## 5 System Model in Two-Hop Case
### Two Proposed Schemes  
In this study, two proposed schemes are assumed. The first scheme (Scheme 1) transmits data depending on preset parameters, which was used in our previous study [20; 21; 22]. On the other hand, in the second scheme (Scheme 2), coding rates are varied with the SNR estimated using a preamble signal according to each QoS requirement (e.g., desired bit error ratio (BER)), which is introduced for the first time in this manuscript. The operation example is shown in Figure 8. Firstly, the channel SNR is estimated by using the preamble of the beacon or the T-Poll received from the hub or the relaying node. Next, the relayed node or the relaying node determines the coding rate according to the estimated channel SNR and transmits data to the relaying node or the hub. If a bit error is detected, elements of the encoded data (codeword) are transmitted to increase error correcting capability after receiving negative-acknowledgement (NACK). Then, if data are transmitted successfully, the channel SNR is estimated by using the returned acknowledgement (ACK) preamble, the coding rate is determined, and the next data are sent. Since Scheme 2 uses an existing preamble, extra overhead is not required. Figure 8: Operation example of Scheme 2. ACK, NACK, and QoS are abbreviations of acknowledgement, negative-acknowledgement, and quality of service respectively. Then, the channel SNR is estimated using the following equations [32]:  
\[\Gamma=\frac{\left|\rho\right|^{2}}{1-\left|\rho\right|^{2}} \tag{3}\]  
\[\rho=\frac{\mathbf{x}^{H}\mathbf{r}}{\sqrt{\mathbf{x}^{H}\mathbf{x}\sqrt{\mathbf{r}^{H}\mathbf{r}}}} \tag{4}\]  
\[0\leq\rho\leq 1 \tag{5}\]  
\[\mathbf{x}=\mathbf{s}+\mathbf{\eta} \tag{6}\]  
\[\mathbf{r}=c\mathbf{s} \tag{7}\]  
Here, \(\Gamma\) is the estimated SNR, \(\rho\) is a correlation coefficient, \(\mathbf{x}\) is a preamble signal with noise \(\mathbf{\eta}\), and \(\mathbf{r}\) is a preamble signal that consists of a signal \(\mathbf{s}\) and unknown constant \(c\) without noise or interference. Then, we explain why Equation (3) becomes the SNR. Let's substitute Equation (4) into Equation (3):  
\[\Gamma=\frac{\left|\mathbf{x}^{H}\mathbf{r}\right|^{2}}{\mathbf{x}^{H}\mathbf{x}^{H}\mathbf{r}- \left|\mathbf{x}^{H}\mathbf{r}\right|^{2}}\;. \tag{8}\]  
Here, \(\left|\mathbf{x}^{H}\mathbf{r}\right|^{2}\) can be expanded as follows:  
\[\left|\mathbf{x}^{H}\mathbf{r}\right|^{2}=\left|\mathbf{s}^{H}\mathbf{r}+\mathbf{\eta}^{H}\mathbf{r} \right|^{2}=\left(\mathbf{s}^{H}\mathbf{r}+\mathbf{\eta}^{H}\mathbf{r}\right)\left(\mathbf{s}^{H} \mathbf{r}+\mathbf{\eta}^{H}\mathbf{r}\right)^{*}=\left(\mathbf{s}^{H}\mathbf{r}+\mathbf{\eta}^{H}\mathbf{ r}\right)\left(\mathbf{r}^{H}\mathbf{s}+\mathbf{r}^{H}\mathbf{\eta}\right)=\mathbf{s}^{H}\mathbf{r}^{H} \mathbf{s}\;. \tag{9}\]  
Here, \(\mathbf{\eta}^{H}\mathbf{r}=\mathbf{r}^{H}\mathbf{\eta}=0\) since noise and a preamble signal are uncorrelated. Then, \(\mathbf{x}^{H}\mathbf{x}^{H}\mathbf{r}\) can be expanded as follows:  
\[\mathbf{x}^{H}\mathbf{x}^{H}\mathbf{r}=\mathbf{s}^{H}\mathbf{s}^{H}\mathbf{r}+\mathbf{\eta}^{H}\mathbf{\eta}^ {H}\mathbf{r}. \tag{10}\]  
For the same reason, \(\mathbf{\eta}^{H}\mathbf{s}=\mathbf{s}^{H}\mathbf{\eta}=0\). Then, Equation (8) can be modified from Equations (9) and (10) as follows:  
Figure 8:0  
Additionally, \(\mathbf{s}^{H}\mathbf{s}^{H}\mathbf{r}=\mathbf{s}^{H}\mathbf{r}^{H}\mathbf{s}\) as follows:  
Figure 8:1  
Finally, Equation (11) is summarized as follows, indicating that the SNR can be derived:  
Figure 8:2  
where \(P_{\rm s}\) and \(P_{\rm\eta}\) are signal power and noise power, respectively. In Scheme 2, the criteria to determine the coding rate are expressed as follows:  
Figure 8:3  
where \(L_{info}\) is length of information bits. Hence, the desired BER is calculated from the desired PER such as Table 1 and \(L_{info}\).