# Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise

**ABSTRACT** The error probability is derived analytically for differential phase-shift keying (DPSK) signals contaminated by both self- and cross-phase modulation (SPM and XPM)-induced nonlinear phase noise. XPM-induced nonlinear phase noise is modeled as Gaussian distributed phase noise. When fiber dispersion is compensated perfectly in each fiber span, XPM-induced nonlinear phase is summed coherently span after span and is the dominant nonlinear phase noise for typical wavelength-division-multiplexed (WDM) DPSK systems. For systems without or with XPM-suppressed dispersion compensation, SPM-induced nonlinear phase noise is usually the dominant nonlinear phase noise. With longer walkoff length, for the same mean nonlinear phase shift, 10-Gb/s systems are more sensitive to XPM-induced nonlinear phase noise than 40-Gb/s systems.

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Page 1

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 2, MARCH/APRIL 2004421

Error Probability of DPSK Signals With Cross-Phase

Modulation Induced Nonlinear Phase Noise

Keang-Po Ho, Senior Member, IEEE

Abstract—The error probability is derived analytically for dif-

ferential phase-shift keying (DPSK) signals contaminated by both

self- and cross-phase modulation (SPM and XPM)-induced non-

linear phase noise. XPM-induced nonlinear phase noise is mod-

eled as Gaussian distributed phase noise. When fiber dispersion is

compensated perfectly in each fiber span, XPM-induced nonlinear

phase is summed coherently span after span and is the dominant

nonlinear phase noise for typical wavelength-division-multiplexed

(WDM) DPSK systems. For systems without or with XPM-sup-

pressed dispersion compensation, SPM-induced nonlinear phase

noise is usually the dominant nonlinear phase noise. With longer

walkoff length, for the same mean nonlinear phase shift, 10-Gb/s

systems are more sensitive to XPM-induced nonlinear phase noise

than 40-Gb/s systems.

Index Terms—Cross-phase modulation, differential phase-shift

keying (DPSK), fiber nonlinearities, nonlinear phase noise, phase

modulation.

I. INTRODUCTION

N

amplifiers are used periodically to compensate for fiber loss.

Self-phase modulation (SPM)-induced nonlinear phase noise,

often called the Gordon–Mollenauer effect [1], is the major

degradation of single-channel differential phase-shift keying

(DPSK) signals [1]–[9]. Recently, there has been renewed

interest in using wavelength-division-multiplexed (WDM)

DPSK systems for long-haul transmission systems [9]–[14]

and WDM differential quadrature phase-shift keying (DQPSK)

systems for spectrally efficient transmission [15]–[19]. While

the impact of Gordon–Mollenauer effect on single-channel

DPSK systems is well known [7], [8], [20], to our knowledge,

other than the studies of [21] and [22] on phase noise variance

or frequency response, the impact of XPM-induced nonlinear

phase noise on the error probability of DPSK signals has not

been studied.

Added directly to the signal phase, SPM-induced nonlinear

phase noise is non-Gaussian-distributed [5], [23], [24] but

XPM-induced nonlinear phase noise is Gaussian-distributed.

Induced by many other adjacent WDM channels, the Gaussian

convergence of XPM-induced nonlinear phase noise comes

directly from the central limit theorem [25]. Independent of

both SPM-induced nonlinear phase noise and the additive

ONLINEAR phase noise is induced by the interaction of

fiber Kerr effect and optical amplifier noises when optical

Manuscript received October 3, 2003; revised January 16, 2004. This work

was supported in part by the National Science Council of Taiwan under Grant

NSC-92-2218-E-002-034.

The author is with the Institute of Communication Engineering and Depart-

mentofElectricalEngineering,NationalTaiwanUniversity,106Taipei,Taiwan,

R.O.C. (e-mail: kpho@cc.ee.ntu.edu.tw).

Digital Object Identifier 10.1109/JSTQE.2004.826574

amplifier noises, XPM-induced nonlinear phase noise has the

same mathematical model as laser phase noise for performance

analysis [26].

