Mansoor I. Yousefi

Information Transmission Using the Nonlinear Fourier Transform, Part II: Numerical Methods

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.

Information Transmission Using the Nonlinear Fourier Transform, Part I: Mathematical Tools

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.

Information Transmission Using the Nonlinear Fourier Transform, Part III: Spectrum Modulation

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.

Nonlinear Frequency Division Multiplexed Transmissions Based on NFT

We experimentally demonstrate the successful modulation and error-free detection of three eigenvalue nonlinear frequency division multiplexed (NFDM) signals over 1800 km based on the recently developed nonlinear Fourier transform theoretical framework with digital coherent receivers. The three eigenvalues are located on the upper-half complex plane and are modulated by independent ON-OFF keying signals, thus forming 3-bit NFDM symbols for nonlinear fiber transmissions.

Upper bound on the capacity of a cascade of nonlinear and noisy channels

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.

Multi-eigenvalue communication via the nonlinear Fourier transform

Information transmission using only the discrete component of the nonlinear Fourier transform is studied and multi-eigenvalue signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.

On the Per-Sample Capacity of Nondispersive Optical Fibers

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.

Upper bound on the capacity of the nonlinear Schrödinger channel

It is shown that the capacity of the channel modeled by (a discretized version of) the stochastic nonlinear Schrödinger (NLS) equation is upper-bounded by log(l + SNR) with SNR = P₀/σ²(z), where P₀ is the average input signal power and σ²(z) is the total noise power up to distance z. The result is a consequence of the fact that the deterministic NLS equation is a Hamiltonian energy-preserving dynamical system.

The Kolmogorov–Zakharov Model for Optical Fiber Communication

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.

Communication over fiber-optic channels using the nonlinear Fourier transform

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.