# A New LoRa-like Transceiver Suited for LEO Satellite Communications +
## 4 Results and Discussion
### DCCSS Performance Evaluation and Comparison with CSS  
It should be noted that this comparison between DCSS and CSS is done without implementing the aforementioned channel coding. The FEC and the interleaving will be deployed only when presenting the performance of our receiver. Given the principle of DCSS as described in Section 3.1, one can remark that this modulation naturally introduces a degradation of performance compared to CSS. Indeed, two consecutive DCSS symbols must be properly detected for the original symbol carrying the information to be accurately retrieved (see (20)). However, Figure 4 proves that this impairment remains low. For example, if a bit error rate (BER) equals \(10^{-4}\), the loss is only of 0.2 dB for all of the \(SF\)s.  
The second evaluation test of DCSS is to assess its robustness to a linear time varying Doppler shift. To this end, we represent in Figure 5, the packet error rate (PER) of the latter modulation as function of the DR in a perfect synchronization case (i.e., \(\Delta t\), \(\Delta f=0\)) and without considering the noise. To note that the DR is not compensated in these simulations. It can be seen that the decrease of the maximum permissible error of the dichotomy method \(\phi\) leads to an enhancement of the robustness of DCSS against the DR. However, this increase of precision is at the expense of the computational complexity. Indeed, for \(\phi=\frac{1}{M}\), the DR limit that can be naturally supported by DCSS is \(c_{d}^{th}=385\) Hz/s. For \(\phi<\frac{1}{M}\), the robustness of DCSS is clearly degraded and for \(\phi>\frac{1}{M}\), the gain is not important. Based on this simulation, we can confirm that \(\phi=\frac{1}{M}\) is our optimal choice when using the bandwidth \(B_{ref}\). In general, for any bandwidth configuration, the precision parameter \(\phi\) should be equal to \(\frac{\Delta_{h}}{B_{ref}}\), to have the same performance as in Figure 5.  
Figure 4: Comparison of bit error probabilities of LoRa and DCSS technologies before channel decoding. To compare the robustness of DCSS and CSS to Doppler time variation, we represent, in Table 1, the DR limit that can be supported by each waveform, with different \(SF\)s at the bandwidth \(B=B_{ref}\) and \(\phi=\frac{1}{M}\). To ensure the fairness between the two physical layers, the DR is not compensated in both cases and simulations are done in a perfect scenario (i.e., perfect synchronization without the AWGN channel). It can be seen in the above table that DCSS associated with the interpolation technique, as described in Section 3.1, is much more robust to DR than CSS. This result is explained by the capacity of the differential process combined with the interpolation to retrieve the effective symbols in the presence of fractional offsets when estimating the frequencies of the main peaks in the FFTs. In LoRa, the received signal can be accurately decoded only if the variation, cased by the DR, between each consecutive symbols is lower than the half of the frequency separation between each two adjacent bins, which is not the case in the DCSS technique. We note that the latter results are not dependent on the \(SF\) only, but also on the bandwidth \(B\). In other terms, the frequency separation \(\Delta_{b}\) is the parameter that defines the robustness of DCSS technology to DR. For instance, the robustness to DR of \(SF=12\) with \(B=B_{ref}\) is the same as \(SF=7\) with \(B=\frac{B_{ref}}{2}\). Finally, we represent in Figure 6 the BER evolution of DCSS and CSS for \(SF=12\), \(B=B_{ref}\) and a payload size \(N_{pay}=51\) bytes (According to LoRaWAN protocol, the \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \(SF\) & 7 & 8 & 9 & 10 & 11 & 12 \\ Bin separation \(\Delta_{b}=\frac{B}{2^{SF}}\) (Hz) & 976.56 & 488.28 & 244.14 & 122.07 & 61.03 & 30.51 \\ DCSS & 394,235 & 100,605 & 25,150 & 6260 & 1600 & 385 \\ CSS & 9585 & 2664 & 713 & 192 & 50 & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DR thresholds (\(c_{d}^{th}\) Hz/s) of DCSS and CSS, \(B=125\) kHz.