effect of nonlinear phase noise on the performance of $m$-ary psk signals in optical fiber links

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 3, FEBRUARY 1, 2013 447

Effect of Nonlinear Phase Noise on the Performanceof -Ary PSK Signals in Optical Fiber Links

Nimal Ekanayake, Member, IEEE, and H. M. Vijitha R. Herath, Member, IEEE

Abstract—The effects of amplified spontaneous emission (ASE)noise and nonlinear phase (NLP) noise on the symbol error rateperformance of -ary coherent phase shift keying (CPSK) anddifferential phase shift keying (DPSK) are analyzed. Expressionsfor the symbol error probability (SEP) are derived for the two casesof independent ASE and NLP noises and dependent ASE and NLPnoises. Numerical results are presented for 2-, 4-, 8-, and 16-levelsignaling to demonstrate the applicability of the SEP expressions.Our results indicate that the SEPs for DPSK for the two cases areapproximately the same except for . The SEPs for CPSKdiffer significantly for the two cases for all , implying that inde-pendent phase noise assumption is not valid for CPSK.

Index Terms—Differential phase shift keying,Kerr effect, opticalfiber communication, optical fibers, performance analysis, phaseshift keying.

I. INTRODUCTION

A MONG many sources of noise that cause performancedegradation in optical fiber communication systems,

the amplified spontaneous emission (ASE) noise generatedin optical amplifiers (OA) and the nonlinear phase (NLP)noise generated due to interaction of the ASE noise with Kerrnonlinearity in fibers [1], [2] are considered to be the two mainsources of noise that limit the performance of phase modulatedsystems. The latter is also known as the Gordon-Mollenauereffect [3]. The ASE noise field can be modeled as a bandpassGaussian process that can be expressed in terms of inphaseand quadrature-phase (I-Q) noise components. The ASE noisecauses random amplitude and phase variations in the opticalsignal. This random phase variation is referred to as phase ofamplifier noise or linear phase (LP) noise in the literature. In thispaper, we favor the latter term for simplicity. The NLP noiseis directly dependent on the composite signal power, or equiv-alently, on the square of the amplitude of the signal-plus-ASEnoise, thereby, making it a non-Gaussian process that is statis-tically dependent on the LP noise. The analytical evaluationof the effect of NLP noise on the performance of digitallymodulated signals, therefore, is considered to be a relativelydifficult problem to solve.Based on the previous work of Gordon and Mollenauer [3],

Mecozzi [4] has analyzed the bit error rate (BER) for coherentphase shift keying (CPSK) treating NLP noise as a distributed

Manuscript received July 17, 2012; revised September 18, 2012; acceptedNovember 29, 2012. Date of publication December 10, 2012; date of currentversion January 09, 2013.The authors are with the Department of Electrical and Electronic Engineering,

University of Peradeniya, Peradeniya, Sri Lanka (e-mail: [email protected];[email protected]).Digital Object Identifier 10.1109/JLT.2012.2233195

process. Nearly a decade later, Ho has published a series of pa-pers [5]–[11] analyzing the statistical properties of NLP noiseand deriving its characteristic function (CHF). Subsequently,Ho [6] expressed the CHF in a very compact form using thedistributed model for NLP noise making the performance eval-uation more tractable. The distributed model yields accurate re-sults only for systems containing a large number of spans. Ho[8]–[11] has used the CHF to evaluate the BER for binary CPSK(BPSK) and binary differential phase shift keying (BDPSK) forthe case of independent phase noises and the case of depen-dent phase noises. In [17], Ho has extended the BER analysisto include quaternary DPSK (QDPSK) for the case of depen-dent phase noises. The design of decision boundaries for max-imum likelihood reception together with phase compensationfor -ary CPSK and differential phase shift keying (DPSK)has been treated in [12].This paper can be considered as extensions and generaliza-

tions of Ho’s previous work on binary PSK schemes [9], [10] toinclude symbol error probability (SEP) performance of -aryCPSK andDPSK signals transmitted through a long optical fiberlink. We analyze the effects of ASE noise and NLP noise only.The effects of signal dispersion or any other distortion are notconsidered. The rest of the paper is organized as follows. InSection II we detail the system model and derive an expressionfor the received signal and its phase for both -ary CPSK andDPSK. The evaluation of the probability density function (pdf)for the received signal phase for -ary signaling is consideredin Section III. Expressions for the SEP that are valid for anyare derived in Sections IV and V, and numerical results are

presented in Section VI. Finally, we present the conclusions inSection VII.

