Wideband TimeDomain Digital Backpropagation via
Subband
Processing and Deep Learning
\setstretch
1.07
\thesection Introduction
Realtime digital backpropagation (DBP) based on the splitstep Fourier method (SSFM) is widely considered to be impractical due to the complexity of the chromatic dispersion (CD) steps. To address this problem, finite impulse response (FIR) filters may be used instead of fast Fourier transforms (FFTs) to perform timedomain CD filtering^{3, 4, 5, 6, 7, 8, 9}. Indeed, the FIR filters can be as short as taps per SSFM step, provided that the step size is sufficiently small (i.e., many steps are used) and the filters in all steps are jointly optimized^{8}.
The complexity of timedomain DBP (TDDBP) is dominated by the total number of CD filter taps in all steps. Recent work has focused on relatively narrowband signals (e.g., Gbaud in^{8} and Gbaud in^{7, 5, 6}) for which the overall CD memory is low. Since the memory increases quadratically with bandwidth, it is not clear if TDDBP can be scaled gracefully also to more wideband signals.
In this paper, we consider a Gbaud signal where the delay spread per km amounts to symbol periods. It is shown that TDDBP can still offer a good performance–complexity tradeoff by leveraging digital subband processing. In particular, the group delay difference in different subbands can be compensated almost entirely using delay elements. A fractional delay filter is only needed after the last SSFM step.
\thesection Subband Processing and Related Work
Subband processing has been previously studied for both linear^{10, 11, 12, 13} and nonlinear^{14, 15, 16} impairment compensation. The idea is to split the received signal into parallel signals using a filter bank. Assuming a bandwidth reduction by , the delay spread per subband signal is reduced by . This can allow for significant complexity savings.
We consider a uniformly modulated filter bank as shown in Fig. 1. The subband signals are obtained by filtering a downconverted version of with a prototype filter , where is the frequency shift. The signals are then downsampled by and jointly processed. Finally, a synthesis filter bank reassembles the signal . Certain subbands may be inactive if they do not contain useful signal components. Active subbands are indexed symmetrically around the central subband according to .
Example 1: Consider a Gbaud signal sampled at GHz. For subbands, most of the spectrum falls within the central subbands, see Fig. 1. Thus, one may set .
\thesection Proposed DSP Architecture
A theoretical foundation for DBP based on subband processing can be obtained by inserting the splitsignal assumption into the NLSE. This leads to a set of coupled equations which can then be solved numerically. Our approach is based on the SSFM proposed in^{17}. The method is essentially equivalent to the standard SSFM for each subband, except that all sampled intensity signals are jointly processed with a multipleinput multipleoutput (MIMO) filter prior to the nonlinear phase rotation step. This accounts for crossphase modulation (XPM) between subbands but not fourwave mixing (FWM) because no phase information is exchanged. The method can also be used for DBP of noncoherent subband signals, e.g., in wavelength division multiplexing scenarios with different local oscillators. This was done in ^{14}.
Fig. 2 shows the proposed architecture, which consists of the following three main components:

Short filters compensate for pulse broadening in each subband and step .

Delay elements are used to compensate the group delay difference in different subbands.

