Site-Specific Fair SSB Codebook Design for mmWave MIMO-OFDM Communications

In millimeter wave communications, the initial access stage involves a trade-off between beam sweeping overhead and base station coverage. To balance this, we consider site-specific fair synchronization signal block codebook design, formulated as an optimization problem, aiming to maximize the minimum average signal-to-noise ratio (SNR) across user clusters, subject to constant modulus constraints. To solve this, we propose a double-loop framework, where the outer loop relaxes the original problem into sub-problems using an augmented Lagrangian (AL) algorithm, and the inner loop solves these sub-problems using a hybrid variance-reduced (HVR) stochastic gradient descent (SGD) algorithm. In each inner iteration, the non-smooth objective function of the sub-problem, comprising numerous user-wise SNR functions, is approximated by a differentiable function, with its gradient estimated using HVR estimators. The estimated gradient is then used to update the codebook. Simulation results demonstrate the efficiency of the proposed AL-HVR-SGD scheme.