We introduce an efficient and accurate nonlinear compensator (NLC) for digital back-propagation (DBP) of coherent optical OFDM receivers, based on a factorization procedure for the Volterra Series Transfer Function (VSTF) with 3N degrees of freedom for N frequency samples. The O(N2) nonlinear compensation complexity of generic Volterra evaluation (normalized per-subcarrier) is reduced to 28 + 6logN. Our analysis and simulations indicate that this NLC system outperforms previous VSTF-based non-linear compensation methods. Compared to a most recent VSTF-based method, the new method incurs 52% extra computational complexity in return for improved nonlinear tolerance of ~2 dB for the particular analyzed link.
© 2013 OSA
The nonlinear impairment is a dominant one in long-haul fiber-optic transmission. Multiple studies of Nonlinear (NL) propagation and transmission capacity have been conducted and various NL compensation (NLC) methods have been proposed and demonstrated. A prominent NLC method is digital back-propagation (DBP) [1–5] wherein the distorted, typically oversampled, received signal feeds a digital-domain compensator which counteracts the Nonlinear Interference (NLI) accrued by the signal as it propagated along the optical fiber. The NLC emulates optical propagation through an inverse fiber link (with reversed order segments and opposite sign parameters). The DBP NLC may either be realized by means of a Split Step Fourier (SSF) fiber emulator, which is the most widespread method or alternatively by means of a Volterra Series (VS) emulator.
A major challenge of all NLC realizations is computational complexity. For VS based NLC the number of Complex Multipliers (CM) normalized per sample is , with N the size of the frequency-domain record (for OFDM transmission this is the number of time-domain samples per OFDM symbol or equivalently the number of subcarriers), whereas for an SSF-like DBP NLC the complexity is where is the number of fiber spans and is the number of computational segments per span, which must be substantial for high accuracy. The challenge still withstands to improve the complexity vs. performance tradeoffs of these methods in order to eventually enable real-time hardware realizations.
Volterra-based nonlinear modeling may be realized either in the time-domain, formulating the VS model based on the so-called Volterra kernel (VK) or in the frequency-domain based on the Volterra Series Transfer Function (VSTF) . VS-like NL propagation models [7–10] and VS-based nonlinear compensation methods have been increasingly researched in the last few years, either in the time-domain [11,12] or in frequency-domain [13–28] which is our particular focus in this paper.
Briefly reviewing the most recent VSTF-based frequency-domain NLC approaches which are directly relevant to our paper , investigate a VSTF-based “non-iterative” method, in the sense that the DBP sub-system is not modeled as a serial cascade of processing steps, but rather its end-to-end VSTF is modeled at once, albeit with complexity. In contrast , reverts to partitioning the VSTF-based NLC into multiple parallel paths, each of which is associated with a particular fiber span, adopting a simplified frequency-flat NL model for each processing span, reducing complexity per span down to per sample.
It is our objective here to further improve the NLC performance, while retain complexity, enhancing the NL tolerance of the end-to-end link by replacing the frequency-flat approximation of  with a more precise frequency-dependent representation of each span, while marginally increasing complexity.
We have already outlined in our previous work [17–19] an efficient Decision Feedback driven NLC realization for OFDM transmission, based on a factorized approximation for the VSTF, which lowered the computational complexity per sample from down to . However, our previous approach [17–19] performed less favorably especially over multi-span dispersion unmanaged fiber links.
