# Quadratic Matrix Inequality Approach to Robust Adaptive Beamforming for General-Rank Signal Model

## TL;DR

This paper introduces a novel approach to robust adaptive beamforming for general-rank signals by reformulating the problem as a quadratic matrix inequality and employing convex relaxation techniques, leading to improved performance.

## Contribution

It presents the first direct solution to the original nonconvex beamforming problem using QMI reformulation and LMI relaxation, with new optimality conditions and an approximate algorithm.

## Key findings

- Enhanced beamforming performance demonstrated in simulations
- The proposed method outperforms existing approximate solutions
- The approach effectively handles the nonconvexity of the problem

## Abstract

The worst-case robust adaptive beamforming problem for general-rank signal model is considered. This is a nonconvex problem, and an approximate version of it (obtained by introducing a matrix decomposition on the presumed covariance matrix of the desired signal) has been well studied in the literature. Different from the existing literature, herein however the original beamforming problem is tackled. Resorting to the strong duality of linear conic programming, the robust adaptive beamforming problem for general-rank signal model is reformulated into an equivalent quadratic matrix inequality (QMI) problem. By employing a linear matrix inequality (LMI) relaxation technique, the QMI problem is turned into a convex semidefinite programming problem. Using the fact that there is often a positive gap between the QMI problem and its LMI relaxation, an approximate algorithm is proposed to solve the robust adaptive beamforming in the QMI form. Besides, several sufficient optimality conditions for the nonconvex QMI problem are built. To validate our results, simulation examples are presented, which also demonstrate the improved performance of the new robust beamformer in terms of the output signal-to-interference-plus-noise ratio.

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Source: https://tomesphere.com/paper/1905.10519