## 6 Performance Analysis  
If \(P_{s}\) is the source power and \(\sigma_{SR}\) is the noise power corresponding to \(n_{SR}\), then the SNR for the THz link with the presence of pointing errors can be expressed as  
\[\gamma_{T}=\frac{P_{s}|H_{THz}|^{2}}{\sigma_{SR}^{2}}. \tag{11}\]  
Considering a scenario with no pointing errors in the THz link, the SNR of the THz link can be expressed as  
\[\gamma_{T_{NP}}=\frac{P_{s}|h_{p}h_{a}|^{2}}{\sigma_{SR}^{2}}. \tag{12}\]  
If \(P_{r}\) is the relay power and \(\sigma_{RD}\) is the noise power, then the SNR for the VLC link can be expressed as  
\[\gamma_{VLC}=\frac{P_{r}|H_{VLC}|^{2}}{\sigma_{RD}^{2}} \tag{13}\]  
**Lemma 1**: _Consider a dual-hop decode-and-forward relay-assisted THz-VLC link. Assuming the presence of pointing errors in the THz link and deterministic fading in the VLC link, and utilizing Binary Phase-Shift Keying (BPSK) modulation for the THz link along with on-off keying (OOK) modulation for the VLC link, the E2E BER can be expressed as in Equation (14):_  
\[BER_{E2E}= \frac{\xi^{2}}{4\Big{(}\
u A_{0}^{2}\big{|}\sqrt{P_{s}}h_{p}h_{a} \big{|}^{2}\Big{)}^{\frac{\xi^{2}}{2}}}\Gamma\Big{(}\
u A_{0}^{2}\big{|}\sqrt {P_{s}}h_{p}h_{a}\big{|}^{2},\frac{\xi^{2}}{2}\Big{)} \tag{14}\] \[+\frac{1}{2}\text{erfc}\bigg{(}\sqrt{\frac{\gamma_{VLC}}{2}} \bigg{)}-2\Bigg{(}\frac{\xi^{2}}{4\Big{(}\
u A_{0}^{2}\big{|}\sqrt{P_{s}}h_{p }h_{a}\big{|}^{2}\Big{)}^{\frac{\xi^{2}}{2}}}\Gamma\Big{(}\
u A_{0}^{2}\big{|} \sqrt{P_{s}}h_{p}h_{a}\big{|}^{2},\frac{\xi^{2}}{2}\Big{)}\cdot\frac{1}{2} \text{erfc}\bigg{(}\sqrt{\frac{\gamma_{VLC}}{2}}\bigg{)}\Bigg{)}.\]Proof. : Given that misalignment fading stands as the sole random fading component, assuming \(\
u\) represents the average SNR of an uncoded communication system, as referenced in [5], the instantaneous BER of the THz link can be expressed as:  
\[BER_{THz}=E_{f_{h_{m}}}\bigg{[}Q\bigg{(}\sqrt{2|\sqrt{P_{s}}h_{p}h_{a}|^{2}|h_{m} |^{2}\
u}\bigg{)}\bigg{]} \tag{15}\]  
Here, \(Q(. )\) denotes the tail distribution function of the Gaussian distribution, and \(E_{f_{h_{m}}}[. ]\) signifies the expected value of the misalignment PDF. Consequently, the BER expression can be formulated as  
\[BER_{THz}=\frac{\xi^{2}}{A_{0}^{\xi^{2}}}\int_{0}^{A_{0}}Q\bigg{(}\sqrt{2\bar{ \
u}|\sqrt{P_{s}}h_{p}h_{a}|^{2}x^{2}}\bigg{)}x^{\xi^{2}-1}dx \tag{16}\]  
Drawing parallels with [5], this study employs the Chernoff exponential bound to approximate the Gaussian \(Q\) function, expressed as \(Q(x)\leq\frac{1}{2}\exp(-\frac{1}{2}x^{2}),x>0\). Incorporating this approximation, the BER expression can be reformulated as  
\[BER_{THz}=\frac{\xi^{2}}{4\Big{(}\
u A_{0}^{2}|h_{p}h_{a}|^{2}\Big{)}^{\frac{ \xi^{2}}{2}}}\Gamma\bigg{(}\
u A_{0}^{2}|h_{p}h_{a}|^{2},\frac{\xi^{2}}{2} \bigg{)} \tag{17}\]  
where \(\Gamma(.,. )\) is the lower incomplete gamma function. Given that the VLC link uses a deterministic path loss model, the BER can be expressed as  
\[BER_{VLC}=\frac{1}{2}erfc\sqrt{\frac{\gamma_{VLC}}{2}} \tag{18}\]  
The E2E BER for the THz-VLC link can be obtained through the unified error rate expression in [27] as follows:  
\[BER_{THz-VLC}=BER_{THz}+BER_{VLC}-2\cdot BER_{THz}\cdot BER_{VLC}. \tag{19}\]  
Substituting Equations (17) and (18) into (19) proves (14). This way, the E2E BER for a dual-hop decode-and-forward relay-assisted THz-VLC link with pointing errors in the THz link and deterministic fading in the VLC link, using BPSK and OOK modulation, is derived using the expected value of the misalignment PDF and the Chernoff exponential bound approximation for the Gaussian Q function. Lemma 2.: _Consider a dual-hop decode-and-forward relay-assisted THz-VLC link. Assuming deterministic fading in the entire THz-VLC link, and utilizing BPSK modulation for the THz link along with OOK modulation for the VLC link, the E2E BER can be expressed as_  
\[\gamma_{T_{NP}}=\frac{P_{s}|h_{p}h_{a}|^{2}}{\sigma_{SR}^{2}}. \tag{12}\]0  
Proof.