The Impact of Transceiver Noise on Digital Nonlinearity Compensation
Daniel Semrau, Domanic Lavery, Lidia Galdino, Robert I. Killey, Polina Bayvel
OOctober 3, 2017 1
The Impact of Transceiver Noise on DigitalNonlinearity Compensation
Daniel Semrau,
Student Member, IEEE,
Domanic¸ Lavery,
Member, IEEE,
Lidia Galdino,
Member, IEEE,
Robert I. Killey,
Senior Member, IEEE, and Polina Bayvel,
Fellow, IEEE, Fellow, OSA
Abstract —The efficiency of digital nonlinearity compensation(NLC) is analyzed in the presence of noise arising from amplifiedspontaneous emission noise (ASE) as well as from a non-idealtransceiver subsystem. Its impact on signal-to-noise ratio (SNR)and reach increase is studied with particular emphasis on splitNLC, where the digital back-propagation algorithm is dividedbetween transmitter and receiver. An analytical model is pre-sented to compute the SNR’s for non-ideal transmission systemswith arbitrary split NLC configurations. When signal-signalnonlinearities are compensated, the performance limitation arisesfrom residual signal-noise interactions. These interactions consistof nonlinear beating between the signal and co-propagating ASEand transceiver noise. While transceiver noise-signal beating isusually dominant for short transmission distances, ASE noise-signal beating is dominant for larger transmission distances. Itis shown that both regimes behave differently with respect tothe optimal NLC split ratio and their respective reach gains.Additionally, simple formulas for the prediction of the optimalNLC split ratio and the reach increase in those two regimesare reported. It is found that split NLC offers negligible gainwith respect to conventional digital back-propagation (DBP) fordistances less than 1000 km using standard single-mode fibersand a transceiver (back-to-back) SNR of 26 dB, when transmitterand receiver inject the same amount of noise. However, whentransmitter and receiver inject an unequal amount of noise, reachgains of 56% on top of DBP are achievable by properly tailoringthe split NLC algorithm. The theoretical findings are confirmedby numerical simulations.
Index Terms —Digital nonlinearity compensation, Optical fibercommunications, Gaussian noise model, Nonlinear interference,Transceiver noise, Digital back propagation, Split nonlinearitycompensation
I. I
NTRODUCTION D IGITAL nonlinearity compensation (NLC) offers a greatpotential in overcoming the limit in optical communi-cation systems imposed by fiber nonlinearity [1]–[3]. Mostdigital nonlinearity compensation techniques extend the phys-ical link with a virtual link in the digital signal processing(DSP) stage using an inverted propagation equation. To date,three different implementations have been proposed in theliterature, depending on whether this virtual link is placed atthe transmitter, receiver or evenly split between them.Receiver-side NLC, also called digital back-propagation(DBP), has been proposed in numerous research papers to
This work was supported by a UK EPSRC programme grant UNLOC(EP/J017582/1) and a Doctoral Training Partnership (DTP) studentship forDaniel Semrau. D. Lavery is supported by the Royal Academy of Engineeringunder the Research Fellowships scheme.D. Semrau, D. Lavery, Lidia Galdino, R. I. Killey, and P. Bayvel are withthe Optical Networks Group, University College London, London WC1E 7JE,U.K. (e-mail: { uceedfs; d.lavery; l.galdino; r.killey; p.bayvel } @ucl.ac.uk.) reduce the impact of fiber nonlinearities and achieve improvedtransmission performance [4]–[9]. Reach increases of around100% (from km to km) and 150% (from km to km) have been experimentally demonstrated, when NLCis applied jointly to all received channels [10], [11]. For shorterdistances, even a threefold increase in transmission distancewas experimentally achieved (233% from km to km)[11]. Overcoming the relatively small bandwidths of digital-to-analog converters and the use of mutually coherent sourcesenabled the application of transmitter-side NLC, sometimesreferred to as transmitter-side DBP or digital precompensation(DPC) [12]. Reach gains of 100% (from km to km)and 200% (from km to km) for shorter distanceshave been shown experimentally [13], [14].The performance difference