Radix-2 Self-Recursive Sparse Factorizations of Delay Vandermonde Matrices for Wideband Multi-Beam Antenna Arrays
Open accessAbstract
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 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.
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Although the algorithms proposed in [8,10] are much more efficient than brute-force calculation, their order of arithmetic complexity is not O(NlogN).
“Although the algorithms proposed in [8,10] are much more efficient than brute-force calculation, their order of arithmetic complexity is not O(NlogN).”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [10]
Section IV (page unavailable) says this paper’s proposed radix-2 algorithms use O(N log N) GDBs, contrary to the claim that the proposed algorithms are not O(N log N). The paper distinguishes this analog GDB-count metric from digital arithmetic complexity, so its result should not be described without that qualification. (LLM, conf=0.88)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe 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 [10]
Section III describes low-complexity delay Vandermonde matrix algorithms, so the paper supports the citing sentence’s reference to prior DVM work. But Section VII says its proposed algorithms reduce the complexity of analog beamforming; Section II distinguishes those analog implementations from digital ones. The passages do not support attributing a digital algorithm using Thiran fractional delays to this paper. Page numbers were not provided. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe 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 [10]
Section III identifies e−jωτ as the delay-dependent factor in the analog beamforming matrix. Section II says true-time delays can be realized with all-pass networks, but it does not specify CMOS all-pass filters or establish that they efficiently approximate the factor on chip. No page numbers are available in our copy. (LLM, conf=0.84)
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 [10]
The exact calculation of the DVM vector product via the DVM algorithm in [10] can be used to reduce the complexity of RF N-beam analog beamforming systems.
“The exact calculation of the DVM vector product via the DVM algorithm in [10] can be used to reduce the complexity of RF N-beam analog beamforming systems.”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [10]
The ddvm algorithm exhibits a higher count of adders and gain-delay blocks compared to the algorithms presented in [10,11].
“The ddvm algorithm exhibits a higher count of adders and gain–delay blocks compared to the algorithms presented in [10,11].”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [10]
The algorithms in [10,11] have adders and gain-delay block counts of the order O(NlogN), whereas the ddvm algorithm maintains these counts of less than O(N^2).
“This is because the algorithms in those papers have adders and gain–delay block counts of the order O(NlogN), whereas the ddvm algorithm maintains these counts of less than O(N2).”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [10]
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- SIRANI M. PERERA received the B.Sc. (Hons.) degree in mathematics from the University of Sri Jayewardenepura, Sri Lanka, in 2004, the Part III Mathematical Tripos (Hons.) degree from the University of Cambridge, U.K., in 2006, and the Ph.D. degree in mathematics from the Univer- sity of Connecticut, USA, in 2012. She was the first Sri Lankan to attend the Center for Mathe- matical Sciences at the University of Cambridge, U.K., through the full Shell Centenary Scholarship (under the Cambridge Commonwealth Trust). Since 2015, she has been an Assistant Professor in mathematics with Embry-Riddle Aeronautical Uni- versity (ERAU), USA. She is working in the field of numerical linear alge- bra. Her research interests include scientific computing, structured matri- ces, including Vandermonde, quasiseparable, semiseparable, Bezoutian, and banded, matrices with displacement structure, DCT, DST, FFT, fast and stable algorithms, complexity and performance of algorithms, signal process- ing, and image processing. She is a member of the SIAM.
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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 Access
- Published
- 2020
- Volume
- 8
- Pages
- 25498-25508
- ISSN
- 2169-3536 (ISSN-L)
- OpenAlex
- W3004147897
- Cited by
- 18
- References
- 24
- Citations by year
- 2025: 4 · 2024: 4 · 2023: 2 · 2022: 3 · 2021: 4 · 2020: 1
- FWCI
- 1.36
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- IEEE Access (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 (1711625, 1711395, 1902283)
- Conselho Nacional de Desenvolvimento Científico e Tecnológico (306437/2015-5)
- Topics
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Computer Science › Signal Processing › Direction-of-Arrival Estimation Techniques
- Physical Sciences › Engineering › Aerospace Engineering › Antenna Design and Optimization
- Keywords
- Vandermonde matrix, true-time delay, multibeam antenna arrays, sparse factorization, radix-2 algorithm, orthogonal matrices, circuit complexity reduction, numerical stability, error bounds, signal flow graph, analog circuit implementation
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