Effect of inter-sample spacing constraint on spectrum estimation with irregular sampling

TL;DR

This paper investigates how a minimum inter-sample spacing constraint affects spectrum estimation from irregular samples, showing limitations for alias-free sampling and highlighting the need for new estimators for band-limited spectra.

Contribution

It demonstrates that under the spacing constraint, no point process sampling scheme is alias-free for all spectra, and discusses the implications for spectrum estimation methods.

Findings

No alias-free point process sampling scheme exists for all spectra under the constraint.

Sampling can be alias-free for band-limited spectra despite the constraint.

Common estimators perform poorly when the minimum spacing constraint is applied.

Abstract

A practical constraint that comes in the way of spectrum estimation of a continuous time stationary stochastic process is the minimum separation between successively observed samples of the process. When the underlying process is not band-limited, sampling at any uniform rate leads to aliasing, while certain stochastic sampling schemes, including Poisson process sampling, are rendered infeasible by the constraint of minimum separation. It is shown in this paper that, subject to this constraint, no point process sampling scheme is alias-free for the class of all spectra. It turns out that point process sampling under this constraint can be alias-free for band-limited spectra. However, the usual construction of a consistent spectrum estimator does not work in such a case. Simulations indicate that a commonly used estimator, which is consistent in the absence of this constraint, performs poorly when the constraint is present. These results should help practitioners in rationalizing their expectations from point process sampling as far as spectrum estimation is concerned, and motivate researchers to look for appropriate estimators of bandlimited spectra.