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The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes
Abstract
The quaternion least mean square (QLMS) algorithm is introduced for adaptive filtering of three- and four-dimensional processes, such as those observed in atmospheric modeling (wind, vector fields). These processes exhibit complex nonlinear dynamics and coupling between the dimensions, which make their component-wise processing by multiple univariate LMS, bivariate complex LMS (CLMS), or multichannel LMS (MLMS) algorithms inadequate. The QLMS accounts for these problems naturally, as it is derived directly in the quaternion domain. The analysis shows that QLMS operates inherently based on the so called ldquoaugmentedrdquo statistics, that is, both the covarianceE{xxH} and pseudocovarianceE{xxT} of the tap input vectorxare taken into account. In addition, the operation in the quaternion domain facilitates fusion of heterogeneous data sources, for instance, the three vector dimensions of the wind field and air temperature. Simulations on both benchmark and real world data support the approach.
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For circular signals, WL-QLMS reduces to QLMS, and QLMS converges faster than WL-QLMS.
“Importantly, for circular signals WL-QLMS reduces to QLMS [44] and QLMS converges faster than WL-QLMS [45].”
From A Quaternion Weighted Fourier Linear Combiner for Modeling Physiological Tremor (2016) · cites this paper as [45]
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Cited by
Sorted by citations
Quaternion-valued short-term joint forecasting of three-dimensional wind and atmospheric parameters
Took, C. C., Strbac, G., Aihara, K., & Mandic, D. P. (2011) · Renewable Energy · doi:10.1016/j.renene.2010.12.013 · Cited by 87
Self-citationA Quaternion Weighted Fourier Linear Combiner for Modeling Physiological Tremor
Adhikari, K., Tatinati, S., Ang, W. T., Veluvolu, K. C., & Nazarpour, K. (2016) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2016.2530564 · Cited by 48
Metadata
- Authors
- Took, C. C., & Mandic, D. P.
- Affiliations
- Imperial College London (GB)
- Venue
- IEEE Transactions on Signal Processing
- Published
- 2009-04
- Volume
- 57
- Issue
- 4
- Pages
- 1316-1327
- ISSN
- 1053-587X (ISSN-L)
- 1941-0476
- OpenAlex
- W2099057098
- Cited by
- 332
- References
- 50
- Citations by year
- 2026: 16 · 2025: 16 · 2024: 27 · 2023: 28 · 2022: 18 · 2021: 19 · 2020: 17 · 2019: 38 · 2018: 14 · 2017: 27 · 2016: 25 · 2015: 19 · 2014: 14 · 2013: 15 · 2012: 9
- FWCI
- 10.48
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Signal Processing (publishedVersion)
- Surrey Research Insight Open Access (The University of Surrey) (submittedVersion)
- Open Research (University of Surrey) (submittedVersion)
- Funders
- Imperial College London
- Engineering and Physical Sciences Research Council (EP/D061709/1)
- Topics
- Physical Sciences › Computer Science › Signal Processing › Advanced Adaptive Filtering Techniques
- Physical Sciences › Mathematics › Applied Mathematics › Algebraic and Geometric Analysis
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Keywords
- adaptive filtering, wind field estimation, atmospheric modeling, vector fields, heterogeneous data fusion, nonlinear dynamics
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