A consistent nonparametric spectral estimator for randomly sampled signals

In this paper, the nonparametric spectral analysis of a randomly sampled signal is discussed. For this purpose, the general form of Masryʹs recursive spectral estimators is considered and improved. The corresponding algorithms require the knowledge of the sampling times, the samples, and the sampling law and that the high-frequency decay of the spectrum be at least roughly known. The minimization of the asymptotic equivalent of the mean square estimation error leads to an estimator that is not only consistent but also asymptotically optimal. Theoretical and simulation results regarding this new estimator are given and compared with the standard methods. Besides, the developed theoretical framework strictly includes the case where the sampling time process is Poisson and the signal is Gaussian.