z
u
in Eq. (17) with
the end of the fiber span as
sL
κ
. The optical field
()
ls
uL
κ
+
, we obtain the optical field at
()
2
l
*
,
κ
2
,,, , ,
p q r l
2
l
2
,
*
r
*
r
,,, , , ,
p q r l
(, )
κ
()exp()(,)
2
exp(
∑
)
2
( )()()(,).
lslsspqrss
p q r l
+ − =
lls
ppqqss
p q r l
+ − =
uMLuLjLjuu u Y
κ
L ML
κ
unjL
jununu n YL ML
κ
κ
κ
κκκ
β
κ ω κ
β
ω κ
+++++
=−+
=+−+
+++
∑
(21)
Ignoring the higher order term of
ln , we have
()
2
l
2
,
*
r
*
r
*
r
*
r
,,,,,,,, , , ,
p q r l
,
(, )
κ
∑
exp()
2
u
()(, ).
lslls
pqpqqppqss
p q r l
+ − =
uMLunjL
juuun u
+
n uun uuYL ML
κ
κ
κκκκκκκκ
κ
β
ω κ
+
≈+−+
++
(22)
From Eq. (22), we have
2
l
2
,,
(, )
κ
exp()( , ),
κ
2
lslsFWM ls
uMLujLuu ML
δ
κκ
β
ω κ
+
=−++
(23)
where
,
FWM
u
κ is the deterministic distortion caused by FWM, expressed as
∑
*
r
,,,, , ,
p q r l
,
(,).
FWMpqss
p q r l
+ − =
ujuuuYL ML
κ
κκκ
κ
=
(24)
This distortion can be compensated with the digital backward propagation, and thus it has
no impact on the nonlinear phase noise. The third term on the RHS of Eq. (23)
describes the ASEFWM interaction and can be expanded as
(, )
κ
lu M
δ
()
2
l
*
q
2
1
(, )
κ
exp(),
2
N
lslsqqq
q
u ML
δ
njLjn An B
β
ω κ
=
=−++
∑
(25)
where
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/2 1
−
∑
*
p
,,, ,
/2
2(,),,
N
qp l q
+ −
q p l q p l
u Y
κ
ss
pN
AuL ML
κ
p q lplq
+ −
=−
=≠≠+ −
(26)
/2 1
−
∑
, , , ,
/2
(,),,
N
qq l p
+ −
p q l p p q l
u Y
κ
+ −
ss
pN
Bu L ML
κ
p q lqlp
=−
=≠≠ + −
(27)
From Eq. (25), we have
(
)
222
22
1
,
N
llqqq
q
unnAB
δ
=
=++
∑
(28)
/2 1
−
2
2
22
l
2
/2
2exp()2.
2
N
lllsqqq
qN
u j nBjLnA B
β
δ ω κ
=−
=−− ∑
(29)
After the digital backward propagation removes the deterministic distortions, the phase
noise of the received field is
*
,
Im()
,
2
lll
l
ll
uuu
uj u
κ
δδδ
δ
−
Φ≈=
(30)
where Im( ) ⋅ denotes the imaginary part. As
,
0
l κ
δΦ=
, we can calculate the variance of the
phase noise as
()
()
2
2
2 *2
*
2
l
,
22
2
.
44
lll
ll
ll
uuu
uu
uu
κ
δδδ
δ
δ
δ
−+
−
Φ≈ −=
(31)
Insert Eqs. (28) and (29) into (31) and use Eq. (9), we have:
2
2
l
2
l
2
,
1
24Imexp().
242
N
ASEASE
qqls
q
sc block
P T
sc block
P T
ABBjL
κ
ρρβ
δ ω κ
=
Φ=+++−
∑
(32)
The first term on the RHS of Eq. (32) is the variance of the linear phase noise, and the second
term on the RHS of Eq. (32) is the variance of the nonlinear phase noise caused by FWM.
Summing Eq. (32) over all amplifiers in the fiber system, we obtain the phase noise variance
caused by linear phase noise and FWM as follows:
2
linear l
2
linear l
, , ,
1
.
2
M
ASE
sc block
P T
M
κ
κ
ρ
δ
δ
=
Φ=Φ=
∑
(33)
2
2
FWM l
2
FWM l
,, ,
111
2
l
2
1
2
4
4Im exp().
2
MMN
ASE
qq
q
sc block
P T
M
ls
AB
BjL
κ
κκ
κ
ρ
δ
δ
β
ω κ
===
=
Φ=Φ=+
+−
∑∑∑
∑
(34)
The first term on the RHS of Eq. (34) is the nonlinear phase noise induced by FWM, and the
second term on the RHS of Eq. (34) is the interaction between the linear and nonlinear phase
noise.
