This research is developing a new and significantly better method for the design of a wide variety of digital filters. The new method is based on a successive approximation algorithm called Iteratively Reweighted Least Squares (IRLS). One form of IRLS, Lawson's algorithm, has been used before but not extensively because of slow and inconsistent convergence. Initial results with the new method using a second order IRLS with a form of path- following or homotopy modification have produced a robust, quadratically converging, efficient, versatile filter design technique. The method finds general L^p approximations with 1<=p. It also allows different criteria in different bands for the design of mixed criteria optimal filters. The new method solves, not only the standard filter design problems, but many that the Parks- McClellan Remez algorithm or windowed least wquares will not. Initial results on designing with constrained L^p criteria, complex approximation, and two dimensions show great promise. Robust convergence is crucial for the practical success of any iterative algorithm. We feel that we have a potential breakthrough on that issue. This research is doing a theoretical analysis of the convergence for the more general optimization criteria, developing a practical adaptive filter design program, and presenting it in a form accessible to the practicing engineer. The algorithm will be applied to three filter design problems: a) the constrained least-squares and, more generally, the constrained L^p approximation problem. This is often what people really want when they use a straight Chebyshev or window design. b) The important problems of complex approximation which is important in equalization adaptive filter design. c) The multidimensional filter design and image processing problems. This new approach shows excellent promise and a possible great improvement over present techniques.
