We examine here what type of predictive modelling, classification, or regression, using neural networks (NN), fits better the task of soft-demapping based post-processing in coherent optical communications, where the transmission channel is nonlinear and dispersive. For the first time, we present possible drawbacks in using each type of predictive task in a machine learning context, considering the nonlinear coherent optical channel equalization/soft-demapping problem. We study two types of equalizers based on the feed-forward and recurrent NNs, for several transmission scenarios, in linear and nonlinear regimes of the optical channel. We point out that even though from the information theory perspective the cross-entropy loss (classification) is the most suitable option for our problem, the NN models based on the cross-entropy loss function can severely suffer from learning problems. The latter translates into the fact that regression-based learning is typically superior in terms of delivering higher Q-factor and achievable information rates. In short, we show by empirical evidence that loss functions based on cross-entropy may not be necessarily the most suitable option for training communication systems in practical scenarios when overfitting- and vanishing gradients-related problems come into play.

We consider spatially coupled low-density parity-check codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing the amount of coupling between spatial positions, we show that we can construct codes with improved decoding speed compared with conventional, uniform smoothing constructions. This leads to a significantly better performance under decoder complexity constraints while keeping the degree distribution regular. We optimize smoothing configurations using differential evolution and illustrate the performance gains by means of a simulation.

Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourier domain. While this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive with classic Nyquist-based schemes. This is especially due to the observation that soliton pulses generally exhibit a large time-bandwidth product. We consider the phase modulation of spectral amplitudes of higher order solitons, taking into account their varying spectral and temporal behavior when propagating along the fiber. For second- and third-order solitons, we numerically optimize the pulse shapes to minimize the time-bandwidth product. We study the behavior of multi-soliton pulse duration and bandwidth, and generally observe two corner cases where we approximate them analytically. We use these results to give an estimate on the minimal achievable time-bandwidth product per eigenvalue.

Soliton pulses are special waveforms to account for nonlinearity in fiber optical communication. They can be represented in a nonlinear spectrum by eigenvalues and spectral amplitudes which have simple transformation equations along the ideal link. This motivates to encode data in the nonlinear spectrum. In this paper, we consider dual-polarization modulation of spectral amplitudes, and show that they become highly correlated during propagation along a noisy fiber link. Thus, joint equalization is generally needed for detection at the receiver. We propose a simple precoding scheme that almost removes these correlations. This allows to significantly improve the detection performance even without any complex equalization. The spectral amplitudes are transformed into pairs of common and differential information part. We show that the differential part is almost preserved along the link, even in the presence of noise. Thus, it can be directly detected from the received spectral amplitudes with high reliability. Exploiting the differential gain of the precoding allows much higher bit rates at comparable error rates. We analyze our precoding scheme and verify its performance gain in split-step-Fourier simulations by comparing it to the conventional independent modulation of spectral amplitudes for first and second order solitons.

In this work, we formulate the fixed-length distribution matching as a Bayesian inference problem. Our proposed solution is inspired from the compressed sensing paradigm and the sparse superposition (SS) codes. First, we introduce sparsity in the binary source via position modulation (PM). We then present a simple and exact matcher based on Gaussian signal quantization. At the receiver, the dematcher exploits the sparsity in the source and performs low-complexity dematching based on generalized approximate message-passing (GAMP). We show that GAMP dematcher and spatial coupling lead to an asymptotically optimal performance, in the sense that the rate tends to the entropy of the target distribution with vanishing reconstruction error in a proper limit. Furthermore, we assess the performance of the dematcher on practical Hadamard-based operators. A remarkable inherent feature of our proposed solution is the possibility to: i) perform matching at the symbol level (nonbinary); ii) perform joint channel coding and matching.

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability (PB) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability p, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.

We consider spatially coupled low-density parity-check codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing the amount of coupling between spatial positions, we show that we can construct codes with excellent thresholds and small rate loss, even with the lowest possible smoothing parameter and large variable node degrees, which are required for low error floors. We also establish that the decoding convergence speed is faster with non-uniformly coupled codes, which we verify by density evolution of windowed decoding with a finite number of iterations. We also show that by only slightly increasing the smoothing parameter, practical codes with potentially low error floors and thresholds close to capacity can be constructed. Finally, we give some indications on protograph designs.

Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourier domain. While this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive with the classic Nyquist-based schemes. In this paper, we study the evolution of the time-bandwidth product of multi-solitons as they propagate along the optical fiber. For second and third order soliton pulses, we numerically optimize the pulse shapes to achieve the smallest time-bandwidth product when the phase of the spectral amplitudes is used for modulation. Moreover, we analytically estimate the pulse-duration and bandwidth of multi-solitons in some practically important cases. Those estimations enable us to approximate the time-bandwidth product for higher order solitons.

