Proc. IEEE International Symposium on Circuits and Systems, vol. 4, pp. 77-80, May, 2000

# Joint Optimization of Multiple Behavioral and Implementation Properties of Digital IIR Filter Designs

Magesh Valliappan (1), Brian L. Evans (1), Mohamed Gzara (1), Miroslav D. Lutovac (2), and Dejan V. Tosic (2)

(1) Embedded Signal Processing Laboratory, Department of Electrical and Computer Engineering, Engineering Science Building, The University of Texas at Austin, Austin, TX 78712-1084 USA
magesh@ece.utexas.edu - bevans@ece.utexas.edu - gzara@ece.utexas.edu

(2) School of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, 11000 Belgrade, Yugoslavia
etosicde@ubbg.etf.bg.ac.yu - lutovac@iritel.bg.ac.yu

## Abstract

This paper presents an extensible framework for the simultaneous constrained optimization of multiple properties of digital IIR filters. The framework optimizes the pole-zero locations for behavioral properties of magnitude and phase response, and the implementation property of quality factors, subject to constraints on the same properties. We formulate the constrained nonlinear optimization problem as a sequential quadratic programming (SQP) problem. SQP solvers are robust when provided formulas for the gradients of the cost function and constraints. We program Mathematica to compute the gradient formulas and convert the formulas into Matlab programs to perform the optimization. The automated approach eliminates errors in manipulating the algebraic equations and transcribing equations into software. The key contributions are (1) an automated, extensible, multicriteria filter optimization framework, and (2) two novel filter designs. We have released the source code on the Internet.

The following questions and answers were given during the presentation of the paper (the session chair was Prof. S. C. Pei of the National Taiwan University):
1. Question (Miroslav Vlcek): Why |H|=1 when the filter order is even?

2. Question (Miroslav Vlcek): Do we assume odd order?
Answer: No. The approach works for even and odd orders. We give the formula for the even case.

3. Question (Miroslav Vlcek): What is the magnitude response type after optimization?
Answer: It really depends on the cost function, constraints, and initial guess.

4. Question (Miroslav Vlcek): What is the group delay?
Answer: The derivative of the phase response.

5. Question (Akinori Nishihara): Why do you use Q-factor [quality factor] as an implementation property?"
Answer: The Q-factor is a technology-independent measure of the sensitivity of a real pole or conjugate pole pair to perturbation (e.g. roundoff or truncation). The larger the Q-factor, the more likely the pole will oscillate.
Note: Nishihara published a 1998 paper in Electronics Letters, in which he defines a Q-factor for digital IIR filters.

6. Question (Andre Gunst): Do you have similar software for analog filters? If yes, is it user friendly?