Weighted Fock Spaces Research Articles

A generalized Fock space framework for nonlinear system and signal analysis

A unified framework is presented for nonlinear (and, in particular, linear) system and signal analysis, whereby a number of problems involving approximation and inversion of nonlinear functions, nonlinear functionals, and nonlinear operators, are cast in a reproducing kernel Hilbert space (RKHS) setting, and solved by orthogonal projection methods. The RKHS's used for the above purpose are the "arbitrarily weighted Fock spaces" introduced by de Figueiredo and Dwyer in II] and called in the present paper "generalized Fock (GF) spaces". These spaces consist of polynomials or power series in one or more scalar variables, or of finite (polynomic case) or infinite Volterra functional series in one or more functions, or of finite or infinite Volterra operator series in one or more functions. In each case, the space is equipped with an appropriate weighted inner product, with the option of making the choice of weights depend on the particular problem under consideration. These developments are illustrated by means of various applications, in particular, the modeling of semiconductor device characteristics, best approximation of nonlinear systems, and cancellation of large nonlinear distortion in signals propagating through electronic equipment.

A best approximation framework and implementation for simulation of large-scale nonlinear systems

A conceptual and mathematical framework is presented for optimally approximating a large-scale continuous-time-parameter nonlinear dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_C</tex> by a continuous-time-parameter model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_C</tex> as well as a discrete-time-parameter model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_D</tex> , which can be readily simulated respectively on an analog and on a digital computer. A reproducing kernel Hilbert space approach in appropriate weighted Fock spaces is used in the problem formulation and solution. Assuming that the input-output map of the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_C</tex> can be represented by a Volterra functional series <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_t</tex> , belonging to a Fock space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_{\underline{\rho}}(E)</tex> , the input-output maps for the simulators <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_C</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_D</tex> are obtained as "best approximations" in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_{\underline{\rho}}(E)</tex> for the entire (untruncated) series <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_t</tex> . Each of these models has the following features: (a) It is adaptive because it is based on a set of test input-output pairs which can be incorporated in the system by on-line multiplexing, (b) it is optimal in the sense of being a projection in a Hilbert space of nonlinear operators, (c) it is easily implementable by means of a set of interconnected linear dynamical systems and zero-memory nonlinear functions of single variables, and (d) unlike polynomic (truncated Volterra series) approximations, it constitutes a global approximation and thus is valid under both small- and large-signal operating conditions.