# Hybrid Chirp Signal Design for Improved Long-Range (LoRa) Communications
## 5 Numerical Results and Discussion  
In this section, we present the performance evaluation of our proposed LoRa modulation and demodulation scheme. To evaluate the performance of our proposed scheme, a LoRa simulation environment was built in Matlab. The simulation parameters are summarized in Table 1. In the simulation, we set \(B=125\) KHz and a packet size of 200 bits. In Figure 3, we present the performance comparison between our proposed scheme and the traditional scheme in the presence of additive white Gaussian noise (AWGN). The bit error rate (BER) performance is presented for \(S=\{6,8,10\}\). As expected, the improved BERperformance is observed as \(\mathcal{S}\) increases. We observe that the proposed LoRa modulation scheme outperforms the traditional LoRa modulation scheme for all scenarios. The proposed scheme exhibits over 3 dB improvement at a BER of \(10^{-4}\) compared to the traditional scheme. A further improvement over the proposed FFT-based demodulation is observed by applying the proposed hybrid demodulation scheme, since the correlation performed in the hybrid demodulation eliminates the ambiguity in nature (up/down) of the received signal. For different spreading factors, in Table 2, we present the required signal-to-noise ratio (SNR) to obtain a packet error rate \(\leq\) 0.01. For each case, we observe a significant improvement in the SNR threshold with our proposed modulation and demodulation schemes. The following examples demonstrate the significance of the above performance gain. Let us consider a LoRa system with operating frequency = 900 MHz, bandwidth = 125 kHz, transmission power = 2 dBm, and noise figure = 10 dBm. We calculate the received power based on the free space pathloss model [26]. With the conventional LoRa modulation scheme, one can achieve a communication range of 29 km and 58 km for \(\mathcal{S}=7\) and \(\mathcal{S}=9\), respectively, while meeting the packet error rate threshold of \(\leq\)0.01. On the other hand, our proposed modulation schemes can achieve a communication range of 52 km and 93 km for \(\mathcal{S}=7\) and \(\mathcal{S}=9\), respectively. Now, we present the performance of the proposed schemes in the presence of interference, where the desired LoRa signal collides with another LoRa signal. For interference signals with spreading factors \(\mathcal{S}=\{6,8,10\}\), the BER performance comparison is presented in Figure 4. In the simulation, we set the noise as Gaussian distributed with variance \(\sigma^{2}=1\). We consider the unit received power for the desired LoRa signal, while varying the received power of the interference LoRa signal to vary the signal-to-interference and noise ratio (SINR). Again, the BER performance improves as \(\mathcal{S}\) increases, except in the scenario where the interference signal has the same spreading factor as the desired signal, for which the worst performance is observed. Similar to the results shown in Figure 3, we observe a performance improvement for all the scenarios with the proposed schemes. For all combinations of reference signals and interference signals, the SINR threshold comparison is presented in Table 3. Again, we find the minimum SINR required for each combination to achieve a packet error rate of 0.01. From the table, we observe that, \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Packet size & 200 Bits \\ \hline Bandwidth & 125 kHz \\ \hline Packet error rate threshold & 0.01 \\ \hline Carrier frequency & 915 MHz \\ \hline Transmit power & 2 dBm \\ \hline Noise figure & 10 dBm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters.