# Multiple Importance Sampling for Efficient Symbol Error Rate Estimation

## TL;DR

This paper introduces a novel application of the ALOE multiple importance sampling technique to efficiently estimate symbol error rates in digital communication systems with complex constellations, significantly reducing simulation time.

## Contribution

It adapts the ALOE multiple importance sampling method to the problem of SER estimation for non-square constellations, improving efficiency over traditional Monte Carlo methods.

## Key findings

- ALOE provides unbiased SER estimates.
- Simulation times are orders of magnitude shorter.
- Effective for complex digital constellations.

## Abstract

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.