between transmitter-side andreceiver-side NLC lies only in the periodic arrangement ofthe optical amplifiers along the link [15]. This is due to over-/under-compensated ASE noise-signal interactions (hereafter“ASE noise beating”) that strongly depend on the specificlocation where each ASE noise contribution is introduced. Forconventional links, where an optical amplifier is located aftereach span, DPC improves the transmission performance by upto one additional span. The gain in signal-to-noise ratio (SNR)decreases with distance and is approximately . dB after spans and less than . dB after more than spans [16].Apart from transmitter and receiver-side NLC, an imple-mentation has been proposed where the virtual link is equallydivided between transmitter and receiver, which is referredto as split NLC or split DBP [15]–[17]. This approach min-imizes the residual ASE noise beating and yields at least . dB improvement in SNR compared to conventional DBP,assuming full-field compensation (NLC applied jointly to allchannels) and the absence of transceiver noise. However, todate there is no experimental demonstration of this potentiallyadvantageous scheme.All theoretical considerations above only assume ASE noiseinjected by optical amplifiers. Therefore, they might not gen-erally apply for real transmission systems that further ex-hibit noise originating from non-ideal transceivers. Transceivernoise (TRX noise) is related to the back-to-back perfor-mance; that is, the maximal achievable SNR in a transmissionsystem. This phenomenological quantity combines all noisecontributions from transmitter and receiver such as quantiza-tion noise of analogue-to-digital (ADC) or digital-to-analogue(DAC) converters and noise from linear electrical amplifiersand optical components. We recently showed that resultingTRX noise-signal interactions (hereafter “TRX noise beating”) a r X i v : . [ ee ss . SP ] S e p ctober 3, 2017 2 significantly reduce the gains of (receiver-side) digital back-propagation [11]. Due to the adverse impact of transceivernoise-signal beating, the performance analysis of transmitter-side, receiver-side and split NLC must be substantially revisedfor practical transmission systems with realistic transceiversub-systems.In this paper, digital nonlinearity compensation in the pres-ence of transceiver noise is studied with particular emphasison split NLC. We refer to split NLC as an arbitrary splitof the virtual link between transmitter and receiver, includingDPC and DBP as special cases. The contributions of thepaper are twofold. First, an analytical model is presented thatpredicts the received SNR for any arbitrary NLC split ratio (inSec. II). Second, using this model, two regimes are defineddepending on the negligibility of either TRX or ASE noisebeating. The two regimes are studied separately as they bothexhibit a different behavior with respect to the optimal NLCsplit ratio and the achievable reach gain. Simple expressionsfor their computations are reported and the implications ofsplit NLC for realistic systems are deduced (in Sec. II-Aand II-B). Finally, the theoretical findings are confirmed bynumerical simulations for two cases (in Sec. III): One wherethe transmitter and receiver introduce an equal amount of noiseand another where an unequal amount of noise is introduced.II. A NALYTICAL M ODEL
In this section the impact of transceiver noise on splitnonlinearity compensation is studied analytically. Split NLCmeans that nonlinearity compensation is performed over X spans at the transmitter and over the remaining N − X spans at the receiver, where N is the total number of spansin the physical link. The transceiver noise resulting from alimited transceiver SNR (i.e., the back-to-back SNR) is dividedbetween transmitter and receiver according to a ratio κ R ,where κ R = 0 means that all the TRX noise is injected atthe transmitter and κ R = 1 means that all the TRX noiseis injected at the receiver. In other words, the transmitterimposes a maximum achievable SNR of SNR TX = SNR
TRX − κ R and the receiver imposes a maximum achievable SNR ofSNR RX = SNR
TRX κ R , where SNR TRX is the transceiver SNR. Thetransmission set-up is schematically illustrated in Fig. 1.
