Efficient Creation of Population of Stable Biquad Sections with Predefined Stability Margin for Evolutionary Digital Filter Design Methods

Abstract

Evolutionary methods are very often used to digital filters design (especially for digital filters with non-standard amplitude characteristics). In practical digital filter implementation, we can use a cascade of biquad sections. Each biquad section will be stable if all poles of transfer function are located into unitary circle in the z-plane. To generate $k-th$ stable biquad section, we can use an existing equations which are generated for the case when the coefficient $a_{k,0}=1$. The problem is complicated if we want to have the vary values of all filter coefficients from the continuous range $[-1; 1]$ or from the discrete range $[-1; 1-2^{-M}]$ (if filter will be implemented into Q.M fixed-point format). In this paper we have presented an efficient method for generation of stable biquad sections. The proposed method can be used in any evolutionary digital filter design method for increase its efficiency. Using proposed approach we can fast generate the population of stable biquad sections with prescribed stability margin. Due to presented approach, we can also very fast evaluate the stability of the given biquad section (the methods for polynomial roots generation are not needed).