Low-complexity Voronoi shaping for the Gaussian channel

TL;DR

This work designs low-complexity Voronoi constellations using a cubic coding lattice and applies pseudo-Gray labeling to minimize BER while enabling finer spectral efficiencies. It adopts Kurkoski's encoding/decoding with Gray-like labeling and augments lattice design with rotations and scalings to achieve additional bits per symbol-pair, trading some coding gain for practical shaping gains up to $g_s \,\approx\,1.03\text{ dB}$. A novel mutual information estimation framework based on step-wise importance sampling is developed to analyze very large constellations, together with an LLR approximation method that makes AIR analysis feasible for high-dimensional VCs. Empirical results show shaping gains up to about $0.935\text{ dB}$ at high SNR and that the proposed VCs can achieve higher AIR than conventional scaled VCs when paired with LDPC codes, due to improved pseudo-Gray labeling and reduced decoding complexity.

Abstract

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. 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 achievable information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity.

Figures (6)

• Figure 1: Example: different integer mapping rules for a two-dimensional VC. The blue filled points are encoded into points in the shifted Voronoi region $\boldsymbol{a}+\Omega(\Lambda_\mathrm{s})$ (the light blue region) in encoding.
• Figure 2: The APE gain $g$ as a function of $\beta$ for VCs with a cubic coding lattice. The smaller markers on top of the lines represent the cases in which the scaling factor is not a power of $2$. The black dashed lines are the asymptotic shaping gains $g_{\text{s}}(\Lambda_\mathrm{s})$ for these shaping lattices stated in Table \ref{['tab:gammas']}.
• Figure 3: The estimated value $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ as a function of $D$ for different SNRs (solid curves with markers). The black dashed lines are the corresponding benchmark values $pf_{\boldsymbol{Y}}(\boldsymbol{y})$. The number after the '/' is the corresponding $|\mathcal{B}(\boldsymbol{y},\sqrt{D-1})|$.
• Figure 4: The estimated value $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ as a function of $D$ for different SNRs. Solid lines without markers are estimated with $K_{d}=|\mathcal{I}_d(\boldsymbol{y})|$ for all subsets. The markers are estimated with $K_{d}=10^4$ uniform samples from $\mathcal{I}_d(\boldsymbol{y})$ for subsets with $d>8$ and $K_{r}=|\mathcal{I}_d(\boldsymbol{y})|$ for subsets with $1 \leq d\leq 8$. The black dashed lines are the corresponding benchmark values $pf_{\boldsymbol{Y}}(\boldsymbol{y})$.
• Figure 5: The estimated MI as a function of the SNR for multidimensional VCs (solid curves without markers). The markers are the MI estimated using the exact $f_{\boldsymbol{Y}}(\boldsymbol{y})$ by \ref{['eq:fy']}.