Phased-array multibeam RF beamformers require calibration of receivers used in an array of elements before the signals can be applied to an analog beamforming network. This article presents a fast O(n2) algorithm for solving linear systems having a delay Vandermonde matrix (DVM) for the analog beamforming matrix. The structure of the DVM enables to obtain a fast algorithm to efficiently reverse multibeams at the DVM output such that calibration of the input low noise amplifiers can be achieved. The arithmetic complexity of the proposed DVM system solving algorithm is preferable over the standard matrix inversion consuming O(n3) operations. Numerical experiments are presented for forward accuracy of the proposed algorithm computed at the calibration frequency. Moreover, numerical results are shown to compare the order of the arithmetic complexity and the execution time of the proposed algorithm. Signal flow graphs are presented for four- and eight-element multibeam phased arrays.

中文翻译：

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一种解决相控阵数字接收机延迟 Vandermonde 系统的快速算法


相控阵多波束射频波束形成器需要先对元件阵列中使用的接收器进行校准，然后才能将信号应用于模拟波束形成网络。本文提出了一种快速 O(n2) 算法，用于求解具有模拟波束形成矩阵的延迟 Vandermonde 矩阵 (DVM) 的线性系统。 DVM 的结构能够获得快速算法，以在 DVM 输出处有效地反转多波束，从而实现输入低噪声放大器的校准。所提出的 DVM 系统求解算法的算术复杂度优于消耗 O(n3) 运算的标准矩阵求逆。数值实验给出了在校准频率下计算的所提出算法的前向精度。此外，还给出了数值结果，以比较所提出算法的算术复杂度和执行时间。给出了四元件和八元件多波束相控阵的信号流图。