## 2 System Model
### Computation of the LLRs for the 4-PAM and the QPSK Modulation Schemes  
In this paper, the NB-IoT transmitted data is encoded with a convolutional coding scheme. Furthermore, for Case 3, the encapsulated NB-IoT frames are protected using the LDPC code of the DVB-S2(X) standard. In both cases, soft-decoding is used and due to that, the LLRs has to be determined. However, the IoT devices and the satellite node may be limited in their power consumption. Fortunately, for both modulation schemes used in this paper (i.e., 4-PAM and QPSK), it is possible to derive reduced complexity closed-form formulas for computing the optimal LLRs. The 4-PAM modulation scheme is used in the optical feeder link, whereas the QPSK modulation is used in the radio access link. We assume that Gray-mapping is used in both cases, and that their corresponding modulation symbols are \(\mathbf{s}_{4-\mathrm{pam}}=\{-3,-1,1,3\}/\sqrt{5}\) for 4-PAM and \(\mathbf{s}_{\mathbf{QPSK}}=\{-1-j,1-j,1+j,-1+j\}/\sqrt{2}\) for QPSK (see Figure 4). Since both modulation schemes transport two bits per modulated symbol, it is necessary to compute two LLRs per modulated symbol. Thus, the closed-form expression for computing these LLRs is given by:  
\[LLR_{b_{m}}=-\log\left(\frac{\left(\sum_{p=0;s_{p}|b_{m}=0}^{M/2}\mathrm{e}^{- \frac{|z-h_{p}|^{2}}{2h_{m}^{2}}}\right)}{\left(\sum_{q=0;s_{q}|b_{m}=1}^{M/2} \mathrm{e}^{-\frac{|z-h_{q}|^{2}}{(2h_{m}^{2}})}\right)}\right), \tag{8}\]  
where \(LLR_{b_{m}}\) is the LLR that corresponds the \(m\)-th bit of the modulated symbol, \(\sigma_{n}^{2}\) is the noise power, \(M\) is the number of constellation symbols, \(h\) is the communication channel, \(z\) is the received data at the input of the de-mapper, and \(s_{p}\) (\(s_{q}\)) symbolizes the constellation symbols in which the \(m\)-th bit is 0 (1). Thus, according to (8) and the Gray mapping proposed in Figure 4, the LLR of the first bit \(b_{0}\) for both 4-PAM and QPSK modulation schemes is given by:\[LLR_{b_{0}}=-\log\left(\frac{\left(\mathrm{e}^{-\frac{\left|-k_{b}s_{0}\right|^{2} }{2\pi_{n}^{2}}}+\mathrm{e}^{-\frac{\left|-k_{b}s_{1}\right|^{2}}{2\pi_{n}^{2}} }\right)}{\left(\mathrm{e}^{-\frac{\left|-k_{b}s_{2}\right|^{2}}{2\pi_{n}^{2}} }+\mathrm{e}^{-\frac{\left|-k_{b}s_{1}\right|^{2}}{2\pi_{n}^{2}}}\right)} \right), \tag{9}\]  
whereas LLR of the second bit \(b_{1}\) attains the form:  
\[LLR_{b_{0}}=-\log\left(\frac{\left(\mathrm{e}^{-\frac{\left|-k_{b}s_{0}\right| ^{2}}{2\pi_{n}^{2}}}+\mathrm{e}^{-\frac{\left|-k_{b}s_{1}\right|^{2}}{2\pi_{n} ^{2}}}\right)}{\left(\mathrm{e}^{-\frac{\left|-k_{b}s_{1}\right|^{2}}{2\pi_{n }^{2}}}+\mathrm{e}^{-\frac{\left|-k_{b}s_{2}\right|^{2}}{2\pi_{n}^{2}}}\right) }\right). \tag{10}\]  
If (9) and (10) are computed for 4-PAM and QPSK modulation schemes, the closed-form expressions for \(LLR_{b_{0}}\) and \(LLR_{b_{1}}\) are shown in Table 1. In these closed-form formulas, we have that:  
\[a=\left(2\,z\,s_{1}\right)/(2\,\sigma_{n}^{2}),\qquad b=(4\,s_{1}^{2})/(2\sigma _{n}^{2}),\qquad c=3\cdot a, \tag{11}\]  
where \(s_{1}\) is the second symbol of the 4-PAM constellation, i.e., \(s_{1}=-1/\sqrt{5}\) (see Figure 4) and signal \(z\) represents the data after equalizing the time-packed signal (see Section 2.3). For _Case 2_, this signal \(z\) corresponds to the NB-IoT one, and its LLRs are introduced in the soft-convolutional decoder for recovering the transmitted message to regenerate the NB-IoT signal. For _Case 3_, the signal \(z\) corresponds to the DVB-S2(X) after the time-packing equalization. The computed LLRs for this case is introduced to the LDPC decoder of the DVB-S2(X) receiver [33]. After that, the encoded NB-IoT bits are QPSK mapped, OFDM modulated, and forwarded to the IoT-device. Finally, at the IoT receiver, the LLRs of the received QPSK modulation symbols are computed for all cases under study. These LLRs are used by the soft-Viterbi decoder to recover the message transmitted from the satellite gateway.