II. DERIVATION OF PHASE OF THE RECEIVED SIGNAL

We consider the transmission of an -ary optical PSK signal,either coherent or differential, over a long single mode fiber linkthat contains repeater amplifiers at regular spans as shown inFig. 1(a). The PSK signal transmitted during the time interval

can be written as

(1)

where is the carrier amplitude, is the carrier frequency,and is the information bearing phaseangle. The -ary symbols can take on any value in the set

with equal probability. For DPSK,we may assume that these symbols are differentially encoded toenable differential detection. After propagated through span ,the modulated laser beam is fed to the first repeater OA,i.e., in Fig. 1, which linearly amplifies it to compensatefor the power loss in the fiber and raise the power level to thetransmitted power level. In the amplification process,

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448 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 3, FEBRUARY 1, 2013

Fig. 1. (a) -Span Fiber Link, (b) Lowpass Model for the Optical Amplifierand Fiber Span .

adds ASE noise field to the transmitted signal as illustrated inFig. 1(b). Using the bandpass representation for the ASE noise,the optical signal at the output of can be expressedas

(2)

where and are the inphase and quadrature-phasecomponents of the ASE noise field each having a variance of

with being the strength of the power spectraldensity of white ASE noise and the bandwidth of an OA. Wehave tacitly assumed that NLP noise in the first span causesthe PSK signal to undergo only a constant phase shift as theGaussian noise added to the signal in this span is negligible.Since the receiver can track the constant phase shift, we havenot included it in (2).The Kerr nonlinearity causes the signal-plus-noise field

to suffer NLP noise as it propagates through fiberspan . In order to facilitate analysis, it is convenient torepresent the signal and noise fields in lowpass form as

, and ,where we have adopted lower case variables to denote lowpasssignals with denoting . Since NLP noise is directlydependent on the signal power [3], the NLP noisegenerated in can be expressed as

(3)

where is the nonlinear coefficient of the fiber and is theeffective length of a span. The effect of NLP noise is to ro-tate the signal vector propagating in the fiber by in theanti-clockwise direction. The exact relation between the inputsignal and the output signal for a fiber is given by the nonlinearSchroedinger equation.When the dispersion introduced by Kerrnonlinearity is disregarded, the nonlinear Schroedinger equa-tion has a simple solution [16] relating input and output signals,which when applied to span , the output signal becomes

(4)The amplifier amplifies this signal and adds ASE noise

in the process. As the laser propagates in the fiber, eachOA adds ASE noise while each fiber span introduces NLP noisebased on the power in the signal-plus-noise propagating through

it. Assuming that all fiber spans and OAs have identical char-acteristics, the total NLP noise introduced can be expressed, onsuppressing the time variable for notational convenience, as

(5)

The exact CHF of has been evaluated by Ho [6] by mod-eling the generation of ASE noise as a uniformly distributedprocess taking place along a single fiber span of length con-sisting of all spans [4]. Consequently, the analytical model inFig. 1(b) is applicable to the distributed model and the signalreceived at the receiver can be expressed as

(6)

where

(7a)

(7b)

(7c)

(7d)

in (7c) is the intensity of the optical signal and in(7d) is the linear phase noise component. The Gaussian noiseprocesses and have equal variance . From (6),we obtain the received phase angle in the signaling interval

(8)

The optical receiver makes use of the I-Q components for thedetection of PSK signals, but it is mathematically equivalent tothe detector which employs phase of the received signalto decide on the transmitted phase. When NLP noise is present,the optimal detection needs modification of decision boundaries[12], which we do not consider here. The simple phase detec-tion for the present case becomes sub-optimal unless completephase compensation is achieved. Nevertheless, we consider thephase detection receiver in our analysis as it has been used in thepast literature. The probability density function (pdf) of ,therefore, is needed for evaluation of the SEP for phase detec-tion. In the following Sections, we treat the evaluation of pdfof the received signal phase in detail. The coherent detection ofPSK signals requires carrier recovery and the carrier recoverymethods in optical fiber systems including phase-locked loopshave been discussed in detail in [2].

