# New Results for the Error Rate Performance of LoRa Systems over Fading Channels
## 1 Introduction
Under the assumption of Nakagami-\(m\) and Rice fading channels, we present approximate analytical expressions for the SER performance of LoRa systems. These expressions yield accurate results in the entire signal to noise ratio (SNR) region that are practically indistinguishable from the exact solution;
2. For the special case of Nakagami-\(m\) fading, using a moment matching method, a simple yet tight approximation to the SER is obtained in closed form;
3. For all fading scenarios, exact analytical SER expressions in terms of a single integral are presented;
4. A novel, accurate analytical expression for the SER of LoRa systems operating in the presence of Hoyt fading is presented. To this end, a new integral involving exponentials, modified Bessel functions and the Marcum-Q function, whose second argument is a linear function of the integration variable, is evaluated;
5. An exact single integral expression for the SER of LoRa systems operating over \(\eta\)-\(\mu\) fading channel is presented, assuming a propagation environment consisting of a finite number of multi-path clusters;
6. An exact single integral expression for the SER of LoRa systems operating over generalized fading channels is presented, by approximating the PDF of the SNR with a mixture gamma distribution. As a test case, SER results of LoRa systems operating in the presence of \(\kappa\)-\(\mu\) fading channels are presented. In order to validate the correctness of the proposed mathematical analysis, all analytical results are substantiated by means of Monte-Carlo simulations. Note that the proposed analytical framework provides accurate results in the entire SNR range, thus circumventing the need for evaluating system performance via time consuming Monte-Carlo simulations (It is a common practice to use Monte-Carlo simulations in order to verify the correctness of analytical results. Please note that although Monte-Carlo simulations may also be used to obtain performance evaluation results, they suffer from two significant disadvantages, as compared to analytical results. First, one has to specify the system model using software defined components, i.e., to simulate channel, noise, modulation, demodulation and detection. Although this process provides further insights on the system structure, it is computationally very intensive, time consuming and requires large amounts of memory to achieve a given accuracy. Specifically, as a rule of thumb, in order to obtain exact BER results of the order of \(10^{-6}\), random samples of two orders of magnitude larger, namely \(10^{8}\), are required. Such large vectors are difficult to be handled by software tools such as Matlab. On the other hand, analytical results in the form of equations yield accurate results within a large range of system level parameters). The remainder of this work is structured as follows. Section 2 presents an overview of the LoRa modulation and its BER performance. Section 3 presents the main results of this work. Numerical results are presented in Section 4 whereas Section 5 concludes the work. _Notations_: A list of mathematical notations used in this work is available in Table 1.