Bias-Free Estimation of the Auto- and Cross-Covariance and the Corresponding Power Spectral Densities from Gappy Data

Signal processing of uniformly spaced data from stationary stochastic processes with missing samples is investigated. Besides randomly and independently occurring outliers also correlated data gaps are investigated. Non-parametric estimators for the mean value, the signal variance, the autocovariance and cross-covariance functions and the corresponding power spectral densities are given, which are bias-free, independent of the spectral composition of the data gaps. Bias-free estimation is obtained by averaging over valid samples only from the data set. The procedures abstain from interpolation of missing samples. An appropriate bias correction is used for cases where the estimated mean value is subtracted out from the data. Spectral estimates are obtained from covariance functions using Wiener-Khinchin's theorem.