Volterra LMS Filter implementations.
Volterra series are widely used in nonlinear system modeling. The Volterra series expansion of a nonlinear system consists of a nonrecursive series in which the output signal is related to the input signal as
d'(k) is the desired system output, and woi are the Volterra Kernels of the system. In order to apply the LMS algorithm to a nonlinear LMS filter, we have to interpret the input vector x and the weight vector w in a different manner. For the N-th order filter with 2nd order Volterra series, we have:
The Decomposable Volterra Model (DVM) is described in detail in [2]. There is a decomposability condition, and through it the algorithm reduces the number of computations for the Volterra kernels from O(MK) to O(KM), where K is the series order and M is the filter order.
26/08/2020 - There are still some problems being investigated regarding to higher order filters and series. The algorithms do not run for particular filter/series orders.
[1] Diniz, Paulo S. R. - Adaptive filtering: algorithms and practical implementation (2013).
[2] Pinheiro, F. C. & Lopes, C. G. - A Low-Complexity Nonlinear Least Mean Squares Filter Based on a Decomposable Volterra Model. IEEE Trans. Signal Process. 67, 5463–5478 (2019).