  In this paper, a generic extension of variational mode decomposition (VMD)
algorithm for multivariate or multichannel data sets is presented. We first
define a model for multivariate modulated oscillations that is based on the
presence of a joint or common frequency component among all channels of input
data. Using that model for multivariate oscillations, we construct a
variational optimization problem that aims to extract an ensemble of
band-limited modes containing inherent multivariate modulated oscillations
present in multivariate input signal. The cost function to be minimized is the
sum of bandwidths of all signal modes across all input data channels, which is
a generic extension of the cost function used in standard VMD to multivariate
data. Minimization of the resulting variational model is achieved through the
alternate direction method of multipliers (ADMM) approach. That yields an
optimal set of multivariate modes in terms of narrow bandwidth and
corresponding center frequencies that are assumed to be commonly present among
all channels of a multivariate modulated oscillation. We demonstrate the
effectiveness of the proposed method through results obtained from extensive
simulations involving test (synthetic) and real world multivariate data sets.
Specifically, we focus on the ability of the proposed method to yield joint
oscillatory modes in multivariate data which is a prerequisite in many real
world applications involving nonstationary multivariate data. We also highlight
the utility of the proposed method in two real world applications which include
the separation of alpha rhythms in multivariate electroencephalogram (EEG) data
and the decomposition of bivariate cardiotocographic signals that consist of
fetal heart rate and maternal uterine contraction (FHR-UC) as its two channels.
