Carrier Phase Estimation in Dispersion-Unmanaged Optical Transmission Systems

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1 Carrier Phase Estimatio i Dispersio-Umaaged Optical rasmissio ystems iahua Xu, Polia Bayvel Dept. Electroic & Electrical Egieerig Uiversity College Lodo (UCL) Lodo, UK tiahua.xu@ucl.ac.uk Guar Jacobse, Jie Li Networkig ad rasmissio Lab Acreo wedish IC AB Kista, wede guar.jacobse@acreo.se iege Liu, Yimo Zhag Dept. Optical Egieerig iaji Uiversity iaji, Chia tgliu@tju.edu.c ergei Popov Optics & Photoics Group Royal Istitute of echology (KH) tockholm, wede sergeip@kth.se (Ivited) Abstract he study o carrier phase estimatio (CPE) approaches, ivolvig a oe-tap ormalized least-mea-square () algorithm, a block-wise average algorithm, ad a Viterbi-Viterbi algorithm has bee carried out i the log-haul high-capacity dispersio-umaaged coheret optical systems. he close-form expressios ad aalytical predictios for biterror-rate behaviors i these CPE methods have bee aalyzed by cosiderig both the laser phase oise ad the equalizatio ehaced phase oise. It is foud that the Viterbi-Viterbi algorithm outperforms the oe-tap ad the block-wise average algorithms for a small phase oise variace (or effective phase oise variace), while the three CPE methods coverge to a similar performace for a large phase oise variace (or effective phase oise variace). I additio, the differeces betwee the three CPE approaches become smaller for higher-level modulatio formats. Keywords optical fibre commuicatio; digital sigal processig; chromatic dispersio compesatio; carrier phase estimatio I. INRODUCION Optical fibre etworks costitute the substatial part of curret commuicatio ifrastructure by carryig most of digital traffic data. Log-haul high-capacity optical fibre etworks ca be seriously deteriorated by physical impairmets i the trasmissio systems, e.g. chromatic dispersio (CD), polarizatio mode dispersio, laser phase oise (PN) ad fibre oliearities [-0]. ice the compesatio ad mitigatio of these system impairmets ca be implemeted i electrical domai, the combiatio of coheret optical detectio, advaced modulatio formats ad digital sigal processig (DP) has become oe of the most popular ad promisig solutio for ew-geeratio log-haul ad metro optical commuicatio etworks to perform close to the limit of hao capacity, based o the kowledge of sigal amplitude ad phase [-9]. o compesate the PN comig from the trasmitter (x) ad local oscillator () lasers, some research has bee carried out o the feed-forward ad feed-back carrier phase estimatio (CPE) to estimate the phase of optical carriers [30-38]. Amog these, the oe-tap ormalized least-measquare (O-), the block-wise average (), ad the Viterbi-Viterbi () CPE approaches have bee justified for compesatig the carrier PN effectively, which are cosidered as promisig sigal processig methods for ext-geeratio high-capacity fibre commuicatio systems [36-38]. I fibre commuicatio etworks employig DP based CD compesatio, a pheomeo so-called equalizatio ehaced phase oise () is iduced from the iterplay betwee dispersio compesatio module ad carrier PN [39,40]. he optical fibre commuicatio systems will be more severely degraded by, with the icrease of chromatic dispersio, symbol rate, system badwidth, laser 3-dB liewidth (x or ), ad modulatio format [4-44]. he effects of have bee studied for sigle-chael, WDM, OFDM, pre-distortio of dispersio, ad multi-mode fibre trasmissio etworks [45-49]. Also, research has bee implemeted to study the performace of i the CPE i log-haul optical trasmissio systems [50-56]. Cosiderig the impact of, traditioal aalyses of the CPE algorithms are ot appropriate ay more for the desig of DP based highcapacity commuicatio systems. Correspodigly, it would be very iterestig ad useful to study the performace i the O-, the, ad the CPE approaches, whe the impact of both laser PN ad is cosidered. I this paper, theoretical assessmets o the CPE i loghaul fibre commuicatio etworks usig the O-, the, ad the CPE are preseted ad discussed. he close-form expressios of the estimated carrier PN withi these CPE approaches are derived, ad the performace, e.g. the s, has bee aalytically ivestigated. For differet PN variace (or effective PN variace cosiderig

