Beyond Diagonal RIS Design for Parameter Estimation With and Without Eavesdropping

In this letter, we investigate the transmission of a complex-valued parameter vector from a transmitter to an intended receiver, considering both the presence and absence of an eavesdropper. The direct links from the transmitter to both the intended receiver and the eavesdropper are assumed to be blocked, and communications occur solely through cascaded channels facilitated by a beyond-diagonal reconfigurable intelligent surface (BD-RIS). While previous research has considered this system under conventional (diagonal) RIS assistance, we extend the setup to incorporate BD-RIS and quantify the resulting improvement in estimation performance at the intended receiver. This performance is measured by the trace of the Fisher information matrix (FIM), or equivalently, the average Fisher information, while simultaneously limiting the estimation capability of the eavesdropper. We propose solutions and algorithms for optimizing the BD-RIS response matrix and demonstrate their effectiveness. Numerical results reveal that the BD-RIS provides a significant enhancement in estimation quality compared to conventional diagonal RIS architectures.