While probabilistic constellation shaping (PCS) enables rate and reach adaption with finer granularity [1] (Cho and Winzer, 2009), it imposes signal processing challenges at the receiver. Since the distribution of PCS-quadrature amplitude modulation (QAM) signals tends to be Gaussian, conventional blind polarization demultiplexing algorithms are not suitable for them [2] (Johnson et al., 1998). It is known that independently and identically distributed (iid) Gaussian signals, when mixed, cannot be recovered/separated from their mixture. For PCS-QAM signals, there are algorithms such as [3] and [4] Dris et al. (2019) and Athuraliya et al. (2004) which are designed by extending conventional blind algorithms used for uniform QAM signals. In these algorithms, an initialization point is obtained by processing only a part of the mixed signal, which have non-Gaussian statistics. In this article, we propose an alternative method wherein we add temporal correlations at the transmitter, which are subsequently exploited at the receiver in order to separate the polarizations. We will refer to the proposed method as frequency domain (FD) joint diagonalization (JD) probability aware-multi modulus algorithm (pr-MMA), and it is suited to channels with moderate polarization mode dispersion (PMD) effects. Furthermore, we extend our previously proposed JD-MMA [5] (Bajaj et al., 2022) by replacing the standard MMA with a pr-MMA, improving its performance. Both FDJD-pr-MMA and JD-pr-MMA are evaluated for a diverse range of PCS (entropy $\mathcal {H}$) of 64-QAM over a first-order PMD channel that is simulated in a proof-of-concept setup. A MMA initialized with a memoryless constant modulus algorithm (CMA) is used as a benchmark. We show that at a differential group delay (DGD) of 10% of symbol period T$_{\text{symb}}$ and 18 dB SNR/pol., JD-pr-MMA successfully demultiplexes the PCS signals, while CMA-MMA fails drastically. Furthermore, we demonstrate that the newly proposed FDJD-pr-MMA is robust against moderate PMD effects by evaluating it over a DGD of up to 40% of T$_{\text{symb}}$. Our results show that the proposed FDJD-pr-MMA successfully equalizes PMD channels with a DGD up to 20% of T$_{\text{symb}}$.

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Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained.@en

Lax-integrable partial differential equations (PDEs) can by definition be described through a compatibility condition between two linear operators. These operators are said to form a Lax pair for the PDE, which itself is usually nonlinear. Lax pairs are a very useful tool, but unfortunately finding them is a difficult problem in practice. In this paper, we propose a method that determines the spectral operator of an AKNS-type Lax pair such that the corresponding PDE fits given measurement data as well as possible. The spectral operator then enables practitioners to solve or analyze the underlying PDE using the induced nonlinear Fourier transform. The underlying PDE only has to be approximately Lax-integrable; the method will find the spectral operator that explains the data best. Together with the dispersion relation, the spectral operator of AKNS type completely determines an integrable PDE that approximates the true underlying PDE. We identify the most suitable spectral operator by matching PDE-dependent quantities that should be conserved during evolution. The method is automatic and only requires recordings of solutions at two different values of the evolution variable, which do not have to be close.

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The shallow waters off the coast of Norderney in the southern North Sea are characterised by a higher frequency of rogue wave occurrences than expected. Here, rogue waves refer to waves exceeding twice the significant wave height. The role of nonlinear processes in the generation of rogue waves at this location is currently unclear. Within the framework of the Korteweg–de Vries (KdV) equation, we investigated the discrete soliton spectra of measured time series at Norderney to determine differences between time series with and without rogue waves. For this purpose, we applied a nonlinear Fourier transform (NLFT) based on the Korteweg–de Vries equation with vanishing boundary conditions (vKdV-NLFT). At measurement sites where the propagation of waves can be described by the KdV equation, the solitons in the discrete nonlinear vKdV-NLFT spectrum correspond to physical solitons. We do not know whether this is the case at the considered measurement site. In this paper, we use the nonlinear spectrum to classify rogue wave and non-rogue wave time series. More specifically, we investigate if the discrete nonlinear spectra of measured time series with visible rogue waves differ from those without rogue waves. Whether or not the discrete part of the nonlinear spectrum corresponds to solitons with respect to the conditions at the measurement site is not relevant in this case, as we are not concerned with how these spectra change during propagation. For each time series containing a rogue wave, we were able to identify at least one soliton in the nonlinear spectrum that contributed to the occurrence of the rogue wave in that time series. The amplitudes of these solitons were found to be smaller than the crest height of the corresponding rogue wave, and interaction with the continuous wave spectrum is needed to fully explain the observed rogue wave. Time series with and without rogue waves showed different characteristic soliton spectra. In most of the spectra calculated from rogue wave time series, most of the solitons clustered around similar heights, but the largest soliton was outstanding, with an amplitude significantly larger than all other solitons. The presence of a clearly outstanding soliton in the spectrum was found to be an indicator pointing towards the enhanced probability of the occurrence of a rogue wave in the time series. Similarly, when the discrete spectrum appears as a cluster of solitons without the presence of a clearly outstanding soliton, the presence of a rogue wave in the observed time series is unlikely. These results suggest that soliton-like and nonlinear processes substantially contribute to the enhanced occurrence of rogue waves off Norderney.@en

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.

