On the symbol error probability of regular polytopes

An exact expression for the symbol error probability of the four-dimensional 24-cell in Gaussian noise is derived. Corresponding expressions for other regular convex polytopes are summarized. Numerically stable versions of these error probabilities are also obtained

Low-Complexity Geometric Shaping

Approaching Shannon's capacity via geometric shaping has usually been regarded as challenging due to modulation and demodulation complexity, requiring look-up tables to store the constellation points and constellation bit labeling. To overcome these challenges, in this paper, we study lattice-based geometrically shaped modulation formats in multidimensional Euclidean space. We describe and evaluate fast and low complexity modulation and demodulation algorithms that make these modulation formats practical, even with extremely high constellation sizes with more than $10^{28}$ points. The uncoded bit error rate performance of these constellations is compared with the conventional QAM formats in the additive white Gaussian noise and nonlinear fiber channels. At a spectral efficiency of 2 bits/sym/polarization, compared with 4-QAM format, transmission reach improvement of more than 38% is shown at the hard-decision forward error correction threshold of $2.26\times 10^{-4}$

Lattice-based geometric shaping

Geometrically shaped multidimensional constellations with more than 1028 points are simulated using fast and low-complexity algorithms without any look-up tables to store the constellation points. At the same symbol error rate, more than 78% and 114% reach improvement are demonstrated compared with 4- and 16-QAM, respectively

Low-Complexity Voronoi Shaping for the Gaussian Channel

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity

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

Designing Voronoi Constellations to Minimize Bit Error Rate

In a classical 1983 paper, Conway and Sloane presented fast encoding and decoding algorithms for a special case of Voronoi constellations (VCs), for which the shaping lattice is a scaled copy of the coding lattice. Feng generalized their encoding and decoding methods to arbitrary VCs. Less general algorithms were also proposed by Kurkoski and Ferdinand, respectively, for VCs with some constraints on their coding and shaping lattices. In this work, we design VCs with a cubic coding lattice based on Kurkoski\u27s encoding and decoding algorithms. The designed VCs achieve up to 1.03 dB shaping gains with a lower complexity than Conway and Sloane\u27s scaled VCs. To minimize the bit error rate (BER), pseudo-Gray labeling of constellation points is applied. In uncoded systems, the designed VCs reduce the required SNR by up to 1.1 dB at the same BER, compared with the same VCs using Feng\u27s and Ferdinand\u27s algorithms. In coded systems, the designed VCs are able to achieve lower BER than the scaled VCs at the same SNR. In addition, a Gray penalty estimation method for such VCs of very large size is introduced