Optimum Design of Discrete-Time Differentiators via Semi-Infinite Programming Approach

In this paper, a general optimum full-band, high-order discrete-time differentiator design problem is formulated as a peak-constrained least squares optimization problem. That is, the objective of the optimization problem is to minimize the total weighted square error of the magnitude response subject to the peak constraint of the weighted error function. This problem formulation provides great flexibility for the tradeoff between the ripple energy and the ripple magnitude of the discrete-time differentiator. The optimization problem is actually a semi-infinite programming problem. Our recently developed dual parameterization algorithm is applied to solve the problem. The main advantages of employing the dual parameterization algorithm to solve the problem are as follows: (1) the guarantee of the convergence of the algorithm and (2) the obtained solution being the global optimal solution that satisfies the corresponding continuous constraints. Moreover, the computational cost of the algorithm is lower than that of algorithms that are implementing the semidefinite programming approach.