846 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 6,...

846 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 6, MARCH 15, 2013

Range of Influence and Impact of PhysicalImpairments in Long-Haul DWDM Systems

Houbing Song, Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE

Abstract—In long-haul dense wavelength-division multiplexing(DWDM) systems with periodic dispersion compensation andamplification, system performance is adversely affected by severephysical impairments due to fiber losses, dispersion and nonlin-earity. Fiber modeling is a prerequisite for the development ofphysical impairment mitigation techniques to improve systemperformance. The distance between two interacting symbols intime and wavelength, i.e., the range of influence (RoI) of eachphysical impairment, plays an important role in the developmentof these mitigation techniques. In this paper, we use the Volterraseries transfer function (VSTF) method to define impairmentcharacteristic coefficients that capture intersymbol interference(ISI), self phase modulation (SPM), intrachannel cross phasemodulation (IXPM), intrachannel four wave mixing (IFWM),cross phase modulation (XPM) and four wave mixing (FWM), tocharacterize the impact of these impairments individually on thesystem output. We then investigate the impact of system parame-ters, namely, duty cycle, spectral efficiency, frequency chirp, andspan length, on the RoI for long-haul DWDM systems.

Index Terms—Chromatic dispersion, Kerr effect, nonlinearfiber optics, optical propagation, wavelength-division multi-plexing (WDM).

I. INTRODUCTION

T O support dramatically increasing demand for internetdata traffic over long distance, most long-haul fiber-optic

communication systems are implemented as multispan systems,with periodic dispersion compensation and amplification, andusing dense wavelength-division multiplexing (DWDM). Inthese systems, severe physical layer impairments (PLIs) areinevitable due to fiber losses, dispersion and nonlinearity. Inter-symbol interference (ISI), interchannel interference (ICI), selfphase modulation (SPM), intrachannel cross phase modulation(IXPM), intrachannel four wave mixing (IFWM), cross phasemodulation (XPM) and four wave mixing (FWM), togetherwith amplified spontaneous emission (ASE) noise, adverselyaffect the system performance [1]–[4]. Fiber modeling is aprerequisite to understanding PLIs and developing mitigation

Manuscript received August 03, 2012; revised October 22, 2012; acceptedDecember 03, 2012. Date of publication December 19, 2012; date of currentversion January 23, 2013.H. Song was with the Charles L. Brown Department of Electrical and

Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743USA. He is now with the Department of Electrical and Computer Engi-neering, West Virginia University Institute of Technology, Montgomery, WV25136-2437 USA (e-mail: [email protected]; [email protected]).M. Brandt-Pearce is with the Charles L. Brown Department of Electrical and

Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JLT.2012.2235409

techniques to improve system performance and exploit theadvantages of digital communications and digital signal pro-cessing (DSP). A desirable mathematical model of long-haulDWDM systems is two-dimensional (time and wavelength),discrete-time, and able to characterize PLIs individually. Themodel described in [5] satisfies these criteria, and forms thebasis for the analysis in this paper.The PLIs acting on a symbol in a long-haul DWDM system

are caused by other symbols located in neighboring channelsand neighboring time slots. If two symbols are close to eachother in time and/or wavelength, there are strong PLIs imposedon one another. If two symbols are far enough from each other intime and/or wavelength, these impairments all weaken. There-fore, there exists a range of influence (RoI) for PLIs that must becarefully determined to establish an appropriate level of com-plexity of signal processing. The notion of RoI is similar tothe concept of coherence time/bandwidth, except that the RoIis determined by both linear and nonlinear effects. The RoImeasures the distance (in number of symbols or channels) overwhich crosstalk (due to the presence of multiple pulses and/orchannels) can affect a pulse, unlike coherence time/bandwidth,which measures the width of the autocorrelation of a single op-tical field.In [6], we developed a two-dimensional (2-D) discrete-time