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When a large number of solitons dominates the dynamics of a system, scientists describe this collective behaviour of solitons as a soliton gas. Soliton gases are currently the subject of intense practical and theoretical investigations. The existence of soliton gases has been confirmed in experiments, but is not clear what kind of sea states might lead to soliton gases. Therefore, in order to determine the wave parameters for sea states that lead to soliton gases, large numbers of surface wave elevations are generated by the well-known JOSNWAP model in this paper. Here, we only discuss soliton gases in deep water governed by the nonlinear Schrödinger (NLS) equation. The nonlinear Fourier transform (NFT) with vanishing boundary conditions is applied to the simulated ocean surface waves. The resulting nonlinear Fourier spectrum is used to calculate the energy of radiation waves and solitons. We investigate which JONSWAP parameters result in sea states that can be characterized as soliton gases, and find that a large Phillip’s parameter α, a large peak enhancement parameter γ and a short peak period T_P are important factors for soliton gas conditions. The results allow researchers to estimate how likely soliton gases are in deep waters. Furthermore, we find that the appearance of rogue waves is slightly increased in highly nonlinear sea states with soliton gas-like conditions.

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Recently, a novel parameter identification method for partial differential equations based on the Koopman operator framework has been proposed. We evaluate its suitability for the identification of single span optical fiber links of various lengths in simulations.@en

We experimentally investigate the problem of monitoring the Kerr-nonlinearity coefficient $\gamma$ from transmitted and received data for a single-mode fiber link of 1600 km length. We compare the accuracy and speed of three different approaches. First, a standard split-step Fourier method is used to predict the output at various $\gamma$ values, which are then compared to the measured output. Second, a recently proposed nonlinear Fourier transform (NFT)-based method, which matches solitonic eigenvalues in the transmitted and received signals for various $\gamma$ values. Third, a novel fast version of the NFT-based method, which only matches the highest few eigenvalues. Although the NFT-based methods do not scale with link length, we demonstrate that the SSFM-based method is significantly faster than the basic NFT-based method for the considered link of 1600 km, and outperforms even the faster version. However, for a simulated link of 8000 km, the fast NFT-based method is shown to be faster than the SSMF-based method, although at the cost of a small loss in accuracy.

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We present a fast method to calculate the significantly large solitonic components of signals with large time-bandwidth products governed by the nonlinear Schrödinger equation, for which the computation typically becomes prohibitively expensive and/or numerically unstable. We partition the full signal in both frequency and time to obtain short signals with a constant number of samples, independent of the size of the full signal. The solitons within each short signal are computed using a conventional nonlinear Fourier transform (NFT) algorithm. The partitioning in general leads to spurious solitons not present in the full signal. We therefore design an acceptance scheme that removes spurious solitons. The remaining solitons are attributed to the full signal. Solitons that are too wide to fit into the short signals cannot be detected by this approach, but since wide solitons must be of low amplitude, the significant solitons will be found. This approach only requires O(N) floating point operations, with N the number of signal samples. It can furthermore be applied to signals with large time-bandwidth products for which conventional NFT algorithms become unreliable or even fail. When applying our proposed method to a signal of 15,000 samples, the significant solitonic components were computed 14 times faster than when considering the whole signal, for which the conventional algorithm furthermore provided wrong results. We found that time-partitioning yields accurate results, while frequency-partitioning causes a small loss in accuracy. Combined frequency-time partitioning leads to the fastest computation, but also suffers from the same loss in accuracy as with frequency-partitioning. As time-partitioning yields a significant speed-up at nearly no loss in accuracy, we regard this as the method of choice in most practical scenarios.

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We present a simple, efficient “direct learning” approach to train Volterra series-based pre-distortion filters using neural networks. We show its superior performance over conventional training methods using a 64-QAM 64 GBaud simulated transmitter with varying transmitter nonlinearity and noisy conditions.@en

Fast millimeter wave (mmWave) channel estimation techniques based on compressed sensing (CS) suffer from low signal-to-noise ratio (SNR) in the channel measurements, due to the use of wide beams. To address this problem, we develop an in-sector CS-based mmWave channel estimation technique that focuses energy on a sector in the angle domain. Specifically, we construct a new class of structured CS matrices to estimate the channel within the sector of interest. To this end, we first determine an optimal sampling pattern when the number of measurements is equal to the sector dimension and then use its subsampled version in the sub-Nyquist regime. Our approach results in low aliasing artifacts in the sector of interest and better channel estimates than benchmark algorithms.@en

Recently, a nonlinear Fourier transform-based Kerr-nonlinearity identification algorithm was demonstrated for a 1000 km NZDSF link with accuracy of 75%. Here, we demonstrate an accuracy of 99% over 1600 km SSMF. Reasons for improved accuracy are discussed.@en

We present the first method for the joint modulation of the continuous and the discrete nonlinear Fourier spectrum of finite duration signals.@en

Recently, a nonlinear Fourier transform-based Kerr-nonlinearity identification algorithm was demonstrated for a 1000 km NZDSF link with accuracy of 75%. Here, we demonstrate an accuracy of 99% over 1600 km SSMF. Reasons for improved accuracy are discussed.

