Joint Angle Estimation Error Analysis and 3-D Positioning Algorithm Design for mmWave Positioning System

This paper presents a comprehensive framework for jointly analyzing the angle estimation error and designing a three-dimensional (3D) positioning algorithm for an Internet of Things (IoT) millimeter wave (mmWave) positioning system. Initially, the azimuth and elevation angles of arrival (AoAs) at the anchors are estimated by applying the two-dimensional discrete Fourier transform (2D-DFT) algorithm. The angle estimation error is then analyzed in terms of probability density functions (PDF) by utilizing the properties of the 2D-DFT algorithm and employing challenging derivations and linear approximations. The analysis reveals that the resulting angle estimation error is non-Gaussian, distinguishing it from previous studies. Next, the complex expression of the PDF for the AoA estimation error is simplified using the first-order linear approximation of triangle functions. Subsequently, a complex expression for the variance is derived based on the obtained PDF. Specifically, the variance for the azimuth estimation error is integrated separately according to the different non-zero intervals of the obtained PDF. Additionally, the closed-form expressions of the variances are formulated using generalized hypergeometric series. Finally, the two-stage weighted least square (TSWLS) algorithm is employed to estimate the 3D position of the mobile user (MU) using the estimated AoAs and the obtained non-Gaussian variance. Extensive simulation results confirm the non-Gaussian nature of the derived angle estimation error and demonstrate the superiority of the proposed framework.