Synthesis of Sparse Arrays With Frequency-Invariant-Focused Beam Patterns Under Accurate Sidelobe Control by Iterative Second-Order Cone Programming

Abstract

It is shown in this communication that the problem of synthesizing a frequency-invariant (FI) beam pattern with as few elements as possible can be transformed into a sequence of weighted $\ell_1$ optimizations under multiple convex constraints. The objective of the weighted $\ell_1$ optimization is to minimize the number of radiating elements. For the filter-and-sum beamforming structure, this is equivalent to minimizing the number of filters, each with a set of optimized coefficients. The multiple constraints are adopted to individually control the mainlobe and sidelobe pattern characteristics, so as to preserve the FI property in the mainlobe region while satisfying a given upper bound in the sidelobe region. The WNG constraint can also be incorporated into the proposed method to enhance the synthesis robustness. This method can be easily implemented by iterative second-order cone programming (SOCP), and only few iterations are required to reach the convergence. A set of examples for the synthesis of FI patterns with uniform sidelobe level (SLL) or multiple nulls, scannable FI patterns, and the synthesis of FI pattern for an arc array with directive elements, is presented to validate the effectiveness and advantages of the proposed method. The element saving is about 23.3%–50% for the test cases.