Least squares design of IIR filters with arbitrary magnitude and phase responses and specified stability margin

This paper presents an algorithm for the frequency domain design of infinite impulse response (IIR) digital filters with prescribed magnitude and phase responses according to a least squared error criterion. We use the implications of Rouché's Theorem to modify standard algorithms for nonlinear least squares optimization in such a way that a constraint on the maximum pole radius can be incorporated. Hence, the filters obtained by the proposed method can be guaranteed to be stable and, moreover, exhibit an arbitrary stability margin. This helps to avoid undesired behavior of the magnitude response in the transition bands, decreases the sensitivity to coefficient quantization, and helps to reduce the amplitude of small-scale limit cycles. A design example shows the superiority of the proposed algorithm over several other design methods.