Antithetical random sampling: Statistical analysis of fourier transforms estimators

We consider the estimation of the Fourier transform of continuous-time signals from a finite set N of discrete-time nonuniform observations. We introduce a class of antithetical stratified random sampling schemes and we obtain the performance of the corresponding estimates. For functions f(t) with two continuous derivatives, we show that the rate of mean square convergence is l/N5, which is considerably faster that the rate of l/N³ for stratified sampling and the rate of l/N for standard Monte Carlo integration. In addition, we establish joint asymptotic normality for the real and imaginary parts of the estimate. The theoretical results are illustrated by examples for lowpass and highpass signals.