Distribution of the multiple coherence estimate for the detection of periodic signals in noise

Abstract

The sampling distribution of the multiple coherence estimate between one periodic, deterministic signal and a set of N other Gaussian-distributed ones has been derived. This has been accomplished based on a relationship between the estimate and the Hotelling's T2 statistic extended to complex variables and further on another one involving the noncentral F and beta distributions. The obtained probability density function (PDF) was confirmed by means of simulation, which was also used to evaluate the sampling distribution of another multivariate detector. The PDF reduces to that of the univariate case when N is set to one and the zero-coherence case is easily assessed by setting the noncentrality parameter to zero. This expression also holds for the multiple coherence involving only Gaussian signals. Therefore, the multivariate extension of the well-known invariance of simple coherence estimate with respect to second signal provided the first is Gaussian and coherence is zero is now proved. An illustrating example with simulated signals is also provided. The results in this communication may be useful, for example, in investigations involving the multichannel electroencephalogram during sensory stimulation for monitoring of surgeries or in brain-computer interfaces, where one aims at detecting the evoked responses as fast as possible.