A closed-form solution to the least-square design problem of 2-D linear-phase FIR filters

In this paper, the least-square design problem of a general two-dimensional (2-D) linear-phase FIR filter with an arbitrary magnitude response is studied. By minimizing the frequency-domain error function and exploiting some of the properties of the functions and matrices associated with the design problem, a novel closed-form solution is developed. The solution is expressed in terms of the desired magnitude specifications and is eventually presented as an explicit expression for the impulse response of the filter to be designed, making a very fast evaluation of the filter's coefficients possible. It is also shown that when this solution is used to design filters that have centrosymmetric and quadrantally symmetric magnitude responses, some further computational savings can be achieved. Several design examples illustrating the effectiveness of the proposed solution are considered.