We propose a novel tracking algorithm for optical channels suffering from fast state of polarization (SOP) rotations and polarization-dependent loss (PDL). Unlike gradient descent-based algorithms that require step size adjustment when the channel conditions change, our algorithm performs similarly or better without parameter tuning

Agrell, Erik

Farsi, Mohammad

H\ue4ger, Christian

Karlsson, Magnus

English

Chalmers Research

Improved Polarization Tracking in the Presence of PDLDownloaded from: https://research.chalmers.se, 2023-09-08 04:35 UTCCitation for the original published paper (version of record):Farsi, M., Häger, C., Karlsson, M. et al (2022). Improved Polarization Tracking in the Presence ofPDL. European Conference on Optical Communication, ECOCN.B. When citing this work, cite the original published paper.research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology.It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004.research.chalmers.se is administrated and maintained by Chalmers Library(article starts on next page)Improved Polarization Tracking in the Presence of PDLMohammad Farsi(1), Christian Häger(1), Magnus Karlsson(2), and Erik Agrell(1)(1) Dept. of Electrical Engineering, Chalmers Univ. of Technology, Sweden, farsim@chalmers.se(2) Dept. of Microtechnology and Nanoscience, Chalmers Univ. of Technology, SwedenAbstract We propose a novel tracking algorithm for optical channels suffering from fast state of po-larization (SOP) rotations and polarization-dependent loss (PDL). Unlike gradient descent-based algo-rithms that require step size adjustment when the channel conditions change, our algorithm performssimilarly or better without parameter tuning. ©2022 The Author(s)IntroductionIn modern optical communication systems, ad-vanced digital signal processing (DSP) utilizespolarization-division multiplexing and high-ordermodulations to reach high spectral efficiency(SE)[1]. However, high SE comes at the expenseof a lower tolerance to impairments such asstate of polarization (SOP) drift and polarization-dependent loss (PDL)[2]. In a fiber link, theSOP drifts randomly due to external perturbations(e.g., mechanical/thermal stress, weather condi-tions, etc.) of the fiber. Previous studies havereported that the SOP drift can be extremely slow(days) in buried fibers[3] and very fast (microsec-onds) in aerial fibers[4]–[6]. The SOP drift com-bined with PDL results in a time-varying powerimbalance between the polarizations making thepolarization tracking challenging at the receiver.The gradient descent (GD)-based algorithmsare the most commonly used tracking algorithmsin the literature. The most popular ones arethe constant modulus algorithm (CMA)[7], modi-fied CMA (MCMA)[8], and decision-directed leastmean squares (DDLMS)[9],[10]. However, these al-gorithms suffer from phase ambiguity and the sin-gularity problem[11] which might be resolved bypilot-assisted[12] and hybrid[13] algorithms. How-ever, GD-based algorithms are also step size sen-sitive[14], implying that they may require adjustingtheir step size as the SOP drift speed changes.However, in some applications, an estimate of theSOP drift speed is not available at the receiver.In this paper, considering both SOP drift andPDL, we propose a novel tracking algorithm thatshows a higher polarization drift tolerance thanthe conventional GD-based algorithms. More-over, unlike the GD-based algorithms that requirestep size adjustment, the proposed algorithm re-quires no parameter tuning, making it robust tovarying channel conditions.System ModelAssuming negligible (or already compen-sated) nonlinearities, chromatic dispersion, andpolarization-mode dispersion (PMD), we considermemoryless dual-polarization transmission in thepresence of PDL, SOP drift, and amplified spon-taneous emission (ASE) noise at the receiver.The 2-dimensional random vector sk denotes thetransmitted signal at time k, taking values froma set S = {c1, c2, ..., cM} of complex, zero-mean,equiprobable constellation points. The vector ofreceived complex samples isxk = Hksk + zk, (1)where Hk is a 2× 2 complex channel matrix, andzk ∼ CN (0, σ2zI2) is a