The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

• Published in: IEEE transactions on signal processing Vol. 57; no. 4; pp. 1316 - 1327
• Main Authors: Took, C.C., Mandic, D.P.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive filters; Adaptive multistep ahead prediction; Adaptive signal processing; Algorithms; Applied sciences; Benchmarking; data fusion via vector spaces; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Filtering algorithms; Information theory; Information, signal and communications theory; Least mean squares; Least squares approximation; Mathematical analysis; Miscellaneous; multidimensional adaptive filters; Multidimensional signal processing; quaternion signal processing; Quaternions; Radar signal processing; Signal and communications theory; Signal processing; Signal processing algorithms; Signal, noise; Sonar; Statistics; Telecommunications and information theory; Vectors (mathematics); wind modeling; Quaternion; Adaptive filtering; Adaptive filter; Algorithm; Modeling; data fusion via vector spaces; Three dimensional model; wind modeling; Covariance; Vector space; Multistep method; Simulation; multidimensional adaptive filters; quaternion signal processing; Information processing; Signal processing; Data fusion; Adaptive multistep ahead prediction; Multiple channel; Vector field; Least mean squares methods
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2008.2010600

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 covariance E { xx H } and pseudocovariance E { xx T } of the tap input vector x are 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.