On the Error Performance of LoRa-Enabled Aerial Networks Over Shadowed Rician Fading Channels

UAVs can be used as aerial relays to provide communication services in remote uncovered areas or dense environments with occasional high capacity demands. However, due to the low power of Internet-of-Things (IoT) devices, UAV-based IoT applications, such as precision agriculture and environment monitoring, may experience high shadowing or equipment failure, which degrades the communications’ quality between IoT devices and their gateways. To tackle this issue, we consider the long range (LoRa) communication technology. Specifically, we investigate the performance of LoRA-enabled aerial communications, where a LoRa gateway communicates with a distant IoT device through the assistance of an amplify-and-forward (AF) aerial relay. Under the assumption of shadowed Rician fading channels, we characterize at first the end-to-end LoRa communication link. Then, we derive an exact symbol error rate expression for the underlying system model. Finally, numerical results are presented to corroborate the efficacy of our derived expressions and provide valuable insights into the error performance of LoRa-enabled aerial networks.

matured over the last years, and its use for civil applications is actively explored in the research and industry fields [2], [3]. In our context, thanks to their deployment flexibility and mobility, UAVs can be placed at strategic 3D locations to both strengthen the gateway-IoT device communication link and provide extended coverage for gateways. This approach is expected to be more cost-effective than deploying additional fixed gateways in the targeted IoT servicing area.
Although the use of UAVs in the context of IoT applications is interesting, its efficiency depends on the communication protocol. Long range (LoRa) differs from existing protocols, such as Wi-Fi, Bluetooth and ZigBee [4], by its ability to achieve communications over long distances at descent (typically low) data rates. Specifically, it relies on the advantages of spread spectrum, chirp orthogonality, and the specific characteristics of kHz and MHz spectrum bands. A LoRa network exploits a fixed infrastructure of gateways for devices connecting with low power and over a long distance range. Nevertheless, due to environmental conditions, LoRa signals transmitted over long distances experience corruption or loss. In this regard, dedicated channel coding mechanisms, e.g., DaRe, are utilized to ensure reliable LoRa signals transmission [5], [6].
Several works investigated the use of LoRa for UAV-assisted IoT networks. The authors of [7] and [8] proposed to use UAVs as flying LoRa gateways to extend the coverage of IoT networks to rural and remote areas. They presented a number of use cases and preliminary results on the received LoRa signal quality at the UAVs, based on a practical implementation. In [9], the authors reported signal strength measurements for LoRa-enabled UAV networks in urban and sub-urban environments. They demonstrated that the UAV altitude and antenna orientation significantly impact the communication range. Additionally, the authors in [10] designed a LoRabased mesh network protocol where communication between an IoT device and a gateway is realized in a multi-hop UAV network. The proposed protocol extends LoRa transmission range compared with the one-hop links, as well as being robust against network dynamic topology changes. Considering a UAV as an aerial gateway, the authors of [11] optimized UAV's altitude to efficiently collect data from ground IoT devices, in a precision agriculture use case. Finally, the authors in [12] supported a disaster management application where LoRa is used for the UAV-IoT device link, while Wi-Fi handles the UAV-gateway communication. The authors evaluated the system's performance in terms of average end-to-end packet 1558-2558 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. reception rate, found to be above 80% when a sufficient number of UAVs is deployed in the targeted area.
In the aforementioned works, UAVs operate as aerial gateways rather than relays. This mechanism requires additional investment and processing at the UAV, which increases the communication and energy costs. Moreover, state-of-the-art works lack performance evaluation studies and theoretical analysis. Within this context, we propose in this letter a new approach consisting of amplifying-and-forwarding (AF) LoRa signals at the UAV. In addition, we provide a rigorous performance analysis of the LoRa-enabled aerial communication system using AF aerial relaying and accurate channel modelling. Although there are some reported results on the performance of LoRa in the literature [13]- [15], to the best of our knowledge, this is the first work that analytically investigates the performance of such a LoRa-enabled system over shadowed Rician fading channels.

