QR-decomposition based algorithms for adaptive volterra filtering

A QR-RLS adaptive algorithm for nonlinear filtering in presented. The algorithm is based solely on Givens rotation. Hence the algorithm is numerically stable and highly amenable to parallel implementations. The computational complexity of the algorithm is comparable to that of the fast transversal Volterra filters. The algorithm is based on a truncated second-order Volterra series model; however, it can be easily extended to other types of polynomial nonlinearities. The algorithm is derived by transforming the nonlinear filtering problem into an equivalent multichannel linear filtering problem with a different number of coefficients in each channel. Such multichannel algorithms were not available in the past even for adaptive linear filtering applications. The derivation of the algorithm is based on a channel-decomposition strategy which involves processing the channels in a sequential fashion during each iteration. This avoids matrix processing and leads to a scalar implementation. Results of extensive experimental studies demonstrating the properties of the algorithm in finite and "infinite" precision environments are also presented. The results indicate that the algorithm retains the fast convergence behavior of the RLS Volterra filters and is numerically stable.