# Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels
## 1 Introduction  
Low Power Wide Area Networks (LPWAN) are the new trend in the development of wireless communication technologies in the era of the Internet of Things (IoT) [1]. Long Range (LoRa) [2] technology has awakened the interest of the research community due to its growing usage for LPWAN deployments [2; 3; 4; 5].  
As clearly stated in [6; 7; 8; 9], LoRa technology is mainly based on the features of the chirp spread spectrum (CSS) modulation. In [10], it establishes a more solid mathematical representation of the modulation/demodulation process addressing the bit error rate (BER) performance for AWGN and frequency selective channels. In [11], the BER was derived assuming CSS modulation and the use of moment matching to obtain analytical BER approximations for AWGN and Rayleigh scenarios. The BER performance under the perspective of orthogonal signaling described in [12] can be used to assess the performance of chirp spread spectrum modulation, as pointed out in [10].  
The primary motivation of this investigation is to find new expressions for the bit error rate of chirp spread spectrum (CSS) modulation in the essential fading scenarios. For instance, Rayleigh statistics models well the signal power in a multipath environment [13]. On the other hand, Rice appropriately models multipath in situations with a dominant line-of-sight signal. In complex dense urban environments, we expect all kinds of small-scale fading, i.e., Ricean fading for strong, static links, and a smooth transition from Rayleigh fading down to Double-Rayleigh fading for moving links. A mathematically convenient method to approximate these distributions is by using the Nakagami distribution [14]. The use of Nakagami mathematical method is broad and can describe the amplitude of a received signal after maximum ratio diversity combining, since the \(k\)-branch MRC with Rayleigh-fading signals results in a Nakagami with \(m=k\). In this way, it is possible to use the Nakagami-_m_ distribution to investigate the use of multiple antennas (MIMO) in LoRa gateway (or LoRa base station). The Nakagami distribution is also the best match for many fading measured data, as related in [15; 16; 17].  
The main contribution of our work is the derivation of exact closed-form expressions to calculate the CSS BER performance in diverse scenarios such as Rayleigh, Rician, and Nakagami fading channels.  
We believe that our main results extend the knowledge in this topic by including two new scenarios: the performance analysis under the Nakagami-\(m\) case and the Rice case. As it is well known in the literature, as the Nakagami parameter \(m=1\), or the Rice parameter \(\kappa=0\), our results specialize into the Rayleigh case. Also, the authors in [11] provided an approximate expression for the Rayleigh case, whereas we have provided an exact and elegant result.  
We organize the remainder of this paper as follows: Section 2 refers to the underlying system model; Section 3 refers to the exact bit error probability performance derivation. Section 4 confirms the accuracy of the proposed exact expressions. Finally, Section 5 points out the main aspects of our achievements.