A Theoretical Analysis of Performance Improvement in Multi-Frequency and Multi-System Integrated Positioning Based on the Cramér-Rao Lower Bound

With the maturity of the BeiDou Navigation Satellite System (BDS) and the deployment of the BeiDou-3 system (BDS-3), more abundant frequency resources are now available for positioning. In recent years, the construction of Low Earth Orbit (LEO) satellite internet has also become a national priority. Combined with the future requirements for a more ubiquitous Positioning, Navigation, and Timing (PNT) network, the integration of these two systems can provide rich multi-frequency signal data and diverse observation measurements, thereby enhancing the accuracy and reliability of positioning services. However, most current studies on multi-frequency and multi-system observation combinations in positioning focus on simulation results at the phenomenological level, without deeply analyzing the underlying mathematical principles. This paper, based on the definition of the Cramér-Rao lower bound (CRLB), provides a detailed derivation and analysis of how multi-frequency and multi-system observation combinations can improve positioning accuracy. This analysis offers theoretical support for the future construction and application of a more ubiquitous PNT network.