The minimax design of digital all-pass filters with prescribed pole radius constraint using semidefinite programming (SDP)

This paper proposes a new method for designing digital all-pass filters with a minimax design criterion using semidefinite programming (SDP). The frequency specification is first formulated as a set of linear matrix inequalities (LMI), which is a bilinear function of the filter coefficients and the ripple to be minimized. Unlike other all-pass filter design methods, additional linear constraints can be readily incorporated. The overall design problem turns out to be a quasi-convex constrained optimization problem (solved using the SDP) and it can be solved through a series of convex optimization sub-problems and the bisection search algorithm. The convergence of the algorithm is guaranteed. Nonlinear constraints such as the pole radius constraint of the filters can also be formulated as LMI using the Rouche theorem. It was found that the pole radius constraint allows an additional tradeoff between the approximation error and the stability margin in finite wordlength implementation. The effectiveness of the proposed method is demonstrated by several design examples.