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Design Approximation Problems for LinearPhase Nonrecursive Digital Filters
"... this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linearphase nonrecursive digital lters and are distinguished by the type of used trigonometric approximation functions. In case of unconstrained minim ..."
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this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linearphase nonrecursive digital lters and are distinguished by the type of used trigonometric approximation functions. In case of unconstrained minimax designs these problems are normally solved by the ParksMcClellan algorithm, which is a straightforward adaptation of the second algorithm of Remez to these problems and which is one of the most popular tools in lter design. In this paper the four types of approximation problems are investigated for all L
Frequency and Magnitude Response Design Approximation Problems for NonlinearPhase Nonrecursive Digital Filters
"... In this paper 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. T ..."
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In this paper 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 paper concern the completeness of the functions exp( ik!), 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( ik!), 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 coecients) or [ ; ] (in case of complex coffiecients) which almost always is given in lter design,...
FIR filter design problems of simultaneous approximation of magnitude and phase and magnitude and group delay
"... Two of the four central design problems for FIR filters in the frequency domain are the problems of simultaneous approximation of prescribed magnitude and phase responses and prescribed magnitude and group delay responses respectively. In the past, these problems have almost always been approached i ..."
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Two of the four central design problems for FIR filters in the frequency domain are the problems of simultaneous approximation of prescribed magnitude and phase responses and prescribed magnitude and group delay responses respectively. In the past, these problems have almost always been approached in indirect and approximative ways only. Especially (approximate) solutions of the simpler frequency approximation problem have served as substitutes for solutions of the magnitudephase problem. In this paper, at first a rigorous mathematical formulation of both problems is developed and then, for these problems, the existence of solutions and results on the convergence of the approximation errors are proved. (A method to solve both problems is simultaneously suggested in [8].) Also the improvement, obtained by use of a direct solution of the magnitudephase response problem instead of a solution of the frequency response problem, is quantified by computable bounds. In the study, the approxim...