Cramér-Rao Bound for Localization with A Priori Knowledge on Biased Range Measurements

This paper derives a general expression for the Cram\'er-Rao bound (CRB) of wireless localization algorithms using range measurements subject to bias corruption. Specifically, the a priori knowledge about which range measurements are biased, and the probability density functions (PDF) of the biases are assumed to be available. For each range measurement, the error due to estimating the time-of-arrival of the detected signal is modeled as a Gaussian distributed random variable with zero mean and known variance. In general, the derived CRB expression can be evaluated numerically. An approximate CRB expression is also derived when the bias PDF is very informative. Using these CRB expressions, we study the impact of the bias distribution on the mean square error (MSE) bound corresponding to the CRB. The analysis is corroborated by numerical experiments.