This paper presents a self-contained factorization for the delay Vandermonde matrix (DVM), which is the super class of the discrete Fourier transform, using sparse and companion matrices. An efficient DVM algorithm is proposed to reduce the complexity of radio-frequency (RF) $N$-beam analog beamforming systems. There exist applications for wideband multi-beam beamformers in wireless communication networks such as 5G/6G systems, system capacity can be improved by exploiting the improvement of the signal to noise ratio (SNR) using coherent summation of propagating waves based on their directions of propagation. The presence of a multitude of RF beams allows multiple independent wireless links to be established at high SNR, or used in conjunction with multiple-input multiple-output (MIMO) wireless systems, with the overall goal of improving system SNR and therefore capacity. To realize such multi-beam beamformers at acceptable analog circuit complexities, we use sparse factorization of the DVM in order to derive a low arithmetic complexity DVM algorithm. The paper also establishes an error bound and stability analysis of the proposed DVM algorithm. The proposed efficient DVM algorithm is aimed at implementation using analog realizations. For purposes of evaluation, the algorithm can be realized using both digital hardware as well as software defined radio platforms.
翻译:本文介绍了对延迟的Vandermonde 矩阵(DVM)的自足因素化。 Vandermonde 矩阵是离散的Fleier变异的超级类,使用稀少的和相伴的矩阵。建议采用高效的DVM算法来降低无线电频率(RF)的复杂程度,或结合多投入的多输出(MIIMO)无线系统来使用。在5G/6G系统等无线通信网络中,对宽带多波束光仪应用了各种应用,通过利用根据传播方向对传播波进行一致的比对,改进信号对噪音比(SNR)的能力可以提高。多种RF光束的存在允许在高级SR(RR)建立多种独立的无线链接,或与多投入的多输出(MIMO)无线系统一起使用。为了改进系统SNRR,从而提高能力,我们利用可接受的模拟电路路复杂程度,我们利用DVM的稀薄因数化系数来得出低的计算复杂度的DVM算法。本文还确立了一个错误和稳定性分析。拟议的DVM的系统,作为实现的硬件,作为DVV的实现的既定的硬件分析工具。