The MIMO filtering is performed in the time domain using sparse tensor decompositions.
Compared to^{17, 14}, no FFT/IFFT pairs are used. In the following, the individual components (A)–(C) in Fig. 2 are described in more detail.
\thesection (A) PulseBroadening FIR Filters
The frequency response of an ideal CD compensation filter is where and is the propagation distance. Since the subband signals are downconverted relative to the carrier frequency, the filter responses have to be shifted as well. The ideal response for subband is
where with compensates the walkoff relative to the central subband, and . The filters correspond to and compensate for pulse broadening which is independent of . Thus, the same filter can be used in all subbands. Moreover, the filters are symmetric since is symmetric, allowing for a folded DSP implementation with real multiplications (RMs) assuming complex filter taps. Different filters are used in different steps (even if the step size is the same) to avoid accumulating truncation errors that arise from approximating with finitelength filters^{8}.
\thesection (B) Walkoff Delays and SSFM Step Size
The group delay depends linearly on the propagation distance . The step size can thus be chosen such that for all is an integer multiple of the subband sampling interval :
(0) 
where is the group delay difference in two neighboring subbands. Thus, as long as the step size is an integer multiple of , the walkoff can be compensated exactly using delay elements.
Example 2: For the parameters in Ex. 1, , and ps/km, we have km.
It is clear that the transmission distance is not necessarily an integer multiple of . Therefore, a fractional delay filter is inserted after the last step to account for any remaining noninteger delay prior to the synthesis filter bank.
\thesection (C) MIMOFIR Filter
Let be the transform of the intensity signal after the filter in subband and step and define . The nonlinear phase shift is computed based on filtered intensity signals whose transform is
(0) 
where is an polynomial matrix. The order of the (real and nonsymmetric) MIMO filter, i.e., the largest polynomial degree in , is assumed to be equal to the maximum number of walkoff delay elements in step .
Example 3: For the parameters as before and a step size , the group delay difference of the outermost subbands corresponds to . Thus, the order of is .
\thesection Joint Filter Optimization via Deep Learning
“Unrolling” all SSFM steps in Fig. 2 leads to a multilayer computation graph similar to a deep neural network^{7, 8}. Thus, joint optimization of all filters can be achieved using tools from machine learning, in particular deep learning via stochastic gradient descent. The tunable parameters are

the prototype filters and ,

the filters for ,

the MIMO filters for ,

the fractional delay filters for .
The optimization is performed in TensorFlow using the Adam optimizer. The mean squared error between the transmitted and received data symbols is used as a loss function, assuming a matched filter and phase offset rotation after the subband processing. Initially, the prototype filters are raisedcosine filters, the filters are preoptimized using leastsquares methods^{8}, and the filters are 8tap Lagrange interpolation filters. The MIMO filters are randomly initialized.
A potential issue in terms of complexity is the large number of MIMO filter coefficients, e.g., the filter in Ex. 3 is a tensor with real coefficients. We assume that these tensors can be decomposed into a cascade of sparse tensors , where all have dimension and order . To encourage sparsity during training, we employ norm regularization for all MIMO filters.
\thesection Results and Discussion
We consider a Gbaud signal (rootraised cosine, rolloff, Gaussian symbols), km of fiber (dB/km, ps/km, 1/W/km), and amplifiers with dB noise figure. Forward propagation is simulated using the standard SSFM with samples/symbols and logarithmic steps per span (StPS). Subband TDDBP is performed with , , , and a uniform step size of km for the first steps (see Ex. 2). The last step size is km for a total of steps ( StPS on average).
The results after training are shown in Fig. 3. Our method achieves a dB SNR improvement over linear equalization. The loss with respect to full DBP ( samples/symbol, StPS) is mostly due to the incoherent subband processing. To illustrate this, we also show results assuming essentially unrestricted complexity (dashed line), where , , and frequencydomain filtering according to^{17} with StPS is used.
To quantify the complexity, we use RMs focusing on the pulsebroadening and MIMO filters which dominate the requirements. The results were obtained with tap pulsebroadening filters () which can be implemented using RMs. For the MIMO filters, is used. The learned coefficients were thresholded, after which only out of total coefficients were nonzero. This gives RMs per subband and step on average, i.e., RMs in total. A similar analysis for frequencydomain overlapandadd filtering is presented in ^{14}. Following the same arguments, the number of RMs for our scenario is per subband and step with an optimized FFT size of . This is significantly more than required using TDDBP.
\thesection Conclusions
We have proposed a novel DSP architecture for DBP based on subband processing. Our method uses short FIR filters for the CD compensation to achieve computational efficiency. It was shown that a proper step size choice can significantly simplify the walkoff compensation by using delay elements. Lastly, the complexity of the XPM MIMO filters proposed in^{17} can be reduced by applying sparse tensor decomposition.
\thesection Acknowledgements
This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 749798. The work was also supported in part by the National Science Foundation (NSF) under Grant No. 1609327. Any opinions, findings, recommendations, and conclusions expressed in this material are those of the authors and do not necessarily reflect the views of these sponsors.
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