Optimizing sparse MIMO arrays: sumset geometry and answer set programming

Abstract

Future wireless networks will rely heavily on multiple-input multiple-output (MIMO) systems. Integrated Sensing and Communications (ISAC) serves as a prime example application, where strict hardware constraints arise due to antennas and transmit signals being shared by sensing and communications functionalities. Sparse antenna arrays can mitigate these power and cost limitations without sacrificing spatial resolution. However, finding optimal configurations for these arrays presents a significant practical challenge due to the underlying optimization problem often being combinatorial in nature. In particular, MIMO array design typically corresponds to finding two finite sets of integers (corresponding to the transmitters and receivers) that give rise to a desired set of pairwise sums (sumsets). This thesis investigates the design of sparse MIMO array geometries based on such sumsets by deriving fundamental mathematical properties from first principles and validating them through exhaustive numerical searches using Answer Set Programming (ASP), a framework that we suggest for its speed and ease of development.

Specifically, we derive the upper bound for the number of admissible arrays for tuples of sumset and physical sensor array sizes, which allows us to quantify the limits of the design space for a given hardware budget. For non-redundant arrays operating at the maximum bound, an exact analytical enumeration of admissible configurations is derived through polynomial factorization, providing a method to generate optimal array geometries without relying on computationally expensive searches. Additionally, we establish upper and lower bounds for the number of admissible subcovers that can be used to analyze array fault tolerance to sensor deactivation.

The theoretical results are numerically verified and complemented by ASP, which provides insights into problem instances for which analytical bounds are unavailable or loose. These include fully overlapping symmetric arrays (phased arrays), disjoint arrays (co-prime and nested arrays), and virtual arrays with gaps (Golomb’s rulers). Ultimately, this thesis combines theoretical and numerical results to establish a scalable and rigorous framework for designing cost-effective, highresolution MIMO arrays.