III. PROBABILITY DENSITY FUNCTION OF PHASE OFTHE RECEIVED SIGNAL

The pdf of the received signal phase has been derivedby Ho in [7], [17] for evaluating the BERs of binary CPSK andDPSK. In order to derive the Fourier series (FS) of pdfs in [7],the FS coefficients are obtained using the CHFs rather than theFourier transforms (FT) of the pdfs. The conventional FT of apdf equals the complex conjugate of the CHF, thus the use ofCHF affects the sign of the phase of FS coefficients. Although,this may not affect the final result in some specific cases, e.g.,

EKANAYAKE AND HERATH: EFFECT OF NONLINEAR PHASE NOISE 449

DPSK, it can lead to incorrect expressions in some cases, specif-ically for CPSK. Moreover, the transmitted phase has been as-sumed to be zero in pdf derivations in [7], [17]. It is not straight-forward to modify the pdfs in [7] and [17] to use in our analysiswith the aim of deriving general SER expressions for -arysignaling. To this end, we derive the FS of pdfs using the trueFTs of pdfs so that the coefficients have correct magnitudes andphases closely following the analysis in [7]. In what follows, weshall suppress the time variable for notational convenience.

A. Independent Linear and Nonlinear Phase Noises

When LP and NLP noises are statistically independent, thepdf of in (8) is obtained by convolving the pdf of LP noiseand the pdf of NLP noise . The pdf of in (7d) is well

known [15], [14], which is just the pdf of the phase of a sinusoidin narrowband Gaussian noise. For the present problem, we canexpress it, conditional on the transmitted phase , as

(9)

where

(10)

and is the signal-to-noise ratio (SNR). In (10),and denote the gamma function and the confluent

hypergeometric function [13], respectively. The following alter-native form for in terms of the modified Bessel function

is also useful in numerical computations [14]:

(11)

The pdf of NLP noise is not available in the literature,but Ho has derived the CHF of a normalized form of inclosed-form [6], [7]. On removing the normalization, the CHFgiven in [6] can be expressed as

(12)

where . The FT of pdf is, which is just the complex conjugate

of the CHF, i.e., . The -th coefficientof the FS of modulo equals . Itsmagnitude is , which is even sym-metric in , and the phase is ,which is odd symmetric in . The desired pdf of modulothen takes on the form

(13)

Having obtained the pdfs of and , we need to convolvethem to yield the pdf of . For this, we substituteinto in (9), multiply by the pdf in (13), and integrate with re-

spect to from 0 to [14]. The resulting pdf of simpli-fies to

(14)

where we have substituted the value . The pdf in(14) will be used to determine the average SEP for CPSK signalsin the next Section.In order to detect DPSK signals, we need to consider the re-

ceived phases in two successive signaling intervals. To this end,we consider the interval in addition to .The conditional pdf of received phase in this interval also hasthe same form as (14). Therefore, the pdf of is obtained from(14) by replacing by and by , which is the phasetransmitted in this interval, that is,

(15)

B. Dependent Linear and Nonlinear Phase Noises

When the dependence of the LP and NLP noises is taken intoaccount, the evaluation of the conditional pdf of the receivedphase in (8) is rather lengthy. Therefore, the detailed deriva-tion is given in the Appendix. The desired conditional pdf from(A.34) is

(16)

where is defined by (A.30), which is very similar to (10),and the frequency dependent SNR is defined under (A.28) as

. The optical SNR isdefined previously under (10).As the conditional pdf of the received phase in the interval

is also needed for evaluating the SEP of DPSKsignals, we obtain it from the pdf of in (16) by replacingwith the information bearing phase as

(17)

Having obtained the pdfs of the received phase angles for thecases of interest, we proceed to the next Section to evaluateexpressions for the SEP.

IV. AVERAGE SEP FOR CPSK SIGNALS

Let us assume that we need to detect the phase angle trans-mitted in the interval . The CPSK receiver makes acorrect decision if the received phase angle lies in the sector

. The probability of error of thisdetection scheme conditional on is

(18)


Since the -ary signals have equal energy and are symmetri-cally located in the signal space, the average SEP equals theconditional error probability in (18). Next we evaluate the av-erage SEP expressions for the cases of independent phase noisesand dependent phase noises.

