Duality-Based Fixed Point Iteration Algorithm for Beamforming Design in ISAC Systems
Xilai Fan, Ya-Feng Liu
TL;DR
This work tackles joint ISAC beamforming by formulating a power-minimization problem under $K$ SINR constraints and an $E^*(\mathbf{R}) \le \eta$ radar MSE constraint. It leverages SDR to obtain a generalized downlink beamforming (GDB) problem with potentially indefinite weighting, then derives a Lagrangian dual that leads to a fixed-point structure. A novel Dual-FPI algorithm combines an outer subgradient ascent over the dual variables with an inner fixed-point iteration that solves the GDB; the authors prove a necessary and sufficient boundedness condition and establish global convergence of the inner FPI under mild initialization assumptions. Empirical results show that Dual-FPI attains globally optimal solutions with substantially lower complexity than baseline SDP-based methods, offering a scalable and reliable approach for practical ISAC systems.
Abstract
In this paper, we investigate the beamforming design problem in an integrated sensing and communication (ISAC) system, where a multi-antenna base station simultaneously serves multiple communication users while performing radar sensing. We formulate the problem as the minimization of the total transmit power, subject to signal-to-interference-plus-noise ratio (SINR) constraints for communication users and mean-squared-error (MSE) constraints for radar sensing. The core challenge arises from the complex coupling between communication SINR requirements and sensing performance metrics. To efficiently address this challenge, we first establish the equivalence between the original ISAC beamforming problem and its semidefinite relaxation (SDR), derive its Lagrangian dual formulation, and further reformulate it as a generalized downlink beamforming (GDB) problem with potentially indefinite weighting matrices. Compared to the classical DB problem, the presence of indefinite weighting matrices in the GDB problem introduces substantial analytical and computational challenges. Our key technical contributions include (i) a necessary and sufficient condition for the boundedness of the GDB problem, and (ii) a tailored efficient fixed point iteration (FPI) algorithm with a provable convergence guarantee for solving the GDB problem. Building upon these results, we develop a duality-based fixed point iteration (Dual-FPI) algorithm, which integrates an outer subgradient ascent loop with an inner FPI loop. Simulation results demonstrate that the proposed Dual-FPI algorithm achieves globally optimal solutions while significantly reducing computational complexity compared with existing baseline approaches.