Complex Chebyshev approximation for FIR filters using linear semi‐infinite programming method

Abstract

SUMMARYIn the optimum design of an FIR filter by the complex Chebyshev method, it is difficult to limit the increase of computational effort and to guarantee convergence to an optimum solution due to the nonlinearity of the problem and the instability of the frequency points for the optima. In this paper, the complex Chebyshev approximation problem is formulated as a linear semi-infinite programming problem by means of the real rotation theorem. An optimum design method is proposed using the three-phase method, a linear semi-infinite programming method. In the present method, the design problem is first reduced to a linear programming problem with constraints and a provisional solution is derived by the simplex method. After the constraints corresponding to the degenerated optimum frequencies contained in the provisional solution are combined, the solutions are adjusted to satisfy the optimum condition so as to obtain an optimum solution. Since low accuracy is sufficient for the provisional solution, memory requirements and computational effort can be reduced. By means of a design example, it is shown that the proposed method is superior in terms of solution accuracy and computational effort to the technique based on the simplex method.