vector of complex, ran-dom ASE noise samples. The considered dual-polarization PDL (DP-PDL) channel is describedby a combination of the SOP drift model in[15] andN concatenated PDL elements. In particular, theDP-PDL channel matrix isHk = ΓNJk,N · · ·Γ1Jk,1 =N∏n=1ΓnJk,n, (2)where n is the segment index and Jk,n is a ran-dom 2× 2 unitary matrix accounting for SOP drift.It is defined in[15] asJk+1,n = exp(−jαk,n · −→σ )Jk,n, (3)where exp (·) is the matrix exponential andαk,n ∼ N (0, σ2pI3), where σ2p = 2π∆p · T and ∆pis the polarization linewidth determining thespeed of the SOP drift and T is the symbol du-ration. The total polarization linewidth scales withN and can be defined as ∆ptot = N ·∆p. Finally,−→σ = (σ1, σ2, σ3) is a tensor of the Pauli spin matri-ces[16]. Moreover, Γn = diag(√1 + γn,√1− γn)is a 2 × 2 positive real-valued diagonal ma-𝑘𝑘 𝐿𝐿𝐾𝐾�𝐾𝐾𝑠𝑠PilotsPayloadEqualization WindowFig. 1: Data frame with inserted pilot and payload symbols.trix modeling the power imbalance induced byPDL[17], where 0 ≤ γn ≤ 1 is each segment’sPDL ratio. The segment-wise PDL is defined asφn = (1 + γn)/(1− γn), and the average aggre-gated PDL over K transmitted symbols in dB isdefined asρ̄ = 10 log10(EH[1KK−1∑k=0∥λmaxk ∥2∥∥λmink ∥∥2]), (4)where λmaxk and λmink are the eigenvalues of Hk.Proposed AlgorithmWe propose a pilot-aided adaptive algorithmcalled sliding window least squares (SW-LS). Theidea is to use a length L decision-directed slidingwindow called the equalization window and up-date the channel estimate for each window.The data frame construction is shown in Fig. 1,where a pilot sequence of length Kp is insertedat the beginning of each transmission block oflength Ks. The 2×Kp matrix of pilots is definedas D = [d0, d1, . . . , dKp−1], where the pilots arechosen from the quadrature phase-shift keying(QPSK) constellation. The pilots are orthogonalwith respect to each polarization (i.e., DD† = δI2where δ ∈ R is a constant). This coincides withthe optimal pilot selection suggested in[18],[19].Using the previous estimate of the channel Ĥkand the minimum Euclidean distance criterion,the transmitted symbols in an equalization win-dow can be estimated asŝk = argminc∈S∥∥∥Ĥ−1k xk − c∥∥∥2 (5)and the 2× L matrix of estimated symbolscan be formed as Ŝk = [̂sk, ŝk+1, . . . , ŝk+L−1].Defining the 2× L matrix of received symbolsXk = [xk,xk+1, . . . ,xk+L−1], the channel estima-tion problem can be written asargminĜk∥∥∥Xk − ĜkĤkŜk∥∥∥2 . (6)The optimal solution of Eq. (6) is given by the leastsquares (LS) algorithm asĜk = XkŜ†kĤ†k(ĤkŜkŜ†kĤ†k)−1. (7)Initializing the estimated channel matrix as Ĥ0 =I2, the estimated channel Ĥk then is updated byĤk+ν = ĜkĤk, (8)where ν is the sliding stride of the equalizationwindow, meaning that the equalization windowslides ν symbols at a time.Algorithm 1 describes the SW-LS algorithm.Algorithm 1 SW-LSInput: Xk, Ĥk, k, νOutput: Ĥk+ν , S̃k, k1: for l = 0, . . . , L− 1 do2: i = (k + l mod Ks)3: if i ≤ Kp − 1 then4: ŝk+l = di5: else6: ŝk+l = argminc∈S∥∥∥Ĥ−1k xk+l − c∥∥∥2 // Eq. (5)end7: Ĝk = XkŜ†kĤ†k(ĤkŜkŜ†kĤ†k)−1// Eq. (7)8: Ĥk+ν = ĜkĤk9: S̃k = [̂sk, ŝk+1, . . . , ŝk+ν−1]10: k = k + νSimulation SetupPolarization-multiplexed 16 quadrature amplitudemodulation (PM-16-QAM) at a symbol rate ofRs = 28 Gbaud (i.e., T = 1/Rs) is considered.A random sequence of K = 105 symbols is trans-mitted on each polarization, initialized by random,independent, unitary matrices J0,1, . . . ,J0,N . Thepresented results are averaged over 105 such se-quences. The link has N = 20 segments withidentical polarization linewidth ∆p and segment-wise PDL φn = 0.70 dB, resulting in an averageaggregated PDL ρ̄ = 3.0 dB. The launch power ineach polarization is P = E[s†ksk]. The symbol er-ror rate (SER) is evaluated for various setups inthe presence of SOP drift, PDL, and ASE noisequantified by the signal-to-noise ratio (SNR) perpolarization, defined as SNR = P/σ2z . The pilotand transmission block lengths are set to Kp = 16and Ks = 1016 symbols, yielding 1.6% overhead.The SW-LS algorithm is set to use an equalizationwindow of L = 24 with a sliding stride of ν = 6 toget a fair performance while keeping the complex-ity low.We compare to MCMA[8] (blind), LS (pilot-based), and LS-DDLMS (hybrid) benchmarks. InLS-DDLMS, the DDLMS[9],[10] algorithm is initial-ized with a coarse estimate of the channel ob-tained by applying LS to the pilot sequence ineach transmission block. The maximum likelihood10-8 10-7 10-6 10-5 10-4 10-310-310-210-1100∆ptot · TSERML (H is known)MCMA µ = 10−3MCMA opt. µLS-DDLMS µ = 10−3LS-DDLMS opt. µLSSW-LSFig. 