Adaptive Filtering Using Complex Data and Quaternions

We provide an overview of complex-data and quaternion-based nonlinear adaptive filtering. The use of quaternion-valued data has been drawing recent interest in various areas of statistical signal processing, including adaptive filtering, image pattern recognition, and modeling and tracking of motion. Specifically, linear and widely linear adaptive filter designs based on quaternion data have been presented in the literature. The benefit for quaternion-valued processing in particular includes performing data transformations in 3 or 4-dimensional space conveniently compared to vector algebra. The transformations are performed using quaternion addition and multiplication, which differs from real and complex-valued multiplication in that quaternion multiplication is non-commutative. Recently, new algorithms have been developed that outperform others for these kinds of problems. We present an overview and discuss performance analysis results as well as simulations to verify the theory.