Sum and Product of Multivariate Fox-H Functions: Exact Analysis for RIS-Assisted System Over Rician Fading With Imperfect CSI

Existing performance analysis of wireless systems based on reconfigurable intelligent surfaces (RIS) over fading channels has primarily focused on providing statistical insights into the sum and product of random variables, typically represented as a single-variate Fox-H function. A research gap exists for an exact performance analysis when the statistical characterization of wireless systems, including RIS-assisted systems, requires more than a single-variate Fox-H function. In such cases, incorporating a multivariate representation becomes imperative, particularly when addressing generalized fading models and channel estimation errors, even with simpler one-parameter fading models. This paper introduces a novel approach to derive the distribution of the sum and product of independent and nonidentical distributed (i.ni.d) random variables characterized by the multivariate Fox-H function. We also establish a general framework for an exact analysis of ergodic capacity and average signal-to-noise ratio (SNR) when the multivariate Fox-H function describes the statistics of the channel. Applying the derived results, we conduct an exact performance analysis of outage probability and ergodic capacity, exemplified by RIS-assisted communication over Rician fading channels with imperfect phase compensation and channel estimation errors. Computer simulations validate the exact analysis and illustrate the performance of the RIS-assisted system under various practically relevant scenarios, enhancing the overall performance assessment.