On the Sum of Products of Independent Complex Normal Variables: Understanding the Fundamental SNR Gain Limit of MIMO-RIS

The dual objective of this paper is to first characterize the probability distribution of the random variable (RV),
$U$
, equivalent to the sum of multiple products of two independent circularly symmetric complex normal (CN) RVs, in order to then evaluate the fundamental limit of multi-input multi-output (MIMO)-reconfigurable intelligent surface (RIS) communication systems in terms of channel/signal-to-noise ratio (SNR) gain. First, we derive simple closed-form expressions of the joint moment generating function and probability density function (pdf) of the real and imaginary parts of
$U$
, as well as the pdf and mean value of
$|U|$
, and validate their accuracy via numerical simulations. Second, we provide a rigorous analysis of how MIMO-RIS improves channel propagation by relying on our probabilistic characterization of
$U$
. More specifically, we obtain a simple upper bound of the MIMO-RIS capacity, expressed as a function of its channel gain, for a large number of RIS elements. We then derive closed-form expressions of this channel gain, as well as the MIMO-RIS to MIMO SNR gain, and validate their accuracy via numerical simulations. Our analysis leads to the following insights: 1) RIS improvement in channel propagation accounts for a “baseline”/intrinsic improvement due to the amplitude response of each RIS phase-shifter element and a directionality improvement due to the optimization of the RIS phase-shifters used for steering the incident radiation towards its intended direction. 2) The intrinsic improvement is larger than the directionality improvement, and the latter is only significant when the number of RIS elements is very large in comparison with the square root of the product between the number of transmit and receive antennas.