  Although the widely linear least mean square error (WLMMSE) receiver has been
an appealing option for multiple-input-multiple-output (MIMO) wireless systems,
a statistical understanding on its pose-detection
signal-to-interference-plus-noise ratio (SINR) in detail is still missing. To
this end, we consider a WLMMSE MIMO transmission system with rectilinear or
quasi-rectilinear (QR) signals over the uncorrelated Rayleigh fading channel
and investigate the statistical properties of its SINR for an arbitrary antenna
configuration with $N_t$ transmit antennas and $N_r$ receive ones. We first
derive an analytic probability density function (PDF) of the SINR in terms of
the confluent hypergeometric function of the second kind, for WLMMSE MIMO
systems with an arbitrary $N_r$ and $N_t=2, 3$. For a more general case in
practice, i.e., $N_t>3$, we resort to the moment generating function to obtain
an approximate but closed form PDF under some mild conditions, which, as
expected, is more Gaussian-like as $2N_r-N_t$ increases. The so-derived PDFs
are able to provide key insights into the WLMMSE MIMO receiver in terms of the
outage probability, the symbol error rate, and the diversity gain, all
presented in closed form. In particular, its diversity gain and the gain
improvement over the conventional LMMSE one are explicitly quantified as
$N_r-(N_t-1)/2$ and $(N_t-1)/2$, respectively. Finally, Monte Carlo simulations
support the analysis.