When many additive phase noise components are assumed

to be independent of each other, the Fourier coefficients of the

overall phase noise are the product of the Fourier coefficients of

each individual component. This paper combines the models of

[8] and [26] after the variance of XPM-induced nonlinear phase

noise is evaluated, mostly using the methods of [27] and [28].

This paper are organized as follows. Section II derives the

nonlinear phase noise variance due to XPM for multispan

WDM DPSK systems and compares it with that due to SPM.

Section III calculates the error probability of a DPSK signal

with both SPM- and XPM-induced nonlinear phase noise.

Sections IV and V are the discussion and conclusion for this

paper, respectively.

II. NONLINEAR PHASE NOISE VARIANCE DUE TO

CROSS-PHASE MODULATION

This section derives the ratio of the variance of nonlinear

phase noise induced by XPM to that by SPM. The variance is

first derived for the two-channel pump-probe model and later

extended to multichannel and multispan WDM systems.

A. Pump-Probe Model

Based on the pump-probe model of [21] and [27]–[29], the

phase modulation of channel 1 (probe) induced by channel 2

(pump) is

(1)

where

and time ,

is the nonlinear coupling coefficient,

coefficient,

is the fiber length,

walkoff between two channels with wavelength separation of

, andis the dispersion coefficient.

By taking the Fourier transform of the autocorrelation func-

tion, when the power spectral density of

power spectral density of

isthepowerofchannel2asafunctionofposition

is the launched signal at the transmitter,

is the fiber attenuation

is the relative

is, the

is

(2)

where

or [27], [28]

(3)

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422 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 2, MARCH/APRIL 2004

At the DPSK receiver, after the asymmetric Mach–Zehnder

interferometer [10], [11], [20], the differential nonlinear phase

noise of

adds to the differential phase of the signal, where

symbol interval. Similar to [22], the power spectral density of

is

is the

(4)

When the pump (channel 2) has amplifier noises,

, where and

and noise, respectively. In the power of

, the dc term of

phase noise but a constant phase shift, the signal-noise beating

of

givesa noise spectraldensity of

and the noise-noise beating of

sity of

, where is the spectral density of the am-

plifier noise and

is the optical bandwidth of the ampli-

fier noise. The optical signal-to-noise ratio (SNR) over an op-

ticalbandwidth of

is

power of

and a single optical amplifier with a noise variance

of

, we get

phase variance as a function of frequency separation is

are the electric fields for signal

gives no nonlinear

,

gives a noise spectral den-

.For a launched

. The

(5)

where the integration is reduced from

into account only the phase noise over a bandwidth confined

within the bit rate. The dependence of the variance of (5) on

is originated from the dependence of

For SPM-induced nonlinear phase noise, using (5), we get

to1by taking

of (3) on.

, where

is the effective length per span. The factor of 1/4 is

because phase shift induced by XPM is twice as large as that

induced by SPM for the same intensity. For a long fiber span

with

and large walkoff coefficient of

§3.824], we get

, using [30,

(6)

where

for two aligned pump and probe pulses of length

completely walkoff [31]. Like that for stimulated Raman scat-

tering [31], even without approximation, the variances of XPM-

induced nonlinear phase noise depend on the walkoff length of

only.

is the walkoff length that is the distance

becomes

B. Multi-Span WDM Systems

For a (2

the center channel, in the first span, the nonlinear phase noise

variance per span is equal to

1)-channel WDM system, for the worst case of

(7)

where, using the same symbol as the above pump-probe model,

isthewavelengthseparationbetweenadjacentchannels.For

a short walkoff length of

per channel separation of

wavelength separation of

gives a XPM-induced nonlinear

phase noise variance of about a factor of 1

that for a wavelength separation of

channels, using the relationship of

§0.233], we get

, a

smaller than

. For a large number of

[30,

(8)

For a system with

XPM-induced nonlinear phase noise depends on the method of

dispersion compensation. Similar to (4), the power spectrum

density of XPM-induced nonlinear phase noise after

spans is [28]

fiber spans, the variance of the overall

fiber

(9)

where

and

respectively.

