A Low-Complexity Nonlinear Least Mean Squares Filter Based on a Decomposable Volterra Model

Nonlinear signal processing is important in various applications, however it usually requires high computational cost. This paper proposes a low-complexity nonlinear adaptive least mean squares (LMS) filter based on decomposable Volterra kernels. The decomposability condition comes from a well-posed approximation problem, which imposes a rank-one structure on the full Volterra model, resulting in a system equivalent to a product of ordinary linear filters. A mean-square estimation problem is posed over such a decomposable Volterra model (DVM) and a solution via a steepest descent algorithm is introduced, subsequently motivating an adaptive implementation. The resulting DVM-LMS filter is nonlinear in the input signal as well as in its parameters, however leading to an exponential decrease in computational complexity, as compared to the full Volterra kernel. The nonlinearity in the parameters may introduce instabilities, which are tackled via normalizing strategies, introduced together with heuristics for proper initialization. Necessary conditions for stability are derived and serve as guidance to design the step-size, while simulations show that the algorithm is unbiased after convergence. Additionally, an expression for the steady-state excess mean square error is derived. Several simulations show the new filter's performance and competitiveness against celebrated methods in the literature.