Frequency and Magnitude Response Design Approximation Problems for Nonlinear-phase Nonrecursive Digital Filters

In this article special (possibly constrained) problems of linear and nonlinear complex approximation are studied with respect to the existence and uniqueness of solutions and the convergence of the approximation errors, where the errors are measured by an arbitrary L p and l p norm respectively. The problems arise in connection with the frequency and magnitude response approximation at the design of nonrecursive digital filters in the frequency domain. Two main results of the article concern the completeness of the functions exp ( m ik y ), k = 0,1,2,… , with respect to a certain space of continuous functions. These results imply that, under usual assumptions and with increasing number of approximating functions exp ( m ik y ), the errors in the frequency and magnitude response approximation problems converge to zero when the design region is not the total interval [0, ~ ] (in case of real coefficients) and not [ m ~ , ~ ] (in case of complex coefficients) which is given for the majority of filter design problems, but that the frequency response errors may not converge to zero when the design region equals [0, ~ ] or [ m ~ , ~ ] respectively.

Keywords:

• Filter Design
• Digital Filters
• Complex Approximation
• Complete Systems