Statistical Analysis of Fourier Transform Estimates: Monte Carlo and Stratified Sampling

We consider the estimation of the Fourier transform of continuous-time deterministic signals from a finite number N of discrete-time non-uniform observations. The primary focus is to investigate the properties of a class stratified random sampling estimates. We obtain the statistical properties of the estimates including precise expressions and rate of convergence of the mean-square errors. We also optimize over the class 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 first rate of 1/N³. This rate is significantly higher than the rate of 1/N for standard Monte Carlo. The analytical results are illustrated by numerical examples