Here, we propose and simulate a novel extension of our previous VSTF-based NLC [17–19] operated here not in decision feedback mode but rather as a feed-forward DBP NLC, increasing its compensation accuracy for multi-span fiber links with or without dispersion compensation. The proposed approximate factorization for the VSTF triples the number of Degrees Of Freedom (DOF) relative to our previous approach [17–19], representing the VSTF more accurately, enhancing the resulting nonlinear tolerance while incurring some incremental complexity relative to [17–19]. Another new aspect is having the Factorized VSTF (F-VSTF) NLC incorporate a multi-stage structure, emulating NLI build-up over multiple fiber spans in a serial cascaded mode, akin to cascading the processing segments in an SSF-based DBP system. In contrast, in  each of the multiple per-span-compensating modules are combined in parallel, rather than in series and lack frequency-dependency, in effect ignoring the interaction between CD and nonlinearity within each span. Thus, our proposed system essentially accounts for the CD-NL interaction per-span by means of the F-VSTF modeling of the frequency-dependency and further adopts a cascaded topology akin to that of the conventional DBP (albeit replacing the SSF-based per-span model by the Volterra one).
Our simulations reveal that the new F-VSTF NLC improves Modulation Error Ratio (MER) (ratio of the average powers of the signal and the NLI) by ~7dB relative to uncompensated detection for a specific modeled fiber link, while outperforming by ~2 dB relative to a frequency-flat VSTF reference scheme akin to  which we simulated for comparison. Our analysis of the relative computational complexities of the two compared schemes reveals that the CM count in our scheme for the specific link examined must be increased by 52% in order to implement the VSTF factorization and serial cascading of processing spans which enables the significant nonlinear tolerance improvement quoted above.
As this paper is devoted to proof of concept, for the sake of clarity, rather than handling both orthogonal polarizations, the paper effectively treats a single polarization. The extension of our scalar treatment to dual polarization is in principle straightforward (it may be performed as in  by introducing cross-coupling between the two processing paths corresponding to the two polarizations) however this aspect is relegated to future work. The paper is structured as follows: In section 2 we overview the method and design an efficient digital-domain NL fiber link emulator, utilized in section 3 to realize a VS-based DBP NLC wherein we also evaluate its performance and complexity. The results are further placed in perspective in the concluding section 4. Appendix A outlines the optimization of the approximate factorized VSTF representation. Appendix B lists the abbreviations used in this paper.
2. Digital emulation of the NL fiber
2.1 Overview of the proposed nonlinear compensator for a multi-span fiber link
We start with a brief overview of the main objective of this paper – the design and analysis of an efficient NLC consisting of a serial concatenation of partial NLCs, each targeting the compensation of a particular fiber span (Fig. 1) for OFDM transmission. The proposed NLC structure within the OFDM receiver (Rx) is akin to that of current constrained-complexity SSF based DBP systems, except that the NLC segments are each designed here according to the principles of Volterra non-linear analysis, rather than based on chaining up SSF segments. The digital-back-propagation processing is applied in the frequency domain (at the output of the OFDM FFT in the Rx, acting on each sub-carrier). Each NLC segment models reverse linear and nonlinear propagation through a corresponding span based on a nonlinear filter used as FWM emulator adding up its FWM output to the through (identity) linear path, then applying one-tap equalization to each the subcarriers according to transfer factors to be detailed below, accounting for linear Chromatic Dispersion (CD) and for Self- and Cross- Phase Modulation (SPM/XPM) accrued over fiber propagation. For brevity we generically refer to both XPM and SPM as “XPM”. Notice that the multiplicative application of after subtraction of the emulated FWM implies that the emulated XPM is referred to the input of the fiber span (rather than its output). The cascade of NLC sections emulates the span-by-span nonlinear back-propagation through a link with reversed fiber parameters and spans. The rest of this section is devoted to systematically deriving the multi-span FWM emulator based on Volterra nonlinear analysis.
2.2 Nonlinear fiber propagation model – Volterra Series Transfer Function
Our first step is to design an efficient digital NL fiber link emulator modeling non-linear propagation of OFDM signals as faithfully as possible. Briefly reviewing the Volterra-based NL model developed and perfected in [16–19], [27,28], we analyze the propagation of an OFDM symbol containing N frequency-domain tones, , launched into a NL fiber link. Throughout the paper we refer to frequency-domain OFDM sub-carriers as “tones”. This term may also be applied to the frequency components of a Fourier series describing a general (not necessarily OFDM-modulated) signal over a finite time-interval. The ith received tone is expressed as:16–19,27,28], which may be analytically calculated from the link physical parameters (loss, group velocity dispersion and NL parameter) as follows:
Interestingly, the VSTF of Eq. (3) may be expressed as the Fourier-Transform (FT) of , with appropriate scaling. For regular fiber link structures consisting of repetitive spans, the FT expression for the VSTF leads to its formulation as the product of the single-span VSTF and an array factor [16,19] describing the coherent interference of the multiple spans, akin to the radiation pattern from an antenna array.
