Low-Pass-Filter Assisted Digital Backpropagation

• Low-Pass-Filter Assisted Digital Backpropagation is a reduced-complexity method that inserts a Gaussian low-pass filter into each nonlinear compensation step to suppress unwanted high-frequency components.
• It achieves a practical trade-off by slightly sacrificing exact model fidelity in the nonlinear Schrödinger equation to significantly lower implementation costs in ultra-high-baud coherent optical systems.
• LDBP methods, including variations with learned parameters, address intra-channel and inter-channel nonlinear effects, showing measurable performance gains under different amplification and dispersion conditions.

Low-Pass-Filter Assisted Digital Backpropagation (LDBP) is a reduced-complexity form of digital backpropagation in which low-pass filtering is inserted into the nonlinear compensation chain so that the nonlinear phase update is driven by a bandwidth-limited intensity waveform rather than the full-band intensity. In coherent optical transmission, this places LDBP within the broader family of approximate split-step inverse-propagation methods that trade exact fidelity to the nonlinear Schrödinger or Manakov model for implementability. The term is not uniform across the literature: several influential papers use LDBP to mean learned digital back-propagation, namely an SSFM-derived, model-based neural architecture with trainable short filters. The two usages share the objective of practical nonlinear compensation under stringent complexity constraints, but they are not identical in mechanism (Yang et al., 28 Jul 2025, Häger et al., 2017).

1. Terminology and scope

A central source of ambiguity is that the same acronym refers to different but adjacent research lines. In the classical low-pass-filter-assisted sense, LDBP uses an explicit low-pass filter inside the DBP chain. In the learned-DBP sense, the complexity reduction comes from trainable short FIR-like operators, filtered nonlinear steps, or other constrained parameterizations rather than an explicit per-step low-pass field truncation.

Usage in the literature	Core mechanism	Representative paper
Classical low-pass-filter assisted DBP	LPF in each nonlinearity compensation step; Gaussian LPF bandwidth optimized empirically	(Yang et al., 28 Jul 2025)
Learned digital back-propagation	Unrolled SSFM with jointly optimized short symmetric convolution filters	(Häger et al., 2017, Häger et al., 2020)
Related reduced-complexity DBP variants	ESSFM-style filtered nonlinear steps, subbanding, or perturbation-aided nonlinear operators	(Cellini et al., 27 Jan 2026, Shevelev et al., 14 May 2026, Lin et al., 2021)

This distinction matters because many later papers are directly relevant to low-pass-filter-assisted DBP without being classical LPF-assisted DBP papers. A common misconception is therefore that every paper labeled “LDBP” uses an explicit low-pass operator. The record is more heterogeneous: some methods bandlimit the intensity entering the nonlinear update, some constrain the linear dispersive operator to short trainable filters, and some replace the standard memoryless Kerr phase rotation by filtered, perturbative, or subband-coupled nonlinear modules (Cellini et al., 27 Jan 2026, Civelli et al., 2021).

2. Physical basis and the role of low-pass filtering

The underlying propagation model in the directly relevant ultra-high-baud study is the Manakov equation

$$j\frac{\partial \mathbf{E}(t,z)}{\partial z} +\frac{\beta_2}{2}\frac{\partial^2 \mathbf{E}(t,z)}{\partial t^2} = j\gamma |\mathbf{E}(t,z)|^2 \mathbf{E}(t,z) -\alpha \mathbf{E}(t,z),$$

with optical field

$$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$

Ideal DBP is formulated as virtual inverse propagation with linear and nonlinear operators

$$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$

The inversion is implemented span by span and step by step with the same SSFM logic used in forward propagation (Yang et al., 28 Jul 2025).

Classical LDBP modifies this structure procedurally rather than by changing the overall DBP decomposition. The defining operation is that low-pass filtering is performed in each nonlinearity compensation step to eliminate the high frequency components of the intensity, and a Gaussian LPF is used with bandwidth optimized empirically for best compensation performance. The paper does not print a single compact closed-form LDBP operator, but it explicitly describes the nonlinear phase as being computed from a filtered intensity rather than the raw $|\mathbf{E}|^2$. The stated interpretation is twofold. First, the LPF suppresses high-frequency intensity beating terms that are less correlated with the target intra-channel Kerr effect and more associated with inter-channel beating when the received WDM signal is demultiplexed and processed channel by channel. Second, it lowers implementation cost because the nonlinear phase term can be estimated from a smoother, lower-bandwidth intensity waveform (Yang et al., 28 Jul 2025).

