Chebyshev approximation for two-dimensional nonrecursive digital filters

The Remez exchange algorithm is extended for the design of two-dimensional nonrecursive digital filters approximating circularly symmetrical low-pass specifications according to a weighted Chebyshev error norm. Since the approximating function does not satisfy the Haar condition, the optimal solution is not necessarily unique and a straightforward extension of the one-dimensional exchange method may fail to converge. It is shown how the algorithm has to be complemented with a perturbation technique in order to force convergence under all circumstances. In the case of nonuniqueness the solution provided by the algorithm is a vertex of the polyhedron containing all optimal solutions, and a method is given which allows one to compute an adjacent vertex located on the same edge, thereby allowing the successive determination of all the vertices defining the polyhedron. For this case also, a procedure is described which selects, among all optimal solutions, the best one according to some additional criterion. Finally, the efficiency and accuracy aspects of the algorithm are considered and practical conclusions are drawn as an aid for the designer.