# Performance of Narrow Band Wide Area Networks with Gateway Diversity
## 2 Principles of the Lora Modulation
The laboratory measurements are obtained with an SX1261/2 chipset in a conducted setup and are compared to those in the datasheet as well [11]. Additionally, the Bit Error Rate (BER) can be estimated as \(BER=0.5\times Q\Big{(}\sqrt{SNR\;2^{SF+1}}-\sqrt{1.386\times SF+1.154}\Big{)}\)[12]. Then, the Symbol Error Rate (SER) and PER can be calculated as \(SER=\frac{2^{SF}-1}{2^{SF-1}}\times BER\) and \(PER=1-(1-SER)^{N_{symbols}}\). The PER estimation based on these equations is presented in Figure 4.  
Figure 3: PHY symbol duration, PHY bitrate, energy per bit, sensitivity, and range versus SF. The simulation results (Figure 4) versus datasheet numbers are presented in \[\text{Symbol duration:}\ T_{s}=2^{SF}\ T_{c}\ \ [\text{s}] \tag{2}\]0 The results show that the simulation results are within 1 dB accuracy to that published in the datasheet (\[\text{Symbol duration:}\ T_{s}=2^{SF}\ T_{c}\ \ [\text{s}] \tag{2}\]11 in [11]), and hence it is a good tool to use for further performance evaluation. The simulation results that show the resilience to interference is presented in Figure 5 in the form of the Signal-to-Interference Ratio (SIR), assuming 500 kHz bandwidth. As the results versus the spreading factor show, the signals with the same chirpiness (the same chirp rate as shown by Equation (4)) are non-orthogonal to each other [8; 9]. For the same chirp rate, the orthogonality can be achieved by using IQ inversion, (i.e., complex conjugation) of the main chirp used for modulation and demodulation. The IQ inversion means that instead of up-chirping the symbols at the modulator, down-chirping is used and the receivers start demodulating with the opposite up-chirps. The results in Figures 5 and 6 show that close to orthogonal performance can be achieved in between two LoRa signals despite using the same chirp rate as long, as they use the opposite chirp direction. For example, if GWs use the opposite chirp direction for DL versus the end-nodes in the UL, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Spreading** & \begin{tabular}{c} _SNR at 1\% PER_, \\ **[dB]** \\ **(Simulation)** \\ \end{tabular} & \begin{tabular}{c} _BER_ \\ **(Simulation)** \\ \end{tabular} & \
\begin{tabular}{c} _SNR at 1\% PER_, \\ **[dB]**, \\ **(Datasheet)** \\ \end{tabular} &
\begin{tabular}{c} _Simulation to_ \\ **Datasheet Delta,** \\ **[dB]** \\ \end{tabular} \\ \hline
6 & \(-4.5\) & & \(-5\) & \(0.5\) \\ \hline
7 & \(-7\) & & \(-7.5\) & \(0.5\) \\ \hline
8 & \(-9.9\) & & \(-10\) & \(0.1\) \\ \hline
9 & \(-12.75\) & & \(-12.5\) & \(-0.25\) \\ \hline
10 & \(-15.5\) & & \(-15\) & \(-0.5\) \\ \hline
11 & \(-18.4\) & & \(-17.5\) & \(-0.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BER and PER simulation results versus datasheet for AWGN channel (64 bytes payload). Figure 4: The PER estimate with the LoRa modulation with 64 bytes payload. the effects of interference between UL and DL transmissions can be reduced significantly. For example, a SIR \(\geq-20.7\) dB is sufficient instead of a SIR \(\geq 0\) dB with SF-11 for a BER target of \(10^{-4}\) (Figures 5 and 6). Simulation results further show that a worst-case SIR of at least 0 dB is required within UL transmissions to achieve a Bit Error Rate (BER) target of \(10^{-4}\) when using SF = 11 among the end-nodes (Figure 5). A collision does not always result in a packet loss as long as the SIR is above the thresholds presented in Table 2. This table is prepared from the BER versus SIR simulation curves and extracted at a 1% Packet Error Rate (PER) target, corresponding to a BER target of \(10^{-4}\) (Table 1).