Intheworstcaseofperfectdispersioncompensationwith

such that all channels are well aligned when launched to each

fiber span, the variance of (5) is increased by a factor of

afterfiber spans. With perfect dispersion compensation, the

variance of the overall XPM-induced nonlinear phase noise for

an

-span fiber system is

is the fraction of dispersion compensation, i.e.,

for perfect and without dispersion compensation,

(10)

where the first term of

duced from the amplifier noise from the first span, the second

term is the nonlinear phase noise induced from the amplifier

noise from the second span, and so on. Because of perfect dis-

persion compensation, the same noise from the first span is per-

fectly (or coherently) added

phase noise. The noise from the second span

the overall nonlinear phase noise

sumption that all spans have the same configuration,

. The maximum nonlinear phase

noise of (10) is

is the nonlinear phase noise in-

times into the overall nonlinear

adds into

times. With the as-

(11)

For SPM-induced nonlinear phase noise, the amplifier noise

from the first span is also perfectly aligned in all fiber spans af-

terward. The relationship between single- and multispan phase

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HO: ERROR PROBABILITY OF DPSK SIGNALS 423

noise variance is the same as that of (11). With perfect disper-

sioncompensation,theratioofthevarianceofXPM-toSPM-in-

duced nonlinear phase noise is independent of the number of

fiber spans in WDM systems. The ratio of (8) can be used to ap-

proximate the ratio of XPM- to SPM-induced nonlinear phase

noise in the worst case of perfect dispersion compensation.

When dispersion compensation is conducted channel-by-

channel with XPM suppression [32], [33], the factor of

the offset of pump and probe at the output of a fiber span with

respect to the fiber walkoff. If

phase noise induced to every fiber span is independent of each

other, we get

is

such that the nonlinear

(12)

or

(13)

and the ratio of

(14)

In (12), the amplifier noise from the first span induces non-

linear phase noise in each fiber span. However, the nonlinear

phase noises induced to the first and second span are from the

amplifier noise in completely nonoverlapped time interval and

independentofeachother.Inan

linear phase noise is generated from (1/2)

independent amplifier noise. The variance of (12) is also valid

when

and.

The last term of (9) due to multispan effect of

haspeakvaluesof

,whereisapositiveornegativeinteger.The

last term of (4) due to differential operation of

notches at

. If the notches of

peaks of

induced nonlinear phase noise is approximately minimized. If

, the dispersion compensation factor of

-spansystem,theoverallnon-

1) “pieces” of

or when

has

match to the

, XPM-

(15)

approximately gives minimum XPM-induced nonlinear phase

noise. With resonance effect [34], certain combinations of

walkoff length and dispersion compensation factor minimize

the variance of XPM-induced nonlinear phase noise. Numerical

results show that thecompensation factor of (15) approximately

minimizes the variance of XPM-induced nonlinear phase noise.

Fig. 1 shows the ratio of the standard deviation of XPM-

to SPM-induced nonlinear phase noise as a function of the

walkoff length

per channel separation

is calculated numerically and uses the disper-

sion compensation factor of (15) for

. The fiber attenuation coefficientis

The fiber span length is

(labeled as “2” WDM channels), the approximation of (6) is

valid for

less than 50 km.

. The ratio of

and for

.

. For the pump-probe model

Fig. 1.

separation.Thedashedcurvesaretheapproximationof(6)and(8).Theratiofor

the worst case of ?

is independent of the number of spans. The ratios

for the typical cases of ?

and the minimum case of ?

? ? ?? fiber spans and are shown as dotted and solid lines, respectively. The

label of each curve is the number of WDM channels.

The ratio of ???

as a function of walkoff length per channel

are for

For multichannel WDM systems, in both the worst and

independent cases, the approximation of (8) is valid for

less than 100 km and a number of channels larger than 17.

