Sampling with Autoregressive Clock Jitter: Infinite-Dimensional Stability Failure and Finite-Dimensional Conditioning

Abstract

Motivated by recent developments in stochastic modeling of clock jitter in Analog-to-Digital Converters (ADCs) as autoregressive processes of order one (AR(1)), we study the density and stability properties of AR(1)-jittered sampling sets for Paley-Wiener signals. While the stability of sampling sets under i.i.d. perturbations of the integer lattice is classical, no analogous analysis was previously available for the correlated jitter observed in practice. We show that, despite having the correct asymptotic density both on average and almost surely, such sets almost surely fail to be stable sampling sets. We complement this negative result with a finite-dimensional analysis, providing explicit non-asymptotic conditions on the number of samples $2N+1$, the jitter standard deviation ${\sigma }_{\xi }$ and the confidence level $1-\eta$ under which the smallest singular value of the $(2N+1)\times (2N+1)$ jittered sinc matrix exceeds a prescribed threshold $\epsilon$ with probability at least $1-\eta$, so that the matrix is well-conditioned with high probability.

arXiv commentComments are welcome! To appear in Signal Processing. Tightened the statement of one Lemma, fixed minor wording, clarified a couple of points in the proof of Theorem 2