## 2 System Model  
An IRS-assisted bidirectional FD-MTC system is shown in Figure 2, with two source nodes \(S_{1}\) and \(S_{2}\). Each node is assumed to have two antennas. The Line of Sight (LoS) link is weak between the two nodes due to obstacles and fading. Hence, IRS is placed between the two source nodes \(S_{1}\) and \(S_{2}\) to enhance the data exchange between them. IRS is composed of \(N\) reflecting metasurface elements. FD mode is employed in the proposed bidirectional MTC system, for simultaneous transmission and reception. Due to FD operation mode, SI occurs at two source nodes \(S_{1}\) and \(S_{2}\). Binary Phase Shift Keying (BPSK) symbols \(x_{1}\) and \(x_{2}\) are transmitted by source nodes \(S_{1}\) and \(S_{2}\), respectively. The transmit powers of both the source nodes \(S_{1}\) and \(S_{2}\) are assumed to be the same as \(P_{s}\). The physical channels between the source nodes \(S_{1}\), \(S_{2}\), and the IRS follow independent and identical distribution (i.i.d) of Rayleigh fading with mean \(\sigma\sqrt{\pi/2}\) and variance \(((4-\pi)/2)\sigma^{2}\), where \(\sigma\) is the scale parameter of the distribution. Let \(\mathbf{h}^{(1)}=[h_{1}^{(1)}h_{2}^{(1)}\dots h_{N}^{(1)}]^{T}\) and \(\mathbf{h}^{(2)}=[h_{1}^{(2)}h_{2}^{(2)}\dots h_{N}^{(2)}]^{T}\). The individual element in \(h_{i}^{(n)}\) is defined as,  
\[h_{i}^{(n)}=|h_{i}^{(n)}|e^{i\theta_{i}^{(n)}},i=1,2,\dots N\ \ \&\ \ n=1,2 \tag{1}\]  
The incident signal at the \(i^{th}\) element of IRS is given by  
\[y_{IRS}=\sqrt{P_{s}}[h_{i}^{(1)}x_{1}+h_{i}^{(2)}x_{2}]e^{-i\psi_{i}} \tag{2}\]  
Figure 2: IRS-assisted bidirectional FD-MTC system. \(\psi_{i}\) is the phase introduced by the IRS, which is the sum of the phases of the channel coefficients \(h_{i}^{(n)},n=1,2\). It is assumed that the phase of the channel coefficients \(h_{i}^{(n)},n=1,2\) are perfectly known. In an ideal scenario, IRS is configured to choose the optimal (continuous) phase shift \(\psi_{i},i=1,2,\ldots N\) at each element. In a practical scenario, complete phase compensation at IRS is not possible. Hence, it is configured as a discrete phase shifter of different phase levels: one-bit shifter \(b=1\) (0, \(\pi\)), two-bit shifter \(b=2\) (0, \(\pi/2\), \(\pi\), \(3\pi/2\)) and three-bit shifter \(b=3\) (0, \(\pi/4\), \(\pi/2\), \(3\pi/4\), \(\pi\), \(5\pi/4\), \(3\pi/2\), \(7\pi/4\)). It is defined as  
\[\psi_{i}=q_{b}(\theta_{i}^{(1)}+\theta_{i}^{(2)}),i=1,2,\ldots N, \tag{3}\]  
where \(q_{b}(. )\) is the quantized phase level of the discrete phase shifter. The received signal at IRS, \(y_{IRS}\) is reflected to source nodes \(S_{1}\) and \(S_{2}\) simultaneously through their reciprocity channels \(h_{i}^{(n)},n=1,2\), respectively. The received signals at the source nodes \(S_{n},n=1,2\) are expressed as,  
\[y_{s_{n}}=\sum_{i=1}^{N}h_{i}^{(n)}y_{IRS}+w_{n}+SI_{n},n=1,2 \tag{4}\]  
By substituting (3) in (4), the received signal at the source node \(S_{1}\) is rewritten as,  
\[y_{S1}=\sqrt{P_{\rm S}}\sum_{i=1}^{N}|h_{i}^{(1)}|^{2}e^{i(2\theta_{i}^{(1)}- \psi_{i})}x_{1}+\sqrt{P_{\rm S}}\sum_{i=1}^{N}|h_{i}^{(1)}||h_{i}^{(2)}|e^{i( \theta_{i}^{(1)}+\theta_{i}^{(2)}-\psi_{i})}x_{2}+w_{1}+SI_{1} \tag{5}\]  
In the received signal at source node \(S_{1}\), the first term consists of its own symbol \(x_{1}\) and the phase difference between \(h_{i}^{(1)}\) and IRS discrete phase shifter \(\psi_{i}\). The random variable in the second term is defined as \(B=Ae^{i(\theta_{i}^{(1)}+\theta_{i}^{(2)}-\psi_{i})}\), let \(A=\sum_{i=1}^{N}|h_{i}^{(1)}||h_{i}^{(2)}|\), it is the sum of the product of the magnitudes of \(h_{i}^{(1)}\) and \(h_{i}^{(2)}\)\(\forall i\), and it follows a double Rayleigh distribution approximated by the first order term of the Laguerre series [33].