Radix-2 Self-Recursive Sparse Factorizations of Delay Vandermonde Matrices for Wideband Multi-Beam Antenna Arrays

This paper presents a self-contained factorization for the Vandermonde matrices associated with true-time delay based wideband analog multi-beam beamforming using antenna arrays. The proposed factorization contains sparse and orthogonal matrices. Novel self-recursive radix-2 algorithms for Vandermonde matrices associated with true time delay based delay-sum filterbanks are presented to reduce the circuit complexity of multi-beam analog beamforming systems. The proposed algorithms for Vandermonde matrices by a vector attain $\mathcal{O}(N \log N)$ delay-amplifier circuit counts. Error bounds for the Vandermode matrices associated with true-time delay are established and then analyzed for numerical stability. The potential for real-world circuit implementation of the proposed algorithms will be shown through signal flow graphs that are the starting point for high-frequency analog circuit realizations.

Efficient and Self-Recursive Delay Vandermonde Algorithm for Multi-Beam Antenna Arrays

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.