A Fast Algorithm to Solve Delay Vandermonde Systems in Phased-Array Digital Receivers

Abstract

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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) 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.