High-Performance Beampattern Synthesis via Linear Fractional Semidefinite Relaxation and Quasi-Convex Optimization

This paper presents a new method to synthesize high-performance beampatterns, with the aid of linear fractional semidefinite relaxation (LFSDR) technique and a quasi-convex optimization approach. We consider two beampattern synthesis problems. The first one is how to determine the weight vector to maximize the array gain (or equivalently, to minimize the gain loss in mainlobe), under the condition that the amplitude response satisfies specific requirements. The second one is how to minimize the notch level at a given region, on the premise of a permissible gain loss in mainlobe. To these ends, two nonconvex optimization problems are first formulated and then relaxed to their quasi-convex forms, by using the LFSDR technique. To further solve the resultant problems, the bisection method, which is commonly used in quasi-convex optimization, is adopted. Suboptimal solutions to the original nonconvex problems are finally obtained through eigenvalue decomposition or randomization manipulations. The proposed method performs well in both the cases described earlier. Moreover, our method is not limited to the array configurations and/or noise environments. Representative simulations are presented to demonstrate the effectiveness of the proposed method in high-performance beampattern synthesis.