Hypercomplex adaptive filtering

The degree of diffusion of hypercomplex algebras in adaptive and non-adaptive filtering research topics is growing faster and faster. The performance of hypercomplex adaptive filters has been widely experimented during the last decade. Quaternion filters, in particular, have been utilized in systems where the signals to be processed have some form of correlation. Besides correlation, the debate today concerns the usefulness and the benefits of representing multidimensional systems by means of these complicated mathematical structures and the criterions of choice between one algebra or another. One of the goals of this work is to discuss whether the choice of a certain algebra in the description of a problem/environment can play a significant role and determine an adaptive filter performance. That said, 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 calculation outcomes 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. The bulk of study and experiments presented in this work was carried out in a 3-Dimensional (3D) audio context. 3D audio is the new frontier in audio technology and it is quickly taking place in many applications, from cinema to virtual reality, audio surveillance and video games. The large amount of data requires fast and compact solutions for signal processing. With this aim in view, research is moving towards the exploration of hypercomplex algebras in order to find a non-redundant and compact form for the representation of 3D sound fields without loss of information. Quaternion sound fields are currently under investigation and this thesis presents some recent results concerning the integration of hypercomplex (quaternion) adaptive signal processing into a 3D audio environment.