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We propose a novel method to determine the average water depth from shallow, weakly nonlinear water waves that are approximated by the Korteweg-de Vries equation. Our identification method only requires free-surface measurements from two wave gauges aligned in the direction of wave propagation. The method we propose is based on comparing solitonic components in wave packets, which are computed using the nonlinear Fourier transform (NFT) (typical time-series data often contains at least some solitonic components, even when these components are not directly visible). When the correct water depth is used for the normalisation of the wave, the solitonic components found by the NFT remain constant as the wave packet propagates, whereas any other water depth will result in solitonic components that do not remain constant. The basic idea is thus to iteratively determine the water depth that leads to a best fit between the solitonic components of time series measurements at two different gauge positions. We present a proof-of-concept on experimental bore data generated in a wave flume, where the identified water depth is within 5% of the measured value.@en

We propose a novel method for blind polarization-demultiplexing of probabilistically shaped signals for coherent receivers. The method is capable of separating signals with (quasi) Gaussian distributions by exploiting temporal correlations added to the transmit signals. The proposed method is evaluated in challenging mixing scenarios.

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We propose a numerical algorithm that computes the eigenvalues of the Korteweg–de Vries equation (KdV) from sampled input data with vanishing boundary conditions. It can be used as part of the Non-linear Fourier Transform (NFT) for the KdV equation. The algorithm that we propose makes use of Sturm-Liouville (SL) oscillation theory to guaranty that all eigenvalues are found. In comparison to similar available algorithms, we show that our algorithm is more robust to numerical errors and thus more reliable. Furthermore we show that our root finding algorithm, which is based on the Newton–Raphson (NR) algorithm, typically saves computation time compared to the conventional approaches that rely heavily on bisection.@en

In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The experimental flume is extended numerically to an effective relative length of x/h=3000, which allows all far-field solitons to emerge from the undular bore in the simulation data. We apply the periodic KdV-based nonlinear Fourier transform (KdV-NFT) to the time series taken at different numerical gauges and compare the results with those of the conventional Fourier transform. We find that the periodic KdV-NFT reliably predicts the number and the amplitudes of all far-field solitons from the near-field data long before the solitons start to emerge from the bore, even though the propagation is only approximated by the KdV. It is the first time that the predictions of the KdV-NFT are demonstrated over such long distances in a realistic set-up. In contrast, the conventional linear FT is unable to reveal the hidden solitons in the bore. We repeat our analyses using space instead of time series to investigate whether the space or time version of the KdV provides better predictions. Finally, we show how stepwise superposition of the determined solitons, including the nonlinear interactions between individual solitons, returns the analysed initial bore data.@en

Large vessels propagating in narrow, shallow maritime waterways generate a system of ship-induced waves consisting of long-period primary waves and short-period secondary waves. Progressive long-period free-surface wave systems are governed by the Korteweg–de Vries (KdV) equation, and are known to possibly disperse into a train of solitons and trailing oscillatory waves in the far field. By application of the nonlinear Fourier transform based on the KdV equation (KdV-NFT), these far-field solitons can already be revealed in the nonlinear spectra of the near-field data. In this paper, we apply the KdV-NFT to measured ship-wave time series from experiments in order to investigate the solitonic structures of these strongly nonlinear waves. Furthermore, we present qualitative and quantitative relations between the spectral solitons from frequency-domain KdV-NFT and channel, geometry, ship dynamics and primary-wave height as obtained by time-domain analysis of the time series.

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Rogue waves are extreme waves in the ocean that appear from nowhere and disappear without a trace. They are usually modelled by the nonlinear Schrödinger equation (NLS), which describes nonlinear phenomena such as modulational instability and solitons on finite backgrounds. In this study, the periodic nonlinear Fourier transform (NFT) for the NLS equation is applied to simulate ocean surface waves in deep water. The temporal and spatial structures of surface waves are obtained by evolving JONSWAP time series using the NLS equation. Several parameters extracted from the NFT spectra of the initial time series are investigated as predictors for the maximum wave height during evolution. We investigate several parameters from the literature, and find that with suitably optimized coefficients, a NFT-based parameter based on the largest unstable mode has a good correlation with the overall maximum wave amplitude. This new spectral criterion can contribute to rogue wave forecasting under extreme sea states.

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