  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.
