A Recursive Algorithm of Digital Polynomial Filtering

The paper considers a class of digital nonlinear filters defined by a discrete truncated Volterra series. Such filters called Volterra filters or polynomial filters are a natural generalization of linear filters. The computational weight of polynomial filters is widely known to exponentially grow with the nonlinearity degree. This article is devoted to design of effective algorithms of polynomial filtering based on the algebraic theory of signals and systems. The implementation of nonlinear filters of this class is based on the procedure of data segmentation and calculation of nonlinear circular convolutions. In the paper we suggest recursive algorithms for calculating nonlinear circular convolutions based on multivariate polynomial transforms and the Chinese remainder theorem. Unlike the discrete Fourier transform, the calculation of polynomial transforms does not require multiplications and is implemented using additions and shifts. It is shown that the calculation of a nonlinear circular convolution of the m- th order can be reduced to the calculation of the convolution of the (m- 1)- th order, the execution of the operation of transition to a single variable in the polynomial region followed by restoring the result using the Chinese remainder theorem. A modification of the recursive algorithm for calculating nonlinear circular convolutions using fast polynomial transform algorithms is proposed. This algorithm allows to reduce the computational costs by using an effective procedure based on the recursive addition of polynomials with exponentially growing bases for the multiplication of polynomials. The article concludes with the assessment of computational complexity of the proposed recursive algorithm and the recommendations for its application.