The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the signal-to-noise ratio SNR and the number of segments in distance$K$is considered. It is shown that if$K\geq \text {SNR} ^{2/3}$and$\text {SNR} \rightarrow \infty $, the capacity of the resulting continuous-space lossless model is lower bounded by$\frac {1}{2}\log _{2}(1+ \text {SNR}) - \frac {1}{2}+ o(1)$, where$o(1)$tends to zero with SNR. As$K \rightarrow \infty $, the inter-symbol interference (ISI) averages out to zero due to the law of large numbers and the SSFM model tends to a diagonal phase noise model. It follows that, in contrast to the discrete-space model where there is only one signal degree-of-freedom (DoF) at high powers, the number of DoFs in the continuous-space model is at least half of the input dimension$n$. Intensity-modulation and direct detection achieves this rate. The pre-log in the lower bound when$K= \sqrt [\delta]{ \text {SNR}}$is generally characterized in terms of$\delta $. It is shown that if the nonlinearity parameter$\gamma \rightarrow \infty $, the capacity of the continuous-space model is$\frac {1}{2}\log _{2}(1+ \text {SNR})+ o(1)$. The SSFM model when the dispersion matrix does not depend on$K$is considered. It is shown that the capacity of this model when$K= \sqrt [\delta]{ \text {SNR}}$,$\delta >3$, and$\text {SNR} \rightarrow \infty $is$\frac {1}{2n}\log _{2}(1+ \text {SNR})+ O(1)$. Thus, there is only one DoF in this model. Finally, it is found that the maximum achievable information rates (AIRs) of the SSFM model with back-propagation equalization obtained using numerical simulation follows a double-ascent curve. The AIR characteristically increases with SNR, reaching a peak at a certain optimal power, and then decreases as SNR is further increased. The peak is attributed to a balance between noise and stochastic ISI. However, if the power is further increased, the AIR will increase again, approaching the lower bound$\frac {1}{2}\log (1+ \text {SNR})- \frac {1}{2} + o(1)$. The second ascent is because the ISI averages out to zero with$K \rightarrow \infty $sufficiently fast.

Nonlinear channel impairments are a major obstacle in fiber-optic communication systems. To facilitate a higher data rate in these systems, the complexity of the underlying digital signal processing algorithms to compensate for these impairments must be reduced. Deep learning-based methods have proven successful in this area. However, the concept of computational complexity remains an open problem. In this paper, a low-complexity convolutional recurrent neural network (CNN + RNN) is considered for deep learning of the long-haul optical fiber communication systems where the channel is governed by the nonlinear Schrodinger equation. This approach reduces the computational complexity via balancing the computational load by capturing short-temporal distance features using strided convolution layers with ReLU activation, and the long-distance features using a many-to-one recurrent layer. We demonstrate that for a 16-QAM 100 G symbol/s system over 2000 km optical-link of 20 spans, the proposed approach achieves the bit-error-rate of the digital back-propagation (DBP) with substantially fewer floating-point operations (FLOPs) than the recently-proposed learned DBP, as well as the non-model-driven deep learning-based equalization methods using end-to-end MLP, CNN, RNN, and bi-RNN models.

The per-sample zero-dispersion channel model of the optical fiber is considered. It is shown that capacity is uniquely achieved by an input probability distribution that has continuous uniform phase and discrete amplitude that takes on finitely many values. This result holds when the channel is subject to general input cost constraints, that include a peak amplitude constraint and a joint average and peak amplitude constraint.

How many neurons are needed to approximate a target probability distribution using a neural network with a given input distribution and approximation error? This paper examines this question for the case when the input distribution is uniform, and the target distribution belongs to the class of histogram distributions. We obtain a new upper bound on the number of required neurons, which is strictly better than previously existing upper bounds. The key ingredient in this improvement is an efficient construction of the neural nets representing piecewise linear functions. We also obtain a lower bound on the minimum number of neurons needed to approximate the histogram distributions.

