# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 2 System Model  
We consider the communication from a source equipped with \(N_{t}\) antennas to a destination with \(N_{r}\) antennas. The reliable communication from the transmitter to the destination is assumed to be only feasible by means of a relay with \(K_{t}\) transmitter and \(K_{r}\) receiver antennas at the output and input front ends, respectively. This means that the direct link from the transmitter to the destination and the link from the transmitter to the IRS and to the destination has a negligible impact on the throughput. This assumption is realistic for the scenarios where the path loss is high due to the high frequency ranges such as mmWave and Terahertz or due to far distances [35], as well as cases were there are objects that block the direct link between the source and the destination. An IRS consisting of \(M\) elements is established to either cancel the RSI or help enhance one of the transmitter-relay/relay-destination links. The overall system model can be found in Figure 1.  
In this paper, it is assumed that signal delivery over the transmitter-IRS-receiver link is not available. This is mainly due to the power attenuation and the power radiation pattern effects [36]. As the IRS is a passive device, it has some power attenuation in practice, which makes the reflected waves weaker than the received ones. In addition, due to the power radiation pattern, based on the angle of arrival and departure, both received and reflected waves are subject to attenuation, respectively. In our system, as the IRS is established in the vicinity of the relay and is faced towards it, both aforementioned effects cause the source-IRS-destination link to be extremely weaker than the source-relay-destination link.  
Next, we present the achievable throughput rates for the HD and FD relaying. We start with the HD relay in which \(\kappa=0\). In the second case, IRS can be exploited to enhance the quality of the channel between the source and the relay. In such a case, the received signals at the relay and destination can be expressed as  
\[\mathbf{y}_{r} =(\mathbf{H}_{1}+\mathbf{H}_{IR}\mathbf{\Theta}\mathbf{H}_{SI}) \mathbf{x}_{\mathrm{s}}+\kappa\mathbf{H}_{r}\mathbf{x}_{r}+\mathbf{n}_{t}, \tag{1}\] \[\mathbf{y}_{\mathrm{d}} =\mathbf{H}_{2}\mathbf{x}_{r}+\mathbf{n}_{\mathrm{d}}, \tag{2}\]  
where \(\mathbf{H}_{SI}\in\mathbb{C}^{M\times N_{t}}\) is the channel from the source to the IRS. Finally, IRS can be used to help the channel from the relay to the destination. In this case, the received signals are going to be  
\[\mathbf{y}_{r} =\mathbf{H}_{1}\mathbf{x}_{\mathrm{s}}+\kappa\mathbf{H}_{r} \mathbf{x}_{r}+\mathbf{n}_{t}, \tag{3}\] \[\mathbf{y}_{\mathrm{d}} =(\mathbf{H}_{2}+\mathbf{H}_{ID}\mathbf{\Theta}\mathbf{H}_{RI}) \mathbf{x}_{r}+\mathbf{n}_{\mathrm{d}}, \tag{4}\]  
where \(\mathbf{H}_{ID}\in\mathbb{C}^{M\times N_{t}}\) is the channel from the source to the IRS.  
Figure 1: System model of an IRS assisted full-duplex relay. In our model both source and destination are equipped with \(N_{t}\) and \(N_{r}\) antennas, respectively. In addition, the relay is equipped with \(K_{t}\) transmitting and \(K_{t}\) receiving antennas, and the IRS has \(M\) passive elements.  
In what follows, we find the achievable rate for three aforementioned cases and compare them to see under what conditions each of them should be applied. Notation and definitions are summarized in Table 1.