DPC × X OpticalTransmitter
SNR = SNR
TRX − κ R LINK × N OpticalReceiver
SNR = SNR
TRX κ R DBP × ( N − X ) Fig. 1: The transmission model used for investigating theperformance of fiber nonlinearity compensation, where thedigital nonlinearity compensation is arbitrary divided betweentransmitter and receiver.The SNR after full-field nonlinearity compensation is givenby [11, Eq. (6)]SNR = PκP + N P
ASE + 3 η ( κξ TRX P + ξ ASE P ASE ) P , (1)where P is the launch power per channel, κ = SNR
TRX , P ASE is the ASE noise per amplifier, η is the nonlinear interference coefficient for one span, ξ TRX is the TRX noisebeating accumulation factor and ξ ASE is the ASE noise beat-ing accumulation factor. The latter two quantities representuncompensated nonlinear mixing products between signal andnoise and are the only quantities in Eq. (1) that depend on X . Therefore, their minimization plays a fundamental role forsplit nonlinearity compensation. Both accumulation factors aregiven by ξ TRX = (1 − κ R ) X (cid:15) (cid:124) (cid:123)(cid:122) (cid:125) TX beat. + κ R ( N − X ) (cid:15) (cid:124) (cid:123)(cid:122) (cid:125) RX beat. , (2) ξ ASE = X − (cid:88) i =1 i (cid:15) + N − X (cid:88) i =1 i (cid:15) , (3)where (cid:15) is the coherence factor [18], [19]. The coherence factoris a measure for coherent accumulation of nonlinearity alongthe spans of a link. The first term in (2) represents the residualuncompensated beating between signal and transmitter noiseand the second term in (2) represents the residual beatingbetween signal and receiver noise. Eq. (3) represents theresidual beating between signal and ASE noise from theoptical amplifiers. Both beating contributions build up duringthe propagation in the physical link (LINK box in Fig. 1) andare then either reduced or enhanced in the virtual link at thereceiver-side (DBP box in Fig. 1).As described in the following sections, the NLC split ratiothat minimizes the signal-noise interaction is different in thecase of TRX noise beating and ASE noise beating. ASE noisebeating can be typically neglected for short distances whichwe refer to as the transceiver noise beating regime. On theother hand, TRX noise beating can be neglected for very largedistances which we refer to as the ASE noise beating regime.In the following, both regimes are studied separately withrespect to their split NLC gains and approximate inequalitiesare derived that define both regimes. A. The impact of transceiver noise beating
We define the transceiver noise beating regime as theregime, where the TRX noise beating is much strongerthan the ASE noise beating at optimal launch power ( κξ TRX P opt (cid:29) ξ ASE P ASE ) . In the TRX noise beating regimethe general SNR (1) reduces toSNR = PκP + N P
ASE + 3 ηκξ
TRX P . (4)The optimal NLC split X opt is obtained by setting the deriva-tive of (4) with respect to X to zero and solving for X opt . Theoptimal split is found as X opt = N (cid:16) − κ R κ R (cid:17) (cid:15) , (5)with the optimal TRX noise beating accumulation factor ξ TRX,opt = (1 − κ R ) − (cid:15) + κ − (cid:15) R (cid:104) (1 − κ R ) − (cid:15) + κ − (cid:15) R (cid:105) (cid:15) · N (cid:15) , (6) ctober 3, 2017 3 where (cid:98) x (cid:101) denotes the nearest integer function. This functionis the result of the quantization of the number of spans. In thefollowing this rounding is removed for notational convenience.It should be noted that the optimal NLC split ratio X opt N is onlya function of the transceiver noise ratio and the coherencefactor.For comparison, the gain in reach with respect to DBP ( X =0 ) is analyzed. The TRX noise accumulation factor for DBPis ξ TRX,DBP = κ R N (cid:15) [11]. Inserting ξ TRX,opt and ξ TRX,DBP in(4), forcing SNR opt = SNR
