Sequential Constrained Minimization Approaches To Minimax Design Of Stable IIR Digital Filters

This thesis considers the minimax design of infinite impulse response (IIR) digital filters. Inorder to deal with the nonconvexity and enlarge the stability domain of the design problem as wellas to reduce the filter’s passband group delay and transition-band magnitude overshoot, severalsequential constrained minimization based design algorithms have been developed in this thesis.Better filters, especially with smaller passband group delays and smaller transition-band magnitudeovershoots than those of the filters designed by existing algorithms, have been obtained by theproposed algorithms.The sequential constrained least squares (SCLS) method converts a minimax design probleminto a series of constrained least squares (CLS) design problems and has higher opportunity ofobtaining better filters than direct minimax methods. The second-order factor based sequentialminimization (SMSOF) method converts the minimax design of an IIR digital filter into a series ofminimax design subproblems of IIR digital filters with second-order denominators, such that thenecessary, sufficient and linear stability triangles for second-order factors can be used as thestability constraint of the design problem to obtain better filters than existing methods.However, the above methods do not impose any constraints on the frequency responses of thefilters, and thus often obtain filters with large group delays near the passband edges and largemagnitude overshoot in the transition bands. In addition, how to effectively use the characteristicsof the above two methods is also an important issue that needs to be considered. After a brief surveyof IIR digital filter designs and presenting the existing SCLS and SMSOF methods for minimaxdesign of IIR digital filters, this thesis presents several improved design algorithms for IIR digitalfilters, which are summarized as follows:1. A combinative method based on the SCLS and SMSOF methods. This method first convertsa minimax design problem into a series of minimax problems with second-order denominators, Theobtained minimax problems are then further converted into a sequence of constrained least squaressubproblems for the designs of IIR digital filters with second-order denominators, the frequencyresponse error upper bounds of which are produced by a shrinking-and-bisection technique. Designexamples show that the combinative method has obtained smaller frequency response error andpassband group delay error.2. Two sequential constrained minimax (SCMM) methods. Both methods consider the minimi-zation of the maximum passband and stopband frequency response error as well as the minimizationof the maximum transition-band frequency response error, but use different techniques to minimize different errors. While the first method minimizes the maximum transition-band frequency responseerror using the shrinking-and-bisection technique and the maximum passband and stopbandfrequency response error with a direct minimax algorithm, the second method minimizes the formererror with a direct minimax algorithm and the later error by a shrinking-and-bisection technique.Design examples demonstrate that, while the maximum passband and stopband frequency responseerrors are minimized, both methods can obviously reduce the transition-band magnitude overshootsand the passband group delay errors.3. An SMSOF based SCMM method. This method combines the SCMM method with theSMSOF method and converts the minimax design of an IIR digital filter into a sequence ofconstrained minimax sub-problems of IIR digital filters with second-order denominators whosestability constraints are the linear and necessary and sufficient stability triangles. Design examplesshow that the filters’transition-band magnitude overshoots and the passband group delay errors canbe further reduced by the SMSOF based SCMM method.