input-output model of PLIs in long-haul DWDM systems,specifically multichannel multipulse multispan systems withperiodic dispersion management and amplification, and in-troduced the notion of RoI. Yet the impairment characteristiccoefficients defined in [6] do not contain all identifying systemparameters, and the RoI definition resulted in ranges that areunnecessarily long. The purpose of this paper is to redefineimpairment characteristic coefficients to be more general,present a broader definition for the RoI of PLIs in long-haulDWDM systems, and investigate the impact of four systemparameters, namely, duty cycle, spectral efficiency, frequencychirp, and span length, on the RoI of these PLIs. The RoIof PLIs defined in this paper is independent of modulationformats. Although it is assumed that chirped Gaussian pulsesare used at the transmitter, the RoI of interchannel PLIs appliesto non-Gaussian pulse-shapes, such as hyberbolic secant pulsesand super-Gaussian pulses [5]. Intrachannel effects are moresensitive to pulse-shape [5].Over the last decade there has been considerable research

in developing signal processing techniques to combat PLIsto improve the system performance of long-haul DWDMsystems, such as forward error correction [7], constrainedcoding [8]–[10], equalization (or predistortion) [11]. Yet thesealgorithms either use an ad hoc method to find the RoI of

0733-8724/$31.00 © 2012 IEEE

SONG AND BRANDT-PEARCE: RANGE OF INFLUENCE AND IMPACT OF PHYSICAL IMPAIRMENTS IN LONG-HAUL DWDM SYSTEMS 847

Fig. 1. Schematic of a typical long-haul DWDM system with periodic dispersion compensation and amplification. is the length of one span.

the impairments or ignore the RoI altogether, which couldresult in significantly overestimating or underestimating thedegradation. Based on the defi


The output field on the th channel at discrete times is givenby1

( 3 )

T h i s c h a n n e l m o d e l c a n b e u s e d f o r t h e d e v e l o p m e n t o f P L I

m i t i g a t i o n t e c h n i q u e s a s i t c h a r a c t e r i z e t h e v a r i o u s P L I s i n d i -v i d u a l l y . I n ( 3 ) , w e m a k e u s e o f P L I c h a r a c t e r i s t i c c o e f fic i e n t s ,

w h i c h q u a n t i f y t h e i m p a c t o f t h e s e i m p a i r m e n t s o n t h e

fib e r

o u t p u t a s f o l l o w s .1 ) I S I c o e f fic i e n t

( 4 )2 ) I n t r a c h a n n e l n o n l i n e a r i m p a i r m e n t c o e f fic i e n t s ( S P M ,I X P M , a n d I F W M )

( 5 )w h e r e

( 6 )

a n d

( 7 )

1 P l e a s e s e e d e t a i l s i n [ 5 ] , w h i c h i n c l u d e s a b a n k o f o p t i c a l fil t e r s . T h e c u r r e n t

m o d e l c a n b e o b t a i n e d b y s e t t i n g t h efil t e r t i m e c o n s t a n t s t o z e r o a n d a m p l i t u d e

g a i n s t o u n i t y .

3 ) I n t e r c h a n n e l n o n l i n e a r i m p a i r m e n t c o e f fic i e n t s ( X P M

a n d F W M )

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T h e A p p e n d i x c o n t a i n s t h e d efi n i t i o n s o f t h e s i m p l i f y i n g f u n c - t i o n s

,a n du s e d i n ( 7 ) a n d ( 1 0 ) .T h i s m o d e l e s t a b l i s h e s a m a p p i n g f r o m t h e d a t a i n p u tt o t h e fi b e r o u t p u tt h a t a p p l i e s t o a n y m o d u l a t i o n f o r m a t . A v e r s i o n o f t h i s m o d e l t h a t i n c l u d e s t h e W D M d e m u l - t i p l e x e r a n d t h e p h o t o d e t e c t o r i s p r e s e n t e d i n [ 5 ] . 2 T h e r e p r e - s e n t a t i o n i n ( 3 ) i n c l u d e s t h e s y m b o l o f i n t e r e s t a s t h e fi r s t t e r m ,

l i n e a r i m p a i r m e n t s , a n d n o n l i n e a r i m p a i r m e n t s . T h e l i n e a r i m - p a i r m e n t s i n c l u d e I S I , c a p t u r e d i n t h e s e c o n d t e r m . T h e l a s t t e r m i n ( 3 ) i s d u e t o t h e l i n e a r o v e r l a p o f o t h e r c h a n n e l s o n

t h e c h a n n e l o n i n t e r e s t ,

. O n c e o p t i c a l fi l t e r i n g i s a d d e d t o t h e m o d e l , t h e l i n e a r a d j a c e n t c h a n n e l i n t e r f e r e n c e e x p e r i e n c e d b y t h e s i g n a l c o u l d b e c o m p u t e d f r o m t h e l a s t t e r m . W i t h o u t o p t i c a l

