Design of nearly linear-phase recursive digital filters by constrained optimization

• Main Author: Guindon, David Leo
• Other Authors: Antoniou, Andreas
• Language: English; en
• Published: 2007
• Subjects: optimization; digital filter; constrained optimization; recursive digital filter; linear-phase; IIR; quadratic programming; filter stability; gradient; hessian; equalizer; UVic Subject Index::Sciences and Engineering::Engineering::Electrical engineering
• Online Access: http://hdl.handle.net/1828/296

The design of nearly linear-phase recursive digital filters using constrained optimization is investigated. The design technique proposed is expected to be useful in applications where both magnitude and phase response specifications need to be satisfied. The overall constrained optimization method is formulated as a quadratic programming problem based on Newton’s method. The objective function, its gradient vector and Hessian matrix as well as a set of linear constraints are derived. In this analysis, the independent variables are assumed to be the transfer function coefficients. The filter stability issue and convergence efficiency, as well as a ‘real axis attraction’ problem are solved by integrating the corresponding bounds into the linear constraints of the optimization method. Also, two initialization techniques for providing efficient starting points for the optimization are investigated and the relation between the zero and pole positions and the group delay are examined. Based on these ideas, a new objective function is formulated in terms of the zeros and poles of the transfer function expressed in polar form and integrated into the optimization process. The coefficient-based and polar-based objective functions are tested and compared and it is shown that designs using the polar-based objective function produce improved results. Finally, several other modern methods for the design of nearly linear-phase recursive filters are compared with the proposed method. These include an elliptic design combined with an optimal equalization technique that uses a prescribed group delay, an optimal design method with robust stability using conic-quadratic-programming updates, and an unconstrained optimization technique that uses parameterization to guarantee filter stability. It was found that the proposed method generates similar or improved results in all comparative examples suggesting that the new method is an attractive alternative for linear-phase recursive filters of orders up to about 30.