Design Problems for Nonrecursive Digital Filters Ii

While the rst part of the paper dealt with the problem of approximatinga prescribed frequency response by nonrecursive linear-phase and nonlinear-phase lters, in this part the three other central design problems for such lters in the frequency domain are studied in regard to the existence of solutions and the convergence of the approximation errors. These are the problems of approximatinga magnituderesponse and the problems of simultaneously approximatinga magnitude and phase response and a magnitude and group delay response respectively. The last two problems of simultaneous approximation rst are formulated in a proper mathematical way. The approximation errors are measured again by an arbitrary L p-resp. l p-norm, 1 p 1; and constraints on the lter coeecients are permitted. 5. Approximation of a magnitude response. The main deenitions needed in the following are given in x2 of the rst part of the paper. Topic of this section is the study of the (possibly constrained) approximation of a magnitude response leading to the problem p;N := inf x2KN kW (jDj ? jH N (x;)j)k ;p (5.1) for 1 p 1 which especially is of interest in regard to magnitude only designs. (For references on this problem for the cases p = 2 resp. p = 1 see e.g. 3] and 9]) We begin by modifying Assumption 4.7.