A Unified Framework for Adaptive Filtering

Having recently shown that the LMS adaptive filtering technique can be viewed as an iterative linear equation solver applied to a time varying linear equation set directly related to the Wiener-Hopf equation, we address in this paper the question if a larger class of adaptive filtering algorithms can be formulated within the framework of the theory of iterative linear equation solvers. We show that this is indeed the case and present the Least Mean Square (LMS), Normalized Least Mean Square (NLMS), Affine Projection Algorithm (APA) as well as the Recursive Least Squares (RLS) algorithms as algorithms that can all be seen as special cases within this general framework. Three important consequences are: 1) The theory of adaptive filtering can be presented in a unified and greatly simplified manner to DSP students. 2) The extensive body of available theory for iterative linear equation solvers becomes directly relevant to the field of adaptive filtering. 3) Performance results can be obtained within the unified framework and be made directly applicable to most known adaptive filtering algorithms through the specification of a few parameters of the general model.