The NL fiber channel may be probed by means of a training sequence, estimating the VSTF applying a NL system identification procedure recently developed in .
The NL transmission model of Eq. (1) may be used to digitally emulate the NLI accrued by the OFDM signal as it propagates through any given fiber link. The evaluation of the tri-linear superposition of Eq. (2) over all subcarriers, , is computationally intensive, requiring CMs (per N samples), posing for large values of N a prohibitive complexity bottleneck of VS-based NLC. We next introduce an efficient method reducing NLC complexity to CMs. To this end, we first derive in subsection 2.3 a factorized approximation for the VSTF, utilized in subsection 2.4 to synthesize efficient single-span link emulators which are finally cascaded to form an efficient multi-span link emulator in subsection 2.5.
Finally, let us comment on the limits of validity of the third-order Volterra model. For input power larger than some threshold we must extend the model to also account for fifth-order terms, however the threshold is link dependent. For the exemplary link treated in this paper, the third-order model is still accurate up to a few dBm input power.
2.3 VSTF factorization with 3N DOFs
We introduce a factorized representation for the VSTF, expressed as the product of four independently selected sampled transfer functions, , each with N samples approximately representing the O(N3) samples VSTF with just 4N DOFs (here the superscripts evidently do not denote powers but rather index the four transfer function factors):Eq. (2)) evaluated with the exact VSTF whereas is Eq. (2) evaluated with the factorized VSTF of Eq. (4):17–19] that the optimization of Eq. (5) over the VSTF factors, , is approximately equivalent to the following optimization problem:Eq. (4), as suggested here, is an extension of the factorized representation initially introduced in [17–19], based on just N rather than 4N DOFs, replacing Eq. (4) by, i.e. performing the factorization in terms of the samples of a single linear factor, rather than using four different factors as suggested here. That N-DOFs factorization may be viewed as a degenerate case of our 4N-DOFs factorization, obtained by setting). The extra DOFs in our current factorization of Eq. (4) enhance the approximation accuracy. Appendix A presents an approximate solution of the optimization problem of Eq. (7) in particular establishing that the resulting solution satisfies, allowing to express the VSTF with 3N rather than 4N DOFs.
2.4 Efficient NL link emulator design
Based on the factorized VSTF approximation of Eq. (4) (or Eq. (8)) we now present an efficient method approximately evaluating the trilinear summation of Eq. (2) with a substantially reduced computational effort, requiring just complex multiplications to compensate N samples. The following five computational stages specify the block diagram of Fig. 2 for the efficient NL emulator:
Stage 1: Frequency-shape the OFDM tones:
Stage 2: 4-fold interpolate the time-domain transformed signals by zero-padding the records to length 4N - realized by inserting 3N zeroes after the first N/2 “positive frequency” samples, followed by the N/2 “negative frequency” samples, and taking an Inverse 4N-points Discrete-Fourier Transform (IDFT):
Stage 3: Generate the conjugate triple-product:Eq. (11)):Stage 4: The interpolated signal (Eq. (12)) is 4-fold decimated and converted back to the frequency domain, by taking its DFT and retaining just the low-frequency coefficients, , discarding the 3N Out-Of-Band (OOB) frequency components in the center of the record (the opposite of the ZP operation in the interpolation stage 2), yielding the signal at the original sampling rate. Finally, the resulting tone complex amplitudes are multiplied by yielding the following signal:
Comparing Eq. (14) for and Eq. (6) for we note that they only differ in their respective summation index sets (), hence the difference between the two expressions is the contribution to the sum (Eq. (14)) resulting from all the elements of the difference between the two sets:Eq. (8), which may allow eliminating the middle path leading to the triple multiplier while placing a squarer on the IFFT output of the first path in a more efficient implementation (as depicted in Fig. 1(b)). The evaluation of the computational complexity of the processing stages (Fig. 2) is deferred to section 3. At this point we note that the processing comprises computationally-efficient (I)FFTs and complex multiplications by the factors, such that the resulting complexity (for N samples) is , substantially lower than the complexity originally incurred in evaluating the tri-linear superposition (Eq. (2)).