The approximation of (8) can be used to model typical WDM

systems. For example, a typical 10-Gb/s 50-GHz spacing

system using nonzero dispersion-shifted fiber (NZDSF) with

has a walkoff length of 62.5 km. Typical

40-Gb/s systems have a walkoff length of 7.8 km (100-GHz

spacing and

When the walkoff effect is weak with long walkoff length

(

), from the pump-probe model, the nonlinear

phase noise standard deviation due to XPM approaches twice

that due to SPM. With long walkoff length, nonlinear phase

noiseinducedbyXPMismuchlargerthanthatinducedbySPM.

For a multispan system, the variance of the SPM-induced

nonlinear phase noise is equal to [1]

).

(16)

withthe exactresults givenin [35],where

for optical matched filter and a single polarization [7], [8]. The

variance of (16) is twice that of [1], [35] for differential signal.

Combining (16) with the ratio in Fig. 1, the variance of XPM-

induced nonlinear phase noise

is theSNR defined

can be calculated.

III. ERROR PROBABILITY

From [8], with only SPM-induced nonlinear phase noise, the

error probability of a DPSK signal is

(17)

where

Bessel function of the first kind, and

is a summation index,is ath-order modified

is the character-

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424IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 2, MARCH/APRIL 2004

Fig. 2.

(b) independent, and (c) minimum variance of XPM-induced nonlinear phase noise.

Error probability as a function of SNR for WDM DPSK systems with various dispersion compensation schemes, corresponding to (a) maximum,

istic function of SPM-induced nonlinear phase noise given by

[23]. Simulation by error counting confirms the validity of the

error probability of (17) [8].

When independent phase noises from different sources are

summed together, the coefficients of the Fourier series of the

overall probability density function are the product of the corre-

sponding Fourier coefficients of each individual component. If

theXPM-inducednonlinearphasenoiseisGaussiandistributed,

the error probability of the DPSK signal is

(18)

The formulas of (17) and (18) are similar to that with noisy

reference [36], laser phase noise [26], phase error [37], or laser

phase noise together with phase error [38]. The terms within the

summation of (18) are the product of that due to SPM-induced

nonlinear phase noise [8] and laser phase noise [26].

Because XPM-induced nonlinear phase noise is generated by

the interaction of many bits or WDM channels, the Gaussian

approximation is valid from the central limit theorem [25]. If

the walkoff length of

is small, the nonlinear phase noise is

induced by at least about

two adjacent channels [31], [39]. If the walkoff length is large,

manyadjacentchannelsinduce more orlessthesame amountof

nonlinear phase noise [31], [39], [40]. In both cases, the central

limit theorem leads to Gaussian distribution.

Fig. 2 shows the error probability of DPSK signal as a func-

tion of SNR

. The error probability is calculated by (17) and

(18) for a system with and without XPM-induced nonlinear

phasenoise,respectively.ThesysteminFig.2has

tical fiber spans, mean nonlinear phase shift of

and 65 WDM channels. The mean nonlinear phase shift of 1 rad

is the same as the estimation from [1] and close to the optimal

independent bits from

iden-

,

operating point from [8]. Taking into account only SPM-in-

duced nonlinear phase shift, the mean nonlinear phase shift can

be calculated approximately from [1] and accurately from [35].

Fig. 2 also plots the error probability of (1/2)

linear phase noise [7], [8], [36], [37].

The walkoff length of Fig. 2 forms a geometric series and

corresponds to typical 10- and 40-Gb/s systems in NZDSF and

standard single-mode fiber (SSMF) with dispersion coefficients

of

and ps/km/nm, respectively. For example, the

walkoff length

km is that of a 10-Gb/s 100-GHz

spacing

systeminSSMFand40-Gb/s100-GHz

spacing system in NZDSF.

Fig. 2(a)–(c) is calculated using the XPM-induced nonlinear

phase noise variances of

in (18), respectively, corresponding to the worst, independent,

andbestcasedispersioncompensation,respectively.Comparing

Fig. 2(b) and (c) with Fig. 2(a), the DPSK system requires the

reduction of XPM-induced nonlinear phase noise using, for ex-

ample, XPM suppressor [32], [33] or optimal dispersion com-

pensation factor of (15). With perfect dispersion compensation,

XPM-induced cross-phase modulation is negligible if

km. Without dispersion compensation or with XPM sup-

pressor, XPM-induced cross-phase modulation is negligible if

km.

