Widely linear SIMO filtering for hypercomplex numbers

The starting point of any derivation is a suitable representation of the given model. Hypercomplex numbers sometimes provide a more compact representation and more insight into a problem's structure than the reals or the complex numbers. Hence, efficient filters are needed for hypercomplex numbers as well. As there is a large zoo of different hypercomplex numbers obeying different algebras it is cumbersome to do derivations for each of them individually. Hence, our contribution is to show how to abstract the concept of hypercomplex numbers and their algebras. We give an insight into the questions how the algebra works in general. Furthermore, we propose two extensions of the concept of widely linear filters for hypercomplex numbers. The first widely linear filter abstraction presumes several properties of the algebra, but can be computed directly in the respective hypercomplex domain. The second abstract solution of widely linear filters does all calculations in the real domain. The latter imposes much less restrictions on the algebra than the first one which leads to a more generic type of widely linear filters.