Parallel and sequential block Kalman filtering and their implementations using systolic arrays

Two sets of block Kalman filtering equations that differ in the manner of generating the initial and updated estimates are derived. Parallel and sequential schemes for generating these estimates are adopted. It is shown that the parallel implementation inherently leads to a block Kalman estimator which provides filtered estimates at the vector (block) level and fixed-lag smoother estimates at the sample level. The sequential implementation scheme, on the other hand, generates the estimates of each sample recursively, leading naturally to a scalar (filter) estimator. These scalar estimates are arranged in a vector form, resulting in a block estimator which solely generates filtered estimates both at the vector and sample levels. Simulation results on a speech signal are presented which indicate the advantages of the sequential block Kalman filter. An algorithm for iterative calculation of Kalman gain and error covariance matrices is given which does not require any matrix inversion operation. The implementation of this algorithm using available systolic array processors is presented. A ring systolic array which can be used to implement the state update part of the block Kalman filter is suggested