Closed-Form Approximation of the EE-SE Trade-off for Multi-hop MIMO-RIS Communication Systems

Energy efficiency (EE) is one of the main design criterion for the current and next generation of communication systems. Whereas, reflective intelligent surface (RIS) is foreseen to be a key enabler of the next generation of communication systems by facilitating the propagation of radio frequency signals, and in turn, possibly improving its spectral efficiency (SE) and/or EE. This paper investigates both the EE and SE of a multi-hop multi-antenna RIS-aided communication system through its fundamental trade-off. To this end, we first provide a generic and accurate closed-form approximation (CFA) of the SE (ergodic capacity) for multi-hop multi-antenna RIS-aided systems and verify its accuracy through simulations for various numbers of antennas/phase shifters and hops. Based on this expression, we then derive a novel and accurate CFA of the fundamental EE-SE trade-off for multi-hop multi-antenna RIS-aided systems. We subsequently use our CFA to analyse the variations of the EE as a function of the number of antennas/phase shifters and hops when considering a realistic power consumption model. It turns out that increasing the number of hops is more energy efficient than increasing the number of antennas/phase shifters and that multi-hop communication with RIS is not necessarily always more energy efficient than classic multi-antenna communication, as it is for instance the case in a simple device-to-device communication scenario.