# New Results for the Error Rate Performance of LoRa Systems over Fading Channels
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c} \hline
**Authors** & **Title** & **Source** & **Findings** \\ \hline \multirow{3}{*}{[PERSON]} & Frequency shift chirp & & Introduced the LoRa modulation system \\  & modulation: The LoRa & [7] & and provided initial results on its \\  & modulation & & performance over AWGN channels by \\  & & & means of a single integral. \\ \hline \multirow{3}{*}{[PERSON]; [PERSON]} & Closed-form & & Provided simple closed-form expressions of \\  & approximation of LoRa & & LoRa systems in the presence of AWGN \\  & modulation BER & & and Rayleigh fading. \\  & performance & & \\ \hline \multirow{3}{*}{[PERSON]; [PERSON]; [PERSON]} & Bit error rate closed-form & & Provided an exact closed-form expression for the BER of LoRa systems under \\  & expressions for LoRa & & Rayleigh fading as well as analytical \\  & systems under Nakagami & & expressions for the BER under Nakagami-_m_ \\  & and Rice fading channels. & & and Rice fading in terms of a finite sum. \\ \hline \multirow{3}{*}{[PERSON]; [PERSON]; [PERSON]; [PERSON]} & A Computable Form for & & Authors elaborate on the properties of the \\  & LoRa Performance & & generalized Marcum Q-function to provide \\  & Estimation: Application & & accurate expressions for the BER of LoRa \\  & to Ricean and Nakagami & & systems in the presence of Rice and \\  & Fading. & & Nakagami-_m_ fading. \\ \hline \multirow{3}{*}{[PERSON]; et al.} & Analysis and & & Authors addressed the performance of \\  & Performance & & LoRa systems operating in the presence of \\  & Optimization of LoRa & & Rayleigh fading, enhanced with antenna \\  & Networks With Time and & & and time diversity techniques. The \\  & Antenna Diversity & & optimization of the performance of such \\  & & & systems has further been addressed. \\ \hline \multirow{3}{*}{[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]} & Design and Performance & & Authors have proposed a new STBC MIMO \\  & Analysis of a New & & LoRa system architecture. Its theoretical \\  & STBC-MIMO LoRa & & performance was analyzed in the presence \\  & System & & of Rayleigh fading. A closed-form \\  & & & approximate BER expression of the \\  & System & & proposed system under perfect and \\  & & & imperfect channel state information (CSI) \\  & & & was proposed. \\ \hline \multirow{3}{*}{[PERSON]; [PERSON]; [PERSON],} & Performance analysis of a & & Authors studied a two-hop opportunistic \\  & two-hop relaying LoRa & & amplify-and-forward relaying LoRa system \\ \cline{1-1}  & system & & employing a best relay-selection protocol \\ \cline{1-1}  & system & & and operating over Nakagami-_m_ fading. \\ \hline \end{tabular}
\end{table} 
Table 1: Related works on theoretical performance of LoRa systems in the presence of fading and noise. as Matlab or Mathematica [26]. Moreover, for the numerical evaluation of the underlying mathematical expressions, the computation of an approximation threshold parameter is required. Nevertheless, the exact computation of this threshold is rather complicated and therefore, a heuristic method for its computation was proposed by the authors in the same work. The above facts motivate simpler, yet accurate expressions for the evaluation of the BER of LoRa systems in the presence of noise and fading. On the other hand, analytical results for the error performance of LoRa systems in the presence of fading channels other than Rayleigh, Rice and Nakagami-\(m\), are--to the best of our knowledge--not available in the open technical literature. Indeed, as it was pointed out in [27], the above mentioned classical fading models do not always fit well measured data, especially at the tail portion. This motivates research on performance evaluation over generalized fading models that include the classical ones as special cases. Motivated by the above facts, in this study we present new analytical expressions for the average bit error rate evaluation of LoRa systems in the presence of fading. More specifically, the novel research contributions of this work can be summarized as follows. 1.