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2.3 Total phase noise
In summary, the total phase noise for the lth subcarrier in an OFDM system including the
linear phase noise and nonlinear phase noise (induced by interaction between ASE and SPM,
XPM and FWM) is as follows:
2
l
2
linear l
2
SPM
2
FWM l
,, ,
XPM
δδδδ
+
Φ=Φ+Φ+Φ
(35)
where the first, second, and third terms on the RHS of Eq. (35) are given by Eqs. (33), (15),
and (34), respectively.
3. Results and discussions
In this section, we validate our analytical model for the variance of the total phase noise in
OFDM systems given by Eq. (35) with numerical simulation. The following parameters are
used throughout the paper unless otherwise specified: the bit rate is 10 Gb/s, the amplifier
spacing is 100 km, and the noise figure (NF) is 6 dB. A single type of fiber is used between
amplifiers. To separate the deterministic (although bit pattern dependent) distortions due to
nonlinear effects from the ASEinduced nonlinear noise effects, we use digital backward
propagation [14–16]. Since digital backward propagation compensates for both dispersion and
deterministic nonlinear effects, we do not use the cyclic prefix. 2048 OFDM frames are used
to get a good Monte Carlo statistics. Each OFDM subcarrier is modulated with binaryphase
shiftkeying (BPSK) data. Figure 1 shows the coherent OFDM system structure in our
simulation.
Fig. 1. Structure of coherent OFDM transmission systems
For Figs. 23, we choose a fiber dispersion D of 1 ps/nm/km and a total launch power of 0
dBm. Here we use only one subcarrier (Ne = 1) to carry data while the total number of
subcarriers is 8 (8thfolder oversampling), so that the nonlinear phase noise model that
includes SPM effects alone can be validated. The subcarrier carrying data is located at the
central of the OFDM spectrum, corresponding to the first subcarrier U0 in Fig. 1 due to FFT
operation. The signal spectrum before entering into the fiber span is shown in Fig. 2. And in
Fig. 3, the solid lines show the analytical linear phase noise and nonlinear phase noise
variance induced by SPM only, the dashed line with circles show the numerical simulation
results for the variance of linear phase noise and SPM induced nonlinear phase noise, as a
function of fiber propagation distance. As can be seen, the agreement is quite good.
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Fig. 2. OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is
8, with one subcarrier carrying data.
Fig. 3. Variance of the total phase noise as a function of propagation distance for SPM effect
only. Total number of subcarrier is 8 with only one subcarrier carrying data.
In order to validate the nonlinear phase noise model including the ASE interaction with
SPM, XPM, and FWM effects in Eq. (35), we use 8 subcarriers to carry data for an OFDM
system with 64 subcarriers. The subcarrier carrying data is located at the central of the OFDM
spectrum, corresponding to the subcarriers U0~U3 and U60~U63 in Fig. 1. Figure 4 shows the
OFDM signal spectrum, and Fig. 5 shows the variance of the linear phase noise and nonlinear
phase noise for numerical simulation (dashed line with circles) and analytical calculation
(solid line), respectively. We see that very good agreement is achieved, which validates our
model for the nonlinear phase noise considering SPM, XPM, and FWM effects.
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Fig. 4. OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is
64, with 8 subcarriers carrying data.
Fig. 5. Variance of the total phase noise as a function of propagation distance considering the
ASE interaction with SPM, XPM and FWM effects. Total number of subcarriers is 64 with 8
subcarriers carrying data.
In Ref [7], the authors showed that the nonlinear degradation due to FWM effects in OFDM
systems is nearly independent of the number of ODFM subcarriers used in the system in the
absence of chromatic dispersion, while in Ref [8]. the authors studied the chromatic
dispersion effects on the FWM and showed that chromatic dispersion could decrease the
FWM effects significantly. However, both of these analyses focused on the deterministic
nonlinear effects. In this section, we will study the dependence of the nonlinear phase noise
effects on fiber dispersion and bit rate in an OFDM system with digital backward propagation.
In Figs. 68, we fix the transmission distance to be 1000 km, the total number of
subcarriers is 128 with 64 subcarriers carrying data (twofold oversampling). Figure 6 shows
the OFDM signal spectrum. Figure 7 shows the variance of the total phase noise (linear +
nonlinear phase noise) as a function of the launch power, for D = 17 ps/nm/km and D = 0
ps/nm/km. Solid lines and circles show the analytical results and the numerical simulation
results, respectively. As can be seen from Fig. 7, the nonlinear tolerance increases
significantly as the fiber chromatic dispersion parameter increases. It is also shown in Fig. 7
that the total variance of the phase noise initially decreases with launch power since the linear
phase noise is dominant at low launch powers. However, as the launch power increases
beyond −2 dBm, the variance increases with D = 0 ps/nm/km since nonlinear phase noise
becomes dominant at higher powers.
