Minimax Design Of IIR Digital Integrators And Differentiators With Nearly Linear Phases

Digital integrators and differentiators are used in a wide range of applications such as digital image processing,radar engineering,biomedical and control systems.Digital integrators and differentiators belong to digital filters.Both digital filters can be classified into two types based on its impulse response,Finite Impulse Response(FIR)digital filter and Infinite Impulse Response(IIR)digital filter.With FIR filters,exactly linear phase responses can be easily designed for the digital filter.However,in many practical applications,exactly linear phase responses are not required.Instead,IIR filters with nearly linear phase response are more attractive than FIR filter because they can satisfy the prescribed filter specifications with much lower filter order and usually have much smaller group delays.In this thesis,we consider the design IIR digital integrators and differentiators with nearly phase using minimax optimization algorithm.Digital integrators and differentiators have been designed by various methods in the literature.These methods can be broadly divided into three types: analytical method,heuristic optimization algorithm and mathematical-programming.The first two types of methods are difficult for obtaining the nearly linear phase method in the design of IIR digital filters.Thus,in this paper,mathematical-programming based are applied to design the nearly linear-phase IIR digital integrators and differentiators.In the design,there are two mainly challenges.First,causal and stable linear-phase IIR filters cannot have exactly linear phase.Second,the design problems of IIR digital integrators and differentiators are usually non-convex.In order to solve these problems we focus on the following two aspects:1?The design of nearly linear-phase IIR digital integrators and differentiators based on an elliptic and V-shaped error constraint model.At any frequency point,the constraint domain of the magnitude and phase error is non-convex.We first transform this non-convex domain into a convex one by replacing the magnitude error constraint in the complex frequency response domain with an elliptic error constraint.Then the error constraint function are further transformed into convex functions in the filter coefficient space.Using a modified Gauss-Newton method with a variable step length is firstly designed by simultaneously minimizing magnitude and phase error.An integrator and differentiator satisfying the magnitude error and stability requirements with the resultant integrator and differentiator as an initial solution,the presented method then minimizes the maximum phase error subject to the magnitude error and stability constraints iteratively.By incorporating the modified GN algorithms,the proposed weighted minimax phase error methods have obtained better differentiators and integrators than the competing methods.2.The design of nearly linear-phase IIR digital integrators and differentiators based on a double-ellipse constraint model.In this design method,the phase error constraint is replaced by another ellipse constraint in order to better control the phase error independently.The design examples show that the filters obtained by the double elliptic model based method sometimes have smaller phase error than those by the method based on the elliptic and V-shaped constraint model.