DBP and solving for ∆ N max = N opt N DBP yields the reach increase of split NLC with respect to DBP.The result is ∆ N max = κ R (cid:104) (1 − κ R ) − (cid:15) + κ − (cid:15) R (cid:105) (cid:15) (1 − κ R ) − (cid:15) + κ − (cid:15) R (cid:15) . (7)Similar to the optimal NLC split ratio, the gain in reach is onlydependent on the transceiver noise ratio and the coherencefactor. Eq. (7) yields the gain with respect to DBP. In orderto obtain the reach gain compared to DPC ( X = N ), κ R mustbe replaced by − κ R .Typical transmission systems in optical communicationsexhibit a high dispersion coefficient and wide optical band-widths that result in a small coherence factor. For dispersionparameters D > pskm · nm , attenuation coefficients α > . dBkm ,and optical bandwidths > GHz, the coherence factor is (cid:15) < . for km spans and EDFA amplification [18, Fig.10]. Coherence factors for backward pumped Raman-amplifiedsystems are slightly higher yielding (cid:15) < . for the sameparameters [19, Fig. 3]. For (cid:15) (cid:28) the optimal NLC splitreduces to X opt = if κ R < . , N if κ R = 0 . ,N if κ R > . , (8)and the TRX noise beating accumulation factor reduces to ξ TRX,opt = (cid:40) (cid:15) · N (cid:15) if κ R = 0 . , min [1 − κ R , κ R ] · N (cid:15) otherwise. (9)Eq. (8) shows that transmission systems with low coherencefactors and higher transmitter noise than receiver noise shoulddeploy transmitter-side NLC for maximum performance andvice versa when there is more transmitter noise. In otherwords, the virtual link should be placed at the one end whereless noise is injected. This, perhaps surprising, result is dueto the fact that only transceiver noise beating is considered inthis section.The split NLC gain in reach with respect to DBP for (cid:15) (cid:28) yields ∆ N max = if κ R ≤ . , (cid:16) κ R − κ R (cid:17) (cid:15) if κ R > . . (10)There is no split NLC reach gain with respect to DBP for κ R < . as DBP is already the optimum itself. When κ R isreplaced by − κ R , (10) gives the split NLC reach gain withrespect to DPC due to symmetry reasons. It is apparent from 0 0.05 0.1 0.15 0.2 0.25 0.3020406080100 κ R = 0 . κ R = 0 . κ R = 0 . κ R = 0 . κ R = 0 . Coherence factor (cid:15) R eac hg a i nov e r D B P ∆ N m a x [ % ] Actual gain, Eq. (7)Approx. gain, Eq. (10)
Fig. 2: Gain in reach of split NLC with respect to DBP asfunction of the coherence factor for a variety of transceivernoise ratios. Shown are the exact gain from Eq. (7) and itsapproximation for small (cid:15) from Eq. (10).(10) that transmission systems with low coherence and equallydivided transceiver noise ( κ R = 0 . ) exhibit no gain comparedto DBP in the TRX noise beating regime. However, split NLCgains are significant, when the transceiver noise is unequallydivided between transmitter and receiver (e.g. in the case thatthe ADC introduces more noise than the DAC).The split NLC reach gain with respect to DBP (Eq. (7)and its approximation Eq. (10)) are shown in Fig. 2 as afunction of coherence factor for a variety of transceiver noiseratios. Only transceiver ratios κ R ≥ . are shown. For lowertransceiver noise ratios the plot can be interpreted as the splitNLC gain with respect to DPC when κ R is replaced by − κ R .Fig. 2 is sufficient to estimate whether the coherence factorcan be considered small and the approximation (10) can beused. Eq. (10) serves as an excellent approximation for mostof the cases except for high coherence factors combined witha transceiver noise ratio close to . . The plot also shows thatthe split NLC reach gain is larger for systems with a largerunbalance between the amount of noise injected by transmitterand receiver. For example, when more noise is injected atthe receiver and (cid:15) (cid:28) , the receiver noise beating (occurringin the DBP box in Fig. 1) can be fully removed by placingthe complete virtual link at the transmitter. This will result intransmitter noise beating (occurring in the physical link) whichwill be smaller than the removed receiver noise beating.In the following, a simple inequality is derived to determinewhether a transmission system is operated in the TRX noisebeating regime. First, we start with the condition that ASEnoise beating is negligible compared to TRX noise beating atoptimal launch power ξ ASE P ASE (cid:28) ξ TR κP opt . (11)Inequality (11) is then expanded as ξ ASE P ASE ≤ ξ ASE,DBP P ASE (cid:28) ξ TR,opt κP opt ≤ ξ TR κP opt , (12) ctober 3, 2017 4 with ξ ASE,DBP = (cid:80) Ni =1 i (cid:15) . It is sufficient to consider theinner inequality in (12) in order to show that (11) holds, whichyields (cf. [11, Appendix])SNR EDC,ideal [ dB ] (cid:29) (cid:18) SNR
