Error Analysis of the Joint Localization and Synchronization of RIS-Assisted mm-Wave MISO-OFDM Under the Effect of Hardware Impairments

This work investigates the theoretical bounds of the joint localization and synchronization processes in a reconfigurable intelligent surface (RIS)-assisted system. We address the case of millimeter-wave (
$mm$
-Wave) multiple-input single-output (MISO) orthogonal frequency-division multiplexing (OFDM) with non-ideal transceivers. Considering a single antenna mobile station (MS) aims to estimate the parameters of the downlinks from the base station (BS) and the RIS by observing a known sequence received by the MS directly from the BS and indirectly through the RIS. The theoretical bounds of the estimation process are assessed by using the Fisher information matrix (FIM). A transformation matrix is then used to convert the FIM of the downlink channel parameters to the FIM of the MS joint localization and synchronization parameters. Specifically, the transformation matrix is derived based on the geometric relationships that convert the estimated downlink channels’ parameters to the position coordinates and clock offset. Next, the Cramer-Rao lower bound (CRLB) matrix of the joint localization and synchronization process is obtained by using the pseudo-inverse of the FIM. Thus, the position error bound (PEB), as well as the synchronization error bound (SEB), are calculated. Computer simulation results are provided to illustrate the adverse effects of the hardware impairments (HWIs) on the accuracy of localization and synchronization. These results are given in proportion to the effective signal-to-noise ratio (SNR), the number of pilot transmissions, and the number of the RIS elements.