# New Results for the Error Rate Performance of LoRa Systems over Fading Channels
## 4 Numerical Results
Figure 6 depicts the estimated BER of the considered propagation scenario as a function of the link distance. Again, an excellent match of exact and approximate results was observed for all test cases under consideration. Figure 5: BER of LoRa systems operating in the presence of Rice fading as a function of the SNR, \(\gamma\), for SF = 12 and various values of \(K\). Figure 6: BER estimation of LoRa systems operating in the presence of Rice fading in an agricultural environment using a measurement campaign. Figure 9 depicts the SER of LoRa modulation in the presence of \(\eta\)-\(\mu\) fading as a function of \(\gamma\) for \(\text{SF}\in\{7,9,10\}\). The fading parameters are assumed to be \(\mu=2.065\) and \(\eta=0.00847518\), obtained through a measurement campaign in an indoor environment, as reported in [27]. In order to apply the analytical results obtained in Proposition 8, a _physical_\(\eta\)-\(\mu\) model with \(\mu=2\) was assumed. Exact results were obtained using Monte-Carlo simulation. As can be observed, analytical results closely approximate exact ones, for all considered values of SF, especially for low and medium values of \(\gamma\), thus demonstrating the usefulness of the proposed analysis. In the following, we consider LoRa systems operating in the presence of \(\kappa\)-\(\mu\) fading as a function of \(\gamma\). Note that the \(\kappa\)-\(\mu\) distribution is a two-parameter fading model that well describes wireless propagation in the presence of a line-of-sight (LoS) component [27]. The PDF of \(h^{2}\) is given by [27]  
\[f_{h^{2}}(y)=\frac{\mu(1+\kappa)^{0.5(\mu+1)}y^{0.5(\mu_{X,k}-1)}}{ \kappa^{0.5(\mu-1)}\exp(\mu\kappa)}\] \[\times\exp[-\mu(1+\kappa)y]I_{\mu-1}\bigg{[}2\mu\sqrt{\kappa(1+ \kappa)y}\bigg{]} \tag{45}\]  
where \(\kappa\) and \(\mu\) account for the intensity of the LoS component and the Nakagami-\(m\) component, respectively. Note that the \(\kappa\)-\(\mu\) distribution includes both the Nakagami-\(m\)\((\kappa=0)\) and the Rice \((m=1)\) distributions as special cases. Using [40] (Equation (19)), the parameters of its mixture gamma approximation can be expressed as  
\[a_{i}=\psi(\theta_{i},\beta_{i},\zeta_{i}),\ \beta_{i}=\mu+i-1,\ \zeta_{i}=\mu(1+\kappa) \tag{46a}\] \[\theta_{i}=\frac{\mu(1+\kappa)^{0.5(\mu+1)}\exp(\mu\kappa)}{\kappa^{0.5(\mu-1) }\exp(\mu\kappa)}\frac{\mu^{2i+\mu-3}[\kappa(1+\kappa)]^{0.5(2i+\mu-3)}}{(i-1)!\Gamma(\mu+i-1)}\] (46b) \[\psi(\theta_{i},\beta_{i},\zeta_{i})=\frac{\theta_{i}}{\sum_{j=1}^{N_{\text{ Nems}}}\theta_{j}\Gamma(\beta_{j})\zeta_{j}^{-\beta_{j}}}. \tag{46c}\]  
Figure 4:0 SER of LoRa systems operating in the presence of \(\eta\)-\(\mu\) fading as a function of the SNR, \(\gamma\), in an indoor environment, as reported in [27] and various values of SF. Figure 10 depicts the SER of LoRa systems over \(\kappa\)-\(\mu\) fading, assuming \(\mu=2.1\), \(\kappa=10\) and \(\text{SF}\in\{7,9,10\}\). The \(\kappa\)-\(\mu\) distribution was approximated with a mixture gamma distribution using 37 terms. As it is evident, the results obtained using the mixture gamma approximation match very well with the exact ones, obtained using (9) and (45), for all considered values of SF. Finally, it is worth pointing out that our newly derived formulae for Rice and Nakagami-\(m\) fading were tested against the ones proposed in [23] and a close match was reported. Nevertheless, as also mentioned in the introduction section, the proposed analytical framework still provides accurate results with much lower complexity than those reported in [23].