Synesthesia of Machines (SoM)-Aided FDD Precoding with Sensing Heterogeneity: A Vertical Federated Learning Approach

High complexity in precoding design for frequency division duplex systems necessitates streamlined solutions. Guided by Synesthesia of Machines (SoM), this paper introduces a heterogeneous multi-vehicle, multi-modal sensing aided precoding scheme within a vertical federated learning (VFL) framework, which significantly minimizes pilot sequence length while optimizing the system's sum rate. We address the challenges posed by local data heterogeneity due to varying on-board sensor configurations through a meticulously designed VFL training procedure. To extract valuable channel features from multi-modal sensing, we employ three distinct data preprocessing methods that convert raw data into informative representations relevant for precoding. Additionally, we propose an online training strategy based on VFL framework, enabling the scheme to adapt dynamically to fluctuations in user numbers. Numerical results indicate that our approach, utilizing short pilot sequences, closely approximates the performance of traditional optimization methods with perfect channel state information.