The same paper draws a sharp distinction between ideal and practical intra-channel compensation. “Ideal” means that deterministic intra-channel fiber nonlinearity is fully removed by sufficiently fine-step backpropagation. Practical LDBP captures only part of that gain because it uses finite steps per span and filtered nonlinear updates. This distinction is important because the utility of low-pass filtering is not to reproduce ideal DBP exactly, but to recover a substantial fraction of the useful intra-channel gain under realistic complexity limits (Yang et al., 28 Jul 2025).

3. Ultra-high-baud operating regime

The most explicit recent treatment of classical low-pass-filter-assisted DBP considers 4 THz C-band WDM systems at 100, 200, and 300 GBaud over 10 spans of 80 km SSMF, with 16-QAM, RRC roll-off 0.02, and both EDFA-amplified and backward DRA-amplified cases. The study’s central motivation is that, when total WDM bandwidth is fixed and symbol rate increases, the number of WDM channels decreases and the fraction of nonlinear interference that is self-channel interference (SCI) rises strongly. In an EDFA-amplified 4 THz SSMF system, the SCI proportion of the center channel rises from 45.4% at 100 GBaud to 66.5% at 300 GBaud, making intra-channel compensation progressively more attractive (Yang et al., 28 Jul 2025).

For practical LDBP, the explicitly reported step-per-span settings are 2, 4, 8, and 20, with 20-stps used as the main practical high-complexity operating point. The reported 20-stps SNR gains relative to uncompensated transmission are:

Symbol rate	EDFA-amplified	DRA-amplified
100 GBaud	0.53 dB	0.89 dB
200 GBaud	about 0.84 dB	1.16 dB
300 GBaud	0.87 dB	1.30 dB

These values show that the practical gain of LDBP increases with symbol rate, and that DRA links provide larger gains than EDFA links because the distributed power profile leaves more deterministic nonlinear distortion available to compensate. The same study also notes that the 200 and 300 GBaud gains are quite similar in EDFA links, indicating some saturation of LDBP performance at ultra-high baud due to difficulty estimating the fast-varying high-frequency nonlinear components (Yang et al., 28 Jul 2025).

The PMD-limited upper bound is equally important. At 300 GBaud with PMD parameter $0.05\ \mathrm{ps}/\mathrm{km}^{1/2}$, the gain of ideal DBP decreases by 3.85 dB in EDFA-amplified links and 5.09 dB in DRA-amplified links relative to the PMD-free idealized case. In WDM, ideal intra-channel NLC still improves with baud rate: for the same PMD parameter, the SNR gain of ideal DBP rises from 0.65 dB at 100 GBaud to 1.36 dB at 300 GBaud in EDFA links, and from 1.02 dB to 1.76 dB in DRA links. Practical 20-stps LDBP therefore recovers a large fraction, but not all, of the ideal intra-channel gain. This suggests that the value of classical LPF-assisted DBP is greatest precisely in regimes where SCI has become a majority contribution but full-field or ideal DBP remains impractical (Yang et al., 28 Jul 2025).

4. Learned and filter-constrained DBP as an adjacent lineage

A major neighboring lineage recasts DBP as a trainable deep architecture. An early precursor explicitly called the method learned digital backpropagation (LDBP) and reinterpreted the unrolled symmetric SSFM as a deep network whose layers alternate linear chromatic-dispersion compensation and physically meaningful per-sample nonlinear phase rotation. The key reduced-complexity device is that each linear operator is restricted to a circular convolution with a symmetric filter of length $2K+1$, reducing free parameters per matrix from $n^2$ to $K+1$. In a 32$\times$100 km, 20 Gbaud, 16-QAM single-channel link with 35 GHz front-end filtering and 2 samples/symbol receiver processing, learned DBP with 1 step/span achieved a Q-factor of $16.8$ dB, while conventional DBP with 2 steps/span achieved $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$0 dB; using steps per span as a rough complexity measure, the paper interpreted this as a 50% complexity reduction for comparable performance. It also reported that learned DBP with 3 steps/span slightly outperformed 50-step/span DBP in the nonlinear regime because the learned filters and nonlinearities could adapt to receiver bandwidth limitations (Häger et al., 2017).