2: SER of various algorithms for a fiber with N = 20segments at SNR = 18 dB where ρ̄ = 3.0 dB is obtained bysetting the φn = 0.7 dB for all the segments.(ML) detection for a known channel at the re-ceiver is also used as a benchmark. The step sizeµ of the MCMA and DDLMS algorithms is opti-mized as a function of SOP drift speed, SNR, andPDL. For MCMA, the phase ambiguity is resolvedby coherent differential coding and the singularityproblem is assumed to be ideally mitigated.ResultsPolarization Drift Tolerance: We start by evaluat-ing the channel tracking ability of SW-LS and theconventional algorithms for the DP-PDL channel.The polarization drift sensitivity is measured bysweeping ∆p while the SNR is fixed.Fig. 2 shows the SER versus ∆ptot · T for anaverage aggregated PDL ρ̄ = 3.0 dB, where theSNR is set such that SER = 10−3 is achieved withML detection. It can be seen that the LS-DDLMSand SW-LS algorithms behave roughly the sameat low SOP drifts (∆ptot · T < 10−6); however,at higher drift speeds (∆ptot · T ≥ 10−6), SW-LSshows the best polarization drift tolerance at theexpense of higher complexity.Moreover, to show the sensitivity of the GD-based algorithms to proper step size µ adjust-ment, the SER of MCMA and LS-DDLMS for afixed µ is also plotted (see blue and yellow dot-ted curves). In particular, we set µ = 10−3,which gives the best performance of MCMA andDDLMS for slowly varying channels. Clearly, theperformance of MCMA and LS-DDLMS is highlydependent on the proper adjustment of µ. How-ever, the proposed algorithm requires no parame-ter tuning, which might be advantageous in someapplications. For instance, in bursty channels, theSOP drift speed does not remain constant duringthe whole transmission. Thus, an algorithm that12 14 16 18 2010-310-210-1100SNR (dB)SERML (H is known)MCMALS-DDLMSLSSW-LSFig. 3: SER of various algorithms for a DP-PDL channel withρ̄ = 3.0 dB and ∆ptot · T = 3.57 · 10−6.is tuned for a specific SOP drift speed may fail totrack sudden changes in the channel.Additive Noise Tolerance: Fig. 3 shows the SERversus SNR of various algorithms for a DP-PDLchannel with ρ̄ = 3.0 dB and ∆ptot ·T = 3.57·10−6.Results show that the proposed SW-LS algorithmoutperforms the benchmarks for the range of con-sidered SNRs. An even more significant perfor-mance gap between SW-LS and the benchmarkshas been observed for faster channels; however,those results are omitted from the paper to savespace.The superior performance of SW-LS comes atthe expense of approximately L/ν times highercomplexity than the benchmarks. Despite thehigher tracking complexity, SW-LS has no param-eter adjustment complexity and is robust for alarge range of drift speeds where the benchmarksbehave poorly. Besides, lower complexity can beobtained by reducing the L/ν ratio, but it also de-grades the performance.ConclusionWe have proposed the SW-LS algorithm to trackmemoryless DP-PDL channels. Numerical simu-lations show that the proposed algorithm is morerobust to polarization drift than the benchmarks,including MCMA as the most popular tracking al-gorithm in the literature. While parameter ad-justment is required for the GD-based algorithms(e.g., the CMA, MCMA, DDLMS, etc.), the pro-posed algorithm outperforms the benchmarkswithout needing such adjustments. The effectof polarization-mode dispersion is left for futurework.Acknowledgements: This work was supported by the Knutand Alice Wallenberg Foundation, grant No. 2018.0090, andthe Swedish Research Council under grants No. 2020-04718and 2021-03709.References[1] K. Kikuchi, “Fundamentals of coherent optical fibercommunications”, Journal of Lightwave Technology,vol. 34, no. 1, pp. 157–179, 2016. DOI: 10.1109/JLT.2015.2463719.[2] K. Roberts, M. O’Sullivan, K.