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Nonlinear adaptive filtering allows for modeling of some additional aspectsof a general system and usually relies on highly complex algorithms, such asthose based on the Volterra series. Through the use of the Kronecker productand some basic facts of tensor algebra, we propose a simple model ofnonlinearity, one that can be interpreted as a product of the outputs of K FIRlinear filters, and compute its cost function together with its gradient, whichallows for some analysis of the optimization problem. We use these results itin a stochastic gradient framework, from which we derive an LMS-like algorithmand investigate the problems of multi-modality in the mean-square error surfaceand the choice of adequate initial conditions. Its computational complexity iscalculated. The new algorithm is tested in a system identification setup and iscompared with other polynomial algorithms from the literature, presentingfavorable convergence and/or computational complexity.

A Nonlinear Adaptive Filter Based on the Model of Simple Multilinear Functionals