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By incorporating reconfigurable intelligent surface (RIS) intocommunication-assisted localization systems, the issue of signal blockagecaused by obstacles can be addressed, and passive beamforming can be employedto enhance localization accuracy. However, existing works mainly consider idealchannels and do not account for the effects of realistic impairments likecarrier frequency offset (CFO) and phase noise (PN) on localization. This paperproposes an iterative joint estimation algorithm for CFO, PN, and user positionbased on maximum a posteriori (MAP) criterion and gradient descent (GD)algorithm. Closed-form expressions for CFO and PN updates are provided. Thehybrid Cram\'{e}r-Rao lower bound (HCRLB) for the estimation parameters isderived, and the ambiguity in CFO and PN estimation is analyzed. To minimizethe HCRLB, a non-convex RIS shift optimization problem is formulated and istransformed into a convex semidefinite programming (SDP) problem using thetechnique of semidefinite relaxation (SDR) and Schur complement. Afteroptimizing the RIS phase shift, the theoretical positioning accuracy within thearea of interest (AOI) can be improved by two orders of magnitude, with amaximum positioning root mean square error (RMSE) lower than $\rm 10^{-2}m$.

Near-Field Localization and Phase Shift Optimization for RIS-Assisted Non-Ideal OFDM Systems