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

TL;DR

This paper introduces a novel self-recursive radix-2 factorization for Vandermonde matrices used in wideband multi-beam antenna arrays, significantly reducing circuit complexity for analog beamforming.

Contribution

It presents a new sparse, orthogonal, self-recursive radix-2 algorithm for Vandermonde matrices, enabling efficient analog multi-beam beamforming with reduced circuit complexity.

Findings

Achieves $ ext{O}(N ext{log} N)$ circuit complexity

Provides error bounds and stability analysis

Demonstrates potential for high-frequency circuit implementation

Abstract

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.