Advanced Topics and New Directions - A Rapid Introduction to Adaptive Filtering - page 112

Chapter 6
Advanced Topics and New Directions
Abstract In this final chapter we provide a concise and brief discussion of other
topics not covered in the previous chapters. These topics are more advanced or are
the object of active research in the area of adaptive filtering. A brief introduction to
each topic and several relevant references for the interested reader are provided.
6.1 Optimum H ∞ Algorithms
The LMS solves approximately the optimum linear mean square problem with a
simple algorithm. The RLS has, under certain conditions, a better performance in
terms of convergence speed and steady state error. However, both algorithms have
to deal with perturbations in real-world environments. The concept of robustness is
linked to the sensitivity of an algorithm to such perturbations. In particular, we focus
now on the idea that an algorithm is robust if it does not amplify the energy of the
perturbations [ 1 ] (this is not the same idea as the one used later in Sect. 6.4 ).
In the mid 1990s, Hassibi et al. presented a relationship between H ∞ optimal
filters and Kalman filtering in Krein spaces [ 2 ]. They also proved that the robustness
of the LMS algorithm is based on the fact that it is a minimax algorithm, i.e., an
H ∞ optimal algorithm [ 3 ]. In addition, it turns out that the RLS is H ∞ suboptimal.
H ∞ optimal filtering has also been done with different approaches [ 4 ]. However,
the approach used by Hassibi et al. relies on an elegant theory revealing the relation
between H ∞ optimal filtering and Kalman filtering in Krein spaces (which are
indefinite metric spaces) [ 2 ].
The H ∞ optimality can be interpreted as this: the ratio between the energy of
the estimation errors and the energy of the perturbations (and model uncertainties) is
upper-bounded for every possible realization of the perturbations. Therefore, small
perturbations lead to small estimation errors. By a proper definition of the errors
and perturbation the regularized APA was found to be H ∞ optimal [ 5 ]. As the H ∞
optimality might be an infinite horizon problem, by following the ideas introduced