Kernel Density Estimation-Based Secrecy Performance Analysis of Cooperative Beamforming for Arbitrary Distributed 3-D Networks

Analog cooperative beamforming (ACB) improves physical-layer security (PLS) by enabling phase-coordinated transmissions among spatially distributed nodes without exchanging channel state information (CSI). However, previous studies have relied on predefined node distributions and two-dimensional (2D) network models, limiting the practical applicability of ACB. Motivated by this limitation, we propose a three-dimensional (3D) ACB-based PLS framework that predicts the secrecy performance of ACB under arbitrary node distributions by explicitly considering both LoS and NLoS channel components. Using kernel density estimation (KDE), we estimate the unknown spatial node distribution and derive closed-form expressions for the secrecy rate by analyzing the array gain statistics at the legitimate receiver and the eavesdropper. We verify the proposed framework through simulations and conduct a detailed analysis to examine the impact of network parameters, such as cluster size and the number of nodes.