# Performance Evaluation of Random Access Methods for IoT-over-Satellite
## 3 Method
### LoRa Emulator
#### 3.4.2 LoRa Demodulation  
Before reaching the receiver LoRa waveform is passed through the Ground-to-Satellite channel model mentioned in Section 3.2. The demodulation at the receiver involves a two-step process where the demodulator will first dechirp the received signal and then demodulate the resulting Multiple frequency-shift keying (MFSK) signal. In order to perform the dechirping, the received signal is mixed with a train of inverted chirps with zero frequency offset as follows,  
\[r(t) =y(t)\times z(t)\] \[=g\{x_{i}(t)\}z(t)+z(t)\sum_{j\
eq i}g\{x_{j}(t)\}+z(t)n(t), \tag{10}\]  
where \(z(t)\) represents the train of inverted chirps. Focusing on the target signal \(x_{i}(t)\), the dechirped signal, i.e., the FSK signal will take the following form,  
\[r_{o}(t)=G\;l\;\zeta\;\sum_{k=1}^{N_{\text{m}}}\exp{(j2\pi\gamma_{k}(t-kT_{ \text{s}}))}, \tag{11}\]  
where \(\gamma_{k}=m\Delta f\) is the frequency offset of the \(k^{\text{th}}\) symbol of the received LoRa packet with content value \(m\), \(\Delta f\) is the frequency offset parameter, and \(T_{\text{s}}\) is the symbol duration. Note that symbol synchronization is critical to ensure the accuracy of the demodulation. After the dechirping process, the obtained MFSK signal is then demodulated by a non-coherent demodulator. The demodulator implements the energy detection of the MFSK signal using fast Fourier transform (FFT) to extract the dominant component from the power spectrum density (PSD), where the dominant component represents the original shift containing the encoded LoRa symbols. And thus, the LoRa symbols can be retrieved as follows [42],  
\[\hat{m}=\left\lfloor\frac{1}{\Delta_{t}}\underset{f}{\text{argmax}}(|Y(f)|) \right\rfloor, \tag{12}\]  
where \(Y(f)\) is the fast Fourier transform of the dechirped signal \(r(t)\), \(\lfloor.\rceil\) is the rounding function and \(\hat{m}\in\{0,1,...,M-1\}\) is the encoded LoRa symbol value. In this work, we investigate the impact of intra-SF interference, i.e., all devices share the same SF. The interference generated by other SF is not included due to the high orthogonality between different SF and the main focus of this paper is the impact of different random access, and satellite constellation parameters [53]. The interference generated by other SF is not included due to the high orthogonality between different SF. The reason for using the fixed SF7 is because SF7 requires the most reliable link when compared to other SF settings. If all the ground devices use SF7, the neglecting impact of using a single fixed SF can also be observed if there are any destructive colliding packets. In order terms, we can use SF7 to emulate the worst-case scenario. Secondly, SF7 is energy-friendly compared to other SFs, and in order to minimize the Doppler shift impact, the frequency shift should not exceed \(\Delta F\), where \(\Delta F=\frac{B}{2^{\Delta F}}\). The higher the SF, the smaller the \(\Delta F\) is. Moreover, LEO constellations provide high Doppler, rendering the usage of high SF becomes inefficient. The main focus of this paper is the impact of different random access, and satellite constellation parameters, as thus, the impact of intra-SF interference and the impact of the LoRa parameter is out of the scope of this paper.