V-Vector Algebra and Volterra Filters

Alberto Carini1; Enzo Mumolo2; Giovanni L. Sicuranza2    1 TELIT Mobile Terminals S.p.A., I-34010 Sgonico, Trieste, Italy
2 Department of Electrical, Electronic and Computer Engineering (DEEI), University of Trieste, I-34127 Trieste, Italy

I INTRODUCTION

This article describes an algebraic structure which is usefully applied to the representation of the input-output relationship of the class of polynomial filters known as discrete Volterra filters. Such filters are essentially based on the truncated discrete Volterra series expansion, which is obtained by suitably sampling the continuous Volterra series expansion, which is widely applied for representation and analysis of continuous nonlinear systems.

Vito Volterra, an Italian mathematician bom in Ancona in 1860, introduced the concept of functionals and devised the series, named after him, as an extension of the Taylor series expansion. His first works on these topics were published in 1887. Besides devising the theory of functionals, he made relevant contributions to integral and integrodifferential equations and in other fields of physical and biological sciences. A complete list of his 270 publications is reported in the book published in 1959 in which his works on the theory of functionals were reprinted in English (Volterra, 1959). Other seminal contributions related to the Volterra series expansion can be found in Fréchet (1910), where it is shown that the set of Volterra functionals is complete. The main result of all this work was the finding that every continuous functional of a signal x(t) can be approximated with arbitrary precision as a sum of a finite number of Volterra functionals in x(t). This result can be seen as a generalization of the Stone-Weierstrass theorem, which states that every continuous function of a variable x can be approximated with arbitrary precision by means of a polynomial operator in x.

The first use of Volterra’s theory in nonlinear system theory was proposed by Norbert Wiener in the early 1940s. Wiener’s method of analyzing continuous nonlinear systems employed the so-called G-functionals to determine the coefficients of the nonlinear model. The relevant property of G-functionals is that they are mutually orthogonal when the input signal to the system is white and Gaussian. An almost complete account of his work in this area is available in Wiener (1958).

These works stimulated a number of studies on Volterra and Wiener theories. Complete accounts of the fundamentals of Volterra system theory and of the developments that occurred until the late 1970s can be found in the survey papers by Billings (1980) and Schetzen (1993) and in the books by Marmarelis and Marmarelis (1978), Rugh (1981), and Schetzen (1989). The first book is primarily devoted to the applications in biomedical engineering.

The development of digital signal-processing techniques and the facilities offered by powerful computers and digital signal processors stimulated a number of studies on discrete nonlinear systems in the 1980s. The model used was often the discrete version of the Volterra series expansion. As a result, a new class of filters , polynomial filters, including Volterra filters, was introduced and widely applied. A number of applications were considered in different fields, from system theory to communications and biology, to mention only a few. Particular interest was devoted to adaptive filters and adaptation algorithms because these devices are employed in many applications. An account of pertinent activities in digital signal processing can be found in the survey papers by Mathews (1991) and Sicuranza (1992) and in the book by Mathews and Sicuranza (2000). This book contains the first complete account of the discrete Volterra series expansion and the whole class of polynomial filters, together with a large number of references.

As already mentioned, adaptive filters based on discrete Volterra models play a relevant role in nonlinear digital signal processing because they are used in many tasks such as nonlinear system identification, compensation for nonlinear distortions, equalization of communication channels, nonlinear echo cancellation, and so forth. In this respect, the V-vector algebra presented in this article constitutes a powerful tool for describing Volterra filters and their adaptation algorithms. In fact, adaptation algorithms for Volterra filters are usually obtained by extending classical algorithms proposed for linear filters. However, what makes this task complicated is the loss of the time-shift property in the input vector. This property is the key factor for deriving many fast adaptation algorithms. It consists of the fact that, in the linear case, passage from the vector collecting the N most recent samples of the input signal at time n to that at time n + 1 requires the last element of the vector to be discarded and then the new input sample to be added as the first element. This property does not apply to the input vector of Volterra filters, which is formed by different products of input samples. V-vector algebra has been accordingly designed to preserve the time-shift property of the input vectors of the linear case. V-vector algebra can thus be viewed as a simple formalism which is suitable for the derivation of adaptation algorithms for Volterra filters by simple reformulation of the well-known adaptation algorithms applied to linear filters. In particular, the vectors of linear algebra are replaced by V-vectors, which can be viewed as nonrectangular matrices. Using the V-vector formalism allows fast and numerically stable adaptation algorithms for Volterra filters to be easily derived from known linear theory. As an additional feature, it is possible to show that V-vector algebra can be usefully exploited to describe multichannel linear adaptive filters with channels of different memory lengths.

The first part of this article provides a brief introductory account of the Volterra series expansions and discrete Volterra filters. The remaining sections are essentially based on a chapter of the doctoral thesis by Carini (1997) and a paper by Carini et al. (2000) that address the main definitions of V-vector algebra. Another paper by Carini et al. (1999) presents, in addition, some applications of V-vector algebra for the derivation of fast and stable adaptation algorithms for Volterra filters. New material in this area is presented in Sections V and VI.

II VOLTERRA SERIES EXPANSIONS AND VOLTERRA FILTERS

Volterra series expansions form the basis of the theory of polynomial nonlinear systems (or filters), including Volterra filters. In this section the Volterra series expansions for both continuous and discrete systems are introduced and their main properties reviewed. A complete account of these arguments can be found in Mathews and Sicuranza (2000).

A Volterra Series Expansions for Continuous Nonlinear Systems

A continuous-time nonlinear system in which the output signal at time t, y(t), depends on only the input signal at time t, x(t), can be described, with some restrictions, by means of an appropriate power series expansion such as the Taylor series expansion. The output of such a system, which is called a memoryless system, is thus described by the input–output relation

(t)=∑p=0∞cpxp(t)

A continuous-time nonlinear system in which the output signal at time t, y(t), depends also on the input signal at any time x different from τ, is said to be a system with memory. Such a system can be represented by means of an extension of expression (1) known as the Volterra series expansion (Rugh, 1981; Schetzen, 1989; Volterra, 1887, 1913,...