## 2 System Model
### Architectures for Forwarding NB-IoT Frames over a Satellite Relaying Node
#### 2.2.1 Case 1: Detect-and-Forward Relaying of NB-IoT Frames  
This strategy is similar to the one used in transparent (bent-pipe) satellite architectures, but with the difference that the NB-IoT amplitude-modulated symbols in the optical feeder link are time-packed. The Inter-Symbol Interference (ISI) that time-packing introduces is removed in the satellite (see Section 2.3) and, after that, the 4-PAM modulated symbols of the NB-IoT signal are detected, remapped to QPSK symbols, OFDM modulated, and re-sent to the IoT terminal over the radio access link. Finally, the IoT terminal computes the LLRs of the QPSK symbols (see Section 2.4) and feeds them in the soft convolutional decoder to estimate the transmitted NB-IoT frames from the gateway (see Figure 3).  
Let \(s_{\text{ntp}}[k]\) be the \(k\)-th NB-IoT modulated symbols, \(g_{\text{tx}}(\cdot)\) the transmit pulse-shaping square-root raised-cosine filter, \(T_{s}\) the Nyquist symbol time of the NB-IoT signal, and \(\delta\) the time-packing overlapping factor. Then, the time-packed signal that is generated by the gateway can be written as:  
\[s_{\text{tp}}(t)=\sum_{k}s_{\text{ntp}}[k]\,g_{\text{tx}}\big{(}t-k(1-\delta)T _{\text{s}}\big{)}. \tag{1}\]  
Next, the Electrical-to-Optical (E/O) conversion of \(s_{\text{tp}}(t)\) is done with a MZM driven by voltage:  
\[v_{\text{mzm}}(t)\ =\ V_{\text{B}}+\beta\widetilde{s}_{\text{tp}}(t)\,(V_{\pi}/ \pi), \tag{2}\]  
where \(V_{\text{B}}\) and \(V_{\pi}\) are the bias and half-wavelength voltages of the MZM, \(\beta\) is the intensity modulation index (scaling factor) that is selected for the communication in the optical feeder link, and:  
\[\widetilde{s}_{\text{tp}}(t)\ =\ s_{\text{tp}}(t)/\sqrt{\mathbb{E}\{|s_{\text{tp }}(t)|^{2}\}}, \tag{3}\]  
where \(\mathbb{E}\{\cdot\}\) is the mathematical expectation operator that determines the mean value of the signal.  
After that, the time-packed signal is transmitted through the optical feeder link. At the satellite, the Optical-to-Electrical (O/E) conversion is performed with the aid of a Photodetector [28], obtaining:  
\[r_{\text{tp,sat}}[n]=\sqrt{\left(\frac{E_{\text{b}}}{N_{0}}\right)_{\text{fl} }}\,h_{\text{fl}}\,s_{\text{tp}}[n]+\eta_{\text{fl}}[n], \tag{4}\]  
where \(h_{\text{fl}}\) is the equivalent channel gain of the optical feeder link (see Section 3), \(s_{\text{tp}}[n]\) is the \(n\)-th discrete sample of the time-packed signal in the gateway, and \(\eta_{\text{fl}}[n]\) is the resulting noise signal of unit power after the O/E conversion in the satellite (see Section 4.1). The value of \((E_{\text{b}}/N_{0})_{\text{fl}}\) represents the equivalent bit-to-noise-energy ratio of the optical feeder link.  
The estimations of the non-time packed NB-IoT symbols at the satellite (i.e., \(\widehat{s}_{\text{ntp,sat}}\)) are obtained after the received signal samples in (4) are first matched-filtered and then equalized (see Section 2.3). Then, the 4-PAM non-time packed signals are re-mapped into QPSK symbols, OFDM modulated, and finally transmitted to the corresponding IoT device. Due to that, the \(k\)-th sample of the signal that the IoT device receives in the radio access link attains the form:  
\[r_{\text{ntp,tot}}[k]=\sqrt{\left(\frac{E_{\text{b}}}{N_{0}}\right)_{\text{al }}}\,h_{a}\,\widehat{s}_{\text{ntp,sat}}[k]+\eta_{\text{al}}[k], \tag{5}\]  
where \(\eta_{\text{al}}[k]\) is the unit power noise signal, \(h_{\text{al}}\) is the gain of the radio access channel, and \((E_{\text{b}}/N_{0})_{\text{al}}\) is the bit-to-noise-energy ratio of the radio access link. Finally, the LLR of the received QPSK modulated symbols are computed, and soft-decoding it is conducted to detect the transmitted bits from the gateway, \(b_{\text{gw}}\) (see Figure 3).