# Beacon Success Rate versus Gateway Density in Sub-GHz Sensor Networks
## 3 PHY Model with GFSK for Low Bit-Rate IoT Applications
The traces are measured with the settings presented in Table 1. For instance, when the modulation index is set to 1, the green and yellow traces demonstrate that the second ACLR can be improved by 5 dB when the pulse shaper filter BT is tightened from 1 to 0.5. The accompanying sensitivity measurements indicate that there is no significant difference in Rx sensitivity between a BT value of 1 and 0.5.  
coefficient of BT = 0.5. The Rx filter bandwidth is set to a value larger than the modulation bandwidth (50 kHz) to accommodate CFO impairment that can vary across process, voltage, temperature, and frequency (PVTF). However, a larger Rx filter bandwidth passes more thermal noise and causes a degradation in Rx sensitivity, but simplifies the hardware. A modulation index of 1 is selected for such a narrowband modulation to improve robustness to RF impairments such as CFO. The receiver sensitivity of the device can be related to the \(RSS\) level required to achieve a bit error rate of \(BER\leq 10^{-4}\). This corresponds to an un-coded PER of 1% with a 64-bytes payload. From the results, it can be seen that this device can achieve an un-coded sensitivity of \(-107\) dBm (at 155.4 kHz Rx bandwidth) and requires an SNR of 6 dB with 9 dB NF. The simulation results presented in the AWGN channel are compared with the conducted laboratory measurements, demonstrating a strong agreement. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline  & **Pulse Shaper BT Parameter** & **Modulation Index, \(h\)** & **Frequency Deviation, kHz** & **Bitrate** \\ \hline Trace-1 (Yellow trace) & & 1 & \(\pm 25\) & \\ \hline Trace-2 (Blue trace) & BT = 0.5 & 0.76 & \(\pm 19\) & \\ \hline Trace-3 (Magenta trace) & & 0.5 & \(\pm 12.5\) & 50 kbps \\ \hline Trace-4 (Green trace) & BT = 1 & 1 & \(\pm 25\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modulation parameters tested in the laboratory using TI-EVB for GFSK 50 kbps IoT modulation. Figure 10: Power Spectral Density measured in relation to the modulation index and pulse shaping filter coefficient for GFSK modulation. baselining, with \(r=0.5\), and \(r=0.25\) for comparison. The system setup presented in Figure 13 is used for the conducted measurements, where there is one interfering user for link-level system characterization. The term \(\tau\) represents the duration of the overlapping section of the interferer over the desired packet, where \(r=\tau/T\) represents the overlap ratio presented in Equation (1). The measurements are conducted for different values of \(\tau\) and, hence, different values of \(r\). The conducted lab measurements with this test setup are presented in Figure 14. In the lab, GW\({}_{1}\) and GW\({}_{2}\) represent a signal generator playing a GFSK waveform created by SX-1262 EVB [31]. The waveform is created by using a modulation index of \(h=1\), BT = 0.5, and assuming a 50 kbps bit rate (ref. Table 1). The waveforms are scaled relative to each other to achieve the desired C/I level at full overlap. The second waveform from the signal generator is delayed by a time equal to \(\tau\) seconds relative to the first signal. Then, both waveforms are added to each other after the scaling and delaying of one of them. The resulting waveform is then input into the EN's receiver, using another SX-1262 EVB. The measurement results are presented in Figure 14. On the \(x\)-axis, \(\frac{C}{T}_{packet}\) is calculated based on the partial interfering overlap duration \(\tau\) and packet duration \(T\) by using Equation (1). The results illustrate that Equation (1) represents the performance well, where smaller overlap ratios help to improve the performance. In other words, the required C/I threshold for successful packet reception reduces by as much as \(10\times log_{10}(r)\), dB. The measurements show that a worst case C/I threshold of at least \(\gamma_{th}=7\) dB is required when there is a full overlap, where \(r=1\).