Beampattern Synthesis for Discrete Phase RIS in Communication and Sensing Systems

Extensive research on Reconfigurable Intelligent Surfaces (RIS) has primarily focused on optimizing reflective coefficients for passive beamforming in specific target directions. This optimization typically assumes prior knowledge of the target direction, which is unavailable before the target is detected. To enhance direction estimation, it is critical to develop array pattern synthesis techniques that yield a wider beam by maximizing the received power over the entire target area. Although this challenge has been addressed with active antennas, RIS systems pose a unique challenge due to their inherent phase constraints, which can be continuous or discrete. This work addresses this challenge through a novel array pattern synthesis method tailored for discrete phase constraints in RIS. We introduce a penalty method that pushes these constraints to the boundary of the convex hull. Then, the Minorization-Maximization (MM) method is utilized to reformulate the problem into a convex one. Our numerical results show that our algorithm can generate a wide beam pattern comparable to that achievable with per-power constraints, with both the amplitudes and phases being adjustable. We compare our method with a traditional beam sweeping technique, showing a) several orders of magnitude reduction of the MSE of Angle of Arrival (AOA) at low to medium Signal-to-Noise Ratio (SNR)s; and b) $8$~dB SNR reduction to achieve a high probability of detection.