Spatio-temporal generalised frequency response functions over bounded spatial domains

In a companion paper (Guo, Guo, Billings, Coca, and Lang 2010), the concept of frequency response functions (GFRFs) has been extended to describe the characteristics of spatio-temporal dynamical systems over an unbounded spatial domain from a frequency domain point of view. In this paper, a similar point of view will be taken to investigate spatial-temporal dynamical systems over a bounded spatial domain in the frequency domain. The main difference is that due to the bounded spatial domain, the property of translation invariance with respect to the spatial domain is not valid any more. In order to overcome this, the paper provides a new definition of impulse response functions, which is different from the standard impulse definitions. The definitions and interpretation of spatio-temporal generalised frequency response functions are given for linear and nonlinear spatio-temporal systems based on this newly defined impulse response function. Examples are provided to illustrate the new frequency domain methods.