Abstract
We estimate the Fourier transform of continuous-time signals on the basis of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> discrete-time nonuniform observations. We introduce a class of antithetical stratified random sampling schemes and we obtain the performance of the corresponding estimates. We show that when the underlying function <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ) has a continuous second-order derivative, the rate of mean square convergence is 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> , which is considerably faster that the rate of 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> for stratified sampling and the rate of 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> for standard Monte Carlo integration. In addition, we establish joint asymptotic normality for the real and imaginary parts of the estimate and give an explicit expression for the asymptotic covariance matrix. The theoretical results are illustrated by examples for low-pass and high-pass signals.
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