Relativistic Cramér-Rao Bound Scaling for Device-Based and Device-Free Sensing

Abstract

This letter investigates range and velocity estimation under relativistic motion for device-based (DB) and device-free (DF) sensing. By deriving the exact time-scaling and time-shift relations induced by one-way and two-way propagation, both sensing modes are cast into a unified affine signal model. Closed-form Cramér--Rao bounds (CRBs) are obtained as explicit functions of normalized velocity, root-mean-squared (RMS) bandwidth, and RMS duration. The bounds recover the classical low-speed results but exhibit distinct velocity scaling in the ultrarelativistic regime. For rapidly receding motion, the range CRB diverges while the velocity CRB vanishes. For rapidly approaching motion, both CRBs vanish. The DB and DF modes further exhibit different asymptotic orders in the two directions, showing that relativistic motion changes not only the signal model but also the fundamental scaling laws governing sensing accuracy.