Statistical evaluation of coherence estimated from optimally beamformed signals

In this paper we investigate a situation where we want to perform a coherence analysis of two signal sources, one of which is measured directly, and the other is measured through a sensor array affected by noise. To extract the latter signal, we suggest the use of the optimal beamforming with reference. We note, however, that this approach results in a coherence estimate that is noticeably biased, and cannot be evaluated by the known statistical tests. We therefore derive a new statistical test, that allows the evaluation of the biased coherence estimate. We illustrate the applicability of our methodology on the coherence analysis of EEG and EMG signals. We note that the suggested approach has several advantages over the surface Laplacian, which is a spatial filter commonly used in the EEG–EMG coherence analysis.