Fig.3showstheSNRpenaltyfor anerror probability of10

as a function of mean nonlinear phase shift of

same system as Fig. 2. Fig. 3(a)–(c) is calculated using the

XPM-induced nonlinear phase noise variances of

, and in (18), respectively, corresponding to

the worst, independent, and best case dispersion compensation,

respectively.

For the systems in Fig. 3(a), with perfect dispersion compen-

sationsuchthatXPM-inducednonlinearphasenoiseaddscoher-

ently, XPM-induced nonlinear phase noise gives the same SNR

penaltyasSPM-inducednonlinearphasenoisewhenthewalkoff

length is about

km for

a 10-Gb/s WDM system in SSMF, the channel spacing must be

without non-

, , and

for the

,

less than 1 rad. For

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HO: ERROR PROBABILITY OF DPSK SIGNALS 425

Fig. 3.

to (a) maximum, (b) independent, and (c) minimum variance of XPM-induced nonlinear phase noise.

SNR penalty as a function of mean nonlinear phase shift ??

? for WDM DPSK systems with various dispersion compensation schemes, corresponding

largerthanorequalto100GHz(or0.8nm).Fora10-Gb/sWDM

system in NZDSF, the channel spacing must be larger than or

equal to 400GHz (or 3.2 nm). For a 10-Gb/sWDM system with

typicalchannelspacingof50or100GHz,withperfectdispersion

compensation,theeffectofXPM-inducednonlinearphasenoise

is larger thanthat induced bySPM. Fora 40-Gb/s WDMsystem

in NZDSF, the channel spacing must be larger than or equal to

100 GHz. With channel spacing larger than 100 GHz, typical

40-Gb/s WDM systems have an SNR penalty from XPM-in-

duced nonlinear phase noise about that from SPM.

For the systems in Fig. 3(b) and (c), with dispersion com-

pensation factor of (15) or with XPMsuppressor, XPM-induced

nonlinear phase noise gives the same SNR penalty as SPM-in-

duced nonlinear phase noise when the walkoff length is about

km. For a 10-Gb/s WDM system in SSMF, the

channel spacing must be larger than or equal to 12.5 GHz (or

0.1 nm). For a 10-Gb/s WDM system in NZDSF, the channel

spacing must be larger than or equal to 50 GHz (or 0.4 nm).

For a 10-Gb/s WDM system with typical channel spacing of 50

or 100 GHz, the effect of XPM-induced nonlinear phase noise

is much smaller than that induced by SPM. Typical 40-Gb/s

WDM systems have an SNR penalty from XPM-induced non-

linear phase noise far less than that from SPM.

IV. DISCUSSION

Inthederivationof(5),thetermof

return-to-zero(NRZ)DPSKsignal,

ignored. For the more popular return-to-zero (RZ) DPSK signal

[10], [13],

is a periodic function with a period of

powerspectraldensityistonesat

differential transfer function of 2

and cancels all nonlinear phase noise due to

RZsignal.ForanRZ-DPSKsignalwithpulsebroadeningdueto

fiber dispersion, if the dipersion is assumed to be a linear effect,

for a system without pulse overlapping, the differential transfer

function can also completely eliminate XPM-induced nonlinear

phase noise. Due to interferometer phase error [41]–[43], small

residual nonlinear phase noise may be induced by those tones.

The numerical results of this paper are also valid for flattop

RZ-DPSK signals by increasing the mean nonlinear phase shift

by a factor equal to the inverse of the RZ pulse duty cycle.

For the same channel spacing and fiber type, 10-Gb/s sys-

tems have a walkoff length four times that of 40-Gb/s systems.