fi l t e r i n g , t h i s i n t e r f e r e n c e c a n n o t b e m e a n i n g f u l l y i s o l a t e d . T h e r e m a i n i n g t e r m s a r e n o n l i n e a r i m p a i r m e n t s . T h e k e y t o u s i n g t h i s m o d e l i s t o fi n d t h e i n d e x c o m b i n a t i o n s t h a t m e e t t h e r e q u i r e m e n t s f o r v a r i o u s n o n l i n e a r P L I s , a n d t o c a l c u l a t e t h e i m p a i r m e n t c o e f fi c i e n t s c o r r e s p o n d i n g t o t h e s e i n d e x c o m b i - n a t i o n s . I n t r a c h a n n e l i m p a i r m e n t s ( I X P M a n d I F W M ) r e q u i r e

, p l u s t h e t i m e l o c a t i o n o f t h e n o n l i n e a r i n - t e r a c t i o n m u s t s a t i s f y

; i n t e r c h a n n e l i m p a i r -m e n t s ( X P M a n d F W M ) r e q u i r e

a n d t h ef r e q u e n c y l o c a t i o n o f t h e n o n li n e a r i n t e r a c t i o n t h a t s a t i sfie s t h e

p h a s e - m a t c h i n g c o n d i t i o n

. S P M r e q u i r e s a l l

o f t h e s e c o n d i t i o n s t o h o l d . F u r t h e r m o r e , f o r I X P M ,

o r; f o r I F W M ,o r; f o r

X P M ,

o r; a n d f o r F W M ,o r.

2

Note th at th e im p airm en t coef fi cients are de fi ned somewhat differently in

th is paper c ompare d to [ 5].


TABLE IIINDEX TRIPLETS FOR A QUINTUPLE PULSE CASE

We illustrate the various intrachannel impairments by con-sidering a five-pulse case ( ). There areindex triplets and 13 possible time locations because

. For example, the output field atis affected by 19 different triplets. The triplet

contributes to the SPM impairment. The IXPM impair-ment is caused by eight triplets: , , , , ,

, , and , whereas the triplets for IFWM impair-ment are , , , , , , , ,

, and . The type and location of the intrachannelnonlinear impairments generated by a pulse quintuple locatedat , , , , and are summarized in Table II.Using the parameters given in Table I, the absolute values ofthe corresponding intrachannel impairment coefficients aresummarized in Table III. Due to limited space, only nine validtime locations are presented. Consider the first four tripletsresponsible for IXPM at


TABLE IIIINTRACHANNEL COEFFICIENTS FOR A QUINTUPLE PULSE CASE IN W

TABLE IVINTERCHANNEL COEFFICIENTS FOR A QUINTUPLE CHANNEL CASE IN

When the parameters take the values as given in Table I, thepercentages of total cumulative impairments captured, as thenumber of adjacent symbols varies, are given in Fig. 3. TheRoI of ISI is 1 for any percentage. This result has been veri-fied using the split-step Fourier method by varying the numberof pulses and eliminating nonlinearity. Intrachannel effects arefar-reaching, with and ; y e t b o t h a r e

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w h e n t h e p a r a m e t e r s t a k e t h ev a l u e s g i v e n i n T a b l e I a r e g i v e n i n F i g . 4 . T h e m a x i m u m R o I f o rX P M i

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t h eK e r r e f f e c t i s c o n s i d e r e d a n d st i m u l a t e d s c a t t e r i n g i s i g n o r e d

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.


Fig. 2. Concept of RoI of PLIs in long-haul DWDM systems.

Fig. 3. RoI of ISI, IXPM, and IFWM.

A. Relative and Total Degradation due to PLIs

In addition to the RoI of each individual PLI, knowing therelative degradation due to PLIs is very desirable. In this paper,the worst-case total degradation due to PLIs is considered, ascaptured by the cumulative impairment expressions defined inSection III-A. Assuming the same launched power for all chan-nels, , , and using the above cumu-lative PLIs and (3), we define the total cumulative degradationdue to PLIs when adjacent symbols and adjacent channelsare included as


Fig. 6. Impact of the launched power level and the number of spans on the totalcumulative degradation.

how the total cumulative degradation for long-haul DWDM sys-tems increases with an increase in the product . Becausethe contribution of ISI to the total cumulative degradation is in-dependent of the number of spans, the total cumulative degra-dation increases nonlinearly.

