Instantaneous oscillatory direction and phase for multivariate timeseries

This text describes a generalization of the analytic signal (Gabor, 1946) approach for the definition of instantaneous amplitude and phase to the case of multivariate signals. It was originally written as an appendix for another paper, where the determination of the locally dominant oscillatory direction (the instantaneous amplitude) described here is used as a preprocessing step for another kind of data analysis. The text is reproduced in a ‘standalone’ form because the procedure might prove useful in other contexts too, especially for the purpose of phase synchronization analysis (Rosenblum et al., 1996) between two (or more) multivariate sets of time series (Pascual-Marqui, 2007). The local oscillatory behavior of a real-valued univariate signal x(t) is commonly characterized using the corresponding complex-valued analytic signal z(t). It is obtained by combining x(t) with an imaginary part, z(t) = x(t) + i y(t), which is defined as the Hilbert transform of x, y(t) = H x(t) = 1 π P.V. ∫