## 4 Ionospheric Channels Analysis
Figure 12 displays the distribution of the delay spread of both ionospheric modes independently. Another fundamental parameter to study for our NVIS system was the frequency offset that our link was affected by. This frequency variation depends on the movement between the transmitter and the receiver, which in our scenario should be produced by the displacement of the physical layers of the ionosphere. Notwithstanding this, the doppler shift caused by the ionosphere was negligible compared to the frequency variation that the clocks of the Red Pitaya platform produced. This shift was directly related to the temperature of the platform, which affected the clock's stability. The channel study performed in this research was implemented with rather cheap nodes, in a system where low-stability clocks are usual. Accordingly, a good data frame design and a good post-processing of the signals received was key to mitigate the negative effects of the usage of low-cost technologies. Figure 13 exhibits a boxplot for all the hours throughout the day when the ionospheric channels are active. It can be appreciated that the maximum frequency offset received was \(-19.5\) Hz and the minimum was \(-15\) Hz, values that were remarkably higher than the ionospheric layer's Doppler shift (a value that can reach a maximum of 4 Hz, approximately [31]). Finally, we present the study of the combination of the ordinary and extraordinary wave's received. Tests on the bit error rate (BER) and bit energy-to-noise power spectral  
Figure 12: (a) NVIS delay spread distribution of the ordinary channel; (b) NVIS delay spread distribution of the extraordinary channel. Figure 13: Frequency offset received vs. time. density (E\({}_{\rm b}\)/N\({}_{0}\)) of the system were performed to evaluate the improvement of the robustness achieved. The results of the fourth-order modulations with a transmit power of 50 W are presented below. First of all, we measured the relationship between the E\({}_{\rm b}\)/N\({}_{0}\) received from the individual characteristic waves (ordinary and extraordinary) compared to the E\({}_{\rm b}\)/N\({}_{0}\) received as a result of the application of diversity combining techniques (selection combining and equal-gain combining). The E\({}_{\rm b}\)/N\({}_{0}\) of the signal received was computed as follows:  
\[\rm E_{b}/N_{0}\ (dB)=SNR(dB)+10\cdot\log 10(B)-10\cdot\log 10(Rb), \tag{2}\]  
where Rb is the signal's bitrate (depends on the modulation order under test), B is the noise bandwidth of the measurement (2.3 kHz in our scenario), and SNR is the signal-to-noise ratio of the received signal. Figure 14 presents the behavior of the ionospheric waves and their combining by the mean E\({}_{\rm b}\)/N\({}_{0}\) received throughout the day. Figure 14 only exhibits the 4 QAM results in order to make the graph clearer. The 4 PSK modulation presented almost identical performances in terms of E\({}_{\rm b}\)/N\({}_{0}\), while the 4 FSK presented worse results. All three modulations are studied in terms of the BER in Figure 15. We can state that the selection combining (SC) technique presented a higher E\({}_{\rm b}\)/N\({}_{0}\) value than the individual ionospheric waves in all the studied hours. An improvement of up to 4 dB was achieved (at 12 UTC) while using this method. The equal-gain combining (EGC) technique also improved the performance of the link. An improvement of up to 3 dB was achieved (at 13 UTC), but there were certain times of the day (11 UTC, 14-16 UTC) when this technique did not improve the robustness of the link. Finally, we can also see that the mean E\({}_{\rm b}\)/N\({}_{0}\) received by the O and X modes differed by a maximum of 2 dB. Figure 14 compares the results obtained by using selection combining, equal-gain combining, the ordinary mode, and the extraordinary mode.