TRX min [1 − κ R , κ R ] (cid:19) [ dB ] − . dB , (13)where ( · ) [ dB ] means conversion to decibel scale andSNR EDC,ideal is the SNR at optimal launch power with elec-tronic dispersion compensation only and no transceiver noise,which can be calculated asSNR
EDC,ideal = 1 (cid:113) P ASE ηN (cid:15) . (14)When inequality (13) is satisfied, the corresponding systemis operating in the transceiver noise beating regime and theoptimal split ratio and reach gain reported in this sectionapplies. B. The impact of ASE noise beating
In this section the regime is discussed where the TRXnoise beating is much weaker than ASE noise beating atoptimal launch power ( κξ TRX P opt (cid:28) ξ ASE P ASE ) . This regimehas already been studied in the literature [15]–[17] and istherefore only briefly covered. In the ASE noise beatingregime the general SNR (1) reduces toSNR = PN P
ASE + 3 ηξP P ASE , (15)with the optimal NLC split given as X opt = (cid:6) N (cid:7) , where (cid:100) x (cid:101) denotes the ceiling function with the optimal ASE noisebeating accumulation factor [16, Eq. (7)] ξ ASE,opt = (cid:18) N (cid:19) (cid:15) + 2 N − (cid:88) i =1 i (cid:15) . (16)Similar to section II-A, the gain of split NLC is compared tothe performance of DBP. The gain in reach ∆ N max = N opt N DBP can be expressed as ∆ N max = 2 (cid:15) (cid:15) . (17)The split NLC reach increase is only a function of the coher-ence factor with ∆ N max → % for (cid:15) (cid:28) . This means that areach increase of % is expected for typical high bandwidthtransmission systems in optical fiber communications.Similarly to section II-A, an inequality is derived to de-termine whether a transmission system is operated in theASE noise beating regime. First, we start with the conditionthat TRX noise beating is negligible compared to ASE noisebeating ξ ASE P ASE (cid:29) ξ TR κP opt , (18)which is then expanded to (for (cid:15) (cid:28) ) ξ ASE P ASE > ξ
ASE,opt P ASE ≈ ξ ASE,DBP P ASE (cid:29) ξ TRX,max κP opt > ξ TRX κP opt . (19) Considering only the inner inequality to prove (18) yieldsSNR EDC,ideal [ dB ] (cid:28) (cid:18) SNR
TRX max [1 − κ R , κ R ] (cid:19) [ dB ] − . dB , (20)with SNR EDC,ideal as in (15). When inequality (20) holds, thecorresponding transmission system is operated in the ASEnoise beating regime and the optimal split ratio and the reachgain reported in this section applies.III. S
IMULATION R ESULTS
In this section two optical transmission systems are simu-lated by numerically solving the Manakov equation using thesplit-step Fourier (SSF) algorithm with parameters listed inTable I. Additive white Gaussian noise was added at trans-mitter and receiver to emulate a finite transceiver SNR andnonlinearity compensation was carried out as schematicallyshown in Fig. 1. A matched filter was used to obtain theoutput symbols and the SNR was ideally estimated as the ratiobetween the variance of the transmitted symbols E [ | X | ] andthe variance of the noise σ , where σ = E [ | X − Y | ] and Y represents the received symbols after digital signal processing.The nonlinear interference coefficient and the coherence factorwere obtained in closed-form from [18, Eq. (13) and Eq. (23)]with the modulation format dependent correction from [20, Eq.(2)]. Closed-form expressions for both quantities in the contextof Raman amplification can be found in [19].TABLE I: Simulation Parameters Parameters
