Fair SSB Codebook Design for Multi-Cell mmWave MIMO Communications

For millimeter wave communications, beams used to transmit synchronization signal blocks (SSBs) affect both base station coverage and beam training overhead. We therefore consider fair SSB codebook design, formulated as an optimization problem, aiming to maximize the minimum average signal-to-interference-plus-noise ratio (SINR) across user clusters. This problem is challenging due to the non-smoothness of the objective function, arising from the optimal beam-pair selection function and the minimum operator. To address this, we propose a double-loop framework, where the outer loop constructs approximations for the selection function with iteratively reduced error, and the inner loop solves the resulting approximate problems. In each inner-loop iteration, the objective function of the approximate problem is smoothed with iteratively reduced smoothness, enabling gradient derivation. This gradient is then estimated using variance-reduced estimators based on samples from users, and the result is used to update codebooks. Following this framework, we develop both first-order (FO) and zeroth-order (ZO) oracle schemes. The FO scheme requires full channel state information samples for gradient estimation while the ZO scheme only requires SINR samples. Simulation results show that in given scenarios, both schemes achieve SINR fairness comparable to or better than that of discrete Fourier transform codebooks, but with fewer beams.