Optimal FIR approximate inverse of linear periodic filters

We propose a method for constructing the FIR approximate inverse for discrete-time causal FIR periodic filters in the presence of measurement noise. The objective function to be minimized is the sum of the error variance over one period. The optimization problem is formulated based on the matrix impulse response of the multi-input multi-output (MIMO) time-invariant representation of the periodic filter as one that minimizes the sum of equation errors of a set of overdetermined linear equations. It is shown that the problem is equivalent to a set of least squares problems and a simple closed-form solution is obtained. Numerical examples are used to illustrate the performance of the proposed FIR approximate inverse.