I have more general question about the extended Kalman filter usage. What is not clear to me is why the EKF uses non-linear functions $f$ and $h$ for state prediction and estimate, while in other places the Jacobian of these functions is used.
Why the following is never used?
First calculate the linearized state and measurement models at previous estimate point using Jacobian. Use the linearized state transition and measurements matrix everywhere instead of non-linear in this specific iteration.