IV. IMPACT OF SYSTEM PARAMETERS ON ROI

This section presents the impact of four system parameters,namely, duty cycle, spectral efficiency (channel density), inputpulse frequency chirp, and span length, on the RoI of PLIs inlong-haul DWDM systems. This provides guidelines for systemdesigners to optimize system parameters to improve perfor-mance of long-haul DWDM systems. Some parameter choicescan improve the overall system performance but increase theRoI, thereby imposing larger complexity requirements onsignal processing algorithms used. All baseline system andsignal parameters are as in Table I except for the parametervaried.

A. Impact of Duty Cycle on RoI

The pulse duty cycle is defined as , where is thepulse full-width at half maximum (FWHM) and is the symbolperiod [18]. The relationship between and the half-width ofthe pulse at the intensity point, , is forGaussian pulses [1]. Fig. 7 shows the impact of duty cycle on theRoI of IXPM and IFWM. With the increase in the duty cycle,the RoI of intrachannel PLIs decreases.

B. Impact of Spectral Efficiency on RoI

The spectral efficiency is defined as: , whereis the symbol rate in and is the channel spacing in rad/s

[18]. Fig. 8 shows the impact of spectral efficiency on the RoIof XPM and FWM. With the increase in the spectral efficiency,the RoI of interchannel PLIs increases, except for whichstays constant. Note that the impact of PLI experienced by a real

Fig. 7. Impact of duty cycle on RoI of IXPM and IFWM.

Fig. 8. Impact of spectral efficiency on RoI of XPM and FWM.

system would depend significantly on the WDM demultiplexerand electrical filter used.

C. Impact of Frequency Chirp on RoI

Compared with initially unchirped Gaussian pulses, chirpingthe Gaussian pulses helps to improve the system performanceby optimizing the average power of the input signal [1]. Fig. 9shows the impact of input pulse frequency chirping on the RoIof IXPM and IFWM. With the increase in the absolute value offrequency chirp, the RoI of PLIs increases dramatically.

D. Impact of Span Length on RoI

If the launched power is around 1 mW, the span lengthshould be in the range of 40 to 50 km for long-haul systems withlengths of 6,000 km and 80 km or so for long-haul systems withlengths under 3,000 km [1]. Fig. 10 shows the impact of spanlength on the RoI of IXPM and IFWM. With the increase in the


Fig. 9. Impact of frequency chirp on RoI of IXPM and IFWM.

Fig. 10. Impact of span length on RoI of IXPM and IFWM.

span length, the RoI of PLIs increases but the rate of increasedecreases.

V. CONCLUSION

In this paper, we use our two-dimensional discrete-timeinput-output model of PLIs based on the VSTF method toredefine impairment characteristic coefficients, present a noveldefinition of RoI of PLIs, and investigate the impact of foursystem parameters, namely, duty cycle, spectral efficiency,frequency chirp, and span length, on the RoI of these PLIsin long-haul DWDM systems. These RoIs are of great valueto the development of PLI mitigation techniques for opticalcommunications. The impact of system parameters on RoI of-fers insights into the required complexity of signal processingalgorithms needed for system performance improvement. Weare currently extending this model to include ASE (amplifiedspontaneous emission) noise, polarization mode dispersion, and

postdetection electrical filtering. We are also applying the con-cept of RoI in the development of PLI mitigation techniques,including constrained coding and nonlinear equalization.

APPENDIXSIMPLIFYING FUNCTIONS

This Appendix defines the simplifying functions , , ,, , and used in (7) and (10) as follows:

AC K N O W L E D G M E N T T h e a u t h o r s w o u l d l i k e t o t h a n k P r o f . S . G . W i l s o n o f t h e

U n i v e r s i t y o f V i r g i n i a f o r h i s s u g g e s t i o n t o i n c l u d e t h e t o t a l c u -

m u l a t i v e d e g r a d a t i o n i n o u r a n a l y s i s . RE F E R E N C E S

[ 1 ] G . P . A g r a w a l

, F i b e r - O p t i c C o m m u n i c a t i o n S y s t e m s

, 4 t h e d . N e w Y o r k : W i l e y , 2 0 1 1 1 [ 2 ] E . B a s c h , R . E g o r o v , S . G r i n g e r i , a

n d S . E l b y , “ A r c h i t e c t u r a l t r a d e o f f s f o r r e c o n

Þg u r a b l e d e n s e w a v e l e n g t h - d i v i s i o n m u l t i p l e x i n g s y s t e m s , ” I E E E J . S e l . T o p i c s Q u a n t u m E l e c t r o n . , v o l . 1 2 , n o . 4 , p p . 6 1 5 – 6 2 6 , J u l . – A u g . 2 0 0 6 .


