# An ADMM-Based Approach to Robust Array Pattern Synthesis

## TL;DR

This paper introduces a novel robust array pattern synthesis method that models element-wise amplitude and phase uncertainties, formulates a min-max optimization problem, and solves it efficiently using an ADMM-based algorithm, outperforming traditional approaches.

## Contribution

It proposes a tighter SV uncertainty model and an ADMM-based solution for robust array pattern synthesis, improving accuracy and computational efficiency.

## Key findings

- The proposed method effectively minimizes side lobe response.
- The ADMM algorithm converges rapidly and is computationally efficient.
- Numerical results validate the superiority over existing techniques.

## Abstract

In most existing robust array beam pattern synthesis studies, the bounded-sphere model is used to describe the steering vector (SV) uncertainties. In this letter, instead of bounding the norm of SV perturbations as a whole, we explore the amplitude and phase perturbations of each SV element separately, thereby obtaining a tighter SV uncertainty model. Based on this model, we formulate the robust array pattern synthesis problem from the perspective of the min-max optimization, which aims to minimize the maximum side lobe response, while preserving the main lobe response. However, this problem is difficult due to the infinitely many non-convex constraints. As a remedy, we employ the worst-case criterion and recast the problem as a convex second-order cone program (SOCP). To solve the SOCP, we further develop an alternating direction method of multipliers (ADMM)-based algorithm, which is computationally efficient with each step being computed in closed form. Numerical simulations demonstrate the efficacy and efficiency of the proposed algorithm.