On Symbol Error Probability-based Beamforming in MIMO Gaussian Wiretap Channels

This paper investigates beamforming schemes designed to minimize the symbol error probability (SEP) for an authorized user while guaranteeing that the likelihood of an eavesdropper correctly recovering symbols remains below a predefined threshold. Unlike previous works that focus on maximizing secrecy capacity, our work is centered around finding an optimal beamforming vector for binary antipodal signal detection in multiple-input multiple-output (MIMO) Gaussian wiretap channels. Finding the optimal beamforming vector in this setting is challenging. Computationally efficient algorithms such as convex techniques cannot be applied to find the optimal solution. To that end, our proposed algorithm relies on Karush-Kuhn-Tucker (KKT) conditions and a generalized eigen-decomposition method to find the exact solution. In addition, we also develop an approximate, practical algorithm to find a good beamforming matrix when using M-ary detection schemes. Numerical results are presented to assess the performance of the proposed methods across various scenarios.