Optimal linear filtering with piecewise-constant memory

The paper concerns the optimal linear filtering of stochastic signals associated with the notion of piecewise constant memory. The filter should satisfy a specialized criterion formulated in terms of a so called lower stepped matrix A. To satisfy the special structure of the filter, we propose a new technique based on a block-partition of the lower stepped part of matrix A into lower triangular and rectangular blocks, L_ij and R_ij with i = 1, ..., l; j = 1, ..., s_i where l and si are given. We show that the original error minimization problem in terms of the matrix A is reduced to l individual error minimization problems in terms of blocks L_ij and R_ij. The solution to each problem is provided and a representation of the associated error is given.