Extended Kalman Filter (EKF) Algorithm

The Extended Kalman Filter (EKF) is a state estimation method suitable for nonlinear systems, with significant applications in carrier phase and frequency estimation for high-dynamic direct spread spectrum signals. Unlike traditional Kalman filters that only apply to linear systems, the EKF addresses nonlinearity through local linearization, maintaining high estimation accuracy even in complex signal environments. In carrier phase estimation, the EKF performs joint estimation of phase and frequency using system state and observation models. Its core concept involves recursive prediction and correction of state variables—including phase offset, frequency offset, and their rates of change. This process effectively overcomes challenges posed by rapidly changing signal parameters in high-dynamic environments. Implementation typically requires defining state transition and measurement Jacobian matrices to linearize nonlinear functions at each time step. Furthermore, EKF outputs can be utilized in carrier-aiding techniques, where precise carrier phase variations correct pseudocode delay loops. Since Doppler shift directly affects pseudocode delay accuracy, carrier-aiding correction significantly mitigates this impact, enhancing the dynamic tracking performance of delay lock loops (DLL). Key advantages of this approach include: Carrier phase measurement accuracy is substantially higher than pseudocode phase accuracy, enabling improved overall system stability through aiding correction. EKF's dynamic parameter estimation capability adapts to high-acceleration, high-dynamic scenarios, preventing traditional phase-locked loops (PLL) or delay lock loops (DLL) from losing lock under rapid changes. Code implementation often involves covariance propagation and Kalman gain computation to update state estimates recursively. By integrating EKF with carrier-aiding technology, systems achieve more accurate delay estimation. This method notably enhances tracking loop robustness and precision in high-dynamic applications such as satellite navigation and missile guidance, where algorithms must handle nonlinear dynamics through iterative linearization and noise covariance management.