Random sampling of deterministic signals: statistical analysis of Fourier transform estimates

This paper considers the estimation of the Fourier transform of continuous-time deterministic signals from a finite number N of discrete-time nonuniform observations. We first extend the recent results (mean and variance) of Tarczynski and Allay by providing both distributional properties as well as rates of almost sure convergence for the simple random sampling scheme of Tarczynski and Allay. The primary focus is to consider and investigate the properties of a class of stratified random sampling estimates. We establish the statistical properties of the estimates, including precise expressions and rate of convergence of the mean-square errors. We also establish the asymptotic normality of the estimates useful for confidence interval calculations. Further, we optimize over the class of the estimates in order to obtain the best performance. In particular, we show that for functions with a first-order continuous derivative, the mean-square estimation error decays precisely at the fast rate of 1/N³. This rate is significantly higher than the rate of 1/N for simple random sampling estimates. The analytical results are illustrated by numerical examples. The effect of observation errors on the performance is also investigated.