A. Independent Phase Noises

If LP and NLP noises are assumed to be independent, theaverage SEP is evaluated by substituting the conditional pdf ofthe received phase in (14) into the integrand in (18), and thenevaluating the integral. On performing the above mathematicaloperations, the desired SEP takes on the form

(19)

where is given in (A.35). Letting andin (19), we obtain the SEP for CPSK when

there is no NLP noise as

(20)

which is just the SEP for CPSK in Gaussian noise.

B. Dependent Phase Noises

For this case, we substitute the conditional pdf in (16) into(18) and evaluate the integral to obtain the desired SEP as

(21)

where is given in (A.28).

V. AVERAGE SEP FOR DPSK SIGNALS

In DPSK, the symbol transmitted in the intervalis determined by the phase difference . Since thesignal set is symmetrical, can have equally likely valuesthat are symmetrically spaced with spacing . In orderto determine the transmitted phase difference , the DPSKreceiver employs phase change of the receivedsignal from to . The receiver makesa correct decision if phase difference falls within the sector

. The conditional probability oferror associated with this decision is

(22)

We consider the evaluation of conditional pdfbelow for the two cases of interest.

A. Independent Phase Noises

Since the received phase angles in distinct signaling intervalsare independent, the pdf of phase difference is obtained byconvolving pdfs in (14) and in (15). To thisend, we replace in (15) by ( ), multiply the resultingexpression by (14), and carry out the integration over from0 to . The resulting pdf for conditional on and is

(23)

Next, we let in (23), substitute the resulting pdfinto (22), and evaluate the integral to obtain the desired SEP for-ary DPSK signals

(24)where is given in (A.35). For , (24) simpli-fies to the SEP expression given in [9] for BDPSK. If we let

in (24), it reduces to the well known expressionfor the SEP of DPSK signals in Gaussian noise.

B. Dependent Phase Noises

For this case, we convolve the pdf of in (16) with thepdf of in (17) by repeating the procedure outlined above inSection V-A for the case of independent phase noises to obtain

(25)

Next, we substitute this into (22) and evaluate the integral toyield the desired average SEP

(26)where is given in (A.35) and in (A.36). It can beobserved that when , (26) reduces to (24) indi-cating that the effect of dependence of two phase processes is tochange SNR to .

VI. NUMERICAL RESULTS

A. CPSK Signaling

The SEPs for CPSK can be computed using (19) and (21),but if no phase compensation is applied, the resulting error rateswould be very high except when NLP noise is low. Therefore,we apply phase compensation by adding a constant phaseto the phase of the received signal, effectively rotating thesignal constellation anticlockwise to counteract the effect ofNLP noise. In the past literature, the mean value of NLP noise

has been used for . In [17], the BER has been reduced


Fig. 2. Variation of SEP of CPSK signals with scaling factor for phase com-pensation by a constant phase angle with ; ;

, 4 and 8.

TABLE ISCALING FACTOR FOR PHASE COMPENSATION

further by optimally selecting to minimize the BER. It iseasy to show that the effect of is to add the angleto the arguments of cosine terms in the SEP expressions (19)and (21) modifying them to and

, respectively. Inthis paper, we express as and select the valueof that gives the lowest SEP. The optimal value of isdependent on and SNR . However, the variation ofwith SNR is small, particularly when the SNR is high.The optimum value of for a given is determined

graphically by plotting the SEPs against as shown in Fig. 2in which the three SEP curves are computed for , 4, 8,and for illustrative purpose. We have used thisprocedure to generate the values of tabulated in Table I. TheSNR values are selected to yield SEPs lower than for agiven .We can draw several conclusions from Fig. 2. Firstly, we ob-

serve that NLP noise causes high SEPs when there is no phasecompensation (i.e., ) confirming our previous assertion.Secondly, we observe that the use of a suitably scaled valueof for phase compensation allows the SEP performanceto improve by several orders. Thirdly, we observe that the SEPcurves for the cases of independent phase noises and dependentphase noises differ significantly, indicating that the independentphase noise assumption is not valid for CPSK. In view of the

Fig. 3. Symbol error probability for CPSK signaling when dependence ofphase noises is taken into account and phase compensation is employed;

, 4, 8, and 16.

third observation, we present the SEPs for CPSK when the de-pendence of the phase noises is taken into account only.Fig. 3 depicts the SEPs for CPSK that were computed with

