Abstract
Conventional Kalman filters deal with the estimation of states x(i), and yield estimates x̂(i | j) together with a corresponding estimate variance P(i | j). The information filter deals instead with the information state vector ŷ(i | j) and information matrix Y(i | j) defined as ŷ(i | j) = P−1(i | j)x̂(i | j), Y(i | j) = P−1(i | j). (1) A set of recursion equations for the information state and information matrix can be derived directly from the equations for the Kalman filter. The resulting information filter is mathematically identical to the conventional Kalman filter. We have 1−W(k)H(k) = P(k | k)P−1(k | k − 1), (2) and W(k) = P(k | k)HT (k)R−1(k). (3) Substituting Equations 2 and 3 into the state update equations for the Kalman Filter and premultiplying through by P−1(k | k) gives the update equation for the information-state vector as P−1(k | k)x̂(k | k) = P−1(k | k − 1)x̂(k | k − 1) +HT (k)R−1(k)z(k). (4) A similar expression can be found for the information matrix. Substituting Equations 2 and 3 into the covariance update equation for the Kalman filter and rearranging gives P−1(k | k) = P−1(k | k − 1) +HT (k)R−1(k)H(k). (5) Defining i(k) = HT (k)R−1(k)z(k) (6) as the information-state contribution from an observation z(k), and
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