From Fig. 3, with the same mean nonlinear phase shift, the non-

linear phase-noise-induced SNR penalty of 10-Gb/s systems is

smaller than that for the corresponding 40-Gb/s systems. How-

ever, 40-Gb/s systems have four times the bandwidth and re-

isignored.Foranon-

isadctermandcanbe

and its

,whereisaninteger.The

has notches at

even for an

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426 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 2, MARCH/APRIL 2004

quire four times the power of the corresponding 10-Gb/s sys-

tems having the same SNR

defined in [7] and [8]. Because

the mean nonlinear phase shift is proportional to the launched

power, for the same SNR and system configuration, the mean

nonlinear phase shift of 40-Gb/s systems is four times larger

than that for 10-Gb/s systems.

Whenthedependencebetweennonlinearphasenoiseandam-

plifiernoiseistakenintoaccount,wederivetheexacterrorprob-

ability of DPSK signals with nonlinear phase noise when the

number of fiber spans approaches infinity [20]. With the dis-

tributed assumption, the difference between exact and approx-

imate error probability is less than 0.23 dB in terms of SNR

penalty. Currently, the exact error probability of DPSK signals

with finite number of fiber spans is not known. From the simu-

lation of [8], the model of (17) is very accurate.

SPM-induced nonlinear phase noise correlates with the re-

ceived intensity and can be compensated using the correlation

properties [8], [20], [35], [44], [45]. Other than using simul-

taneous multichannel detection, XPM-induced nonlinear phase

noise cannot be compensated.

Due to fiber dispersion, phase modulation (PM) converts to

amplitude-modulation (AM) noise. Combined with XPM, AM

noise gives nonlinear phase noise to other channels [46]. When

phase modulation converts to amplitude modulation, only

high-frequency AM noise is induced by a transfer function of

, where is a constant depending on fiber length and

dispersion [47]. With the low-pass characteristic of

(3), the combined effects of AM–PM conversion with XPM

should be small.

of

V. CONCLUSION

Closed-form formulas are derived for the error probability of

WDM DPSK signals contaminated by both SPM- and XPM-in-

duced nonlinear phase noise. The error probability is derived

based on the assumption that the phase of amplifier noise is in-

dependent of nonlinear phase noise. While SPM-induced non-

linear phase noise is not Gaussian-distributed, XPM-induced

nonlinear phase noise is assumed to be Gaussian distributed

when either the walkoff length is small or the number of WDM

channels is large.

When fiber dispersion is compensated perfectly in each fiber

span,XPM-inducednonlinearphaseissummedcoherentlyspan

after span and is the dominant nonlinear phase noise for typical

multispan WDM DPSK systems. For a system without disper-

sion compensation or with XPM suppressor, the dominant non-

linear phase noise is typically induced by SPM. In general, with

longer walkoff length, 10-Gb/s systems are more likely to be

dominatedbyXPM-inducednonlinearphasenoisethan40-Gb/s

systems.

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Keang-Po Ho (S’91–M’95–SM’03) received the

B.S. degree from National Taiwan University,

Taipei, Taiwan, R.O.C., in 1991 and the M.S. and

Ph.D. degrees from the University of California

at Berkeley in 1993 and 1995, respectively, all in

electrical engineering.

He was with the IBM T. J. Watson Research

Center, Hawthorne, NJ, in summer 1994, where

he conducted research on all-optical networks. He

was a Research Scientist with Bellcore (currently

Telcordia Technologies), Red Bank, NJ, from 1995

to 1997, where he conducted research on optical networking, high-speed

lightwave systems, and broadband access. He taught in the Department of

Information Engineering, Chinese University of Hong Kong, from 1997 to

2001. He was Chief Technology Officer of StrataLight Communications,

Campbell, CA, from 2000 to 2003, where he developed spectral efficiency

40-Gb/s lightwave transmission systems. He has been with the Institute of

Communication Engineering and Department of Electrical Engineering,

National Taiwan University, since 2003. His research interests include optical

communication systems, multimedia communication systems, combined

source-channel coding, and communication theory. He was among the pioneers

for research on hybrid WDM systems, combined source-channel coding using

multicarrier modulation or turbo codes, and performance evaluation of PSK

and DPSK signals with nonlinear phase noise. He has published more than 130

journal and conference papers on those fields.

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