Trade-off Between Communication and Positioning in Millimeter Wave Systems with Bounded and Unbounded Positioning Errors

Millimeter wave has proven to be effective in the integrated positioning and communication (IPAC) system. In this work, we establish a millimeter wave IPAC system by leveraging the inner coupling relationship between estimated data rate and positioning error. Moreover, we formulate robust power allocation problems by minimizing the positioning performance metric, i.e., the Cram\'er-Rao bound (CRB), subject to the data rate and total transmit power constraints to investigate the trade-off between positioning and communication of the IPAC system. Specifically, in the first case, we characterize the positioning error as a bounded ellipsoidal model to formulate the power allocation problem, which can be solved by the ${\cal{S}}$-Procedure and the alternating optimization (AO) algorithm. In the second case, we formulate the optimization problem by characterizing the positioning error as an unbounded Gaussian model, which can be solved by the Bernstein-type inequality and successive convex approximation (SCA) methods. Numerical results verify the robustness against the positioning error of the proposed robust design by comparing with the non-robust design, and analyze the trade-off between communication and positioning of the integrated system.