The generalized sliding window recursive least-squares (SGW RLS) algorithm

The Recursive Least-Squares algorithm with a Sliding Rectangular Window (SWCRLS) ex- hibits a better tracking ability than the RLS algorithm with an exponential window (WRLS). The exploitation of a certain shift-invariance property that is inherent to the adaptive l- tering problem allows the derivation of fast versions and leads to the Fast Transversal Filter (FTF) and SWCFTF algorithms whose complexities are O(N), N being the lter length. The SWCRLS algorithm has a major drawback which is noise ampli cation whereas the WRLS al- gorithm is less sensible to noise ampli cation because of the larger memory of the exponential window. In our communication, we derive a new RLS algorithm that is the Generalized Sliding Window RLS (SGW RLS) algorithm. This algorithm uses a generalized window which con- sists of the superposition of an exponential window for the L 0 most recent data and the same but attenuated exponential window for the rest of the data. This new window reduces noise ampli cation in the SWCRLS algorithm. Moreover, we prove theoritically that the use of this window leads to a better compromise between estimation noise and lag noise. Furthermore, after providing a fast version to the GSW RLS algorithm that is the GSW FTF algorithm, we apply the Subsampled-Upadating technique to derive the FSU GSW FTF algorithm, a doubly-fast version of the GSW RLS algorithm.