Weighted least squares design of two-dimensional zero-phase FIR filters in the continuous frequency domain

Using matrix-vector formalism, a weighted least squares version of the design problem of zero-phase 2-D FIR filters is presented in the continuous frequency domain with no assumptions of a quadrantally symmetric or antisymmetric frequency response. By minimizing the weighted mean squared error between the desired and designed frequency responses, the resulting two matrix equations are found to decouple under the assumptions of the separability of the 2-D weighting function and the evenness of the 1-D weighting functions. Consequently closed-form expressions for the matrices of the filter coefficients are derived.