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

Abstract

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.INDEX TERMS Delay vandermonde matrix, efficient algorithms, self-recursive algorithms, complexity and performance of algorithms, approximation algorithms, wireless communications, beamforming, software defined radio.

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

“…Also, the direct computation of vanc(z, N), vancr(z, N), vancc(z, N),and vanccr(z, N) with corresponding matrices V N andṼ N varying sizes from 4 × 4 to 4096 × 4096 are shown in Tables 1, 2, and 3. Following Tables 1, 2, and 3, the proposed radix-2 algorithms for the Vandermonde matrices have shown significant arithmetic complexity reduction as opposed to the DVM algorithms presented in [1], [16], [17]. At the same time, we should recall that the DVM algorithms proposed in [1], [16], [17] have no restriction for nodes or delays as in this vanc(z, N) and vancc(z, N)) vs direct computation.…”

“…, where α = e −jω t τ and accounts for the phase rotation associated with the delay τ at frequency f , and ω t = 2π f , have been derived through our previous work [1], [16], [17]. It is important to realize that the matrix elements are integer powers of α = e −jω t τ which are functions of the temporal frequency variable ω t ; this is an important distinction from the DFT matrix where the elements are constants defined as the primitive N th roots of unity.…”

“…Moreover, we have addressed the error bounds and stability of the DVM algorithm in [17] by filling the gaps in [12], [13], [23]. The DVM algorithm in [16] is faster than [17] but does not produce arithmetic complexity of order O(N log N ). On the other hand, there are no constraints for nodes of DVM in [17] as opposed to what we propose here.…”

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

“…We consider the direct computation of Vandermonde matrices V and Ṽ by the vector z ∈ C N with N(N − 1) complex additions and multiplications (note that V and Ṽ have 1's along the first column so we counted the multiplication count as N(N − 1) as opposed to N 2 ). Also, the direct computation of Vandermonde matrices V and Ṽ by the vector z ∈ R N is taken as N(2N − 1) real addi- Following Tables 1, 2, and 3, the proposed radix-2 algorithms for the Vandermonde matrices have shown significant arithmetic complexity reduction as opposed to the DVM algorithms presented in [1,16,17]. At the same time, we should recall that the DVM algorithms proposed in [1,16,17] have no restriction for nodes or delays as in this paper.…”

“…, where α = e − jω t τ and accounts for the phase rotation associated with the delay τ at frequency f , and ω t = 2π f , have been derived through our previous work [1,16,17]. It is important to realize that the matrix elements are integer powers of α = e − jω t τ which are functions of the temporal frequency variable ω t ; this is an important distinction from the DFT matrix where the elements are constants defined as the primitive Nth roots of unity.…”