2.5 Efficient NL multi-span emulator design
Equipped with the non-linear emulator of Fig. 2, we may attempt to directly apply it for NLC of either a Dispersion-Managed (DM) fiber link or a Dispersion Un-Managed (DUM) one. It turns out that the results for DM compensation are quite good, especially for a multi-span DM link, wherein CD is fully compensated in every span. In fact, in this case the VSTF is simply N times that of a single-span (as the spans sum up coherently [16, 19]), thus the equivalent statement is that our factorization technique provides a very good approximation for the exact VSTF of a single span.
Unfortunately, as borne out by numerical simulations (not shown) the proposed factorization of Eq. (8) fails to provide a reasonable approximation for the VSTF of a multi-span DUM link, the underlying reason being that the array factor mentioned above provides a rapid variation of the VSTF over the space of frequency-domain triplets, hence such variation cannot be well tracked by a factorized approximation.
It follows that our proposed factorized NL emulator (subsection 2.4) for the efficient computation of the FWM NLI is only applicable at this point for either a single-span short link or for a multi-span DM link. To extend its applicability to DUM fiber links, which constitute the trend for coherent transmission (leading to lower uncompensated NLI overall), we propose a novel multi-segment emulator design consisting of serial cascading of multiple VS-based per-span FWM emulators. Figure 3 depicts the first two segments of a concatenation of multiple segments, each consisting of a FWM emulator. For an -spans fiber link, our multi-span fiber link emulator comprises processing segments, each corresponding to a particular fiber span. This VS-based serial design is alternative to the parallel design proposed in Fig. 5 of , comprising multiple non-linear span emulators embedded within linear CD sections.
Each processing segment in our multi-span NLC comprises an efficient single span FWM NLI emulator of the form of Fig. 2 (with parameters pertinent to the corresponding spans), further concatenated with an SPM/XPM/CD propagator consisting of a complex multiplication by the term, of Eq. (1), corresponding to a single span.
The concatenation of of these computational segments accurately models the serial propagation of the nonlinearity through the overall multi-span link, even accounting for the higher-order perturbative FWM-FWM interactions amongst various spans (i.e. the FWM interference generated in one span acting as a source for FWM interactions occurring in subsequent spans). This is in contrast with the parallel model of Fig. 5 in  which just addresses the lowest order of the interaction, merely accounting for the NL generation within each span, as linearly excited via CD-propagation through the previous spans and linearly propagated to the fiber output via the subsequent spans.
Our multi-span emulator (Fig. 3) estimates the overall signal at the fiber link far-end as per Eq. (1), , which may be readily solved for the FWM NLI component (which will turn out to be useful for our purposes in the sequel):
2.6 NL factorized Volterra emulator simulation results
We now validate our multi-span emulation scheme by calculating the FWM NLI signal component and removing it from a nonlinearly-distorted signal recorded at a receiver, for a fiber link accurately modeled by an SSF simulation as depicted in Fig. 4.
