Multistage IIR filter design using convex stability domains defined by positive realness

In this paper, we consider infinite impulse response (IIR) filter design where both magnitude and phase are optimized using a weighted and sampled least-squares criterion. We propose a new convex stability domain defined by positive realness for ensuring the stability of the filter and adapt the Steiglitz-McBride (SM), Gauss-Newton (GN), and classical descent methods to the new stability domain. We show how to describe the stability domain such that the description is suited to semidefinite programming and is implementable exactly; in addition, we prove that this domain contains the domain given by Rouche´´s theorem. Finally, we give experimental evidence that the best designs are usually obtained with a multistage algorithm, where the three above methods are used in succession, each one being initialized with the result of the previous and where the positive realness stability domain is used instead of that defined by Rouche´´s theorem.