Error bound for two-dimensional DOA joint estimation in RIS assisted wireless network

Reconfigurable intelligent surface (RIS) has been a crucial enabler for improving wireless localization accuracy through effectively controlling radio propagation environment. This paper investigates the performance bound for RIS-assisted two-dimensional (2D) direction of arrival (DOA) joint estimation. While the Cramér–Rao lower bound (CRLB) serves as the fundamental performance benchmark for mean square error, it is only asymptotically tight. To this end, an information-theory performance bound termed 2D DOA entropy error (2D-DEE) is proposed through statistical characterization of angle estimation uncertainty. Specifically, the joint a posteriori probability density function (PDF) of 2D DOA is first derived incorporating the uniform and independent a priori distributions of DOAs. Based on this joint a posteriori PDF, the a posteriori entropy is then normalized for different signal-to-noise ratio (SNR) to derive an explicit expression for 2D-DEE. For further insight, the asymptotic expression for entropy errors of 1D DOA and 2D DOA are analyzed in high SNR region. Extensive numerical results validate the accuracy of theoretical analysis and demonstrate that the derived 2D-DEE is able to maintain tight over wider range of SNR in evaluating and predicting 2D DOA estimation performance.