We prepare a sequence of 16-QAM transmission data-symbols and SSF-simulate its propagation along the multi-span NL fiber link. The received symbols are first equalized to remove the effects of XPM and CD (multiplying by the inverse of the transfer factor of Eq. (1)) generating linearly processed which are now only distorted by the FWM NLI additive components. Next, we evaluate the FWM NLI using our efficient NL emulation scheme (Fig. 3 and Eq. (18)) based on the sequence of transmitted symbols, compensating the received signal by subtracting out the emulated nonlinearity:. This process is schematically illustrated in Fig. 4 indicating transmitted 16-QAM constellation and the received NLI-affected constellations pre- and post- NLC.
To assess the effectiveness of our NLC, we determine the MER for the distorted signal and compare it to that of our compensated signal (the MER, alternatively called EVM, is used as a “non-linear SNR” measure, expressed as the ratio of the average power of the received constellation and the variance of the NLI distortion). As a benchmark, we repeat the compensation procedure, now evaluating the FWM NLI using the ‘exact’ Volterra tri-linear model of Eq. (2), monitoring the MER for this compensation method. Figure 5 plots MER vs. tone index for the uncompensated signal (blue) vs. the factorized-Volterra compensated signal (green) and the full-Volterra compensated signal (red).
Evidently, while full-Volterra NLC, with O(N3) complexity per N samples (Eq. (2)) (O(N2) complexity per sample) provides the highest accuracy compensation, our reduced-complexity compensator is just ~1 dB inferior to a full-Volterra one, indicative of the validity of the succession of steps leading to the scheme of Fig. 3, as well as the high accuracy of the factorized approximation per-span, combined with multi-span concatenation.
Notice that both the full-Volterra and the reduced complexity NLCs roll off near the band edges. We conjecture this edge effect to be attributed to our Volterra trilinear summation model not accounting for: (i) for cyclic prefix insertion. (ii) Finite OFDM symbol duration and Root Raised Cosine shaping at the transmitter. Moreover, as discussed in , the SSF simulation itself, used to determine the uncompensated propagation, is affected by finite 4x upsampling, whereas the trilinear model is not affected by upsampling related effects, further contributing to the discrepancy between the uncompensated propagation model and the model of NL compensation generation. Mitigating the edge-effect in the current Volterra-based NL modeling and compensation approach is going to be addressed in future research. Operationally, even with the current method, the few degraded OFDM subcarriers at the edges may still be usable for data transmission by reverting to a lower constellation, e.g., downgrading 16-QAM to QPSK, resulting in a few percent loss in average spectral efficiency.
Equipped with the novel high-precision low-complexity emulator of a multi-span fiber link we may now proceed to apply this key DSP building block to DBP NL compensation.
3. Volterra-based DBP compensation of OFDM
It was shown [1–5] that the Nonlinear Schroedinger Equation (NLSE), governing NL signal propagation through optical fibers, may be inverted by cascading it with a virtual link with opposite-sign parameters and reversed geometry. The DBP NL compensation method digitally propagates the received NL-distorted signal through an inverse nonlinear fiber-link model, counteracting the NLI generated in the original fiber link. Traditionally, a plethora of constrained-complexity emulators have been used for the DBP propagator, typically derived from the SSF propagation method by adopting various simplifying assumptions, taking coarser processing steps in order to reduce the prohibitive computational efficiency of the original SSF method. In our approach, similarly to [21–28], we abandon SSF-related propagation for the DBP sub-system altogether, instead adopting for the NLC realization a Volterra-based NL emulator as derived in section 2, albeit used with inverted fiber parameters. Our VSTF-based approach stands out relative to similar ones [21–28] by achieving significantly improved performance for the same order of complexity.
An important aspect affecting the complexity of DBP-NLC evaluation is the sampling rate, relative to the symbol rate of the single channel being compensated. It is well-known that bandwidth tends to be broadened by nonlinear interactions, either physical ones in the fiber, or synthetic digital ones, in the NLC DSP. We should mention that this paper is restricted to the NL compensation of a single WDM channel, ignoring the cross-channel NL interactions.
Even for a single channel to get perfect compensation a necessary condition would be to use at least 3-fold ADC oversampling of the received signal. Nevertheless, it has been shown [2, 26] that 2x the baudrate and 1x baudrate operation is still feasible, discarding the excessive bandwidth information, albeit at the expense of some reduced non-linear tolerance.
