A unified view of adaptive algorithms for finite impulse response filters using the H∞ framework

The normalized least mean squares (NLMS) and recursive least squares (RLS) algorithms are widely used for adaptive filtering. Interestingly, the NLMS algorithm has been shown to be strictly optimal in the sense of H∞ filtering, whereas the forgetting factor RLS algorithm has not been clearly related to a solution to the H∞ filtering problem. This paper describes a method for further optimizing the solutions to the ordinary H∞ filtering problem over an assumed system model set and a predetermined norm weight set. The extended H∞ filtering problem offers a framework for constructing a unified view of adaptive algorithms for finite impulse response (FIR) filters. The framework enables a discussion of the relationships among the NLMS algorithm, the forgetting factor RLS algorithm, and the H∞ filter over the common parameter space, and facilitates the development of new fast adaptive algorithms that outperform the existing algorithms, such as the NLMS and the fast RLS algorithms. The validity of the discussion based on the H∞ framework is verified using numerical examples.