# Reducing the Cost of Implementing Filters in LoRa Devices
## 3 Simulation Results
### Occupied Bandwidth  
The occupied bandwidth measurements were taken on a Keysight N9030A Spectrum Analyzer [10] using the 99% power bandwidth method detailed in Section 6.9.3 of the ANSI standard for Compliance Testing of Unlicensed Wireless Devices [11]. The ANSI standard was referenced \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{_SF_} & \multicolumn{5}{c}{**Number of Fraction Bits (B)**} \\ \cline{2-6}  & **2** & **4** & **6** & **8** & **10** \\ \hline
6 & 196 & 688 & 934 & 975 & 1016 \\
7 & 228 & 720 & 1663 & 1950 & 1991 \\
8 & 292 & 784 & 2465 & 3736 & 3982 \\
9 & 420 & 912 & 2880 & 6775 & 7800 \\
10 & 676 & 1168 & 3136 & 10,024 & 14,903 \\
11 & 1188 & 1680 & 3648 & 11,520 & 27,100 \\
12 & 2212 & 2704 & 4672 & 12,544 & 40,096 \\ \hline All & 4228 & 4720 & 6688 & 14,560 & 46,048 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Total number of samples (\(N_{\text{TX}}\)) for a system with a length-81 filter, chirp segmentation, and quantization. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{_SF_} & \multirow{2}{*}{_M_} & \multicolumn{5}{c}{**Fractional Bit Precision (B)**} \\ \cline{3-6}  & & **2** & **4** & **6** & **8** & **10** \\ \hline
6 & 64 & 5 & 17 & 23 & 24 & 25 \\
7 & 128 & 5 & 17 & 40 & 47 & 48 \\
8 & 256 & 5 & 17 & 58 & 89 & 95 \\
9 & 512 & 5 & 17 & 65 & 160 & 185 \\
10 & 1024 & 5 & 17 & 65 & 233 & 352 \\
11 & 2048 & 5 & 17 & 65 & 257 & 637 \\
12 & 4096 & 5 & 17 & 65 & 257 & 929 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{_SF_} & All & 5 & 17 & 65 & 257 & 1025 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(N_{\text{in}}\) for LoRa systems with various levels of quantization. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{_SF_} & \multicolumn{5}{c}{**Functional Bit Precision (B)**} \\ \cline{2-6}  & **2** & **4** & **6** & **8** & **10** \\ \hline
6 & 68 & 176 & 230 & 239 & 248 \\
7 & 100 & 208 & 415 & 478 & 487 \\
8 & 164 & 272 & 641 & 920 & 974 \\
9 & 292 & 400 & 832 & 1687 & 1912 \\
10 & 548 & 656 & 1088 & 2600 & 3671 \\
11 & 1060 & 1168 & 1600 & 3328 & 6748 \\
12 & 2084 & 2192 & 2624 & 4352 & 10,400 \\ \hline All & 4100 & 4208 & 4640 & 6368 & 13,280 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Total number of samples (\(N_{\text{TX}}\)) for a system with a length-17 filter, chirp segmentation, and quantization. alongside the measurement guidelines for LoRa devices provided by Semtech [12] in order to comply with FCC regulations as well. Each set of transmitted signals consists of 10 preamble symbols and 250 modulated chirp symbols. The signals were generated in Matlab and then sent to a Keysight N5182B Signal Generator [13]. The central carrier frequency was set to 915 MHz and the transmit power was set to 0 dBm. Since the upsampling factor is equal to 4, the sampling frequency of both the signal generator and spectrum analyzer was set to \(F_{s}=L\times BW=4 BW\). Measurements were obtained with the occupied BW mode using the peak detector and max hold traces on the spectrum analyzer. The values were recorded once the traces had stabilized after sweeping across over 350 points. The spectrum analyzer settings, which are summarized in Table 7, were varied with the LoRa bandwidth to comply with the standard [11]. The LoRa spreading factor was set to 10, while the specified bandwidth was set to 125, 250, and 500 kHz. The measured OBW results can be found in Table 8 for each tested bandwidth setting and fractional-bit precision. It is clear from these results that most tested levels of quantization do not impact the measured OBW. Even in the worst case, the difference is only a little over 100 Hz. Additionally, consider sample screenshots taken from the spectrum analyzer shown in Figure 3. There is some noticeable distortion in the passband of Figure 3c and a small change in OBW due to the quantization error.