System identification in dynamical sampling

Abstract

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 $a \in \ell ^{1}(\mathbb {Z})$ and an unknown initial state x modeled as a vector in $\ell ^{2}(\mathbb {Z})$. Traditionally, under appropriate hypotheses, 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 sampling scheme to recover A and x that allows us to sample the evolving states x,A x,⋯ ,A ^N−1 x on a sub-lattice of $\mathbb {Z}$, and thus achieve a spatiotemporal trade off. The spatiotemporal trade off is motivated by several industrial applications (Lu and Vetterli, 2249–2252, 2009). 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, thus generalizing the idea of the finite dimensional case in Aldroubi and Krishtal, arXiv:1412.1538 (2014). 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 a perturbation analysis based on the spectral properties of the operator A and initial state x, and verify the results by several numerical experiments. Finally, we provide several other numerical techniques to stabilize the proposed method, with some examples to demonstrate the improvement.