# New Results for the Error Rate Performance of LoRa Systems over Fading Channels
## 3 Main Results
### Symbol Error Probability for Nakagami-\(m\) Fading Channels  
Under Nakagami-\(m\) fading, the RV \(h^{2}\) follows a gamma distribution with PDF given as [28]  
\[f_{h^{2}}(y)=\frac{m^{m}}{\Gamma(m)}y^{m-1}\exp(-my) \tag{11}\]  
where \(m>0\) is the fading parameter. For \(m=1\), i.e., for Rayleigh fading, (11) reduces to the exponential distribution. An exact analytical expression for \(P_{s}\) is given in the following proposition. **Proposition 1**. : _The exact symbol error probability of LoRa systems under Nakagami-m fading in terms of a single integral is given as_  
\[P_{s}^{\rm Nak} =\frac{1}{2}\left(\frac{m}{N\gamma+m}\right)^{m}\int_{0}^{\infty} e^{-x/2}[1-(1-e^{-x/2})^{L}]\] \[\times\mathrm{L}_{-m}\bigg{[}\frac{N\gamma x}{2(N\gamma+m)} \bigg{]}\mathrm{d}x \tag{12}\]  
Proof. : By substituting (11) into (9) and changing the order of integration, a valid operation according to the Fubini theorem because the resulting integrals are convergent, one obtains  
\[P_{s} =\frac{m^{m}}{2\Gamma(m)}\int_{0}^{\infty}e^{-x/2}[1-(1-e^{-x/2})^ {L}]\] \[\times\left[\int_{0}^{\infty}y^{m-1}e^{-(N\gamma+m)y}I_{0}\Big{(} \sqrt{2 Nxy}\Big{)}\mathrm{d}y\right]\mathrm{d}x. \tag{13}\]  
By employing [29] (Equation (3.15.1/2)) and [30] (Equation (8.972/1)), (12) is readily obtained, thus completing the proof. Note that (12) yields the exact value of \(P_{s}\) for arbitrary values of \(m\). In addition, it converges rapidly due to its exponentially decaying kernel and can be evaluated numerically in an efficient manner using built-in routines available in popular mathematical software packages such as Matlab or Mathematica. In what follows, we derive accurate approximations for \(P_{s}\), assuming both arbitrary and integer values of the fading parameter \(m\). The following result holds. **Proposition 2**. : _For arbitrary values of \(m\), an accurate approximation for the \(P_{s}\) of LoRa systems in the presence of Nakagami-m fading is given as_  
\[P_{s}^{\rm Nak} \approx\left(\frac{m}{N\gamma+m}\right)^{m}\exp(-\hat{x}_{N}/2)\] \[\times\sum_{n=1}^{\infty}\frac{\hat{x}_{N}^{n}}{2^{n}\Gamma(n+1)} \mathrm{F}_{1}\bigg{[}m;n+1;\frac{N\gamma\hat{x}_{N}}{2(N\gamma+m)}\bigg{]} \tag{14}\]  
_where as for integer values of \(m\)_  
\[P_{s}^{\rm Nak} \approx 1-\frac{N\gamma}{N\gamma+m}\exp\left[-\frac{m\hat{x}_{N}}{2( N\gamma+m)}\right]\] \[\times\sum_{n=0}^{m-1}\epsilon_{n}\bigg{(}\frac{m}{N\gamma+m} \bigg{)}^{n}\mathrm{L}_{n}\bigg{[}-\frac{N\gamma\hat{x}_{N}}{2(N\gamma+m)} \bigg{]} \tag{15}\]  
_where_  
\[\epsilon_{n}= \begin{cases}1&\text{if }n<m-1\\ 1+\frac{m}{N\gamma}&\text{if }n=m-1\end{cases}  \tag{16}\]  
_and_  
\[\hat{x}_{N}=2\sum_{n=1}^{N-1}n^{-1}. \tag{17}\]  
Proof. : Our starting point to the proof is (5) via which a generic expression for \(P_{s}\) can be obtained. Observe that for large values of SF, i.e., for SF \(\geq 6\), the RV \(\rho^{2}\) can be replaced with its mean with sufficient accuracy. Note that this observation has also been reported in [21]. Consequently, using (5) and (8), \(P_{s}\) can be approximated as\[P_{s}\approx 1-\mathbb{E}_{h^{2}}\bigg{\langle}Q_{1}\bigg{(}\sqrt{2N\gamma h^{2}}, \sqrt{\tilde{x}_{N}}\bigg{)}\bigg{\rangle}, \tag{18}\]  
where \(\tilde{x}_{N}\) is the expectation of the RV \(\rho^{2}\), which, by employing the memoryless property of the exponential distribution [21], can be deduced as (17). Using (11), the expectation in (18) can be further written as  
\[P_{s}\approx 1-\frac{m^{m}}{\Gamma(m)}\int_{0}^{\infty}y^{m-1}e^{-my}Q_{1} \Big{(}\sqrt{2N\gamma y},\sqrt{\tilde{x}_{N}}\Big{)}\mathrm{d}y \tag{19}\]  
Using [31] (Equation (10)) and [31] (Equation (11)) (Note that [31] (Equation (11)) has a typo, i.e., \(N\) should be replaced with \(\Gamma(N)\)), (14) and (15) can be deduced for real and integer values of \(m\), respectively, thus completing the proof. Next, using a moment matching method, a simpler closed-form expression for \(P_{s}\) will be derived, which holds for arbitrary values of \(m\).