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 2-D zero-phase FIR digital filters is presented for the general case where no quadrantal symmetry or antisymmetry is assumed. The matrices of the filter impulse response are derived by minimizing the integral of the weighted squared error between the desired and designed frequency responses. The resulting two matrix equations are found to de-couple under reasonable assumptions on the weight function. Closed-form expressions are derived for the impulse response matrices (the filter coefficients). Those expressions require matrix inversion and possibly numerical integration; and in the special case of no weighting no matrix inversion is required. This technique has the advantage of computational efficiency since no iterations-apart from those needed for numerical integration-are required