Design of recursive digital filters with prescribed stability margin: a parameterization approach

A major problem in the design of recursive digital filters is stability. Although an unstable recursive filter obtained from a design algorithm can be stabilized by reciprocal substitution of the unstable factors of the filter without changing its amplitude response, this stabilization technique cannot be applied if phase response is a part of the design specifications. In this paper, we propose a new method for the design of recursive digital filters with a prescribed stability margin by parameterizing all stable transfer functions and carrying out unconstrained optimization over this class of transfer functions. Three parameterization techniques are described, and closed-form formulas for the gradient functions and Hessian matrices of several typical objective functions are derived. The design technique is expected to be useful in the cases where both amplitude and phase specifications are required in the design.