In the next subsection we introduce a F-VSTF DBP NLC, compensating the FWM NLI introduced by the fiber by using the efficient multi-span NL fiber link emulator of section 2. This NLC also operates at baudrate yet achieves high NL tolerance.
3.1 Volterra based DBP NLCEq. (20) amounts to the form of Eq. (19), with an effective single section VSTF given byEq. (20). However, the DBP NLC description is just within the scope of a third-order perturbation model [16, 19] while in fact the concatenation of single spans is even more accurate than the synthesis of Eq. (20), as it accounts for higher-degree terms in perturbation model. The overall topology is that of Fig. 6, with the DBP NLC realized as per Fig. 3, with each of the “FWM-emulate” sections realized as per Fig. 2, with multiplicative factors given by.
3.2 Comparison to a frequency-flat compensation scheme
We now compare our approach with that of the recent VS-based study [25, 26] which proposed a NL DBP scheme based on a frequency-flat approximation for the single-span VSTF. In our context, the approach of  approximates our Eq. (20) for utilizing a frequency-flat approximation to (Eq. (19)), taken as a fixed term , ignoring the effect of CD within the span, effectively setting .
This highlights the difference between the two respective VS-based methods at the FWM emulation level (e.g. for single spans), however another key difference is in the topology of assembling multiple NL emulation blocks into a full NLC. The overall fiber link FWM NLI is evaluated in Fig. 5 of  by parallel combining ofprocessing units, each realized by means of a frequency-flat third-order nonlinearity, each embedded in preceding and following CD linear filters. This NL emulation scheme then differs from our proposed one in two main respects:
- 1. The frequency flat approximation for a single-span of the inverse fiber-link by means of a constant (a single DOF) is less accurate than our 3N DOF approximation.
- 2. Our design uses serial concatenation of the individual spans, whereby the FWM NLI generated in a specific fiber section participates in the FWM interactions occurring in subsequent fiber sections. In contrast, the frequency-flat scheme  superposes FWM NLI contributions from separate fiber sections, accounting just for SPM/XPM/CD that each FWM NLI component accrues over its propagation through the remaining fiber sections.
A system akin to  at the FWM emulation building block level constructed from the parallel combinations of multiple such blocks (one per span), will henceforth be referred to as “frequency-flat VS-DBP”.
3.3 Comparative simulation of non-linear compensation results
We have simulated our DBP compensation scheme according to the setup depicted in Fig. 6, evaluating the post-DBP NLC MER as a measure of the compensation performance.
Figure 7 plots the received and processed MER per tone for several cases, namely no DBP compensation (lowest blue curve), compensation with a “brute-force” baudrate full-Volterra compensator as per Eq. (2) (top red curve), our efficient factorized serially-concatenated VS-DBP scheme (green curve) and the frequency-flat parallel-combined frequency-flat VS-DBP scheme of , evaluated according to Eq. (18) there (light blue curve).
While our efficient F-VSTF approach provides overall MER improvement ~1dB lower than that achievable via brute-force full-Volterra compensation, it surpasses the performance of the frequency-flat VS-DBP approach by more than ~2dB in return for a small relative complexity increase, as determined next.
