Filter recovery in infinite spatially invariant evolutionary system via spatiotemporal trade off

Abstract

We consider the problem of spatiotemporal sampling in an evolutionary process xn = Anx where an unknown operator A driving an unknown initial state x is to be recovered from a combined set of coarse spatial samples {χ| Ωο , x(1)| Ωι ,· · ·, x(N)| 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 l2(Z) and A is an unknown spatial convolution operator given by a filter α e l1 (Z) so that Ax = a ∗ x. We show that {x|Ω m , x(1)|Ω m , ···, x(N)|Ω m :N≥2m −, Ω m = mZ} contains enough information to recover the Fourier spectrum of a typical low pass filter a, if x is from a dense subset of l2 (Z). The idea is based on a nonlinear, generalized Prony method similar to [2]. We provide an algorithm for the case when a and x are both compactly supported. Finally, We perform the accuracy analysis based on the spectral properties of the operator A and the initial state x and verify them in several numerical experiments.