-T. Wu, et al., “Perfor-mance of dual-polarization QPSK for optical transportsystems”, Journal of Lightwave Technology, vol. 27,no. 16, pp. 3546–3559, Aug. 2009. DOI: 10.1109/JLT.2009.2022484.[3] M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift ininstalled fibers”, Journal of Lightwave Technology,vol. 18, no. 7, pp. 941–951, Jul. 2000. DOI: 10.1109/50.850739.[4] D. Waddy, P. Lu, L. Chen, and X. Bao, “Fast state ofpolarization changes in aerial fiber under different cli-matic conditions”, IEEE Photonics Technology Letters,vol. 13, no. 9, pp. 1035–1037, Sep. 2001. DOI: 10 .1109/68.942685.[5] J. Wuttke, P. Krummrich, and J. Rosch, “Polarization os-cillations in aerial fiber caused by wind and power-linecurrent”, IEEE Photonics Technology Letters, vol. 15,no. 6, pp. 882–884, Jun. 2003. DOI: 10 . 1109 / LPT .2003.811143.[6] P. M. Krummrich, D. Ronnenberg, W. Schairer, D.Wienold, F. Jenau, and M. Herrmann, “Demanding re-sponse time requirements on coherent receivers due tofast polarization rotations caused by lightning events”,Optics Express, vol. 24, no. 11, pp. 12 442–12 457, May2016. DOI: 10.1364/OE.24.012442.[7] D. Godard, “Self-recovering equalization and carriertracking in two-dimensional data communication sys-tems”, IEEE Transactions on Communications, vol. 28,no. 11, pp. 1867–1875, Nov. 1980. DOI: 10.1109/TCOM.1980.1094608.[8] K. N. Oh and Y. O. Chin, “Modified constant modulusalgorithm: Blind equalization and carrier phase recov-ery algorithm”, in Proceedings IEEE International Con-ference on Communications (ICC), IEEE, vol. 1, 1995,pp. 498–502.[9] R. W. Lucky, “Techniques for adaptive equalization ofdigital communication systems”, The Bell System Tech-nical Journal, vol. 45, no. 2, pp. 255–286, 1966. DOI:10.1002/j.1538-7305.1966.tb00020.x.[10] S. J. Savory, “Digital coherent optical receivers: Algo-rithms and subsystems”, IEEE Journal of Selected Top-ics in Quantum Electronics, vol. 16, no. 5, pp. 1164–1179, Sep. 2010. DOI: 10.1109/JSTQE.2010.2044751.[11] K. Kikuchi, “Performance analyses of polarization de-multiplexing based on constant-modulus algorithm indigital coherent optical receivers”, Optics Express,vol. 19, no. 10, pp. 9868–9880, May 2011. DOI: 10 .1364/OE.19.009868.[12] B. Spinnler, S. Calabrò, and M. Kuschnerov, “Pilot-assisted channel estimation methods for coherent re-ceivers”, in Optical Fiber Communication Conferenceand Exposition and the National Fiber Optic EngineersConference (OFC/NFOEC), Jun. 2013, OW4B.3. DOI:10.1364/OFC.2013.OW4B.3.[13] M. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Properpolarization demultiplexing in coherent optical receiverusing constant modulus algorithm with training mode”,in Proceedings OptoElectronics and CommunicationsConference (OECC), 2010, pp. 768–769.[14] W. Yi, Z. Zheng, N. Cui, et al., “Joint equalizationscheme of ultra-fast RSOP and large PMD compensa-tion in presence of residual chromatic dispersion”, Op-tics Express, vol. 27, no. 15, pp. 21 896–21 913, 2019.DOI: 10.1364/OE.27.021896.[15] C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannis-son, “Polarization drift channel model for coherent fibre-optic systems”, Scientific reports, vol. 6, no. 1, pp. 1–11,2016. DOI: 10.1038/srep21217.[16] J. N. Damask, Polarization Optics in Telecommunica-tions. NewYork, NY, USA: Springer, 2005.[17] A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems”,IEEE Photonics Technology Letters, vol. 14, no. 3,pp. 313–315, 2002. DOI: 10.1109/68.986797.[18] X. Ma, G. B. Giannakis, and S. Ohno, “Optimal train-ing for block transmissions over doubly selective wire-less fading channels”, IEEE Transactions Signal Pro-cessesing, vol. 51, no. 5, pp. 1351–1366, 2003. DOI:10.1109/TSP.2003.810304.[19] B. Hassibi and B. Hochwald, “How much trainingis needed in multiple-antenna wireless links?”, IEEETransactions on Information Theory, vol. 49, no. 4,pp. 951–963, Apr. 2003. DOI: 10 . 1109 / TIT . 2003 .809594.

Improved Polarization Tracking in the Presence of PDL

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Improved Polarization Tracking in the Presence of PDL

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