Practical realization of discrete-time Volterra series for high-order nonlinearities

Full realization of all versions of Volterra series like pure, truncated, and doubly finite Volterra series, and especially, the realization of their high orders is an intractable problem. Hence, practical implementation of Volterra series for high-order nonlinearities is not feasible with reasonable computational cost. For this reason, mathematicians, neuroscientists, and especially, biomedical and electrical engineers are forced to use only the low-order Volterra series. In this paper, we provide a full realization of off-repetitive discrete-time Volterra series (ORDVS) by departure from a traditional approach in favor of choosing a hierarchical structure. The proposed method is named fast full tantamount of off-repetitive discrete-time Volterra series (FFT-ORDVS). We have proven that the proposed off-repetitive discrete-time Volterra series approximates the basic discrete-time Volterra series very well and with much less computational complexity. In a conventional method, if M+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M} +1$$\end{document} is considered as the memory length of the ORDVS, around 2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{M}$$\end{document} math operations are needed for the full realization of it. In most cases, M is a large number and consequently, 2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{M}$$\end{document} is too large. To solve this problem, we have proposed a simple polynomial time solution and using the proposed method, the same task is done only by 6M math operations. It means that we have found a shortcut to change an intractable problem (O(2M))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}(2^{M}))$$\end{document} to a simple P problem (O(M)). This achievement enables researchers to use high-order kernels and consequently covers high-order nonlinearities with the lowest possible computational load. We have proven our claims mathematically and validated the performance of the proposed method using two numerical examples and a real problem.