Fundamentals of LEO-Based Localization

In this paper, we derive the fundamental limits of low earth orbit (LEO) enabled localization by analyzing the available information in signals from multiple LEOs during different transmission time slots received on a multiple antennas and evaluate the utility of these signals for 9D localization (3D position, 3D orientation, and 3D velocity estimation). We start by deriving the Fisher Information Matrix (FIM) for the channel parameters that are present in the signals received from LEOs in the same or multiple constellations during multiple transmission time slots. To accomplish this, we define a system model that captures i) time offset between LEOs caused by having relatively cheap clocks, ii) frequency offset between LEOs, iii) the unknown Doppler rate caused by high mobility LEOs, and iv) multiple transmission time slots from a particular LEO. We transform the FIM for the channel parameters to the FIM for the location parameters and determine the required conditions for localization. To do this, we start with the 3D localization cases: i) 3D positioning with known velocity and orientation, ii) 3D orientation estimation with known position and velocity, and iii) 3D velocity estimation with known position and orientation. Subsequently, we derive the FIM for the full 9D localization case (3D position, 3D orientation, and 3D velocity estimation) in terms of the FIM for the 3D localization. Using these results, we determine the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas that allow for different levels of location estimation. We then provide insights into the interaction between the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas. One key result is that in the presence of time and frequency offsets and Doppler rate, it is possible to perform 9D localization (3D position, 3D velocity, and 3D orientation estimation) of a receiver by utilizing the signals from three LEO satellites observed during three transmission time slots received through multiple receive antennas.