V-vector algebra and its application to Volterra-adaptive filtering

In this paper, we describe a new algebraic structure called V-vector algebra, which is a formal basis for the development of Volterra-adaptive filter algorithms as an extension of linear-adaptive techniques. In this way, fast and numerically stable adaptive Volterra filtering algorithms can be easily derived from the known linear theory. V-vector algebra can also be applied to deal with linear multichannel filters with channels of different memory lengths. A reformulation of the Lee-Mathews fast recursive least squares (RLS) algorithm and a new fast and stable Givens rotation-based square root RLS algorithm, both derived using V-vector algebra, are finally presented.