The physics-based deep-learning formulation was sharpened further by parameterizing each SSFM linear step as a symmetric FIR filter and progressively pruning taps during gradient descent. That work reported that filters could be pruned to as few as 3 taps/step without sacrificing performance, and it connected the learned solutions to a cascade-level factorization problem rather than an isolated per-step approximation problem. The same paper also discussed explicit filtered nonlinear steps in enhanced SSM: $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$1 and its learned per-step generalization, which places it very close to the classical intuition of low-pass- or filter-assisted DBP even though the paper’s acronym LDBP means learned digital backpropagation (Häger et al., 2020).

Experimental work confirmed that short jointly optimized filters are not merely a simulation artifact. Learned time-domain DBP was demonstrated experimentally for 64-GBd dual-polarization 64-QAM over 1014 km, using 10 steps/span, 10 distinct FIR filters reused every span, and 270 total complex taps. The reported nonlinear-mitigation gain was about 0.3 dB in SNR, comparable with a much more expensive 50-step/span frequency-domain DBP baseline. A separate 25 Gbaud experimental study showed that a 3-StPS learned architecture with many short filters outperformed a 1-StPS learned architecture with fewer longer filters at the same overall impulse-response length, and that 3-StPS LDBP delivered about 0.2 dB higher effective SNR than standard DBP. These results support the broader filter-assisted design principle that many simple jointly optimized steps can outperform a few more exact but more expensive ones (Sillekens et al., 2019, Oliari et al., 2020).

5. Related reduced-complexity architectures

Several adjacent families target the same performance–complexity frontier without implementing classical low-pass-filter-assisted DBP. One strand keeps the linear step exact and frequency-domain while enriching the nonlinear step. L-ESSFM is a machine-learning-aided parameterization of ESSFM in which each step has a learnable dispersion length $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$2 and a learnable nonlinear phase rotation filter $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$3. The linear operator is

$$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$4

so it is an all-pass quadratic phase filter rather than a low-pass filter. In a 5-channel, 93 GBd, dual-polarization 64-QAM, 170 km unrepeatered link, L-ESSFM at $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$5 and $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$6 samples/symbol achieved a 0.8 dB SNR gain over EDC at 172 RM/2D, whereas ESSFM required 761 RM/2D for the same gain and LDBP required more than 2000 RM/2D for similar gains. The paper explicitly positioned this as a learned, FFT-based alternative to FIR-heavy learned DBP at very high symbol rates (Cellini et al., 27 Jan 2026).

A second strand modifies the nonlinear step for WDM multi-channel compensation. CC-ESSFM extends ESSFM so that each channel’s nonlinear phase depends on filtered intensities from neighboring jointly processed channels and both polarizations: $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$7 with finite-memory coefficients $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$8 truncated to $$\mathbf{E}(t,z)= \begin{bmatrix} E_x(t,z)\ E_y(t,z) \end{bmatrix}.$$9 taps and satisfying $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$0. This is not LPF-assisted DBP in the usual sense, but it is a filtered reduced-order nonlinear operator that targets few-step practical WDM DBP (Civelli et al., 2021).

A third strand addresses wideband complexity through multirate subbanding. SbL-DBP decomposes the received signal into multiple subbands, performs exact per-subband frequency-domain dispersion compensation, and models nonlinear intra- and inter-subband interactions by a trainable MIMO intensity-to-phase coupling filter

$$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$1

In 11$$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$240 Gbaud WDM RRC-16QAM over 20$$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$3100 km, the method was reported to dominate DBP and enhanced DBP in low- and medium-complexity regimes; at about 500 RMpS, SbL-DBP with $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$4, $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$5, and $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$6 gave $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$7SNR $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$8 dB, versus $$\hat{D} = j\frac{\beta_2}{2}\frac{\partial^2}{\partial t^2} + \alpha \qquad\text{and}\qquad \hat{N} = -j\gamma |\mathbf{E}|^2.$$9 dB for enhanced DBP and $|\mathbf{E}|^2$0 dB for conventional DBP (Shevelev et al., 14 May 2026).

