Compensating for Nonlinear Effects in Coherent-Detection Optical Transmission Systems

Fan Zhang

(State Key Laboratory of Advanced Optical Communication Systems&Networks,Peking University,Beijing 100871,P.R.China)

Abstract Fiber nonlinearity is one of the most important limiters of capacity in coherent optical communications.In this paper,we review two nonlinear compensation methods:digital backward propagation(BP)and nonlinear electrical equalizer(NLEE)based on the time-domain Volterra series.These compensation algorithms are implemented in a single-channel 50 Gb/s coherent optical single-carrier frequency-division multiplexed(CO-SCFDM)system transmitting over 10×80 km of standard single-mode fiber(SSMF).

Keyw ords coherent optical communication;fiber nonlinearity;digital signal processing

1 Introduction

T he demands of high-capacity optical transmission systems have led to the rapid development of coherent opticalsystems.A coherent receiver with optical hybrid allows the electrical field in the two fiber polarizations to be recovered.High spectral efficiency can thus be implemented using advanced modulation-format encoding information in all the available degrees of freedom.Digital signal processing(DSP)simplifies coherent detection and gives it greater flexibility and hardware transparency as well as the potential to adaptively compensate for channel impairments in the electricaldomain[1],[2].In coherent opticalcommunications,linear impairments,such as fiber chromatic dispersion(CD)and polarization mode dispersion(PMD)(all orders),can be compensated for in principle.Fiber capacity is limited by nonlinearity and amplified spontaneous emission(ASE)in optical amplifiers.ASEis white noise and cannot be eliminated.The maximum transfer rate for a given noise level is called the Shannon limit.Fiber nonlinearity is caused by the optical Kerr effect during signal propagation,and evolution of the optical pulse is governed by the nonlinear Schrödinger equation(NLSE).Therefore,nonlinear distortions are determinate and can be partially compensated for by DSPin the coherent receiver.

Backward propagation(BP)is proposed as a universal method of inverting nonlinear systems.In direct-detection systems,electrical predistortion(EPD)can be used to equalize intrachannel nonlinearities by modeling channel inversion for nonlinear effects and predistorting transmitted waveform accordingly[3],[4].Digital BPis incorporated into coherent opticalreceivers by using a split-step Fourier(SSF)method[5]-[7]based on a hybrid time and frequency-domain approach.BPcan also be implemented using a split-step finite-impulse response filter(SS-FIR)[8]that operates entirely in the time domain.Coherent detection is usually combined with polarization-division multiplexing(PDM)to increase spectralefficiency.Fiber nonlinearity in PDM coherent systems can be compensated for by BPin the Manakov frame[9],[10],which takes the average effect of fast polarization rotations into account.

BPis an iterative numerical method.Because it is non-recursive,a nonlinear electricalequalizer based on the time-domain Volterra series has been proposed for coherent optical systems[11].If a sufficient number of delay taps are used,the accuracy of BPis comparable to that of an SSF method with one step per span.

Coherent optical orthogonal frequency-division multiplexing(CO-OFDM)has intrinsic advantages,such as flexibility in dividing spectrum,high spectralefficiency(SE),and outstanding tolerance of CD and PMD[12].Nonlinearity compensation has previously been performed in CO-OFDM systems using back propagation[10]-[13]and the Volterra series[14],[15].

Although CO-OFDM has some impressive advantages,its main drawback is the high peak-to-average power ratio(PAPR),which can cause severe nonlinear impairments and inefficient power consumption[16]-[18].Recently,coherent opticalsingle-carrier frequency-division multiplexing(CO-SCFDM),also called discrete Fourier transform-spread OFDM(DFT-spread OFDM),has been proposed[19],[20].This modified form of OFDM that has been used in the uplink single-carrier frequency-division multiple access(SCFDMA)scheme for LTE[21].SCFDM has similar throughput and overallcomplexity as OFDM,but it is more tolerant of nonlinear impairments and has lower PAPR[20],[21].

2 Backward Propagation with Split-Step Fourier Method

2.1 Backward Propagation in the Scalar Nonlinear Schrödinger Equation

Signal propagation in fiber can be described by the scalar nonlinear Schrödinger equation(or NLSE)[22]:

where A is the electrical field.The parametersα,β2andγare the attenuation,CD coefficients,and nonlinear parameter of the fiber,respectively.Separate linear and nonlinear components results in

Where D is the linear operator,given by

and N is the nonlinear operator,given by

Because the analytical solution of the nonlinear Schrödinger equation is difficult to obtain,a symmetric SSFsolution is used.In the absence of noise,exact information about the transmitted signal can be obtained at the receiver by solving the BPequation.BPis modeled by numerically solving the nonlinear Schrödinger equation with negative link parameters and using the symmetric split-step Fourier method[7]-[23].In each step,the optical field,A(z,t),evolves according to

where D-1is the inverse linear operator,N-1is the inverse nonlinear operator,andΔh is the step size.