Span length [km] 80Loss ( α ) [dB/km] 0.2Dispersion ( D ) [ps/nm/km] 17NL coefficient ( γ ) [1/W/km] 1.2Number of channels 3Optical bandwidth ( B ) [GHz] 96Symbol rate ( R b ) [GBd] 32Channel spacing [GHz] 32 (Nyquist)Roll-off factor [%] 0Noise Figure [dB] 4Transceiver SNR (SNR TRX ) [dB] 26Oversampling 3Number of SSF steps per span 800 log-distributedModulation format 256-QAMNLI coeff. ( η ) [dB (cid:0) (cid:1) ] 26.2Coherence factor ( (cid:15) ) 0.108In order to test the theory presented in section II, a system witha transceiver noise that is equally divided between transmitterand receiver ( κ R = 0 . ) and a system with an unequal divisionof transceiver noise ( κ R = 0 . ) are simulated. A. Equal transmitter and receiver noise contribution
An optical transmission system with an equal share oftransceiver noise between transmitter and receiver ( κ R = 0 . )is simulated. The received SNR as a function of distance isshown in Fig. 3. The lines represent the analytical model esti-mated by (1) at optimal launch power for electronic dispersion ctober 3, 2017 5 SNR
TRX = ∞ dBSNR TRX = 26 dB . dB % = 2 (cid:15) (cid:15) %ASE beat. < TRX beat. ASE beat. > TRX beat.
Number of spans r ece i v e d S N R [ d B ] EDCDPC ( X = N ) DBP ( X = 0) split NLC ( X = X opt ) Eq. (13), ASE beating ≈ TRX beating
Fig. 3: SNR at optimum launch power as a function of span number obtained by simulation (markers) and Eq. (1) (lines).The case with an infinite transceiver SNR (solid lines) and a finite transceiver SNR of dB (dashed lines) are shown. Thetransceiver noise is equally divided between transmitter and receiver ( κ R = 0 . ).compensation (EDC), DBP ( X = 0 ), DPC ( X = N ) and splitNLC with the optimal split NLC of X opt = (cid:6) N (cid:7) betweentransmitter and receiver. A split of X = (cid:6) N (cid:7) is the optimumfor a system where the transceiver noise is equally dividedbetween transmitter and receiver. For the EDC case a sum-mand η (cid:15) P is added in the denominator to include signal-signal nonlinearity in (1). Markers represent results obtainedby numerical simulation. Furthermore, the same transmissionsystem without transceiver noise (SNR TRX = ∞ dB) is shownwith dashed lines and the point where ASE noise beatingapproximately equals TRX noise beating is shown with ablack vertical dashed line. For the given system parameters,both beating contributions are approximately equal at spansaccording to (13).The model is in excellent agreement with the simulation re-sults. Fig. 3 shows that transceiver noise significantly reducesthe gains of nonlinearity compensation compared to EDC. Inthe case of finite transceiver SNR, the NLC gains increase withdistance, as the transceiver SNR has less impact for lowervalues of received SNR. This is contrast to the case of notransceiver noise, where the gains of nonlinear compensationdecrease with distance.Further, in the case of a finite transceiver SNR, there isnegligible performance difference between DPC (green line)and DBP (blue line), as they only differ for short distances dueto an advantage of one span in favor of DPC in the ASE noisebeating contribution [16, Fig. 2]. However, short transmissiondistances are dominated by TRX noise beating where bothperform the same (cf. Eq. (2) with κ R = 0 . ).Moreover, as predicted in Section II, there is negligible gainof split NLC when TRX noise beating is dominant. The left-hand side of (13), which defines the TRX noise beating regime,scales as − dB per decade in distance increase (for (cid:15) (cid:28) ).Both beating contributions are approximately equal at 58 spansfor the chosen parameters. Therefore, the TRX noise beatingcontribution is 10 dB higher than the ASE noise beating at ≈ spans and the transmission system is well inside theTRX noise beating regime. At this point, ASE beating startsto be notable and split NLC begins to yield notable gains.At spans the gain of split NLC in reach compared toDBP is %. Even at such a long transmission distance, thegain is not fully converged to the case of SNR TRX = ∞ dB.According to (20), a span number of at least is required forthe TRX