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Compensation improvement of DPSK signal with nonlinear phase noise

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Keang-Po Ho at Institute of Electrical and Electronics Engineers
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When nonlinear phase noise is compensated by the received intensity, simple formulas are derived for the error probability of differential phase-shift keying signals. Simulation is conducted to verify the error probability. The tolerance of nonlinear phase noise is doubled by the compensator, allowing doubling of the transmission distance if nonlinear phase noise is the dominant impairment.
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1216 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 9, SEPTEMBER 2003
Compensation Improvement of DPSK Signal With
Nonlinear Phase Noise
Keang-Po Ho, Member, IEEE
Abstract—When nonlinear phase noise is compensated by the
received intensity, simple formulas are derived for the error prob-
ability of differential phase-shift keying signals. Simulation is con-
ducted to verify the error probability. The tolerance of nonlinear
phase noise is doubled by the compensator, allowing doubling of
the transmission distance if nonlinear phase noise is the dominant
impairment.
IndexTerms—Differentialphase-shiftkeying(DPSK), fibernon-
linearities, nonlinear phase noise, phase modulation.
I. INTRODUCTION
N
ONLINEAR phase noise, often called the Gordon–Mol-
lenauer effect[1], is induced by the interaction of the fiber
Kerr effect and optical amplifier noise. Both phase-shift keying
(PSK) and differential phase-shift keying (DPSK) signals are
degraded by nonlinear phase noise [1]–[6]. DPSK signaling has
renewed interests recently for both long-haul and spectral effi-
ciency transmission systems [7]–[11]. Correlated with the non-
linear phase noise,the receivedintensity can beused to compen-
sate the nonlinear phase noise [12]–[15] to about half its stan-
dard deviation [12]–[14].
Recently, Kim and Gnauck [5] show experimentally and the
author [16], [17] shows theoretically that the nonlinear phase
noise is not Gaussian distributed. The variance or
factor [1],
[4], [12]–[14] is not sufficient to characterize the system perfor-
mance. Using the probability density function (pdf) from [16],
this letter evaluates the error probabilityfor DPSK systems with
and without compensation.
This letter assumes that both the nonlinear phase noise
without compensation and the residual nonlinear phase noise
after compensation are independent of the phase component
of the amplifier noise. Using the Fourier series expansion of
the pdf of the phase noise [18]–[20], closed-form formulas are
derived to calculate the error probability of DPSK signals with
either nonlinear phase noise or residual nonlinear phase noise.
The signal-to-noise ratio (SNR) penalty is also calculated for an
error probability of
. Monte Carlo simulation is conducted
to verify the formulas.
Manuscript received March 11, 2003; revised May 19, 2003.
The author is with the Graduate Institute of Communication Engi-
neering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail:
kpho@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/LPT.2003.816703
II. CLOSED-FORM ERROR PROBABILITY
With uncompensated nonlinear phase noise, a DPSK signal
is demodulated using the differential phase of
(1)
where
, , , and are the received phase,
the transmitted phase, the phase of amplifier noise, and the non-
linear phase noise as a function of time, and
is the symbol
time. The phases at
and are independent identically dis-
tributed random variables. Error probability is calculated using
the pdf of
. The pdf of the phase of amplifier noise can be
found in [6] and [18]–[20]. The characteristic function of the
nonlinear phase noise can be found in [16] and [17].
Having a period of
, the pdf of the phase of amplifier noise
of
has a Fourier series of [18]–[20]
(2)
with coefficients of
(3)
where
is the SNR, is the Gamma function,
is the confluent hypergeometric function of the first kind, and
is the th order modified Bessel function of the first kind.
Using a series similar to (2), the error probability is found for
DPSK signals with noisy reference [19], phase error [20], or
laser phase noise [21].
Assume that the signal phases at
and are the same
with
. If the phase of amplifier noise of is
independent of the nonlinear phase noise of
, the pdf of the
differential phase is
(4)
where
is the characteristic function of the nonlinear
phase noise from [16]. In (4), the coefficients of
and
correspond to the term of
and , respectively, in the differential
phase of (1). In the pdf (4), the Fourier coefficients of the
addition (or subtraction) of the differential phase of (1) give the
multiplication of the corresponding Fourier coefficients.
1041-1135/03$17.00 © 2003 IEEE
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HO: COMPENSATION IMPROVEMENT OF DPSK SIGNAL WITH NONLINEAR PHASE NOISE 1217
Based on interferometer, the direct-detection DPSK receiver
provides a decision variable proportional to
[7],
[9]–[11]. The receiver makes the decision based on whether
is positive or negative that is equivalent to whether
is within or without the angle of . The error proba-
bility for DPSK signal is
(5)
or
(6)
Because
if is an even number, we have
(7)
Efficientlycalculated, the errorprobabilityof (7)givesabout the
same results as that in [6]. With a simpleclosed-form expression
of (7), the more complicated characteristic functions from [16]
can be used in the calculation.
