A generalization of the magnitude squared coherence spectrum for more than two signals: definition, properties and estimation

The coherence spectrum is a well-known measure of the linear statistical relationship between two time series. In this paper, we extend this concept to several processes and define the generalized magnitude squared coherence (GMSC) spectrum as a function of the largest eigenvalue of a matrix containing all the pairwise complex coherence spectra. The GMSC is bounded between zero and one, and attains its maximum when all the processes are perfectly correlated at a given frequency. Furthermore, three different GMSC spectrum estimators, extending those previously proposed for the MSC of two processes, are presented. Specifically, we compare the Welch method, the minimum variance distortionless response (MVDR) estimator and a new estimator based on canonical correlation analysis (CCA).