Can Unsynchronized LEOs Provide 3D Orientation for a Ground Receiver?

Large antenna arrays and reconfigurable intelligent surfaces (RIS) have been made available due to the use of higher frequency bands, and there is the possibility that these arrays can become disturbed. Hence, their orientation could change after deployment. Since low earth orbits (LEO) are being proposed to provide position, navigation, and timing services, and LEOs from different constellations could be unsynchronized in time and frequency and experience a high Doppler rate. We ask, "can unsynchronized LEOs provide 3D orientation for a ground receiver?" To answer this question, we introduce the Fisher information matrix (FIM) and use the FIM to quantify the available information needed for 3D orientation estimation utilizing signals received from LEOs during multiple transmission time slots across multiple receive antennas. We observe by analyzing the positive definitiveness of the FIM for the 3D orientation that irrespective of the presence or absence of both time and frequency offsets, the 3D orientation of the receiver can be estimated through the multiple TOA measurements received across the receive antennas from two LEO satellites during a single transmission time slot. We also observe by analyzing the positive definitiveness of the FIM for the 3D orientation that irrespective of the presence or absence of both time and frequency offsets, the 3D orientation of the receiver can be estimated through the multiple TOA measurements received across the receive antennas during two transmission time slots from a single LEO satellite.