2.2 Manakov Equation

Coherent opticalsystems usually use PDM signals to increase spectralefficiency.The electricalfield of the optical signal is A=[AXAY]T,where AXand AYare the orthogonal polarization components of the electrical field.In a birefringence-free fiber,fiber can be described by the vectorialform of the NLSE[22]:

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In(6),the nonlinear interaction between the two polarizations is described.The fiber randomly scatters the polarization of the electrical field during transmission.The polarization scattering length is much shorter than the nonlinear length,which is typically tens of kilometers.In channels where the nonlinear length is much greater than the length of the random polarization rotations,the Manakov equation can be derived by averaging(6)of the nonlinear operator over fast polarization changes on the Poincaré sphere[24]:

The nonlinear operator is

If the electrical fields of the two polarizations are obtained at the receiver side,the original signal can be approximately recovered by solving the Manakov equation with the inverse fiber link parameters and using the SSFmethod.

3 Nonlinear Electrical Equalizer Based on Volterra Series

3.1 Nonlinear Electrical Equalizer for Scalar Nonlinear Schrödinger Equation

The nonlinear electrical equalizer(NLEE)derives from the theory of Volterra-series expansion[25],which is a powerful tool for analyzing nonlinear systems.The theory states that a time-invariant nonlinear system can be modeled as an infinite sum of multidimensional convolution integrals of increasing order.According the theory,the received optical field,yγ(t),is related to the modulated signal,yγ(t),by

where Hi(τ1,...,τi)is the Volterra kernel determined by the fiber link parameters[26].The first term is the linear intersymbolinterference(ISI),and the others are nonlinear distortions.Because there are no even-order nonlinearities in optical fiber,only odd-order Volterra kernels are considered.At the receiver,an NLEEcan be realized by constructing an inverse transform of(9).The equalized signal,ye(n),can thus be written in the discrete form as

▲Figure 1.Structure of the NLEE.

By solving the nonlinear Schrödinger equation with first-order perturbation theory[27],the output can be viewed as having two parts:the linear solutionμnand the nonlinear optical-field distortion,Δμn.In single-channel systems,only pulses with indices that satisfy the temporal matching condition l+m-k=n induce a noticeable distortion on the pulseμn[28].This is equivalent to the phase-matching of four-wave mixing(FWM)in wavelength-division multiplexing(WDM).Intrachannel self-phase modulation(ISPM)and intrachannel cross-phase modulation(IXPM)are also included by containing the items l=m=k=n,l=n,or m=n(where l≠m).Assuming a nonlinear fiber system andΔμn,the following can be expressed according to the Volterra theory:

Fig.1 shows the structure of the third-order NLEE,and the algorithm is detailed in[11].The in-phase component of the received QPSK signals is I(t),and the quadrature component of the received QPSKsignals is Q(t).The equalized signal for output-carrier phase estimation is ye(t).The complex conjugation of the corresponding signals is()＊.The symbol duration is T,and the sampling space is q T.The discrete coefficient is h.Intrachannel nonlinear distortion is hl,m,k,where the subscripts denote the optical pulse positions.The delay tap number,L=2N+1,is equivalent to the nonlinear ISI length.The third-order items are defined as nonlinear terms,and the equalizer coefficients are adaptively determined using the recursive least square(RLS)algorithm.

3.2 Nonlinear Electrical Equalizer for Manakov Equation

which can be expanded further as[30]

where Cl,m,kis the coefficient determined adaptively using the RLSalgorithm,and uxand uyare the signals from the two orthogonal polarizations.For simplicity,we do not consider the intrachannelfour-wave mixing from the orthogonal polarization.

4 Coherent Optical SCFDM System

Fig.2 shows the DSPblock diagrams for the CO-SCFDM system.At the coder,the transmitted binary data is mapped into QAM or phase-shift keying(PSK)signals and then is grouped into blocks containing M symbols.The first step in the CO-SCFDM system is to perform an M-point DFTto produce a frequency-domain representation of the input symbols.Then,M-point DFToutputs are mapped to N(N≥M)orthogonal subcarriers.After the N-point inverse fast Fourier transform(IFFT),which transforms the subcarriers into a time domain signal,a cyclic prefix(CP)is inserted for each block before the data sequence is transmitted.At the decoder,the N-point FFTtransforms the signals into the frequency domain,and channel equalization is performed.The equalized signalis transformed into the time domain by the M-point inverse discrete Fourier transform(IDFT)for decision.CO-SCFDM is very flexible,and its parameter design can have a high degree of commonality with OFDM.Nonlinear compensation algorithms of the BPand NLEEcan be implemented at different positions(Fig.2b).

▲Figure 2.DSPblock diagram for(a)the CO-SCFDMcoder and(b)the CO-SCFDMdecoder.

