  The free-space optical (FSO) communications can achieve high capacity with
huge unlicensed optical spectrum and low operational costs. The corresponding
performance analysis of FSO systems over turbulence channels is very limited,
especially when using multiple apertures at both transmitter and receiver
sides. This paper aim to provide the ergodic capacity characterization of
multiple-input multiple-output (MIMO) FSO systems over atmospheric
turbulence-induced fading channels. The fluctuations of the irradiance of
optical channels distorted by atmospheric conditions is usually described by a
gamma-gamma ($\Gamma \Gamma$) distribution, and the distribution of the sum of
$\Gamma \Gamma$ random variables (RVs) is required to model the MIMO optical
links. We use an $\alpha$-$\mu$ distribution to efficiently approximate the
probability density function (PDF) of the sum of independent and identical
distributed $\Gamma\Gamma$ RVs through moment-based estimators. Furthermore,
the PDF of the sum of independent, but not necessarily identically distributed
$\Gamma \Gamma$ RVs can be efficiently approximated by a finite weighted sum of
PDFs of $\Gamma \Gamma$ distributions. Based on these reliable approximations,
novel and precise analytical expressions for the ergodic capacity of MIMO FSO
systems are derived. Additionally, we deduce the asymptotic simple expressions
in high signal-to-noise ratio regimes, which provide useful insights into the
impact of the system parameters on the ergodic capacity. Finally, our proposed
results are validated via Monte-Carlo simulations.