As shown in [14], the differential received intensity of
can be used to compensate for the differential
nonlinear phase noise of
using the same
optimal scale factor as that in [14]. Similar to the procedure
from (4) to (7), the error probability with nonlinear phase noise
compensation is
(8)
where
from [16] is the characteristic function of the
residual nonlinear phase noise after compensation.
The formulas of (7) and (8) are similar to that of [19] with
noisy reference and that of [21] with laser phase noise. If the
nonlinear phase noise is assumed to be Gaussian distributed and
independentof thephaseof amplifiernoise,the errorprobability
can be calculated by [21].
III. N
UMERICAL RESULTS
Fig. 1 shows the error probability of DPSK signal as a func-
tion of SNR
. The error probabilitywithout and with compen-
sation is calculated by the formulas of (7) and (8), respectively.
The system in Fig. 1 has 32 identical fiber spans. In Fig. 1, the
error probability with different values of mean nonlinear phase
shift
is shown for comparison. Without nonlinear phase
noise, the error probability is
[6], [19], [20]. When
the nonlinear phase noise is compensated using the received in-
tensity with the optimal scale factor from [14] or the approxi-
mated scale factor from [13], the standard deviation of the non-
linear phase noise is reduced by a factor of about two [12]–[14],
[16]. Comparing Fig. 1(a) with Fig. 1(b), the error probability
(a) (b)
Fig. 1. Error probability of DPSK signal with nonlinear phase noise
(a) without and (b) with compensation.
Fig. 2. SNR penalty of an error probability of as a function of mean
nonlinear phase shift of
.
of DPSK signal without compensation and having a mean non-
linear phase shift of
is slightly larger than that with com-
pensation but having twice the mean nonlinear phase shift of
.
Without compensation, the error probability in Fig. 1(a) is
similar to that in [6] but using the more complicated pdf from
[16] instead of the asymptotic pdf from [17]. From [17], the tail
probability of the pdf with 32 fiber spans is larger than that with
infinitely many fiber spans and the error probability of Fig. 1(a)
is slightly larger than that from [6].
Fig. 2 shows the SNR penalty for an error probability of
as a function of mean nonlinear phase shift of
for the same system of Fig. 1. Mean nonlinear phase shifts of
and rad give a SNR penalty of 1 dB without
and with compensation, respectively. For the same power
penalty, the DPSK system with compensation can tolerate a
mean nonlinear phase shift slightly larger than twice of that of
the system without compensation, confirming the same results
from [12]–[14], [16], and Fig. 1.
Similar to [6], the derivations from (4) to (8) assume that the
phase of amplifier noise
is independent to the nonlinear
phase noise of
or the residual nonlinear phase noise of
.It is obviousthatthephase of amplifiernoiseiscorrelated
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1218 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 9, SEPTEMBER 2003
(a) (b)
Fig. 3. Simulated error probability of DPSK signal withnonlinearphasenoise
(a) without and (b) with compensation.
to neither thenonlinear phasenoise of nor the residualnon-
linear phase noise of
. However, as non-Gaussian random
variables, they may be weakly dependent [6].
Fig. 3 show little difference between the simulated and the-
oretical error probability. There is a 0.2-dB SNR difference for
the same error probability. Fig. 3(a) and (b) has mean nonlinear
phase shifts of
and , without and with com-
pensation, respectively, for an SNR penalty around 1 dB. The
variance of the phase of amplifier noise is about 1.5 times the
variance of the nonlinear phase noise or the residual nonlinear
phase noise [1], [14].
The simulation of Fig. 3 is conducted similar to [6] and [14]
but for DPSK signals. Equivalently speaking, the signal distri-
butionfor DPSK signal (see[14, Fig.2] forPSK signal)is found
and the error probability is calculated by counting the number
of points outside the decision region. We count at least ten er-
rors to ensure a good confident interval [22, Fig. 2].
The SNR of Figs. 1 and 3 is defined the same as that in [6]
and [17]–[21]. When optical SNR (OSNR) is measured using
an optical spectrum analyzer with a bandwidth of
, the
SNR is related to OSNR by
where
is the data rate of the signal and the factor of two assumes a
polarization-insensitive optical spectrum analyzer.
IV. C
ONCLUSION
Closed-form formulas are derived for the error probability of
DPSK signals contaminated by nonlinear phase noise with and
without compensation using the received intensity. The error
probability is derived based on the assumption that the phase of
amplifier noise is independent of both the nonlinear phase noise
without compensation and the residual nonlinear phase noise
after compensation. Simulation shows that the error probability
formulas are very accurate. For the sameSNR penalty, the mean
nonlinear phase shift is double, doubling the transmission dis-
tance if nonlinear phase noise is the dominant impairment.
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