Nonlinear Phase Fir Filter Design According to the L 2 Norm with Constraints for the Complex Error List of Tables

We examine the problem of approximating a complex frequency response by a real-valued FIR lter according to the L 2 norm subject to additional inequality constraints for the complex error function. Starting with the Kuhn-Tucker optimality conditions which specialize to a system of nonlinear equations we deduce an iterative algorithm. These equations are solved by Newton's method in every iteration step. The algorithm allows arbitrary tradeoos between an L 2 and an L 1 design. The L 2 and the L 1 solution result as special cases. Wir untersuchen das Problem der Approximation eines komplexen Frequenz-ganges mittels eines reellwertigen nichtrekursiven Filters nach der L 2-Norm mit zuss atzlichen Ungleichungsbedingungen f ur die komplexe Fehlerfunktion. Ausge-hend von den Kuhn-Tucker Optimalitt atsbedingungen, die auf ein nichtlineares Glei-chungssystem f uhren, leiten wir einen iterativen Algorithmus her. Diese Gleichungen werden in jedem Iterationsschritt mittels des Newton-Verfahrens gell ost. Der Algo-rithmus erlaubt beliebige Kompromisse zwischen einem L 2-und einem L 1-Entwurf. Die L 2-und die L 1-LL osung ergeben sich als Spezialff alle. 1 Filter design results of example 1 for diierent values and weighting W E ((). 15 2 Filter design results of example 2 for diierent values and weighting W E ((). 16 List of Figures 1 Filter design results of Ex. 1-1 (|) and Ex. 1-2 ({ {). (a) Magnitude responses; (b) error functions. 2 Filter design results of Ex. 1-3 (|) and Ex. 1-4 ({ {). (a) Magnitude responses; (b) error functions. 3 Comparison of an NLPFIR lter (Ex. 1-3 |) and an LPFIR lter (Ex. 1-7 { {) according to Adams' algorithm 1]. (a) Magnitude responses; (b) error 4 Filter design results of Ex. 2-1 ({ {) and Ex. 2-2 (|). (a) Magnitude responses; (b) group delay functions; (c) weighted error functions; (d) non weighted error functions and error tolerance (()