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled low-density generator-matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity belief propagation guided decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The belief propagation guided decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a temperature directly related to the noise level of the test-channel. We investigate the links between the algorithmic performance of the belief propagation guided decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular, the dynamical and condensation phase transition temperatures (equivalently test-channel noise thresholds) are computed. We observe that: 1) the dynamical temperature of the spatially coupled construction saturates toward the condensation temperature and 2) for large degrees the condensation temperature approaches the temperature (i.e., noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the belief propagation guided decimation algorithm. This paper contains an introduction to the cavity method.

We investigate an encoding scheme for lossy compression based on spatially coupled Low-Density GeneratorMatrix codes. The degree distributions are regular, or are Poisson on the code-bit side and check-regular which allows use for any compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent, and for large check degrees it gets close to Shannon's rate-distortion limit. We investigate links between the algorithmic performance and the phase diagram of a relevant random Gibbs measure. The associated dynamical and condensation thresholds are computed within the framework of the cavity method. We observe that: (i) the dynamical threshold of the spatially coupled construction saturates towards the condensation threshold; (ii) for large degrees the condensation threshold approaches the information theoretic test-channel parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the BPGD algorithm.

We study a new encoding scheme for lossy source compression based on spatially coupled low-density generatormatrix codes. We develop a belief-propagation guided-decimation algorithm, and show that this algorithm allows to approach the optimal distortion of spatially coupled ensembles. Moreover, using the survey propagation formalism, we also observe that the optimal distortions of the spatially coupled and individual code ensembles are the same. Since regular low-density generatormatrix codes are known to achieve the Shannon rate-distortion bound under optimal encoding as the degrees grow, our results suggest that spatial coupling can be used to reach the rate-distortion bound, under a low complexity belief-propagation guided-decimation algorithm.

It was recently shown that spatial coupling of individual low-density parity-check codes improves the belief-propagation threshold of the coupled ensemble essentially to the maximum a posteriori threshold of the underlying ensemble. We study the performance of spatially coupled low-density generator-matrix ensembles when used for transmission over binary-input memoryless output-symmetric channels. We show by means of density evolution that the threshold saturation phenomenon also takes place in this setting. Our motivation for studying low-density generator-matrix codes is that they can easily be converted into rateless codes. Although there are already several classes of excellent rateless codes known to date, rateless codes constructed via spatial coupling might offer some additional advantages. In particular, by the very nature of the threshold phenomenon one expects that codes constructed on this principle can be made to be universal, i.e., a single construction can uniformly approach capacity over the class of binary-input memoryless output-symmetric channels. We discuss some necessary conditions on the degree distribution which universal rateless codes based on the threshold phenomenon have to fulfill. We then show by means of density evolution and some simulation results that indeed codes constructed in this way perform very well over a whole range of channel types and channel conditions.

We define partial cognitive relay channel as a two-transmitter, two-receiver interference channel with a relay node such that one of the transmitters is a cognitive transmitter. As a more feasible assumption we assume that the cognitive transmitter has non-causally access to a part of the other transmitter's encoded message. The cognitive radio transmitter transmits simultaneously over the same channel instead of waiting for an idle channel which is traditional for a cognitive radio. We calculate the achievable rate region for partial cognitive relay channel based on the combination of partial decode-compress and forward for the relay, rate-splitting and Gel'fand and Pinsker coding. Numerical evaluation of the region in the practical Gaussian case is performed. We show that by defining a proper cooperation strategy among the transmitters, equipping some of them with the cognition capability and exploiting the relay nodes, the effective bandwidth of each transmitter will enhance greatly.

In this paper, we obtain a family of lower bounds for the sum capacity of code-division multiple-access (CDMA) channels assuming binary inputs and binary signature codes in the presence of additive noise with an arbitrary distribution. The envelope of this family gives a relatively tight lower bound in terms of the number of users, spreading gain, and the noise distribution. The derivation methods for the noiseless and the noisy channels are different but when the noise variance goes to zero, the noisy channel bound approaches the noiseless case. The behavior of the lower bound shows that for small noise power, the number of users can be much more than the spreading gain without any significant loss of information (overloaded CDMA). A conjectured upper bound is also derived under the usual assumption that the users send out equally likely binary bits in the presence of additive noise with an arbitrary distribution. As the noise level increases, and/or, the ratio of the number of users and the spreading gain increases, the conjectured upper bound approaches the lower bound. We have also derived asymptotic limits of our bounds that can be compared to a formula that Tanaka obtained using techniques from statistical physics; his bound is close to that of our conjectured upper bound for large scale systems.