noise beating to be one order of magnitude lower thanASE noise beating. Such distances are not of practical interest,which illustrates the importance of transceiver noise beating inreal systems. Inequality (20) can further be used to estimatethe impact of a different transceiver SNR. As an example,to shift the point where ASE noise beating approximatelyequals TRX noise beating to . spans, a transceiver SNRof an extraordinary dB would be needed. Both calculationsunderline the importance of TRX noise beating in relation toASE beating.Fig. 3 shows that realistic systems are usually operatedin the TRX noise beating regime for short, medium andlong-haul distances and in a mixed regime for transatlanticand transpacific distances. Split NLC proves only useful inthe latter case for transmission systems with equally dividedtransceiver noise. B. Unequal transmitter and receiver noise contribution
In this section the same optical transmission system as in theprevious section is simulated but with 20% of the transceivernoise injected at the transmitter and 80% injected at thereceiver ( κ R = 0 . ). Unequal contributions of transceiver noiseare more likely in realistic transmission systems. The SNR atoptimal launch power as a function of distance is shown in Fig.4a). The lines represent the analytical model estimated by (1)at optimal launch power for EDC, DPC and DBP. Further, aNLC split of X = (cid:6) N (cid:7) and the optimal split X opt obtainedby taking the maximum of all possible splits X ∈ [0 , N ] are ctober 3, 2017 6 % = (cid:16) κ R − κ R (cid:17) (cid:15) %ASE beat. < TRX beat. ASE beat. > TRX beat. a) Number of spans r ece i v e d S N R [ d B ] EDCDPC ( X = 0) DBP ( X = N ) split NLC (cid:0) X = (cid:6) N (cid:7)(cid:1) split NLC ( X = X opt ) Eq. (13), ASE beat. ≈ TRX beat. . . . . b) NLC split ratio XN [%] S N R g a i nov e r D B P [ d B ] spans, km spans, km spans, km Fig. 4: a) SNR at optimum launch power as a function of span number and the SNR as a function of NLC split ratio b)obtained by simulation (markers) and Eq. (1) (lines). The transceiver SNR is dB and the transceiver noise is unequallydivided between transmitter and receiver ( κ R = 0 . ).shown. The absolute SNR as well as the SNR gain predictionsof the model are in very good agreement with the simulationresults. Fig. 4a) shows that optimal split NLC yields significantreach gain with respect to DBP throughout all distances. Forinstance, in the TRX noise beating regime a reach gain of 56%is achieved (from 5 to 8 spans). This is in stark contrast tothe case of equal division of transceiver noise in the previoussection and confirms the theory presented in section II. In theTRX noise beating regime the optimal NLC split is X = N which is equivalent to the DPC case. As shown in Fig. 4a)DPC performs optimally up to approximately 30 spans wherethe amount of ASE noise beating becomes comparable to theamount of TRX noise beating. As the coherence factor is quitelow ( (cid:15) = 0 . ), the simple Eq. (10) accurately predicts thereach gain, yielding a reach increase of % for this example.The DPC curve starts to approach the DBP curve with aresidual gap as the TRX noise beating is not negligible upto this point. Consequently, the optimal NLC split ratio at spans is 56% with a gain of . dB in SNR with respectto DBP. Those gains are still not in-line with the theoreticalresults in section II-B as some residual transceiver noise stillaffects the transmission.The split NLC gain with respect to DBP as a function of theNLC split ratio is shown in Fig. 4b) for , and spans.The gain of optimal split NLC is . dB at 16 spans. As theASE noise beating becomes more significant, the optimal splitratio slowly shifts from X = N to X = (cid:6) N (cid:7) . At spans,where the amount of ASE noise beating is approximately equalto the amount of TRX noise beating, the relative optimal NLCsplit ratio is %. For longer distances the ASE noise beatingcontribution becomes more dominant and