An analytical approach for obtaining a closed-form solution to the least-square design problem of 2-D zero-phase FIR filters

A closed-form least-square solution to the design problem of general two-dimensional (2-D) real zero-phase FIR filters is obtained. An in-depth study of the functions and matrices arising from the problem definition reveals some very useful structural properties. It is shown that these properties lead to an optimal analytical solution for the filter coefficients, making it unnecessary to use design procedures involving optimization techniques or matrix inversion operations. The derived closed-form expressions for filter coefficients allow their evaluation directly from the filter´s frequency response specifications, resulting in a greatly reduced computational complexity. It is confirmed through design examples that the proposed technique enjoys a very short design time and it rises very slowly as the filter order is increased