A fourth strand enriches the nonlinear step through perturbation theory rather than explicit LPF insertion. PA-LDBP uses first-order perturbation coefficients in each nonlinear block so that the nonlinear phase depends on a local window of neighboring intensities, thereby including SPM and intra-channel XPM. For 32 Gbaud 64-QAM over 20$|\mathbf{E}|^2$180 km, the reported $|\mathbf{E}|^2$2-factor gains over linear compensation were approximately 3.5 dB, 1.8 dB, 1.4 dB, and 0.5 dB for 1, 2, 4, and 10 spans per step, respectively, and the paper concluded that PA-LDBP attained improved performance gains with reduced complexity when compared to LDBP in the cases of 4 and 10 spans per step (Lin et al., 2021).

Taken together, these families show that “filter assistance” in practical DBP can refer to several different mechanisms: explicit LPF of the nonlinear driving intensity, trainable short dispersive filters, filtered nonlinear phase memories, subband decomposition, or perturbation-based local interaction kernels. Classical low-pass-filter-assisted DBP is therefore best understood as one specific point in a wider reduced-complexity design space rather than the only one.

6. Dispersion-managed links, PMD, and adaptive compensation

The learned/filter-constrained DBP literature also extends into regimes where dispersion management and PMD substantially reshape the inverse-propagation problem. In dual-polarization WDM dispersion-managed systems with 28 spans, each containing 72 km SMF and 13 km DCF, one learned-DBP study reported Q-factor improvements of 1.8 dB over linear equalization and 1.2 dB over a DM-adapted DBP baseline at similar nominal step complexity. It also reported that LDBP with only 2 layers achieved $|\mathbf{E}|^2$3 dB, 1 dB above the linear equalizer at the same launch power, while 7 layers reached the best explicitly reported value $|\mathbf{E}|^2$4 dB at $|\mathbf{E}|^2$5 dBm (Abu-Romoh et al., 2022).

A closely related DM study made the acronym issue explicit: LDBP there again meant learned digital back-propagation, not low-pass-filter-assisted DBP. It adapted DBP to the spanwise dispersion map by averaging the local dispersion over each DBP step and then learning the corresponding linear filters. In single-channel transmission it reported effective-SNR gains of 6.3 dB over linear equalization and 2.5 dB over DBP; in WDM transmission the corresponding Q-factor gains were 1.1 dB and 0.4 dB. Its complexity analysis concluded that a frequency-domain implementation of LDBP and DBP was more favorable than the time-domain implementation in the studied DM systems (Abu-Romoh et al., 2023).

PMD-aware distributed compensation creates another extension. LDBP-PMD parameterizes the Manakov-PMD split-step model using short CD FIR filters, short DGD FIR filters, and per-step polarization rotations. With free DGD filters plus $|\mathbf{E}|^2$6 matrices, the reported peak mean effective SNR was 26.16 dB, only 0.30 dB below PMD-free LDBP, with convergence within 1% of peak dB performance after 428 iterations on average. The paper emphasized that this distributed PMD compensation required no knowledge of the particular PMD realization along the link and no knowledge of the total accumulated PMD (Bütler et al., 2020).

Once PMD compensation is embedded into a learned stage-wise backpropagator, adaptation to PMD drift becomes a practical issue. A later study of DCRNN-PMD, positioned relative to LDBP and LDBP-PMD, showed that freezing the CD and nonlinear blocks and retraining only the PMD-related parameters reduced adaptation effort sharply: transfer from a pretrained base model to new PMD realizations required about 50 epochs on average, versus $|\mathbf{E}|^2$7 epochs for full initial training, and decision-directed online learning could track continuous SOP/PMD drift under a hinge model while maintaining the learned model’s performance advantage over DBP and LDBP baselines (Jain et al., 2022).

These results do not redefine classical low-pass-filter-assisted DBP, but they broaden its encyclopedic context. They show that the practical DBP problem is rarely just “how to invert the Kerr nonlinearity.” It is instead a coupled approximation problem involving bandwidth restriction, step discretization, dispersion-map structure, PMD, WDM observability limits, and hardware cost. Within that broader landscape, classical LDBP remains the explicit strategy that low-pass filters the nonlinear driving intensity at each step, while learned and adjacent schemes explore alternative ways of restricting the inverse-propagation operator to a tractable subspace.