Dynamical hysteresis in communications: a volterra functional approach

A formalism to characterise nonlinear dynamical hysteresis is described for multi-channel input-output physical systems that can have multi-valued solutions. The formalism presented is based on an extension to the Volterra functional representation of nonlinear dynamical input-output processes, an extension that overcomes the single-valued limitation of both the Taylor and Volterra series expansions. One important attribute of the formalism is that the coefficients can be physically significant and another is that the response function values can be determined directly from noisy data thereby offering potential insight into the underlying physics of the observed phenomena. The estimated response function values are the empirical coefficients required by a phenomenological theory of the system and can be used to predict likely behaviour, and to design and precisely control improved systems. The response function values that characterise the multi-channel nonlinear and dynamical hysteresis behaviour are estimated using the simultaneous moment equation method. The coefficients that characterise the hysteretic behaviour are obtained by solving a tractable system of simultaneous moment equations that are generated by operating on a suitably truncated Volterra functional expansion. These simultaneous moment equations enable the unknown constant coefficient values to be determined from noisy multi-channel input-output data. In order to demonstrate the attributes of the formalism under controlled conditions, a numerical example is presented which illustrates how to accurately estimate the coefficients of multi-channel nonlinear dynamical hysteresis phenomena corrupted by additive noise. The problems of time-dependent coefficients and the analysis of real data are to be considered elsewhere.