Optimal Amplitude–Density Synthesis of Linear Aperiodic Arrays

New deterministic procedures for the design of linear aperiodic arrays are described, which permit exploiting, in a combined way, both the positions and the excitation amplitudes of the array elements obtaining a pattern, which optimally fits, in terms of a weighted L2 norm, the pattern of a reference linear continuous aperture. The first numerical procedure is based on alternating optimization of positions and amplitudes by means of closed-form convex projectors. Sufficient conditions for the solution to be optimal are established. A second numerical procedure is based on a domino-like sequential determination of the unknowns, which can be iterated to convergence. In addition, analytical asymptotic expressions for the optimal interelement spacing and amplitude are derived by means of variational techniques. The optimal asymptotic interelement spacing is demonstrated to be proportional to the reference tapering to the power (
$- \frac {2}{3}$
), while the optimal asymptotic array amplitude is proportional to the reference tapering to the power (
$\frac {1}{3}$
). For aperiodic arrays with equiamplitude excitation, it is proven that the optimal interelement spacing is proportional to the reference tapering to the power (−1). Based on these asymptotic dependencies, closed-form analytical solutions are obtained for amplitudes and positions of the elements. Several numerical results confirm the effectiveness and accuracy of the new procedures.