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 covariance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> { <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xx</b> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sup> } and pseudocovariance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> { <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xx</b> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> } of the tap input vector <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</b> 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.