3D Positioning with Unsynchronized LEO Satellites and Minimal Infrastructure

In this paper, we rigorously derive the information in the signals received from low earth orbit (LEO) satellites, which are unsynchronized in time and frequency, and their utility for 3D position estimation. To enable this derivation, we define a system model that captures i) the time offset between LEOs caused by having cheap clocks, ii) the frequency offset between LEOs, and iii) multiple transmission time slots from a particular LEO. After this definition, we derive the Fisher information matrix (FIM) for the relevant channel parameters and transform the FIM for the channel parameters to the FIM for the 3D position. These derivations show the interactions between the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas. Subsequently, these allow us to determine the minimal number of LEOs, the number of transmission time slots, and the number of receive antennas needed to determine the 3D position. One key result is that when the LEOs are unsynchronized in time and frequency and experience a high Doppler rate, the 3D position can be determined by observing a single LEO for four transmission time slots.