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[17] Y. Cai, D. Foursa, J.-X. Cai, C. Davidson, O. Sinkin, A. Pilipetskii,M. Nissov, and N. Bergano, “Experimental study on broadband non-linear phase wandering in coherent detection long-haul transmissions,”in Proc. Opt. Fiber Commun. Conf., Mar. 2010, pp. 1–3.

[18] R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear trans-mission of high-speed TDM signals: 40 and l60 Gb/s,” inOptical FiberTelecommunications IV-B: Systems and Impairments, I. P. Kaminowand T. Li, Eds. New York: Academic, 2002.

Houbing Song (SM’05–M’12) received the Ph.D. degree in electrical engi-neering from the University of Virginia, Charlottesville, in 2012.In 2012, he joined the Department of Electrical and Computer Engineering,

Leonard C. Nelson College of Engineering & Sciences, West Virginia Univer-sity Institute of Technology,Montgomery, where he is currently a Visiting Assis-tant Professor. He worked with Shandong Academy of Sciences as an AssistantResearch Scientist from 2004 to 2005 and with Texas Transportation Institute(TTI), Texas A&M University, as an Engineering Research Associate in 2007.He has been the founding editor-in-chief of the Journal of Cyber-Physical Sys-tems, the associate editor-in-chief of one blue book series on Internet of Thingssince 2011, and an associate editor of International Journal: Network Protocolsand Algorithms since 2012. He has authored and coauthored more than 30 aca-demic papers in peer-reviewed journals and conferences. His research interestsare focused on discrete-time signal processing for dense wavelength-divisionmultiplexing systems.Prof. Song has served on the technical program committee for numerous con-

ferences, including the 2nd IEEE International Workshop of Smart Communi-cation Protocols and Algorithms (SCPA 2012), 2012 IEEE Wireless Commu-nications and Networking Conference (WCNC 2012), 2012 IEEE InternationalConference on Computer and Information Technology (CIT 2012), 2012 IEEESymposium on Industrial Electronics and Applications (ISIEA 2012), and 2012IEEE International Power and Energy Conference (PECON 2012).

Maïté Brandt-Pearce (SM’99) received the B.S. degree in electrical engi-neering, M.E.E. degree, and Ph.D. degree in electrical engineering from RiceUniversity, Houston, TX, in 1985, 1989, and 1993, respectively.She worked with Lockheed in support of NASA Johnson Space Center from

1985 until 1989. In 1993, she joined the Charles L. Brown Department of Elec-trical and Computer Engineering, University of Virginia, Charlottesville, whereshe is currently a Full Professor. In 2005, she spent her sabbatical at the EurécomInstitute in Sophia Antipolis, France. Her research interests lie in the mathe-matical and numerical description and optimization of communication systemswith multiple simultaneous components from different sources. This interesthas found applications in a variety of research projects including spread-spec-trum multiple-access schemes, multiuser demodulation and detection, the studyof nonlinear effects on fiber-optic multiuser/multichannel communications, op-tical networks subject to physical layer degradations, freespace optical multiusercommunications, and radar signal processing and tracking of multiple targets.Dr. Brandt-Pearce is a member of Tau Beta Pi and Eta Kappa Nu. She was

the recipient of a National Science Foundation (NSF) CAREER Award, an NSFRIA, and an ORAU Junior Faculty Enhancement Award. She was a corecip-ient of Best Paper Awards at the ICC 2006 Symposium on Optical Systemsand Networks. She was an associate editor for the IEEE TRANSACTIONS ONCOMMUNICATIONS from 1999 to 2006. She has served on the technical programcommittee for numerous conferences and was the 2009 General Chair for theAsilomar Conference on Signals, Systems, and Computers.

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