Best approximation of the identity mapping: The case of variable finite memory

This paper concerns the best causal operator approximation of the identity mapping subject to a specified variable finite memory constraint. The causality and memory constraints require that the approximating operator takes the form of a lower stepped matrix A. To find the best such matrix, we propose a new technique based on a block-partition into an equivalent collection of smaller blocks, {L0,K1,L1,…,Kℓ,Lℓ} where each Lr is a lower triangular block and each Kr is a rectangular block and where ℓ is known. The sizes of the individual blocks are defined by the memory constraints. We show that the best approximation problem for the lower stepped matrix A can be replaced by an equivalent collection of ℓ independent best approximation problems in terms of the matrices [L0],[K1,L1],…,[Kℓ,Lℓ]. The solution to each individual problem is found and a representation of the overall solution and associated error is given.