The direct solution of nonconvex nonlinear FIR filter design problems by a SIP method (Q5958048)

The purpose of this paper is to give a general and precise formulation of the main FIR (Finite Impulse Response) filter design problems. Existence conditions of the solutions of these problems are provided, and results are presented on the convergence of the approximation errors and on the connection between frequency response and the magnitude/phase response problem. A new method for nonlinear SIP (Semi-Infinite Programming) is described (S. Görner, 1997), employing a dynamic discretization scheme in a first phase and solving a reduced continuous problem in a second phase to accelerate the convergence. The Görner method is shown to be convergent and to produce very good results when applied to a large variety of nonlinear design problems. This new method, together with the robust method of A. Potchinkov and R. Reemtsen (1995) for convex SIP, and with a method for unconstrained least-squares problems provides a set of algorithms that can efficiently solve most FIR filter design problems in the frequency domain. The investigated methods constitute also a promising tool for the solution of IIR (Infinite Impulse Response) filter design problems, which all are strongly nonlinear. The authors' approach supports the generous idea that FIR and IIR filter design problems do not have so many peculiarities so as to justify the development of particular algorithms, but they are usually semi-infinite or, when discretized, finite optimization problems that should and can be solved by well-founded customary methods of general optimization. In particular, the new methods of Görner (1997) are capable of solving rather sophisticated nonlinear filter design SIP problems directly and accurately.