the optimal NLCsplit ratio is with % close to X = (cid:6) N (cid:7) .It might be surprising for the reader that the gain in reachis decreasing with transmission distance (e.g., from % at 5spans to % at 70 spans) but the gain in SNR is increasing with distance (e.g., from . dB at 5 spans to . dB at70 spans). Split NLC seems to yield higher SNR gains forlonger distances and higher reach gains for shorter distances.This effect is due to the linear transceiver noise term κP inEq. (1). Different received SNRs are affected differently bythe linear transceiver noise contribution and as a result theSNR gains for short distances are not visible. Hence, the gainin SNR as a figure of merit may be a misleading quantityto compare nonlinearity compensation techniques in systemsthat are impaired by transceiver noise. A better figure of meritis reach increase evaluated at the same received SNR, as thelinear transceiver noise affects both points equally.The obtained SNR values were further used to estimate themutual information (MI) per polarization as described in [10].Mutual information is defined asMI = 1 m (cid:88) x ∈ X (cid:90) C p ( y | x ) log p ( y | x ) p ( y ) dy, (21)where m is the cardinality of the QAM constellation, x and y are random variables representing the transmitted and receivedsymbols and X is the set of possible transmitted symbols.Assuming an additive white Gaussian noise (AWGN) channel,the channel law is given by p ( y | x ) = 1 πσ n exp (cid:18) − | y − x | σ n (cid:19) . (22)Eq. (21) was then numerically integrated using the Monte-Carlo method. The resulting MI of the simulated transmissionsystem is shown in Fig. 5 as a function of span number.At spans (1920 km) DBP increases the MI comparedto EDC by . bit/symbol and split NLC yields another . bit/symbol on top of the DBP gain. This shows that areasonable throughput increase can be achieved by applyingsplit NLC. ctober 3, 2017 7 ASE beat. < TRX beat. ASE beat. > TRX beat.
Number of spans M I[ b it/ s y m bo l ] EDCDPC ( X = 0) DBP ( X = N ) split NLC (cid:0) X = (cid:6) N (cid:7)(cid:1) split NLC ( X = X opt ) Eq. (13), ASE beat. ≈ TRX beat.
Fig. 5: MI at optimum launch power as a function of spannumber obtained by simulation (markers) and Eq. (1) (lines).A finite transceiver SNR of dB is assumed where thetransceiver noise is unequally divided between transmitter andreceiver ( κ R = 0 . ). IV. C ONCLUSION
The performance of split nonlinearity compensation wasanalyzed in the context of realistic transceiver sub-systems. Itwas demonstrated that the gain of split NLC and the optimalsplit ratio are strongly dependent on whether TRX noise orASE noise beating dominates. Simple formulas were derivedthat can be used for system design and gain prediction. Itwas found that split NLC yields negligible gain comparedto DBP for distances below 800 km, when the transceivernoise is equally distributed between transmitter and receiver.However, when the transceiver noise is unequally distributed,reach increases of 56% on top of digital back-propagation areachievable for the system under test. Alternatively, split NLCcan be applied to increase the mutual information by 0.55bits/symbol for distances larger than 1440 km, compared toDBP. This demonstrates that significant throughput or reachincrease can be achieved by properly tailoring the digitalnonlinearity compensation algorithm to the noise distributionof the underlying optical transmission system. The results ofthis work suggest that split NLC yields greater reach increasesthan current experimental demonstrations using DBP or DPC.This demonstration is left for future work.A
CKNOWLEDGMENT
Financial support from UK EPSRC programme grant UN-LOC (EP/J017582/1) and a Doctoral Training Partnership(DTP) studentship to Daniel Semrau is gratefully acknowl-edged. The authors thank G. Liga from University CollegeLondon for valuable comments on previous drafts of thispaper. R
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