Multiple Importance Sampling for Efficient Symbol Error Rate Estimation Digital constellations formed by hexagonal or other nonsquare twodimensional lattices are often used in advanced digital communication systems. The integrals required to evaluate the symbol error rate (SER) of these constellations in the presence of Gaussian noise are in general difficult to compute in closedform, and therefore Monte Carlo simulation is typically used to estimate the SER. However, naive Monte Carlo simulation can be very inefficient and requires very long simulation runs, especially at high signaltonoise ratios. In this letter, we adapt a recently proposed multiple importance sampling (MIS) technique, called ALOE (for At Least One rare Eventu0027), to this problem. Conditioned to a transmitted symbol, an error (or rare event) occurs when the observation falls in a union of halfspaces or, equivalently, outside a given polytope. The proposal distribution for ALOE samples the system conditionally on an error taking place, which makes it more efficient than other importance sampling techniques ALOE provides unbiased SER estimates with simulation times orders of magnitude shorter than conventional Monte Carlo.

Design of TwoDimensional Nonuniform Constellations with Better Performance Modern communication systems adopt nonuniform constellations (NUCs) to approach Shannon limit. Generally, the design process of NUCs is based on maximizing bitinterleaved coded modulation (BICM) capacity for given channel conditions. However, the traditional Monte Carlo methods for computing BICM capacity cannot ensure the accuracy of NUC optimization with low complexity. This paper proposes a new design method including the derivation of a computable expression about BICM capacity and a new optimization scheme with simplified calculation of gradient in the algorithm. In this way, we obtain a set of 2DNUCs with low complexity and better performance than those adopted by ATSC 3.0.