System Identification in Dynamical Sampling

We consider the problem of spatiotemporal sampling in a discrete infinite dimensional spatially invariant evolutionary process $x^{(n)}=A^{n}x$ to recover an unknown convolution operator $A$ given by a filter $a \in \ell^1($\mathbb{Z}$)$andanunknowninitialstate$x$modeledasavectorin$\ell^2($\mathbb{Z}$)$\mathrm{.}Traditionally,underappropriatehypotheses,any$x$ can be recovered from its samples on $\mathbb{Z}$ and $A$ can be recovered by the classical techniques of deconvolution. In this paper, we will exploit the spatiotemporal correlation and propose a new spatiotemporal sampling scheme to recover $A$ and $x$ that allows to sample the evolving states $x,Ax, \cdots, A^{N-1}x$onasub-latticeof$$\mathbb{Z}$$, and thus achieve the spatiotemporal trade off. The spatiotemporal trade off is motivated by several industrial applications \cite{Lv09}. Specifically, we show that ${x(m$\mathbb{Z}$), Ax(m$\mathbb{Z}$), \cdots, A^{N-1}x(m$\mathbb{Z}$): N \geq 2m}$ contains enough information to recover a typical “low pass filter” $a$ and $x$ almost surely, in which we generalize the idea of the finite dimensional case in \cite{AK14}. In particular, we provide an algorithm based on a generalized Prony method for the case when both $a$ and $x$ are of finite impulse response and an upper bound of their support is known. We also perform the perturbation analysis based on the spectral properties of the operator $A$ and initial state $x$, and verify them by several numerical experiments. Finally, we provide several other numerical methods to stabilize the method and numerical example shows the improvement.