## 2 The System and Channel Models for THz Band
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Definition** \\ \hline \(\Delta f_{m,n}\) & Bandwidth for the \(n\)-th sub-band in the \(m\)-th frequency window \\ \hline \(P_{m,n}\) & Transmit power for the \(n\)-th sub-band in the \(m\)-th frequency window \\ \hline \(M_{m,n}\) & Modulation order for the \(n\)-th sub-band in the \(m\)-th frequency window \\ \hline \(W(d)\) & Number of frequency sub-windows \\ \hline \(N_{m}\) & Number of subcarriers in the \(m\)-th sub-window \\ \hline MQAM & Quadrature Amplitude Modulation with a modulation order of M \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._\(p_{4}=-6.36\times 10^{-3}\)[39]. The parameters \(\varphi\), \(p\), and \(T\) denote the relative humidity, the pressure given in hectopascals, and the temperature given in degrees Celsius, respectively. The beam misalignment fading coefficient \(h_{p}\), which represents the fractional power collected across Rx with area, \(A\) at distance \(d\) is expressed as [48; 49]  
\[h_{p}=A_{0}\exp\left(-\frac{2r^{2}}{w_{eq}^{2}}\right) \tag{3}\]  
where \(r\) is the pointing error expressed as the radial distance of the transmission and reception beams. The parameters \(A_{0}=(erf(u))^{2}\) with the error function \(erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt\), \(w_{eq}^{2}=w_{d}^{2}\frac{erf(u)}{2 iw\exp(-u^{2})}\), \(u=\frac{\sqrt{\pi}a}{\sqrt{2}u_{d}}\). The parameters \(a\) and \(w_{d}\) are the radius of the receiver effective area and the transmit beam footprint at distance \(d\). The multi-path fading coefficient \(h_{f}\) is modeled as \(\alpha\) - \(\mu\) distributed [50],  
\[f_{h_{f}}(x)=\frac{\alpha\mu^{\mu}}{\hat{h}_{f}^{\alpha\mu}\Gamma(\mu)}x^{ \alpha\mu-1}\exp\left(-\frac{ux^{\alpha}}{\hat{h}_{f}^{\alpha}}\right) \tag{4}\]  
where \(\alpha>0\) is a fading parameter, whereas \(\mu\) and \(\hat{h}_{f}^{\alpha}\) denote the fading channel envelope's normalized variance and \(\alpha\)-root mean value, respectively. It is to be noted here that the \(\alpha\) - \(\mu\) distribution is a generalized form of many well-known fading distributions, e.g., Rayleigh (\(\alpha=2\), \(\mu=1\)), Nakagami-\(m\) (\(\alpha=2\), and \(\mu\) is the fading parameter), and so on. According to [48], the joint distribution of \(|h_{fp}|\) is  
\[\mathrm{F}_{|h_{fp}|}(x)=1-\frac{1}{\alpha}\frac{x^{\gamma^{2}}}{\hat{h}_{f}^ {\gamma^{2}}}\frac{\gamma^{2}}{A_{o}^{\gamma^{2}}}\times\sum_{k=0}^{\mu-1} \frac{\mu^{\gamma^{2}}}{k!}\Gamma\left(\frac{\alpha k-\gamma^{2}}{\alpha}, \mu\frac{x^{\alpha}}{\hat{h}_{f}^{\alpha}}A_{o}^{-\alpha}\right) \tag{5}\]  
where the parameter \(\gamma=\frac{w_{eq}}{2\sigma_{s}}\). \(\sigma_{s}^{2}\) is the variance of the pointing error displacement at the receiver. An efficient distance-based available bandwidth is obtained by setting the path loss below the threshold \(L_{th}\). The path loss threshold \(L_{th}\) is obtained [12]  
\[L_{th}=P_{T}+G_{T}+G_{R}-(\gamma+P_{w}) \tag{6}\]  
where highly directional antennas or antenna arrays are advocated to overcome the the significant path loss in THz band communication, we consider \(P_{T}=10\) dBm, \(G_{T}=G_{R}=30\) dB, \(\gamma=-10\) dB and \(P_{w}=-56\) dBm [12]. The resulting path loss value is computed as \(L_{th}=136\) dB, which will be used throughout our analysis. When the available bandwidth windows are established, each frequency window is further divided into multiple non-overlapped sub-windows for multi-carrier modulation. Each sub-window's modulation order, bandwidth, and transmission power are jointly optimized to maximize the data rate.