Consistent Estimation of Non-bandlimited Spectral Density From Uniformly Spaced Samples

In the matter of selection of sample time points for the estimation of the power spectral density of a continuous time stationary stochastic process, irregular sampling schemes such as Poisson sampling are often preferred over regular (uniform) sampling. A major reason for this preference is the well-known problem of inconsistency of estimators based on regular sampling, when the underlying power spectral density is not bandlimited. It is argued in this paper that, in consideration of a large sample property like consistency, it is natural to allow the sampling rate to go to infinity as the sample size goes to infinity. Through appropriate asymptotic calculations under this scenario, it is shown that the smoothed periodogram based on regularly spaced data is a consistent estimator of the spectral density, even when the latter is not band-limited. It transpires that, under similar assumptions, the estimators based on uniformly sampled and Poisson-sampled data have about the same rate of convergence. Apart from providing this reassuring message, the paper also gives a guideline for practitioners regarding appropriate choice of the sampling rate. Theoretical calculations for large samples and Monte-Carlo simulations for small samples indicate that the smoothed periodogram based on uniformly sampled data have less variance and more bias than its counterpart based on Poisson sampled data.

I. Introduction

Estimation of the power spectral density of a continuous time wide sense stationary stochastic process is an old problem. A set of regularly (uniformly) spaced samples is generally used for this purpose. When the process is bandlimited, the spectral density of the original process can be recovered from that of the sampled process, provided that the sampling is fast enough. In such a case, estimation of the spectral density from finitely many observations at an appropriate sampling rate is a well established topic and many useful nonparametric and parametric methods have been developed [1]. If the underlying process is not bandlimited, the spectral density of the original process is not identifiable from regularly spaced samples, because of the problem of wrapping around of the spectral density caused by the process of sampling—also known as aliasing [2]. In such a case, one cannot estimate the spectral density consistently from regularly spaced samples at any fixed sampling rate.