Small-sample unbiased linear coherence estimators for a complex Gaussian random process

The Pearson sample correlation coefficient, the maximum likelihood linear coherence estimator for a jointly Gaussian random process, is ubiquitously used in remote sensing and forecasting systems and dictates their performance. However, the asymptotically optimal properties of this estimator degrade in practical cases due to the finite samples available. Here, we analytically derive small-sample unbiased estimators for the correlation coefficient and its squared modulus of a jointly circular complex Gaussian random process. Each obtained complete and sufficient statistic is the unique small-sample unbiased estimator attaining minimum possible variance, and is expressed as a bijective function of the corresponding maximum likelihood estimator. The unbiased squared modulus estimator function aggressively corrects the positive bias in the squared sample correlation coefficient by extending the function's range to negative values, while the unbiased complex estimator function has little bias-correcting effect. For the absolute modulus of the complex correlation coefficient, we find an unbiased estimator does not exist. Performances of the derived estimators are analyzed, and we propose an estimator that improves both the bias and the mean square error of the squared sample correlation coefficient. Examples are provided, showing the bias in ocean acoustic coherence measurements and its impact on array design.