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.
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.
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.
Polarization-division multiplexed (PDM) transmission based on the nonlinear Fourier transform (NFT) is proposed for optical fiber communication. The NFT algorithms are generalized from the scalar ...nonlinear Schrödinger equation for one polarization to the Manakov system for two polarizations. The transmission performance of the PDM nonlinear frequency-division multiplexing (NFDM) and PDM orthogonal frequency-division multiplexing (OFDM) are determined. It is shown that the transmission performance in terms of Q-factor is approximately the same in PDM-NFDM and single polarization NFDM at twice the data rate and that the polarization-mode dispersion does not seriously degrade system performance. Compared with PDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized to multi-mode fibers in the strong coupling regime, paving the way for the application of the NFT to address the nonlinear effects in space-division multiplexing.
Multieigenvalue Communication Hari, Siddarth; Yousefi, Mansoor I.; Kschischang, Frank R.
Journal of lightwave technology,
07/2016, Letnik:
34, Številka:
13
Journal Article
Recenzirano
In the most general case, all three components-the discrete eigenvalues, the discrete spectral amplitudes, and the continuous spectrum-of the nonlinear Fourier transform of a signal can be ...independently modulated. This paper examines information transmission using only the discrete eigenvalues, and presents heuristic designs for multisoliton signal sets with spectral efficiencies greater than 3 b/s/Hz. The first design, called multieigenvalue position encoding, is based on an exhaustive search followed by pruning of the signal set to remove high pulsewidth or high bandwidth outliers. The second design, called trellis encoding, achieves comparable efficiencies to the fist method at much lower complexity. These multisoliton signals do not undergo any pulse broadening, but are significantly limited by bandwidth expansion if the system length is not much smaller than the dispersion length parameter. This limitation suggests that modulating the eigenvalues alone cannot address the problem of nonlinearity in commercial fiber transmission systems, and that our proposed methods are only meaningful when dispersion is very small and dominated by nonlinearity, e.g., close to the zero-dispersion wavelength at 1300 nm.
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.
Nonlinear frequency-division multiplexing (NFDM) is a communication scheme in which users' signals are multiplexed in the nonlinear Fourier domain. The contributions of this paper are twofold. First, ...the achievable information rates (AIRs) of NFDM based on an integrable model of the optical fiber are summarized. For this ideal model, it is shown that the AIR of the NFDM is greater than the AIR of the wavelength-division multiplexing (WDM) for a given bandwidth and signal power, in a representative system with five users and one symbol per user. The improvement results from nonlinear signal multiplexing. Second, the impact of some of the main perturbations on NFDM are investigated, including the fiber loss, polarization effects, and the third-order dispersion. For a realistic nonideal model, it is shown that the WDM AIR with joint dual-polarization back-propagation and third-order dispersion compensation is approximately equal to the NFDM AIR with two independent single-polarization demodulations and without third-order dispersion compensation. Using a joint dual-polarization receiver and perturbations compensation is expected to increase the NFDM AIR.
2018 IEEE Information Theory Society Paper Award Yousefi, Mansoor I.; Kschischang, Frank R.
IEEE transactions on information theory,
2019-Jan., 2019-1-00, 20190101, Letnik:
65, Številka:
1
Journal Article
Recenzirano
Odprti dostop
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 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.
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.
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