Complex Chebyshev Optimization Using Conventional Linear Programming - A versatile and comprehensive solution by

This paper presents a new practical approach to semi-infinite complex Chebyshev approximation. By using a new technique, the general complex Chebyshev approximation problem can be solved with arbitrary base functions taking advantage of the numerical stability and efficiency of conventional linear programming software packages. Furthermore, the optimization procedure is simple to describe theoretically and straightforward to implement in computer coding. The new design technique is therefore highly accessible. The complex approximation algorithm is general and can be applied to a variety of applications such as conventional FIR filters, narrow-band as well as broad-band beamformers with any geometry, the digital Laguerre networks, and digital FIR equalizers. The new algorithm is formally introduced as the Dual Nested Complex Approximation (DNCA) linear programming algorithm. The design example in limelight is array pattern synthesis of a mobile basestation antenna array. The corresponding design formulation is general and facilitates treatment of the solution of problems with arbitrary array geometry and side-lobe weighting. The complex approximation problem is formulated as a semi-infinite linear program and solved by using a front-end applied on top of a software package for conventional finite-dimensional linear programming. The essence of the new technique, justified by the Caratheodory dimensionality theorem, is to exploit the finiteness of the related Lagrange multipliers by adapting conventional finite-dimensional linear programming to the semi-infinite linear programming problem. The proposed optimization technique is applied to several numerical examples dealing with the design of a narrow-band base-station antenna array for mobile communication. The flexibility and numerical efficiency of the proposed design technique are illustrated with these examples where even hundreds of antenna elements are optimized without numerical difficulties.