Mathematical Approach in Hybrid Beamforming for ISAC Systems

This document serves as supplementary material for a journal paper submission, presenting detailed mathematical proofs and derivations that support the results outlined in the main manuscript. In this work, we formulate an Integrated sensing and communication (ISAC) optimization problem aimed at fairly maximizing beampattern gains across multiple directions of interest, while ensuring that the signal-to-interference-plus-noise ratio (SINR) requirements for communication users and the overall power constraint are satisfied. To solve this problem, we propose an optimization algorithm and provide a formal proof of its convergence.