  Blind source separation (BSS) is a signal processing tool, which is widely
used in various fields. Examples include biomedical signal separation, brain
imaging and economic time series applications. In BSS, one assumes that the
observed $p$ time series are linear combinations of $p$ latent uncorrelated
weakly stationary time series. The aim is then to find an estimate for an
unmixing matrix, which transforms the observed time series back to uncorrelated
latent time series. In SOBI (Second Order Blind Identification) joint
diagonalization of the covariance matrix and autocovariance matrices with
several lags is used to estimate the unmixing matrix. The rows of an unmixing
matrix can be derived either one by one (deflation-based approach) or
simultaneously (symmetric approach). The latter of these approaches is
well-known especially in signal processing literature, however, the rigorous
analysis of its statistical properties has been missing so far. In this paper,
we fill this gap and investigate the statistical properties of the symmetric
SOBI estimate in detail and find its limiting distribution under general
conditions. The asymptotical efficiencies of symmetric SOBI estimate are
compared to those of recently introduced deflation-based SOBI estimate under
general multivariate MA$(\infty)$ processes. The theory is illustrated by some
finite-sample simulation studies as well as a real EEG data example.
