The Design And Application Of Allpass IIR Digital Filter

Allpass IIR digital filters have been used in many digital signal processing applications such asphase equalization and the design of filter banks, halfband filters and notch filters. So the designproblems of allpass filters attracted the attention of many researches.This thesis considers the minimax design problems of allpass IIR digital filters. Firstly, thisthesis builds the design model of the frequency response error of an allpass IIR digital filter; andthen by incorporating design algorithms of general IIR digital filters, this thesis obtains four kindsof new algorithms for the minimax frequency response error design of allpass IIR digital filters;Finally, this thesis presents iterative reweighted minimax method to design allpass filters withnearly uniform group delay error response, then applied to the design of allpass-filter-basedhalfband filters, which can be described in more details as follows:(1) Firstly, This thesis analyzes the relation between the phase response error and the frequencyresponse error of an allpass IIR digital filter, and then builds the design model of the frequencyresponse error of an allpass IIR digital filter. Since an allpass filter has a unit magnitude across thewhole frequency area [0, π], its design reduces to a pure phase approximation problem. Consideringthat the magnitude of the frequency response error of an allpass filter is a monotonic function to themagnitude of its phase response error, this thesis directly applies minimax frequency response errordesign methods to the minimax phase error design of allpass IIR filter.(2) And then by incorporating design algorithms for general IIR digital filters, this thesisobtains four kinds of new algorithms for the minimax frequency response error design of allpass IIRdigital filters. Since the problem formulation of the minimax frequency response error design of anallpass IIR digital filter is the same as that of the minimax frequency response error design of ageneral IIR digital filter, this thesis directly applies minimax frequency response error designmethods for general IIR filters to the minimax design of allpass IIR digital filters. By using thespecific form of the frequency response of the allpass IIR filter in existing design methods such asthe direct-SM, SCLS-SM, direct-GN, SCLS-GN methods for the general IIR filters, this thesisobtains four kinds of new algorithms for the minimax design of allpass IIR digital filters. Designexamples demonstrate the effectiveness of the proposed methods.(3) Finally, this thesis presents iterative reweighted minimax method to design allpass filters,then applied to the design of allpass-filter-based halfband filters. All of the four methods presentedabove obtain equiripple phase error, but the group delay error near the band edge is much largerthan the error elsewhere. In order to reduce the maximum group delay error, the envelope of the group delay error obtained in the previous iteration is used to update the weight function of thefrequency response error in this iteration. Using the reweight technique, the resulting allpass filtershave near equiripple group delay error functions, and then applied to the design of allpass-filter-based halfband filters with equiripple group delay error.