Least-squares design of two-dimensional recursive filters with specified magnitude and linear phase (Q1077367)

A new method for the approximation of two-dimensional (2-D) digital recursive filters is presented. The approximating filter is a rational two-variable function whose coefficients \(a_ i\) and \(b_ i\) are to be found. The ideal filter is prescribed via \(\{g_ i(m,n)\}\), the set of ideal response values. To obtain a one-dimensional problem, the data are arranged according to the inverse lexicographic ordering. The canonical least-squares approximation leads to a set of nonlinear equations for the coefficients \(a_ i\) and \(b_ i\). Descent optimization methods based on iterative procedures can then be used. For the Gauss-Newton method, the Hessian of the error quadratic function must be calculated at each iteration. At the same time this method requires a good set of initial values. To this end a modified least-squares approximation previously derived by the author is used to obtain the initial value set. This last procedure is based on the theory of orthogonal polynomials on the unit disc. As examples of application, the design of the circular filter and that of the fan filter are presented. The results confirm the real advantages of the proposed design procedure.