Optimal approximate inverse of linear periodic filters

We propose a method for constructing optimal causal approximate inverse for discrete-time single-input single-output (SISO) causal periodic filters in the presence of measurement noise. The analysis is based on block signals and multi-input multi-output (MIMO) time-invariant models for periodic filters. The objective function to be minimized is the asymptotic block mean square error. The optimization problem is formulated in terms of transfer matrices as an optimal model-matching problem with nonsquare model and plant. Based on an inner-outer factorization on the transpose of the plant rational matrix, it is shown that the problem can be further reduced to one with a lower dimensional square model and plant, which is then solved in the time-domain, and a closed-form solution is obtained. A lower bound on the objective function is given. It is shown that the lower bound can be asymptotically achieved as the order of the optimal transfer matrix increases. The proposed method is extended to MIMO periodic systems. Numerical examples are used to illustrate the performance of the proposed approximate inverse.