Research On The Design Of Stable IIR Digital Filter Algorithm Based On The Double Elliptical Model

There are three main challenges in the minimax design of IIR digital filter: non-linear phase response, non-convex design model, stability constraints. This paperfocuses on the non-convexity and non-linear phase of filter design, and givescorresponding solutions. The paper first describes the significance and current statusof IIR digital filter, and introduces the basic concepts of digital filter design. In orderto solve design model’s non-convexity problem caused by stability-triangle condition,the paper adopts the existing SMSOF method and applies it in the minimax design ofIIR filter. In order to solve the filter’s non-linear phase problem, the paper adoptsdouble-ellipse model to control the magnitude error. Then, based on the study ofexisting design methods of IIR filter, we combine the double-ellipse model andSMSOF method to design better IIR digital filter. Finally, in order to make the designof filter more efficient, the article improves the above-mentioned double-ellipsemodel and get better filter in the examples. The main contents are as follows:1. The introduction of the basic concepts of digital filter, SMSOF methods andthe double-ellipse model. Stability-triangle condition is used to ensure the stability ofthe IIR digital filter because of its good linearity, sufficiency and necessity. In order tosolve the non-convexity problem caused by the use of stability-triangle condition inthe minimax design of IIR digital filter, this paper introduces the SMSOF method indetail to convert IIR filter design problems into a sequence of sub-problems. Eachsub-problem is a design of a filter with a second-order denominator. Then the paperintroduces the double-ellipse model which can control the phase error and magnitudeerror separately. The double-ellipse model even can sacrifice magnitude error forsmaller phase error if necessary. So, it can get near linear-phase IIR filter.2. Combine the SMSOF method and the double-ellipse model to form theSMSOF algorithm for the minimax design of IIR filter. After the study of the existingdesign methods of IIR digital filter, we found that the double-ellipse model canseparately control the magnitude error and phase error, but the paper which proposesthe double-ellipse model use the generalized positive realness condition which issufficient to ensure the filter’s stability, this may exclude the good stable filters fromthe stability domains. In order to solve this problem, we use the stability-trianglecondition instead of the generalized positive realness condition in this paper to design IIR filter. Since it can expand the feasible region of filter design, we can get better IIRfilter.3. Take an improvement of the double-ellipse model to obviously improve thespeed of IIR filter design and get better filter. Since the up bound of the phaseconstrains is a variable, it will make the procedure of the design more complex. Inorder to solve this problem, this paper uses a new ellipse to constrain the phase errorinstead of the ellipse in doube-ellipse model. Then, we take minimax designs of IIRfilter based on the improved double-ellipse model. Finally, we use the examples todemonstrate that the improved double-ellipse model can improve the efficiency of theIIR filter design.