End-to-End Learning of Geometrical Shaping Maximizing Generalized Mutual Information

GMI-based end-to-end learning is shown to be highly nonconvex. We apply gradient descent initialized with Gray-labeled APSK constellations directly to the constellation coordinates. State-of-the-art constellations in 2D and 4D are found providing reach increases up to 26\% w.r.t. to QAM

Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning

End-to-end learning has become a popular method to optimize a constellation shape of a communication system. When the channel model is differentiable, end-to-end learning can be applied with conventional backpropagation algorithm for optimization of the shape. A variety of optimization algorithms have also been developed for end-to-end learning over a non-differentiable channel model. In this paper, we compare gradient-free optimization method based on the cubature Kalman filter, model-free optimization and backpropagation for end-to-end learning on a fiber-optic channel modeled by the split-step Fourier method. The results indicate that the gradient-free optimization algorithms provide a decent replacement to backpropagation in terms of performance at the expense of computational complexity. Furthermore, the quantization problem of finite bit resolution of the digital-to-analog and analog-to-digital converters is addressed and its impact on geometrically shaped constellations is analysed. Here, the results show that when optimizing a constellation with respect to mutual information, a minimum number of quantization levels is required to achieve shaping gain. For generalized mutual information, the gain is maintained throughout all of the considered quantization levels. Also, the results implied that the autoencoder can adapt the constellation size to the given channel conditions

Polarization-ring-switching for nonlinearity-tolerant geometrically-shaped four-dimensional formats maximizing generalized mutual information

In this paper, a new four-dimensional 64-ary polarization ring switching (4D-64PRS) modulation format with a spectral efficiency of 6 bit/4D-sym is introduced. The format is designed by maximizing the generalized mutual information (GMI) and by imposing a constant-modulus on the 4D structure. The proposed format yields an improved performance with respect to state-of-the-art geometrically shaped modulation formats for bit-interleaved coded modulation systems at the same spectral efficiency. Unlike previously published results, the coordinates of the constellation points and the binary labeling of the constellation are jointly optimized. When compared with polarization-multiplexed 8-ary quadrature-amplitude modulation (PM-8QAM), gains of up to 0.7 dB in signal-to-noise ratio are observed in the additive white Gaussian noise (AWGN) channel. For a long-haul nonlinear optical fiber system of 8,000 km, gains of up to 0.27 bit/4D-sym (5.5% data capacity increase) are observed. These gains translate into a reach increase of approximately 16% (1,100 km). The proposed modulation format is also shown to be more tolerant to nonlinearities than PM-8QAM. Results with LDPC codes are also presented, which confirm the gains predicted by the GMI.Comment: 12 pages, 12 figure

Voronoi Constellations for Coherent Fiber-Optic Communication Systems

The increasing demand for higher data rates is driving the adoption of high-spectral-efficiency (SE) transmission in communication systems. The well-known 1.53 dB gap between Shannon\u27s capacity and the mutual information (MI) of uniform quadrature amplitude modulation (QAM) formats indicates the importance of power efficiency, particularly in high-SE transmission scenarios, such as fiber-optic communication systems and wireless backhaul links. Shaping techniques are the only way to close this gap, by adapting the uniform input distribution to the capacity-achieving distribution. The two categories of shaping are probabilistic shaping (PS) and geometric shaping (GS). Various methods have been proposed for performing PS and GS, each with distinct implementation complexity and performance characteristics. In general, the complexity of these methods grows dramatically with the SE and number of dimensions.Among different methods, multidimensional Voronoi constellations (VCs) provide a good trade-off between high shaping gains and low-complexity encoding/decoding algorithms due to their nice geometric structures. However, VCs with high shaping gains are usually very large and the huge cardinality makes system analysis and design cumbersome, which motives this thesis.In this thesis, we develop a set of methods to make VCs applicable to communication systems with a low complexity. The encoding and decoding, labeling, and coded modulation schemes of VCs are investigated. Various system performance metrics including uncoded/coded bit error rate, MI, and generalized mutual information (GMI) are studied and compared with QAM formats for both the additive white Gaussian noise channel and nonlinear fiber channels. We show that the proposed methods preserve high shaping gains of VCs, enabling significant improvements on system performance for high-SE transmission in both the additive white Gaussian noise channel and nonlinear fiber channel. In addition, we propose general algorithms for estimating the MI and GMI, and approximating the log-likelihood ratios in soft-decision forward error correction codes for very large constellations

High-Cardinality Hybrid Shaping for 4D Modulation Formats in Optical Communications Optimized via End-to-End Learning

In this paper we carry out a joint optimization of probabilistic (PS) and geometric shaping (GS) for four-dimensional (4D) modulation formats in long-haul coherent wavelength division multiplexed (WDM) optical fiber communications using an auto-encoder framework. We propose a 4D 10 bits/symbol constellation which we obtained via end-to-end deep learning over the split-step Fourier model of the fiber channel. The constellation achieved 13.6% reach increase at a data rate of approximately 400 Gbits/second in comparison to the ubiquitously employed polarization multiplexed 32-QAM format at a forward error correction overhead of 20%.Comment: 5 pages, 3 figure