(21) for , 4, 8, and 16 using the phase compensationvalues in Table I. For each value of , the SEPs are shown forthree values of the mean of NLP noise , 1.0, and1.4 rad. The reason for using 1.4 rad for instead of 1.5 radis to separate out the curves and avoid overlapping for clarity.We note that, for , the SEP curve foroverlaps with the curve for zero NLP noise indicating that thephase compensation has almost completely nullified the effectof NLP noise. We have verified that SEPs computed with (19)for the case of independent phase noises for BPSK agree withthose given in [8]. The SNR penalty values relative to the zeroNLP noise case at SEPs 1 and 1 are plotted inFig. 5. This figure shows that the SNR penalty values at the twoerror rates are nearly equal. When , QPSK needs2 dB more than BPSK to compensate for NLP noise. The SNRpenalty for BPSK agrees with that given in [17].

B. DPSK Signaling

The SEPs shown in Fig. 4 for DPSK are computed for thecases of independent phase noises and dependent phase noisesusing (24) and (26), respectively. As for CPSK, the parametervalues used are , 4, 8, and 16, and , 1.0,and 1.4 rad. We observe that the SEPs for the independent phasenoise case and the dependent phase noise case agree very closelywhen ; for , there is a marginal difference, which,however, is not large enough to be of any practical significance.We can, therefore, conclude that LP noise and NLP noise canbe treated as independent processes for error rate analysis inDPSK. The SEPs for BDPSK agree with those given in [8],[10], [17]. The SNR penalty curves shown in Fig. 6 at error


Fig. 4. Symbol error probability for DPSK signaling for dependent phasenoises and independent phase noises; , 4, 8, and 16.

Fig. 5. SNR penalty for CPSKwith phase compensation at error rates 1and 1 ; , 4, 8 and 16.

rates 1 and 1 are virtually identical except forthe binary case where there is a marginal difference betweenthe two curves. The SNR penalties for 8-ary and 16-ary DPSKare approximately equal. Our SNR penalty values for QDPSKagree with those in [17]. We also observe that DPSK suffersmore compared to CPSK due to NLP noise when and4, while both CPSK and DPSK undergo nearly the same SNRpenalty when and 16.An insight to the contrasting error rate performance of CPSK

and DPSK can be gained by comparing their SEP expressions.The SEP for CPSK in (19) for the case of independent phasenoises and that in (21) for the case of dependent phase noiseshave cosine terms whose arguments are and

, respectively, which are thephases of the CHFs ; these two expressions yield significantlydifferent error probabilities. In contrast, the two correspondingSEP expressions in (24) and (26) for DPSK do not have terms

Fig. 6. SNR penalty for DPSK at error rates 1 and 1 ; ,4, 8 and 16.

containing the phases of the CHFs and they yield practically thesame SEPs. We can, therefore, conclude that this contrastingbehavior of CPSK and DPSK is partly due to the influence ofthe phases of CHFs. Indeed, the improvement of performanceby phase compensation in CPSK is achieved by making theangles involved in the cosine terms small as possible, therebyminimizing their influence. On the other hand, the absence ofthese terms makes simple direct phase compensation unneces-sary for DPSK signaling.

VII. CONCLUSION

In this paper, we have analyzed the performance of -aryCPSK signals and DPSK signals that are affected by theGordon-Mollenauer effect. The previous analyzes for binaryPSK in [6]–[9] have been extended to include multilevelPSK signaling. Expressions for the SEPs have been derived,first, assuming that ASE noise and NLP noise are statisticallyindependent, and then taking their dependence into account.For CPSK, the assumption of independent phase noises yieldsSEPs that differ significantly from those obtained when theirdependence is taken into account; in contrast, DPSK yieldspractically the same SEPs for these two cases. We have ob-served that some form of phase compensation is needed forCPSK to be practically useful in optical fiber links that exhibitKerr nonlinearity except when NLP noise is small. We havealso observed that DPSK does not lend itself to simple phasecompensation for reducing the effects of NLP noise.

