Efficient and Self-Recursive Delay Vandermonde Algorithm for Multi-beam Antenna Arrays
Open accessAbstract
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.
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The stable and O(NlogN) algorithms for Vandermonde matrices proposed in [9] can be used for the narrowband communication system.
“We recall here that the stable (well conditioned) and O(NlogN) algorithms for Vandermonde matrices proposed in [9] can be used for the narrowband communication system.”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [9]
The paper proposes delay-Vandermonde algorithms for antenna beamforming, but it does not characterize them as stable, well-conditioned O(N log N) algorithms. Section IV, Lemmas 4.1–4.2, gives operation counts containing N² terms, while Section V, “Theoretical Bounds,” explicitly says the proposed algorithms cannot be computed stably for large matrices. The wideband N-beam section describes use at a fixed input frequency for numerical verification, not the claimed combination of properties for a narrowband communication system (page numbers unavailable in our copy). (LLM, conf=0.90)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe parallel realization of N beams using a network of TTDs with durations set to integer multiples of τ0 leads to an N-beam TTD multi-beam beamformer with an underlying mathematical framework given by the DVM.
“The parallel realization of N beams using a network of TTDs with durations set to integer multiples of τ0 leads to an N-beam TTD multi-beam beamformer with an underlying mathematical framework given by the DVM [7,8,9,10,11].”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [9]
The paper proposes a low-complexity wideband TTD digital DVM algorithm using Thiran fractional delays.
“Thus, with this and from the prior work on DVM [7,8,9,10,11], we propose a low-complexity wideband TTD digital DVM algorithm using Thiran fractional delays.”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [9]
The factor e−jωτ associated with analog DVM beamforming can be efficiently approximated on chip by CMOS APFs.
“The factor e−jωτ is associated with the factorization of the AN and can be efficiently approximated on chip by CMOS APFs [7,8,9,10,11,48,49].”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [9]
The gain-delay blocks of the ddvm algorithm consistently exhibit better performance compared to those in [9] for both N=4 and N=8 configurations.
“On the other hand, the gain–delay blocks of the ddvm algorithm consistently exhibit better performance compared to those in [9] for both N=4 and N=8 configurations.”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [9]
The Numerical Results for the Complexity of DVM Algorithms section (page unavailable) compares DVM algorithms with direct matrix-vector computation. Searches for “ddvm,” “gain–delay blocks,” “[9],” and N=4/N=8 comparisons found no passage addressing the claimed block-level performance comparison. (LLM, conf=0.72)
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4 of the 54 references couldn’t be matched to a known paper.
Research works printed but not linked: 5
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- SIRANI M. PERERA received the B.Sc. degree in mathematics (First Class Hons.) from the University Sri Jayewardenepura, Sri Lanka, in 2004, the Part III Tripos in Mathematics (Hons.) from the University of Cambridge, U.K., in 2006, and the Ph.D. degree in mathematics from the University of Connecticut, USA, in 2012. She was the first Sri Lankan selected by the Cen- ter for Mathematical Sciences at the University of Cambridge, UK with full Shell Centenary Scholarship (under The Cambridge Commonwealth Trust).
- ARJUNA MADANAYAKE (Member, IEEE) received the B.Sc. degree in engineering (First Class Hons.) from the University of Moratuwa, Sri Lanka, in 2001, and the Ph.D. and M.Sc. degrees in electrical engineering from the University of Calgary, Canada, in 2008 and 2004, respectively. He is an Associate Professor of electrical and computer engineering at Florida International University (FIU) in Miami, Florida. His research interests are in one-and multi-dimensional signal processing, antenna arrays, mm-wave systems, microwave engineering, analog circuits, integrated circuit design, FPGA architectures and VLSI realizations of algorithms. His research is supported by the Defense Advanced Research Projects Agency (DARPA), the Office of Naval Research (ONR), and the US National Science Foundation (NSF).
- RENATO J. CINTRA (Senior Member, IEEE) received the D.Sc. degree in electrical engineering from the Universidade Federal de Pernambuco (UFPE), Recife, Brazil. He joined as a Faculty Member of with the Centre for Natural and Exact Sciences, UFPE, in 2005. He has held visiting professor appoint- ments at the University of Akron, Ohio, USA; the Département Informa- tique, INSA Lyon, France; the INRIA/IRISA Lab, University of Rennes 1, France; and the University of Calgary, Canada. He was an Associate Editor for the IEEE GEOSCIENCE AND REMOTE SENSING LETTERS; and Springer Circuits, Systems, and Signal Processing. Currently he serves as an Associate Editor for IET Circuits, Devices and Systems; and Journal of Communication and Information Systems; and as Subject Editor for Electronics Letters. His long-term topics of research include approximation theory for discrete trans- forms, theory and methods for digital signal/image processing, and proba- bilistic methods.
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“On the other hand, the gain–delay blocks of the ddvm algorithm consistently exhibit better performance compared to those in [9] for both N = 4 and N = 8 configurations.”
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Self-citation
- Authors
- Perera, S. M., Madanayake, A., & Cintra, R. J.
- Affiliations
- Embry-Riddle Aeronautical University Daytona Beach Florida Campus (US)
- Florida International University (US)
- University of Calgary (CA)
- Venue
- IEEE Open Journal of Signal Processing
- Published
- 2020
- Volume
- 1
- Pages
- 64-76
- ISSN
- 2644-1322 (ISSN-L)
- OpenAlex
- W3021164854
- Cited by
- 15
- References
- 54
- Citations by year
- 2025: 2 · 2024: 5 · 2023: 3 · 2022: 3 · 2021: 1 · 2020: 1
- FWCI
- 1.08
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- IEEE Open Journal of Signal Processing (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University)
- Greater South Information System
- Funders
- National Science Foundation (1902283)
- Conselho Nacional de Desenvolvimento Científico e Tecnológico (306437/2015-5, 308423/2019-4)
- Topics
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Computer Science › Signal Processing › Direction-of-Arrival Estimation Techniques
- Keywords
- sparse matrix factorization, companion matrix, analog beamforming, multibeam antenna arrays, RF beamforming, arithmetic complexity reduction, error bound analysis, numerical stability, coherent summation, MIMO, software-defined radio, 5G/6G networks
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