End-to-end Optimization of Constellation Shaping for Wiener Phase Noise Channels with a Differentiable Blind Phase Search

As the demand for higher data throughput in coherent optical communication systems increases, we need to find ways to increase capacity in existing and future optical communication links. To address the demand for higher spectral efficiencies, we apply end-to-end optimization for joint geometric and probabilistic constellation shaping in the presence of Wiener phase noise and carrier phase estimation. Our approach follows state-of-the-art bitwise auto-encoders, which require a differentiable implementation of all operations between transmitter and receiver, including the DSP algorithms. In this work, we show how to modify the ubiquitous blind phase search (BPS) algorithm, a popular carrier phase estimation algorithm, to make it differentiable and include it in the end-to-end constellation shaping. By leveraging joint geometric and probabilistic constellation shaping, we are able to obtain a robust and pilot-free modulation scheme improving the performance of 64-ary communication systems by at least 0.1 bit/symbol compared to square QAM constellations with neural demappers and by 0.05 bit/symbol compared to previously presented approaches applying only geometric constellation shaping

End-to-end Learning of a Constellation Shape Robust to Channel Condition Uncertainties

Vendor interoperability is one of the desired future characteristics of optical networks. This means that the transmission system needs to support a variety of hardware with different components, leading to system uncertainties throughout the network. For example, uncertainties in signal-to-noise ratio and laser linewidth can negatively affect the quality of transmission within an optical network due to e.g. mis-parametrization of the transceiver signal processing algorithms. In this paper, we propose to geometrically optimize a constellation shape that is robust to uncertainties in the channel conditions by utilizing end-to-end learning. In the optimization step, the channel model includes additive noise and residual phase noise. In the testing step, the channel model consists of laser phase noise, additive noise and blind phase search as the carrier phase recovery algorithm. Two noise models are considered for the additive noise: white Gaussian noise and nonlinear interference noise model for fiber nonlinearities. The latter models the behavior of an optical fiber channel more accurately because it considers the nonlinear effects of the optical fiber. For this model, the uncertainty in the signal-to-noise ratio can be divided between amplifier noise figures and launch power variations. For both noise models, our results indicate that the learned constellations are more robust to uncertainties in channel conditions compared to a standard constellation scheme such as quadrature amplitude modulation and standard geometric constellation shaping techniques

MINE-based Geometric Constellation Shaping in AWGN Channel

The use of high-order constellation modulations is imperative to improve the spectral efficiency, for both radio frequency/laser-based satellite systems and optical wireless communications. The geometric shaping (GS) optimization as one typical constellation shaping method drives the improvement of communication capacity and system performance. This paper presents a novel mutual information neural estimation (MINE)- based GS method to optimize the high-order constellations in pure additive white Gaussian noise (AWGN) channel, which uses the deep neural network (DNN) to estimate the mutual information (MI) value and maximize the MI to approach the AWGN capacity asymptotically. The proposed system trains both the encoder and MINE networks by back propagation, and does not need to train a decoder for optimization and thus can avoid the loss caused by the decoder. Simulation results show that the MINE-based shaping design outperforms the unshaped M-ary quadrature amplitude modulation (QAM) in terms of MI values. Note that the capacity gain increases slightly as the order M increases. Furthermore, the proposed scheme is promising for constellation design in various channel models, such as the phase noise and the fading channels, once the channel model used in MINE is matched, which can be a future research topic

Constellation Shaping in Optical Communication Systems

Exploiting the full-dimensional capacity of coherent optical communication systems is needed to overcome the increasing bandwidth demands of the future Internet. To achieve capacity, both coding and shaping gains are required, and they are, in principle, independent. Therefore it makes sense to study shaping and how it can be achieved in various dimensions and how various shaping schemes affect the whole performance in real systems. This thesis investigates the performance of constellation shaping methods including geometric shaping (GS) and probabilistic shaping (PS) in coherent fiber-optic systems. To study GS, instead of considering machine learning approaches or optimization of irregular constellations in two dimensions, we have explored multidimensional lattice-based constellations. These constellations provide a regular structure with a fast and low-complexity encoding and decoding. In simulations, we show the possibility of transmitting and detecting constellation with a size of more than 10^{28} points which can be done without a look-up table to store the constellation points. Moreover, improved performance in terms of bit error rate, symbol error rate, and transmission reach are demonstrated over the linear additive white Gaussian noise as well as the nonlinear fiber channel compared to QAM formats.Furthermore, we investigate the performance of PS in two separate scenarios, i.e., transmitter impairments and transmission over hybrid systems with on-off keying channels. In both cases, we find that while PS-QAM outperforms the uniform QAM in the linear regime, uniform QAM can achieve better performance at the optimum power in the presence of transmitter or channel nonlinearities