The limit of a discrete-time model of the optical fiber described by the split-step Fourier method (SSFM) when the number of segments in distance K tends to infinity is considered. It is shown that if $K \geq {\mathcal{P}^{2/3}}$ and $\mathcal{P} \to \infty $, where $\mathcal{P}$ is the average input power, the capacity of the resulting continuous-space lossless model is lower bounded by $\frac{1}{2}{\log _2}\left( {1 + {\text{SNR}}} \right) - \frac{1}{2} + o\left( 1 \right)$, where o(1) tends to zero with the signal-to-noise ratio SNR. This implies that at least half of the signal degrees-of-freedom remain asymptotically in this model.

Two signal multiplexing schemes for optical fiber communication are considered: Wavelength-division multiplexing (WDM) and nonlinear frequency-division multiplexing (NFDM), based on the nonlinear Fourier transform. Achievable information rates (AIRs) of NFDM and WDM are compared in a network scenario with an ideal lossless model of the optical fiber in the defocusing regime. It is shown that the NFDM AIR is greater than the WDM AIR subject to a bandwidth and average power constraint, in a representative system with one symbol per user. The improvement results from nonlinear signal multiplexing.

The per-sample zero-dispersion channel model of the optical fiber is considered. It is shown that capacity is uniquely achieved by an input probability distribution that has continuous uniform phase and discrete amplitude that takes on finitely many values. This result holds when the channel is subject to general input cost constraints, that include a peak amplitude constraint and a joint average and peak amplitude constraint.

The recipients of the 2018 IEEE Information Theory Society Paper Award are Mansoor I. Yousefi and Frank R. Kschischang for the three-part paper “Information Transmission Using the Nonlinear Fourier Transform, I, II, III” which appeared in the IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4312–4328 (Part I), pp. 4329–4345 (Part II), and pp. 4346–4369 (Part III), July 2014.

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes of functions, such as strongly convex functions. Second, an upper bound is established on the difference of two neural networks with identical weights but different activation functions.

A mathematical framework is presented to study the evolution of multi-point cumulants in nonlinear dispersive partial differential equations with random input data, based on the theory of weak wave turbulence (WWT). This framework is used to explain how energy is distributed among Fourier modes in the nonlinear Schrödinger equation. This is achieved by considering interactions among four Fourier modes and studying the role of the resonant, non-resonant, and trivial quartets in the dynamics. As an application, a power spectral density is suggested for calculating the interference power in dense wavelength-division multiplexed optical systems, based on the kinetic equation of the WWT. This power spectrum, termed the Kolmogorov-Zakharov (KZ) model, results in a better estimate of the signal spectrum in optical fiber, compared with the so-called Gaussian noise model. The KZ model is generalized to non-stationary inputs and multi-span optical systems.

An upper bound on the capacity of a cascade of nonlinear and noisy channels is presented. The cascade mimics the split-step Fourier method for computing waveform propagation governed by the stochastic generalized nonlinear Schrödinger equation. It is shown that the spectral efficiency of the cascade is at most log(1+SNR), where SNR is the receiver signal-to-noise ratio. The results may be applied to optical fiber channels. However, the definition of bandwidth is subtle and leaves open interpretations of the bound. Some of these interpretations are discussed.

The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels, such as optical fibers, where pulse propagation is governed by the nonlinear Schrödinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This paper explains the mathematical tools that underlie the method.

In this paper, numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the Zakharov-Shabat system, a Lax operator underlying numerous integrable communication channels including the nonlinear Schrödinger channel, modeling pulse propagation in optical fibers. These methods are subsequently tested and their ability to estimate the spectrum are compared against each other. These methods are used to compute the spectrum of various signals commonly used in the optical fiber communications. It is found that the layer peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy. To illustrate the structure of the spectrum, the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples. It is observed that in some cases, as signal parameters vary, eigenvalues collide and change their course of motion. The real axis is typically the place from which new eigenvalues originate or, are absorbed into after traveling a trajectory in the complex plane.

