  Digital constellations formed by hexagonal or other non-square
two-dimensional 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 closed-form, 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
signal-to-noise ratios. In this letter, we adapt a recently proposed multiple
importance sampling (MIS) technique, called ALOE (for `At Least One rare
Event'), to this problem. Conditioned to a transmitted symbol, an error (or
rare event) occurs when the observation falls in a union of half-spaces 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.
