Practical Realization of Bessel's Correction for a Bias-Free Estimation of the Auto-Covariance and the Cross-Covariance Functions

To derive the auto-covariance function from a sampled and time-limited signal or the cross-covariance function from two such signals, the mean values must be estimated and removed from the signals. If no a priori information about the correct mean values is available and the mean values must be derived from the time series themselves, the estimates will be biased. For the estimation of the variance from independent data the appropriate correction is widely known as Bessel's correction. Similar corrections for the auto-covariance and for the cross-covariance functions are shown here, including individual weighting of the samples. The corrected estimates then can be used to correct also the variance estimate in the case of correlated data. The programs used here are available online at http://sigproc.nambis.de/programs.