Widely linear quaternion adaptive filtering in the frequency domain

Adaptive filtering can be expanded to new numerical systems and unseen sides of physical problems can be highlighted thanks to the mathematical properties of such hypercomplex algebras. Each algebra has its own rules and operation results may not be compatible from one algebra to another. However, such peculiarities diversify algebras in a way that each of them fits specific geometrical/physical problems. In this work we propose a quaternion widely linear adaptive algorithm operating in the frequency domain. The aim is to overcome the problem of high computational cost occurring when time-domain algorithms are used. An analysis of the cost is also supplied. Finally, simulations support our proposal.