## 3 General Formulation
### Signal-to-Interference-Plus-Noise Ratio (SINR)  
From (9), it is easy to obtain SINR since the attenuated mapped vector is separated from the interference and additive noise. Indeed, we have that \(\mathbf{\Lambda}_{\mathbf{A}_{0}}\mathbf{X}_{\zeta,l}\) is the attenuated mapped vector, \(\overline{\mathbf{\Lambda}}_{\mathbf{A}_{0}}\mathbf{X}_{\zeta,l}+\sum_{i=0}^{l }\mathbf{A}_{i}\mathbf{X}_{\zeta,l-1}\) is the interference, and \(\mathbf{G}\mathbf{V}_{l}\) is the additive noise contaminated by the noise-enhancement effect. Therefore, the attenuated mapped vector power matrix is given by  
\[\begin{split}\mathbf{P}_{\text{s}}&=\frac{1}{2N} \mathbf{\Lambda}_{\mathbf{A}_{0}}\mathbb{E}\{\mathbf{X}_{\zeta,l}\mathbf{X}_{ \zeta,l}^{\dagger}\}\mathbf{\Lambda}_{\mathbf{A}_{0}}^{\dagger}\\ &=\mathbf{\Lambda}_{\mathbf{A}_{0}}\mathbf{P}_{\text{x}}\mathbf{ \Lambda}_{\mathbf{A}_{0}}^{\dagger},\end{split} \tag{17}\]  
where \(\mathbf{P}_{\text{x}}=\mathbb{E}\{\mathbf{X}_{\zeta,l}\mathbf{X}_{\zeta,l}^{ \dagger}\}/2N\) is a diagonal matrix with elements equal to the transmission power. Moreover, the interference and additive noise are independent WSS random processes, and, consequently, the interference-plus-noise power matrix is defined by  
\[\mathbf{P}_{i+n}\triangleq\overline{\mathbf{\Lambda}}_{\mathbf{A}_{0}}\mathbf{ P}_{\text{x}}\overline{\mathbf{\Lambda}}_{\mathbf{A}_{0}}^{\dagger}+\sum_{i=-1 \atop i\
eq 0}^{l}\mathbf{A}_{i}\mathbf{P}_{\text{x}}\mathbf{A}_{i}^{\dagger}+ \mathbf{G}\mathbf{P}_{\text{n}}\mathbf{G}^{\dagger}, \tag{18}\]  
with \(\mathbf{P}_{\text{n}}=\mathbb{E}\{\mathbf{V}_{l}\mathbf{V}_{l}^{\dagger}\}/2N\) being a diagonal matrix with entries equal to the noise power. Consequently, the SINR matrix of the \(2N\)-length symbol block is given by  
\[\mathbf{\Lambda}_{\gamma_{\text{SINR}}}=\frac{\mathbf{\Lambda}_{\mathbf{P}_{ \text{s}}}}{\mathbf{\Lambda}_{\mathbf{P}_{\text{i+n}}}}, \tag{19}\]  
meaning that \(\mathbf{\Lambda}_{\gamma_{\text{SINR}}}[k,k]\) is the SINR associated with the \(k^{\text{th}}\) subcarrier, subchirp, subslot, or subtile. As shown by expressions (17) through (19), the interference is harmful since it is directly proportional to the transmission power's strength, meaning that the increase of the transmission power also increases the interference. If the transmission power is extremely high (i.e., \(P_{\text{x}}\rightarrow\infty\)), then we can accept that \(\mathbf{P}_{\text{x}}\approx P_{\text{x}}\mathbf{I}_{2N}\). Consequently, the SINR matrix is expressed as \[\begin{split}\mathbf{\Lambda}_{\gamma_{\text{SINR,max}}}& =\lim_{P_{x}\rightarrow\infty}\frac{\mathbf{\Lambda}_{\mathbf{P}_{\text{s}}}}{\mathbf{ \Lambda}_{\mathbf{P}_{\text{in}}n}}\\ &=\frac{\mathbf{\Lambda}_{|\mathbf{\Lambda}_{0}|^{2}}}{\mathbf{\Lambda}_{| \overline{\mathbf{\Lambda}}_{\mathbf{\Lambda}_{0}}|^{2}}+\sum_{i=-1}^{l}\mathbf{\Lambda}_ {|\mathbf{\Lambda}_{i}|^{2}}}.\end{split} \tag{20}\]  
As (20) does not depend on the PSD of the additive noise, we anticipate that the data rate (upper bound) and the BEP (lower bound) will not depend on the PSD of the additive noise. Finally, without interference, \(\mathbf{\Lambda}_{\gamma_{\text{SINR,max}}}\rightarrow\infty\), as expected.