A Fast DVM Algorithm for Wideband Time-Delay Multi-Beam Beamformers

This paper presents a sparse factorization for the delay Vandermonde matrix (DVM) and a faster, exact, radix-2, and completely recursive DVM algorithm to realize millimeter wave beamformers in wireless communication networks. The proposed algorithm will reduce the complexity of $N$-beam wideband beamformers from $\mathcal{O}(N^2)$ to $\mathcal{O}(N {\rm\: log\:} N)$. The scaled DVM algorithm is at least 97$\%$ faster than the brute-force scale DVM by a vector product. The signal flow graphs of the scaled DVM algorithm are shown to elaborate the simplicity of the proposed algorithm. The proposed lower complexity DVM algorithm can be used to design simple signal flow graph and realize in very large scale integrated circuit architecture with the significant reduction of chip area and power consumption. Moreover, the realization of the faster DVM algorithm through analog integrated circuits will be addressed . Finally, the proposed DVM algorithm will be utilized to obtain a low-complexity approximate transform for beamforming