  A full performance analysis of the widely linear (WL) minimum variance
distortionless response (MVDR) beamformer is introduced. While the WL MVDR is
known to outperform its strictly linear counterpart, the Capon beamformer, for
noncircular complex signals, the existing approaches provide limited physical
insights, since they explicitly or implicitly omit the complementary
second-order (SO) statistics of the output interferences and noise (IN). To
this end, we exploit the full SO statistics of the output IN to introduce a
full SO performance analysis framework for the WL MVDR beamformer. This makes
it possible to separate the overall signal-to-interference plus noise ratio
(SINR) gain of the WL MVDR beamformer w.r.t. the Capon one into the individual
contributions along the in-phase (I) and quadrature (Q) channels. Next, by
considering the reception of the unknown signal of interest (SOI) corrupted by
an arbitrary number of orthogonal noncircular interferences, we further unveil
the distribution of SINR gains in both the I and Q channels, and show that in
almost all the spatial cases, these performance advantages are more pronounced
when the SO noncircularity rate of the interferences increases. Illustrative
numerical simulations are provided to support the theoretical results.
