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Time Varying Kalman Filter Processing to Predict the Future Errors of a GPS Receiver Kalman Filter A distinctive feature of a Kalman filter is that its mathematical formulation is described in terms of state-space concepts. Another novel feature of a Kalman filter is that its solution is computed recursively. In particular, each updated estimation of the state is computed from the previous estimation and the new input data, so only the previous estimation requires storage. The Kalman filter is thus ideally suited for implementation on a digital computer and the system errors model can be expressed in state-space. The Kalman filter procedure includes two main steps known as the prediction or time update, and measurement update. In the first step the algorithm predicts the states and also estimation error covariance for time k+1 using the data available up to k time. In the second step, a new measurement, Zk , is used to generate a posteriori state estimation. In this step the state estimation for time k+1 is based on data available at time k. Basic equations in a Kalman filter form as follows: Xk+1=f k X k+ Wk ; Process equation (1) Where XK is an N-dimensional state vector and fK is an N-by-N state transition matrix relating the states of the system at times k and k+1. The N-by-1 vector Wk represents process noise. The vector Wk is modeled as zero-mean, white noise processes whose correlation matrix is defined by:  The measurement equation is: Zk= Hk.Xk+Vk (3) Where Zk is an M-dimensional vector, denotes the observed data of system, and Hk is a known M-by-N measurement matrix. The M-by-1 vector Vk is called measurement noise. It is modeled as zero-mean, white noise processes whose correlation matrix is: 
The noise vectors Wk and Vk are statistically independent, so we may write: E[WkViT]=0 ; For all k and i (5) |