APPENDIXCONDITIONAL PROBABILITY DENSITY

In the following analysis, the Fourier transform of a pdf isdenoted by and the characteristic function of a randomvariable by . In order to find the conditional pdf of the re-ceived phase , we first find the FT of the pdf of . Since

as given in (8), it follows that

(A.27)


where and are the transform domain variables ofand , respectively. Next, in order to determine

, we begin with the partial CHF of therandom variable , given in integral form in [7] by Ho, as

(A.28)where , and is the SNR definedby (10). In (A.28), the exponential term that involves the cosinefunction is expressed in a series of Bessel functions and thenapply the integration formula (11.4.28) given in [13] to obtain

(A.29)

where

(A.30)

Since the FT of a pdf equals the complex conjugate of the CHF,the partial FT equals . Therefore,letting in (A.29) and taking the complex conjugate, weobtain the FT with respect to

(A.31)

where . The FT of (A.31) withrespect to yields the required two dimensional FT

(A.32)

where is the unit impulse function. First, taking the complexconjugate of in (A.32) with respect to ,and then letting and , we obtain the FT of thepdf of received phase as

(A.33)

Equation (A.33) can be simplified by taking insidethe summation and applying the sampling property of impulsefunctions. The inverse FT of the resulting expression is the de-sired pdf for the received phase, that is,

(A.34)

where, from (12) and (A.28), we have

(A.35)

(A.36)

and is as given in (A.28).

ACKNOWLEDGMENT

The authors would like to thank the reviewers for theirhelpful comments and suggestions which greatly improved themanuscript.

REFERENCES

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[3] J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic commu-nication systems using linear amplifiers,” Opt. Lett., vol. 15, no. 23, p.1351, Dec. 1990.

[4] A. Mecozzi, “Limits of long-haul coherent transmission set by the Kerrnonlinearity and noise of the in-line amplifiers,” J. Lightw. Technol.,vol. 12, no. 11, pp. 1993–1994, Nov. 1994.

[5] K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc.Amer. B, vol. 20, no. 9, pp. 1875–1879, Sep. 2003.

[6] K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,”Opt. Lett., vol. 28, no. 15, pp. 1350–1352, Aug. 2003.

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Nimal Ekanayake (M’79) received the B.Sc.Eng (honors) degree in electricaland electronic engineering from the University of Ceylon, Sri Lanka, in 1968,the M.Sc. degree in electronics from King’s College, University of London,England, in 1973, and the Ph.D. degree in electrical engineering fromMcMasterUniversity, Hamilton, Ont., Canada, in 1979.Prior to his graduate studies, he had been with Standard Telephones and

Cables Ltd, London, England, and GTE International Systems Corporation,Waltham, Massachusetts, USA. From 1979 to 1981, he was a Lecturer with theNational University in Singapore, Singapore. He was an Assistant Professor inthe Faculty of Engineering, Memorial University of Newfoundland, Canada,from 1981 to 1985 and an Associate Professor from 1985 to 1989. He joinedthe Department of Electrical and Electronic Engineering, University of Per-adeniya, Sri Lanka, in 1990 where he had held the positions of Senior Lecturer,Professor, and Senior Professor and retired in 2011. Currently, he is an EmeritusProfessor with the University of Peradeniya, Sri Lanka. His research interestsare mainly in the areas of communication theory with emphasis on fading anddiversity systems, interference analysis, and modulation techniques.

H.M. Vijitha R. Herath (M’08) received the B.Sc.Eng (honors) degree in elec-trical and electronic engineering from the University of Peradeniya, Sri Lanka,in 1998, the MS degree in electrical and computer engineering from the Uni-versity of Miami, FL, in 2002, and the Dr.-Ing. degree in electrical engineeringfrom the University of Paderborn, Germany, in 2009.He was a research assistant at the University of Miami, FL, from 2000 to

2002, and a German Academic Exchange Service (DAAD) doctoral scholar atthe of Paderborn, Germany, from 2004 to 2009. Since June 2009, he has beena Senior Lecturer in electrical and electronic engineering in the Faculty of En-gineering, University of Peradeniya, Sri Lanka. His current research interestsinclude nonlinear effects in optical fiber transmission, higher order modulationformats in optical fiber transmission, and algorithm development for coherentQAM transmission systems.Dr. Herath is the founder Chair of IEEE MTT-S Sri Lanka Chapter. He is a

member of OSA, IEEE MTT-S, and IEEE SSSC.

effect of nonlinear phase noise on the performance of $m$-ary psk signals in optical fiber links

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