System identification in dynamical sampling
- PDF / 817,270 Bytes
- 26 Pages / 439.642 x 666.49 pts Page_size
- 117 Downloads / 189 Views
System identification in dynamical sampling Sui Tang1
Received: 30 August 2015 / Accepted: 8 November 2016 / Published online: 22 November 2016 © Springer Science+Business Media New York 2016
Abstract We consider the problem of spatiotemporal sampling in a discrete infinite dimensional spatially invariant evolutionary process x (n) = An x to recover an unknown convolution operator A given by a filter a ∈ 1 (Z) and an unknown initial state x modeled as a vector in 2 (Z). Traditionally, under appropriate hypotheses, any x can be recovered from its samples on 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, Ax, · · · , AN−1 x on a sub-lattice of 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(mZ), Ax(mZ), · · · , AN−1 x(mZ) : N ≥ 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. Communicated by: Yang Wang The research of this work is supported by NSF Grant DMS-1322099 Sui Tang
[email protected] 1
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
556
S. Tang
Keywords Discrete fourier analysis · Distributed sampling · Reconstruction · Channel estimation Mathematics Subject Classification (2010) Primary 94A20 · 94A12 · 42C15 · 15A29
1 Introduction 1.1 The dynamical sampling problem In many situations of practical interest, physical systems evolve in time under the action of well-studied operators, an example of which is a diffusion process. Sampling of such an evolving system can be done by sensors or measurement devices that are placed at various locations and can be activated at different times. In practice, increasing the spatial sampling density is usually much more expensive than increasing the temporal sampling rate [34]. Given the different costs associated with spatial and temporal sampling, we aim to reconstruct any states in the evolutionary process using as few sensors as possible, but allow one to take samples at different time levels. In some cases, obtaining samples at a sufficient rate at any single time level may not even be possible; however, spatiotemporal sampling may resolve this issue by oversampling in ti
Data Loading...
