A New General Front-End Technique for Complex Quadratic Programming - Applications to Array Pattern Synthesis by

This paper presents a new practical approach to complex quadratic programming which solves the broad class of complex approximation problems employing finitization of semi-infinite formulations. The approximation problem may be general with arbitrarily complex basis functions. By using a new technique, the associated semi-infinite quadratic programming problem can be solved taking advantage of the numerical stability and efficiency of conventional quadratic 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 versatile and can be applied to a variety of applications such as narrow-band as well as broad-band beamformers with any geometry, conventional FIR filters, digital Laguerre networks, and digital FIR equalizers. The new algorithm is formally introduced as the quadratic Dual Nested Complex Approximation (DNCA) algorithm. The essence of the new technique, justified by the Caratheodory’s dimensionality theorem, is to exploit the finiteness of the related Lagrange multipliers by adapting conventional finite-dimensional quadratic programming to the semi-infinite quadratic programming re-formulation of complex approximation problems. The design criterion in our application is to minimize the side-lobe energy of an antenna array when subjected to a specified bound on the peak side-lobe level. Additional linear constraints are used to form the main-lobe. The design problem is formulated as a semi-infinite quadratic program and solved by using the new front-end applied on top of a software package for conventional finite-dimensional quadratic programming. The proposed optimization technique is applied to several numerical examples dealing with the design of a narrow-band base-station antenna 4 Mattias Dahl Sven Nordebo Ingvar Claesson 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.