At the transmitter,the mapped quadrature phase-shift keying(QPSK)signals are first grouped into blocks of 1680 symbols,and 32 pilots are uniformly time-multiplexed into each block.Then,the signal is transformed into the frequency domain by 1712-point DFT.In the frequency domain,336 guard subcarriers are added and allocated at both sides of the band.These subcarriers provide about 20%oversampling so that the aliasing products can be spectrally separated with a low-pass filter.After subcarrier mapping,the signal is converted back to the time domain by 2048-point IFFT.The cyclic prefix and cyclic suffix,each 60 symbols long,are inserted into every data block before transmission.The duration of a CO-SCFDM symbol is 117.1 ns.The preamble is 2.44%and includes synchronization and training symbols(Table 1).Taking into account 12%redundancy for Ethernet overhead and forward-error correction,PDM QPSK is capable of 50 Gb/s net.

▼Table 1.Key CO-SCFDMdesign parameters

The fiber link comprises ten spans of 80 km standard single-mode fiber,each with an average loss of 20 d B.The fiber dispersion is 17 ps/km/nm,and the fiber nonlinear coefficient is 1.32 km/W.An Erbium-doped fiber amplifier with 5 d B noise fully compensates for fiber attenuation.No in-line chromatic dispersion compensation is used.With an average launch power of 5 d Bm,the signalsuffers nonlinearity distortion during propagation.Nonlinearity is compensated for by either the NLEEor the BP.For NLEE,we choose NL[15];for BP,we choose two different split-step sizes,one of which is equal to the span length.The other step is a full multistep approach determined by the maximum nonlinear phase shift in each step,which is the same 0.05 as the simulation of the transmission link.

Fig.3 shows the results of nonlinear compensation.The error vector magnitude(EVM)is the performance indicator.For Fig.3(a)to(d),the EVM is 15.62 d B,18.34 d B,18.63 d B,and 18.75 d B,respectively.Although PDM signals are transmitted and equalized,we only present the results of x polarization for the sake of simplicity.The BPwith one step per span improves EVM by about 3.0 d B,similar to the full multistep approach.The NLEEprovides similar performance but with asymmetric constellations.

5 Conclusion

In this paper,we review nonlinear compensation methods of digital BPpropose an NLEEbased on the Volterra series.In terms of computational complexity,the time-domain approach of NLEErequires one order of magnitude more than BP-SSF[11].However,NLEEis much more convenient for real-time implementation because the pulses are processed in sequence rather than in a block.BPin[8]uses split-step FIRfiltering and thus supports sequential processing.By comparison,the computationalcomplexity of NLEEis one order of magnitude lower.NLEEhas an important advantage over BPin that it does not require prior knowledge of the fiber-link parameters.The equalizer coefficients are adaptively determined,and this provides flexibility when signals are routed differently in optical networks.

An NLEEbased on the Volterra series can also be implemented in a frequency domain with low complexity[15],[31].For both OFDM and SCFDM systems,Volterra series-based nonlinear equalization should be further studied for both time and frequency implementation.

This paper is mainly focused on nonlinear compensation in single-channeloperation.BPin WDM systems has been well covered in[10],[13],and[32].Besides intrachannel nonlinearities,interchannel nonlinear interactions,such as XPM and FWM,should be considered.If Axmand Aymare,respectively,the envelopes of x and y polarization tributaries of the received mth WDM channel field,the reconstructed total optical field is

▲Figure 3.Constellations of the recovery signals after ten-spantransmission(a)without nonlinear compensation,(b)with nonlinearcompensator as NL,(c)with one-step-per-span BP,(d)with multistep BP.

The channel spacing isΔf=Δω/2π.A set of local oscillators(LOs)is used for coherent detection and reconstruction of each WDM channel.To mitigate nonlinear distortions in WDM systems,the reconstructed optical field of the entire WDM band should be backward propagated using(7).In[13],nonlinear compensation for different intrachannel and interchannel effects is discussed with various channel granularities.XPM compensation significantly improves system performance,especially the fine granularity.The coherent nature of FWM requires the phase-locked LOs to preserve the relative phase between channels[13],[32].

The disadvantage of BPis that it needs the information of both the fiber link and signals of the neighboring channels.Therefore,multichannelsignals must be obtained in the coherent receiver,which makes the system design more complex.As a consequence,BPis only effective in point-to-point fiber links in which all WDM channels share the same TX-RXlocations[13].For optical networks with add/drop multiplexers and signals that are routed differently,interchannel nonlinear compensation is ineffective for BP.Further study is required into blind equalization of interchannel nonlinear distortions using some kind of electrical equalization structure.

Acknowledgement

This work was supported by National Natural Science Foundation of China(No.61077053,60932004,and 60877045)and National Basic Research Program of China(No.2010CB328201).