3.4 Compensation complexity
We first evaluate the complexity of realization of our factorized Volterra NL emulator of Fig. 3, stated in terms of the number of complex multiplications, required over a single OFDM symbol period (N samples), normalized by dividing by N, to obtain the number of complex multipliers per sample. To this end, we first evaluate the complexity of the FWM emulator building block of Fig. 2, itemizing the complexity of each of the computational stages (1-5) outlined in section 2. The CM counts for the five stages are respectively given by (the number of CMs for an N-points FFT is ):26], whereby the processing is partitioned into multiple parallel paths, as shown in Fig. 8(a) below. Each of the parallel paths consists of a frequency-flat third-order nonlinearity preceded and followed by dispersion filters (Fig. 8(b)), providing an alternative implementation of the structure shown in Fig. 8 of . Our variant accounts for sufficient oversampling in the DSP domain in order to accommodate the spectral broadening accompanying the third-order time-domain nonlinearity. The overall complexity of the “parallel frequency-flat” scheme of our Fig. 8 is evaluated next:
4. Discussion and epilogue
This work follows up on our previous VS-based NLC for OFDM with decision-feedback [17–19] as well as on the more recent study  of VSTF-based NLC and other frequency-domain Volterra methods [21–24]. We provided an improved accuracy Volterra-based NLC, formulated in the frequency-domain in terms of the VSTF (in contrast to time-domain Volterra-based NLC approaches, e.g , .). The proposed VS-based DBP method is applicable for both DM and DUM multi-span fiber links.
Our proposed VS-based method extends our previous contribution [17–19] on factorized VSTF, augmented here by extending it to three independent DOFs rather than a single independent DOF as in [17–19], which yields improved accuracy per span. Moreover, our approach extended [17–19] by adopting a multi-span serial design, in a “feed-forward” mode rather than operating decision-feedback driven as in [17–19], applicable beyond OFDM, although our presentation has been cast in terms of OFDM transmission.
In contrast, in the approach of  the VSTF per span is replaced by a constant (frequency dependence per span is not accounted for), thus the VSTF of the full link is merely represented by the array factor, whereas in a more precise VS model it would be represented by the VSTF times the array factor. Our work then refines  by replacing the crude constant approximation of the VSTF-per-span by a factorized one, enhancing precision while still retaining relatively low complexity by virtue of the novel factorization of the per-span VSTF.
Another unique aspect of our multi-span NLC method is its serial cascading of non-linearities rather than the parallel combination introduced in . The divergence between the two approaches is that  method yields a lower-degree treatment of inter-span perturbations than our method does. To clarify this point we depict a short three spans link in Fig. 9(a). In the parallel NL emulator of Fig. 9(b), which is essentially that of Fig. 5 of , specialized to spans, each successive span is considered as a non-linear source, embedded in a linear “ether” consisting of CD propagation over the previous and following spans. In contrast our serial cascading method (Fig. 9(c)) may have each of the three spans represented as the parallel cascade of a linear span and a non-linearity emulator. However, such serial cascading may be expanded in a “trellis” of multiple end to end paths effectively running in parallel as described in Fig. 9(d). Notice that some of the spans contribute their NL propagators whereas the others contribute the linear propagators. Evidently the paths described in Fig. 9(b) are a sub-set of the paths described in Fig. 9(c), which provides the more precise description, as it does account for the multiple NL interactions. While it may be argued that NL-NL interactions are weak, notice however that for long-haul links, e.g. with , the number of paths in the trellis with precisely two spans designated “NL” (whereas the rest of the spans are designated “LIN”), representing relatively strong high-order interactions, is given by .
In this paper we have not compared the proposed Volterra-based DBP with SSF-based DBP. Such comparison is outside the scope of the current paper, as there is a plethora of SSF-based variants to compare with. However, our preliminary analysis indicates that the two types of systems have comparable overall complexities, which is indicative that Volterra-based DBP should be further considered for investigation as an alternative to conventional SSF-based DBP.
Appendix A – Optimal VSTF factorization
The non-convex optimization problem of Eq. (7), with solution denoted , may be approximately cast into a sub-optimal pair of convex optimization problems, separately formulated for the angle and for the magnitude of the VSTF, now decoupled from each other, similarly to [17–19]:Eqs. (25), (26) reduce to:Eq. (7)) the optimal solution satisfies, beneficially reducing our number of DOFs from 4N to 3N.
This work was supported in part by the Israeli Science Foundation (ISF) and by the Chief Scientist Office of the Israeli Ministry of Industry, Trade and Labor within the ‘Tera Santa’ Magnet consortium.
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