Design Approximation Problems for Linear-Phase Nonrecursive Digital Filters

The topic of this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linear-phase nonrecursive digital filters and are distinguished by the type of used trigonometric approximation functions. In the case of unconstrained minimax designs these problems are normally solved by the Parks–McClellan algorithm, which is an application of the second algorithm of Remez to these problems and which is one of the most popular tools in filter design. In this paper the four types of approximation problems are investigated for all L^p and l^p norms, respectively. It is especially proved that the assumptions for the Remez algorithm are satisfied in all four cases, which has been claimed, but is not obvious for three of them. Furthermore, results on the existence and uniqueness of solutions and on the convergence and the rate of convergence of the approximation errors are derived.