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Fig. 6. OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is
128 with 2folder oversampling (64 subcarriers carrying data).
Fig. 7. Variance of the total phase noise as a function of channel power. The total number of
subcarriers is 128 with twofold oversampling, bit rate is 10 Gb/s, and transmission distance is
1000 km.
In Fig. 8, we show the impact of the bit rate on the total phase noise for a transmission
fiber with D = 17 ps/nm/km and D = 0 ps/nm/km. The total launch power is −3 dBm. Solid
lines show the analytical results, while filled circles show the numerical simulation results.
From Fig. 8, we see that the variance of the total phase noise scales linearly with the bit rate.
This could be explained by the fact that, with the increase of the bit rate, the OFDM symbol
time
block
T
decreases, which leads to the increase of the total phase noise as described in Eqs.
(15), (33) and (34). The qualitative explanation for the increase in phase noise when the bit
rate increases is as follows: as the bit rate increases, OSNR requirement for a given BER
increases. This is because the receiver filter bandwidth scales with bit rate, which leads to the
increase of the total noise within the receiver bandwidth. The same thing happens with phase
noise: the total phase noise within the receiver bandwidth increases as the bit rate increases.
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Fig. 8. Variance of the total phase noise as a function of bit rate in Gb/s. The total number of
subcarriers is 128 with twofold oversampling, total channel power is −3 dBm, and
transmission distance is 1000 km.
For a BPSK system with coherent detection, from [35], one would obtain the bit error rate
BER as a function of the phase noise variance
σ as
2
2
11
σ
,
2
2
BER erfc
=
(36)
where
( )
1 10−
×
erfc ⋅ is the complementary error function [35]. From Eq. (36), we would get that, for
a BER of
would only be valid when linear phase noise is dominant. In the presence of nonlinear phase
noise, it is hard to evaluate the BER without knowing the probability density function of the
nonlinear phase noise, which would be a subject for future investigation.
In Fig. 9, we show the impact of the number of subcarriers on the variance of total phase
noise, obtained analytically using Eq. (35). Twofolder oversampling is used in the
simulation. The total launch power is −3 dBm, the bit rate is 10 Gb/s. Figure 9 shows that, in
the absence of dispersion, the variance of total phase noise scales linearly with the number of
subcarriers, while with moderate levels of dispersion, the variance of total phase noise is
almost constant because the linear phase noise is dominant for such systems.
5
, the required phase noise variance would be
2
5.5 10−
×
. However, this result
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Fig. 9. Variance of the total phase noise as a function of number of subcarriers, obtained
analytically. Twofolder oversampling is used in the simulation. Bit rate is 10 Gb/s, total
channel power is −3 dBm, and transmission distance is 1000 km.
Finally Fig. 10 shows the variance of the nonlinear phase noise as a function of
propagation distance for SPM induced nonlinear phase noise alone (dashed line), XPM
induced nonlinear phase noise alone (dashed line with ‘x’), and FWM induced nonlinear
phase noise alone for D = 0 ps/nm/km (solid line with circles), D = 10 ps/nm/km (solid line
with triangles) and D = 17 ps/nm/km (solid line with diamonds), obtained analytically using
Eqs. (15) and (34). From Fig. 10, we see that, for an OFDM system with large number of
subcarriers, nonlinear phase noise induced by FWM is significantly larger than that induced
by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it
is found that ASEFWM interaction is negligible in quasilinear systems. This difference is
likely due to the fact that the subcarriers of OFDM system are derived from the same laser
source and interact coherently. We also see that, with moderate levels of fiber chromatic
dispersion, the nonlinear phase noise induced by FWM decreases since the phase matching
becomes more difficult.
Fig. 10. Variance of the nonlinear phase noise due to separate effects of SPM, XPM, and
FWM, as a function of propagation distance, obtained analytically. Total number of subcarriers
is 128 with 2folder oversampling. Bit rate is 10 Gb/s with −3 dBm launch power.
4. Conclusions
In summary, we derive an analytical formula for the variance of the nonlinear phase noise in
an OFDM system, taking into account the interaction of ASE noise with SPM, XPM and
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FWM effects. Our analytical results agree well with the numerical simulation. In addition, we
quantitatively compared the nonlinear phase noise induced by SPM, XPM and FWM, and
found that the nonlinear phase noise induced by FWM is the dominant nonlinear effect in
OFDM systems with digital backward propagation. We also studied the effect of dispersion
and bit rate on the nonlinear phase noise. Our results show that nonlinear phase noise can be
suppressed with fiber chromatic dispersion, approaching the limit of linear phase noise for
large dispersion values. Finally, we show that for OFDM systems, the variance of the total
phase noise scales linearly with the channel bit rate.
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