## 3 Bit Error Rate Performance  
For the wireless coherent THz systems, the SNR can be expressed as \(\mathrm{Y}=\bar{\gamma}h\), where \(\bar{\gamma}\) is the average SNR, and \(h\) is the channel gain. Based on the system model outlined in Section 2, the BER for the BPSK modulation is given by  
\[P_{\rm e}(e|h)=Q(\sqrt{2\bar{\gamma}h})=\frac{1}{2}\operatorname{erfc}(\sqrt{ \bar{\gamma}h}) \tag{13}\]  
The average BER can be expressed as  
\[P_{\rm e}{=}\int_{0}^{\infty}P_{\rm e}(e|h)f_{h}(h)dh \tag{14}\]  
The average BER can be reformulated by substituting Equation (11) into Equation (14) as follows:  
\[P_{\rm e,LN}=\frac{\gamma^{2}\exp(\mu_{a})}{4(A_{0}h_{1})^{\alpha^{2}}}\int_ {0}^{\infty}h^{\gamma^{2}-1}\operatorname{erfc}(\frac{\ln(\frac{\Lambda}{A_{0 }h_{1}\zeta})+\mu_{b}}{\sqrt{8}\delta_{X}})\operatorname{erfc}(\sqrt{\bar{ \gamma}h})dh \tag{15}\]  
By applying a variable transformation, Equation (15) can be written as  
\[P_{\rm e,LN}=\frac{\gamma^{2}\mu_{c}}{4}\exp(\mu_{a}-a^{2}\mu_{b})\int_{- \infty}^{\infty}\exp(\gamma^{2}\mu_{c}x)\operatorname{erfc}(\sqrt{\bar{\gamma }A_{0}h_{1}\mu_{c}\exp(\mu_{c}x-\mu_{b})})\operatorname{erfc}(x)\,dx \tag{16}\]  
By introducing an auxiliary parameter \(B\) and dividing the integration range in Equation (16) into \([-\infty,B]\) and \([B,\infty]\), it is possible to derive  
\[ \begin{split} P_{\rm e,LN}=&\frac{\gamma^{2}\mu_{c} }{4}\exp(\mu_{a}-a^{2}\mu_{b})\int_{-\infty}^{B}\exp(\gamma^{2}\mu_{c}x)\\ &\times\operatorname{erfc}\biggl{(}\sqrt{\bar{\gamma}A_{0}h_{1} \mu_{c}\exp(\mu_{c}x-\mu_{b})}\biggr{)}\operatorname{erfc}(x)\,dx+R_{B}\end{split}  \tag{17}\]  
where \(R_{B}\) represents the error in the approximation, which is defined as\[R_{B}=\frac{\gamma^{2}\mu_{c}}{4}\exp(\mu_{a}-\gamma^{2}\mu_{b})\int_{B}^{\infty} \exp(\gamma^{2}\mu_{c}x)\operatorname{erfc}\biggl{(}\sqrt{\gamma A_{0}h_{1}\mu_{ c}\exp(\mu_{c}x-\mu_{b})}\biggr{)}\operatorname{erfc}(x)\,dx. \tag{18}\]  
The upper bound of the approximation error \(R_{B}\) is proved in reference [26]. \(R_{B}\) can be quantified under various system parameters, allowing us to calculate the corresponding values of \(B\). The findings indicate that the approximation error \(R_{B}\) diminishes quickly as \(B\) grows. By adjusting the value of \(B\), the approximation error can be minimized to an arbitrary level. Consequently, the average BER can be approximated as  
\[ \begin{split} P_{\text{e,LN}}\approx\bar{P}_{\text{e,LN}}=& \frac{\gamma^{2}\mu_{c}}{4}\exp(\mu_{a}-\gamma^{2}\mu_{b})\int_{- \infty}^{B}\exp(\gamma^{2}\mu_{c}x)\\ &\times\operatorname{erfc}\biggl{(}\sqrt{\gamma A_{0}h_{l}\mu_{ c}\exp(\mu_{c}x-\mu_{b})}\biggr{)}\operatorname{erfc}(x)\,dx\end{split}  \tag{19}\]  
In obtaining Equation (19), we use a series expansion of the complementary error function ([27], Equation (06.27.06.0002.01)) and an integral identity ([27], Equation (06.27.21.0011.01)) as  
\[\operatorname{erfc}(z)=\left(-\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{(- 1)^{k}z^{2k+1}}{k!(2k+1)}\right) \tag{20}\]  
and  
\[ \begin{split}\int\exp(bz)\exp(cz)\,dz&=\frac{1}{b} \biggl{[}\exp(bz)\operatorname{erfc}(az)-\exp\biggl{(}\frac{b^{2}}{4a^{2}} \biggr{)}\operatorname{erf}\biggl{(}\frac{b}{2a}-az\biggr{)}\biggr{]}\end{split}  \tag{21}\]  
where  
\[ \begin{split} S_{j}=&\exp((2\gamma^{2}+2k+1)\mu_{c }B/2)\operatorname{erfc}(B)-\exp((2\gamma^{2}+2k+1)^{2}\mu_{c}^{2}/16)\\ &\times erf((2\gamma^{2}+2k+1)\mu_{c}/4-B).\end{split}