On the Distribution of SINR for MIMO Systems with Widely Linear MMSE Receivers

The widely linear least mean square error (WLMMSE) receiver has been an appealing option for multiple-input-multiple-output (MIMO) wireless systems, however, a statistical understanding on its pose-detection signal-to-interference-plus-noise ratio (SINR) in detail is still missing. To this send, we consider a WLMMSE MIMO transmission system 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 show that the SINR expression can be interpreted as the sum of $N_t$ different and independent gamma random variables. Based on this finding, an analytic probability density function (PDF) of the SINR is exactly derived, in terms of the confluent hypergeometric function of the second kind, for WLMMSE MIMO systems with $N_t=2$ and 3, while 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. The so-derived PDFs further provide more physical insights into WLMMSE MIMO receivers in terms of the outage probability, the symbol error rate, and the diversity gain, all presented in closed form. Monte Carlo simulations support the analysis.