The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

Abstract

Abstract-The quaternion least mean square (QLMS) algorithm is introduced for adaptive filtering of three-and fourdimensional 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 "augmented" 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 supports the approach.Index Terms-Quaternion signal processing, multidimensional adaptive filters, adaptive multi-step ahead prediction, wind modeling, data fusion via vector spaces.

User–Parameter–Free Robust Adaptive Beamforming Algorithm for Vector–Sensor Arrayswithin the Hypercomplex Framework

“…Various details about quaternions can be found in articles and books like [18], [36], [37], and part of them that will be used in this paper are shown below:…”

Study of wind profile prediction with a combination of signal processing and computational fluid dynamics

“…For the signal processing side, recently, the hypercomplex concepts have been introduced to solve problems related to three or four-dimensional signals [1], such as vector-sensor array signal processing [2], color image processing [3] and wind profile prediction [4], [5]. In many of the cases, the traditional complex-valued adaptive filtering operation needs to be extended to the quaternion domain to derive the corresponding adaptive algorithms, such as the quaternion-valued Least Mean Square (QLMS) algorithm in [6].…”

Quaternion adaptive line enhancer

“…Recently, a class of quaternion least mean square (QLMS) stochastic gradient adaptive filtering algorithms has been designed in [4] for filtering of hyper-complex processes. Such a system can be applied to both circular and noncircular signals and therefore, exploits the correlation between the real and complex components of a quaternion-valued signal.…”

“…Quaternions have found applications in computer graphics, for the modelling of three-dimensional (3-D) rotations [6], in robotics [7], molecular modelling [8], processing colour images [9], hyper-complex digital filters [10], texture segmentation [11], source separation [12], watermarking [13], spectrum estimation [14] quaternion singular value decomposition and in the MUSIC algorithm to process polarized waves [15], [16], quaternion least squares [8], [17], and quaternion singular spectrum analysis [18]. In [4] the formulation for a quaternion LMS adaptive filtering has also been provided and used for the processing of quaternion valued signals.…”