System identification in dynamical sampling

We consider the problem of spatiotemporal sampling in an evolutionary process x = Ax where an unknown linear operator A driving an unknown initial state x is to be recovered from a combined set of coarse spatial samples {x|Ω0 , x |Ω1 , · · ·, x |ΩN }. In this paper, we will study the case of infinite dimensional spatially invariant evolutionary process, where the unknown initial signals x are modeled as l(Z) and A is an unknown spatial convolution operator given by a filter a ∈ l(Z) so that Ax = a ∗ x. We show that {x|Ωm , x |Ωm , · · ·, x |Ωm : N ≥ 2m− 1,Ωm = mZ} contains enough information to recover the Fourier spectrum of a typical low pass filter a, if the initial signal x is from a dense subset of l(Z). The idea is based on a nonlinear, generalized Prony method similar to [1]. We provide an algorithm for the case when both a and x are compactly supported around the center. Finally, we perform the accuracy analysis based on the spectral properties of the operator A and initial state x, and verify them by several numerical experiments.