# Beacon Success Rate versus Gateway Density in Sub-GHz Sensor Networks
## 3 PHY Model with GFSK for Low Bit-Rate IoT Applications  
The principles of GFSK modulation have been analyzed in detail in the literature, along with detailed analytical models [1; 19; 30]. In this section, GFSK modulation is introduced for IoT applications along with the selected modulator parameters. A PHY model with a 50 kbps bitrate is assumed, which is suitable for indoor and outdoor IoT applications [1; 16]. First, the sensitivity performance is presented, and the simulation results are compared to lab measurements. Second, the co-channel interference measurement method is presented along with conducted lab measurements. Both the co-channel interference and the sensitivity models provide the building blocks for coverage area and network capacity analyses. Such a coverage area analysis will be presented in Section 6. The additional link budget losses arising from interference from other networks are accounted for in the path-loss model. (A) GFSK Parameters selection  
In this section, the crucial PHY parameters are presented for GFSK modulation for narrow-band, low-bitrate IoT applications. For GFSK modulation, the modulation index, \(h\), and the frequency deviation, \(f_{d}\), are defined as follows: \(f_{d}=h\times\frac{1}{2}\times\frac{1}{T}\times(2^{m}-1)\) [Hz]. In this paper, the term \(m\) is set to 1 bits/symbol for 2-GFSK, and \(0<h\leq 1\). The modulation index determines the maximum phase shift between two adjacent symbols in time, given by \(\phi=\pm\pi\times h\). When the modulation index is set to maximum value of 1, the modulator runs with a maximum frequency deviation of \(\pm 25\) kHz with a 50 kbps PHY rate. Adjusting the modulation index to a lower value, such as \(h=0.5\) instead of 1, results in a decreased frequency deviation in the modulated bits. While this reduction can be disadvantageous in the presence of RF impairments like carrier frequency offset (CFO), especially in narrow bandwidth scenarios, a smaller modulation index--such as 0.5--can improve the Adjacent Channel Leakage Ratio (ACLR) performance. This improvement is crucial for applications with higher bit rates in the realm of Internet of Things (IoT), such as those utilizing Bluetooth Low Energy (BLE) with bandwidths up to 2 MHz [19]. Based on the conducted PSR versus RSS measurements presented in Figure 9, it is recommended to keep the modulation index at \(h=1\) for GFSK 50 kbps implementation. The PSR results, as shown in Figure 9, indicate that \(h=1\) achieves a better performance than \(h=0.76\) and \(h=0.5\). This makes sense, as a larger frequency deviation results in more separation among the soft FSK bits which are under a White Gaussian Noise (AWGN). This is especially crucial for very low bit-rate IoT applications that use very small frequency deviations. These results were obtained using the Texas Instruments Evaluation Board (TI-EVB), supporting GFSK 50 kbps modulation. The testing involved a setup where the Rx signal power input to the EVB was varied. Essentially, this result presents the performance under thermal noise and corresponds to AWGN performance. Figure 8: The histogram of the RSSI samples at two different locations. The main factor influencing Rx sensitivity is the modulation index used for GFSK modulation (ref Figure 9). The measurements presented in Figure 9 show that a larger modulation index improves the sensitivity performance for low-bandwidth GFSK modulation. The channel spacing is set to 200 kHz, which is significantly larger than the modulation bandwidth. This allows a larger modulation index to be feasible in terms of ACLR performance. The Gaussian pulse shaper assists with the larger modulation index by limiting the power spectrum density (PSD) (ref. Figure 10). The PSD measurement indicates that the transmitter's Gaussian pulse shaper Bandwidth-Time product (BT) parameter should be set to BT = 0.5 to enhance the ACLR [1]. This analysis is presented in Figure 10, which is obtained from measurements with a TI-EVB.