Scaling Laws for Cooperative Node Localization in Non-Line-of-Sight Wireless Networks

We study the problem of cooperative node localization in non-line- of-sight (NLOS) wireless networks and address design questions such as, "How many anchors and what fraction of line-of-sight (LOS) measurements are needed to achieve a specified target accuracy?". We analytically characterize the performance improvement in localization accuracy as a function of the number of nodes in the network and the fraction of LOS measurements. In particular, we show that the Cramer- Rao Lower Bound (CRLB) can be expressed as a product of two factors - a scalar function that depends only on the parameters of the noise distribution and a matrix that depends only on the geometry of node locations. This holds for arbitrary distance and angle measurement modalities under an additive noise model. Further, a simplified expression is obtained for the CRLB, which provides an insightful understanding of the bound and helps deduce the scaling behavior of the estimation error as a function of the number of agents and anchors in the network. The mean squared error in localization is shown to have an inverse linear relationship with the number of anchors or agents. The error is also shown to have an approximately inverse linear relationship with the fraction of LOS readings except at the extremes. The behavior at the extremes suggests that even a small fraction of LOS measurements can provide significant improvements. Conversely, a small fraction of NLOS measurements can significantly degrade the performance.