Motivated by the looming capacity crunch in fiber-optic networks, information transmission over such systems is revisited. Among numerous distortions, interchannel interference in multiuser wavelength-division multiplexing (WDM) is identified as the seemingly intractable factor limiting the achievable rate at high launch power. However, this distortion and similar ones arising from nonlinearity are primarily due to the use of methods suited for linear systems, namely WDM and linear pulse-train transmission, for the nonlinear optical channel. Exploiting the integrability of the nonlinear Schrödinger (NLS) equation, a nonlinear frequency-division multiplexing (NFDM) scheme is presented, which directly modulates noninteracting signal degrees-of-freedom under NLS propagation. The main distinction between this and previous methods is that NFDM is able to cope with the nonlinearity, and thus, as the signal power or transmission distance is increased, the new method does not suffer from the deterministic crosstalk between signal components, which has degraded the performance of previous approaches. In this paper, emphasis is placed on modulation of the discrete component of the nonlinear Fourier transform of the signal and some simple examples of achievable spectral efficiencies are provided.

This paper considers the transmission of information over integrable channels, a class of (mainly nonlinear) channels described by a Lax operator-pair. For such channels, the nonlinear Fourier transform, a powerful tool in soliton theory and exactly solvable models, plays the same role in “diagonalizing” the channel that the ordinary Fourier transform plays for linear convolutional channels. A transmission strategy encoding information in the nonlinear Fourier spectrum, termed nonlinear frequency-division multiplexing, is proposed for integrable channels that is the nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels. A central and motivating example is fiber-optic data transmission, for which the proposed transmission technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods.

Motivated by the looming “capacity crunch” in current fiber-optic systems, we recently suggested using the nonlinear Fourier transform (NFT) to transmit information over integrable communication channels such as the optical fiber channel, which is governed by the generalized nonlinear Schrödinger equation. In this transmission scheme information is encoded in the nonlinear Fourier transform of the signal, consisting of two components: a discrete and a continuous spectral function. In this paper, we restrict to discrete spectrum modulation and provide some simple examples of achievable spectral efficiencies. With this new method, deterministic distortions arising from the dispersion and nonlinearity, such as inter-symbol and interchannel interference are zero for a single user channel or all users of a multiple user network.

The capacity of the channel defined by the stochastic nonlinear Schrödinger equation, which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise, is considered in the case of zero dispersion. In the absence of dispersion, this channel behaves as a collection of parallel per-sample channels. The conditional probability density function of the nonlinear per-sample channels is derived using both a sum-product and a Fokker-Planck differential equation approach. It is shown that, for a fixed noise power, the per-sample capacity grows unboundedly with input signal. The channel can be partitioned into amplitude and phase subchannels, and it is shown that the contribution to the total capacity of the phase channel declines for large input powers. It is found that a 2-D distribution with a half-Gaussian profile on the amplitude and uniform phase provides a lower bound for the zero-dispersion optical fiber channel, which is simple and asymptotically capacity-achieving at high signal-to-noise ratios (SNRs). A lower bound on the capacity is also derived in the medium-SNR region. The exact capacity subject to peak and average power constraints is numerically quantified using dense multiple ring modulation formats. The differential model underlying the zero-dispersion channel is reduced to an algebraic model, which is more tractable for digital communication studies, and, in particular, it provides a relation between the zero-dispersion optical channel and a 2 × 2 multiple-input multiple-output Rician fading channel. It appears that the structure of the capacity-achieving input distribution resembles that of the Rician fading channel, i.e., it is discrete in amplitude with a finite number of mass points, while continuous and uniform in phase.

Optical fiber channels modeled by the stochastic nonlinear Schrödinger equation and operating at zero dispersion are considered in this paper. As a result of the Kerr nonlinearity and its interaction with amplified spontaneous emission noise, the amplitude and phase channels correlate with each other and the statistics of the received signal are non-Gaussian. In order to find the capacity of such a nonlinear channel, one must find the conditional probability density function (PDF) of the channel output given channel input. The complex zero-dispersion channel (viewed as an instance of the Langevin equation) is transformed to polar coordinates using Itô calculus, where the cubic nonlinearity appears to be more tractable. A method is introduced based on the Fokker-Planck differential equation, known in the statistical physics, to describe the PDF of the received signal.