It is shown in this paper that the problem of reducing the number of elements for multiple-pattern arrays can be solved by a sequence of reweighted ℓ1 optimizations under multiple linear constraints. To do so, conjugate symmetric excitations are assumed so that the upper and lower bounds for each pattern can be formulated as linear inequality constraints. In addition, we introduce an auxiliary variable for each element to define the common upper bound of both the real and imaginary parts of multiple excitations for different patterns, so that only linear inequality constraints are required. The objective function minimizes the reweighted ℓ1-norm of these auxiliary variables for all elements. Thus, the proposed method can be efficiently implemented by the iterative linear programming. For multiple desired patterns, the proposed method can select the common elements with multiple set of optimized amplitudes and phases, consequently reducing the number of elements. The radiation characteristics for each pattern, such as the mainlobe shape, response ripple, sidelobe level and nulling region, can be accurately controlled. Several synthesis examples for linear arrays, rectangular/triangular-grid and randomly spaced planar arrays are presented to validate the effectiveness of the proposed method in the reduction of the number of elements.
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