2 ), the theoretical performace of the O-,, ad carrier phase estimatio has bee compared. It is show that the CPE method outperforms the O- ad the CPE methods for a small PN variace (or effective PN variace), as three carrier phase estimatio algorithms coverge to a similar behavior for a large PN variace (or effective PN variace). Besides, the performace of the three CPE approaches gets closer with the icrease of the level of modulatio format. II. PRINCIPLE OF LAER PN AND A. Laser PN he PN i x ad lasers i optical fibre etworks follow a Loretzia distributio ad the laser PN variace ca be described as [,], f x f () x _ Δf x ad Δf are the 3-dB liewidths of x ad laser respectively, ad is the symbol period. Eq. () shows that the laser PN variace will decrease with the icrease of symbol rate. B. he PN variace of i log-haul optical etworks follows the aalytical descriptio i Ref [39,44], the iteractio betwee the dispersio compesatio ad the laser PN is cosidered, f D L c () D is CD coefficiet, L is fibre legth, λ is wavelegth of optical carrier wave. Whe is cosidered i carrier phase estimatio, total effective PN variace i the -level PK optical fibre etworks ca be expressed as the followig descriptio [4,44], x _ x _ D L f c f x _ x f III. ANALYI OF CARRIER PHAE EIMAION A. O- CPE he trasfer fuctio of the O- CPE i optical etworks ca be expressed as follows [35,44], w y k kxk (3) w (4) k w k ek x k xk (5) e k dk yk (6) x(k) meas iput sigal, k meas symbol idex, y(k) meas output sigal, w (k) represets k-th tap weight, d(k) is desired output sigal, e(k) meas the error betwee output sigal ad desired sigal, ad μ meas the step size i O- approach. he O- CPE method behaves similar to the ideal differetial CPE, ad the i the O- CPE for QPK optical etworks ca be approximated as [35,44]: _ QPK erfc (7) 4 Accordigly, the i O- approach for - level PK optical fibre etworks is calculated as follows: log erfc is the total PN variace for -level PK etworks. B. CPE For -level PK etworks, the estimated carrier phase i each data block usig the approach is described as i Ref [35,37], (8) q N k arg x p (9) p q N k q (0) N k is sigal idex, N is block size of method. he for CPE i -level PK trasmissio etworks is described usig aylor expasio as follows: p N N erfc log p p () 3 3 p 3 p N p 3 N p N 6N is the total PN variace i -level PK etworks. () C. CPE he estimated carrier phase i method for the -level PK optical etworks is described usig the equatio i Ref [35,38],

N k arg x k q (3) qn N is the block size of CPE algorithm.

higher-level modulatio formats. log erfc 6N N (4) is the total PN variace i -level PK etworks. IV. REUL AND DIUCION As show from Fig. to Fig.

formats of the QPK (Fig. ), the 8-PK (Fig. ), the 6-PK (Fig. 3), the 3-PK (Fig. 4) ad the 64-PK (Fig. 5) are applied respectively.

3 N k arg x k q (3) qn N is the block size of CPE algorithm. he i the carrier phase estimatio for - level PK coheret trasmissio etworks is assessed usig followig descriptio: differece betwee the three carrier phase estimatio methods becomes smaller for higher-level modulatio formats. log erfc 6N N (4) is the total PN variace i -level PK etworks. IV. REUL AND DIUCION As show from Fig. to Fig. 5, the s versus PN variaces (or effective PN variace cosiderig ) i the above three (the O-, the, ad the ) CPE methods i log-haul fibre etworks have bee comparatively ivestigated, the modulatio formats of the QPK (Fig. ), the 8-PK (Fig. ), the 6-PK (Fig. 3), the 3-PK (Fig. 4) ad the 64-PK (Fig. 5) are applied respectively. I all these aalytical models, the atteuatio, the PMD, the fibre oliearities are eglected. A block size of 5 is used i both ad carrier phase estimatio methods, sice the AWGN i the chael such as amplified spotaeous emissio (AE) oise from the optical amplifiers should also be take ito cosideratio i practical optical commuicatio systems. he s i the three CPE approaches are evaluated ad discussed comparatively i the rage from 0-6 to 0.5. Fig.. s versus PN variaces i the three CPE approaches i 8- Fig. 3. s versus PN variaces i the three CPE approaches i 6- Fig.. s versus PN variaces i the three CPE approaches i Q It ca be foud i Fig. that, i the QPK trasmissio system, the Viterbi-Viterbi CPE method outperforms the O- ad the methods for a small PN variace (or effective PN variace), as the three CPE algorithms will coverge to a similar behavior for a large PN variace (or effective PN variace). he same treds ca be foud i the 8- PK optical etwork i Fig., the 6-PK optical etwork i Fig. 3, the 3-PK optical etwork i Fig. 4, ad the 64-PK optical etwork i Fig. 5. Meawhile, it is also foud that the Fig. 4. s versus PN variaces i the three CPE approaches i 3- It is oted that the O- algorithm ca also be employed for the -QAM coheret optical systems, while the block-wise average ad the Viterbi-Viterbi methods caot be easily used for the square -QAM coheret systems except the circular costellatio -QAM systems.

4 Fig. 5. s versus PN variaces i the three CPE approaches i 64- V. CONCLUION he aalysis of CPE i log-haul high-capacity optical commuicatio etworks, usig the O-, the, ad the approaches, are ivestigated, both cosiderig laser PN ad. he close-form descriptio for estimatig carrier phase i these carrier phase estimatio approaches are aalyzed, ad the performace i these CPE approaches is also studied i detail usig differet modulatio formats. It is foud that, CPE method outperforms the O- ad approaches for a small PN variace (or effective PN variace), as three CPE algorithms will coverge to a similar behavior for a large PN variace (or effective PN variace). Furthermore, the differece betwee the three CPE methods becomes smaller for higher-level modulatio formats. ACKNOWLEDGMEN Fiacial supports uder UK EPRC program UNC EP/J0758/, EU program GRIFFON 3439, EU program ICONE , ad wedish Veteskapsradet REFERENCE [] I. Kamiow,. Li, ad A.E. 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Analytical Estimation in Differential Optical Transmission Systems Influenced by Equalization Enhanced Phase Noise

Analytical Estimation in Differential Optical Transmission Systems Influenced by Equalization Enhanced Phase Noise Tianhua Xu 1,*,Gunnar Jacobsen 2,3,Sergei Popov 2, Tiegen Liu 4, Yimo Zhang 4, and Polina