Introducing Legendre nonlinear filters

This paper introduces a novel sub-class of linear-in-the-parameters nonlinear filters, the Legendre nonlinear filters. Their basis functions are polynomials, specifically, products of Legendre polynomial expansions of the input signal samples. Legendre nonlinear filters share many of the properties of the recently introduced classes of Fourier nonlinear filters and even mirror Fourier nonlinear filters, which are based on trigonometric basis functions. In fact, Legendre nonlinear filters are universal approximators for causal, time invariant, finite-memory, continuous, nonlinear systems and their basis functions are mutually orthogonal for white uniform input signals. In adaptive applications, gradient descent algorithms with fast convergence speed and efficient nonlinear system identification algorithms can be devised. Experimental results, showing the potentialities of Legendre nonlinear filters in comparison with other linear-in-the-parameters nonlinear filters, are presented and commented.