II. SYSTEM MODEL
We consider a network in which two LoRa-enabled IoT devices communicate with each other, namely a source (S) and a destination (D), through an AF aerial relay (R), as depicted in Fig. 1. Each node is equipped with a single antenna. The transmitted chirp signals from S uses bandwidth B, i.e., a LoRa sample is transmitted every T b = 1/B. Recalling that LoRa relies on the chirp spread spectrum (CSS) technology, with spreading factor SF ∈ {7, 8, . . . , 12}, LoRa signals are modulated by spreading the frequency of chirp signals over M = 2 SF samples, i.e., symbol duration is T s = M T b . Note that the spreading factor indicates the number of chips per bit and hence, it controls the chirp rate. Also, M denotes the total number of LoRa samples. The n th sample of the m th LoRa symbol can be written as [16] x Note that x m [n], m = 0, . . . , M − 1, n = 0, . . . , M − 1, represents the LoRa orthonormal basis functions. Therefore, the received LoRa signal at R is expressed as follows where P T is the transmit power of the LoRa gateway, α is the path-loss exponent, and L 1 = H 2 + d 2 1 is the S-R distance, with H and d 1 denoting the aerial relay altitude and the distance between S and the projection of R on the ground, respectively. The additive white Gaussian noise (AWGN) at R is modeled by z (R) m [n] ∼ CN (0, N 0 /M ), and N 0 represents the noise power spectral density. Also, g 1 denotes the smallscale fading of the S-R link, which is modeled as a shadowed Rician random variable. In this work, we consider the channel model derived in [17], in which the multi-path fading is modeled as a Rayleigh random variable, while the shadowing effect is modeled using the Nakagami-m 1 distribution. It is worth noting that such a versatile model provides an accurate characterization of the random propagation environment, and facilitates the investigation of the system performance under different scenarios.
The probability density function (PDF) of |g 1 | 2 is given by [17] wherem 1 is the fading severity, with (·) j denoting the Pochhammer symbol and and β i = 1/Ω i,1 . Note that Ω i, 1 and Ω i,2 represent the average power of the multi-path and LoS components, respectively. Assuming blind relaying, the aerial relay amplifies the received LoRa waveform by a scaling factor A and forwards it to D. Subsequently, the normalized received signal at the IoT destination can be written as where g 2 represents the R-D channel coefficient, which follows the shadowed Rician distribution. Since we consider AF relaying, the scaling factor at the relay, A, can be given by where L 2 = H 2 + d 2 2 denotes the R-D distance, d 2 represents the distance between D and the projection of R on the ground, P R is the transmission power of R, andγ 1 = P T /N 0 andγ 2 = P R /N 0 are the average received signal-tonoise ratios (SNRs) at R and D, respectively. Finally, E{·} and (·) * are the expectation and conjugate operators, respectively.
LoRa demodulation is achieved by exploiting the orthogonality property between the M LoRa basis functions, and therefore, a correlator is utilized to evaluate the correlation between the received signal and LoRa basis functions. Based on this, and assuming perfect time and frequency synchronization, the detected LoRa symbol can be evaluated as which, alternatively, can be written as where δ[.] is the Kronecker delta function, andZ m [n] ∼ CN (0, [A 2 |g 2 | 2 + 1]). Therefore, the output of the LoRa demodulator can be written as Assuming an energy detection-based receiver, the detected LoRa symbol is given by Kindly note that such a detection scheme is considered sub-optimal.

III. ERROR RATE PERFORMANCE ANALYSIS
In this section, we derive an exact symbol error rate (SER) expression for the underlying system model, in order to quantify the error rate performance of LoRa modulation in aerial networks.
The probability that the m th LoRa symbol is incorrectly detected can be given as Given that |Z l [n]| are independent and identically distributed for l = 0, . . . , M − 1 and l = m, then it can be concluded that Accordingly, the error rate expression in (14) can be rewritten as Theorem 1: The exact SER of LoRa modulation in the underlying system model is given by (17), as shown at the bottom of the page, where Γ(.) and Ψ(.; .; .) are the complete Gamma and the Tricomi functions, respectively [18], and Proof: In order to evaluate the closed-form expression of the SER, we first evaluate the following probability expression Conditioned on g 1 , g 2 , andZ m [n], and given thatZ l [n] follows the complex Gaussian distribution, (19) can be expressed in terms of the cumulative distribution function of a Rayleigh random variable as To obtain an unconditional SER expression, in the following, we first derive the PDF of Δ m . Conditioned on |g 1 | and |g 2 |, Δ m is modeled as a Rician random variable with noncentrality parameter equals toγ 1 A 2 |g 2 | 2 |g 1 | 2 and variance [A 2 |g 2 | 2 +1]. Therefore, the PDF of Δ m can be represented as Using the PDF of |g 1 | 2 , the PDF of Δ m , given in (21) (.; ., .) denotes the confluent hypergeometric function. The derived PDF in (23) is further utilized to evaluate an exact SER expression. Conditioned on |g 1 | 2 , the SER of the m th LoRa symbol can be evaluated by integrating (16) over the PDF in (23), yielding (24), as shown at the bottom of the page. Note that, similar to the PDF of |g 1 | 2 , the PDF of |g 2 | 2 is given by (3), by replacingm 1 →m 2 , Ω 1,1 → Ω 2,1 , and Ω 1,2 → Ω 2,2 . To further simplify the expression in (24), we first resort to the binomial expansion theorem as follows Finally, by leveraging (25) and [20, eq. 2.3.6.9], the conditional SER in (24) can be averaged over the PDF of |g 2 | 2 , and subsequently, a closed-form expression of the SER can be obtained as (17).

IV. NUMERICAL RESULTS
In this section, we present numerical results to validate the derived analytical framework, and to demonstrate the error rate performance of LoRa-enabled aerial networks, when shadowed Rician fading is experienced over the dual-hop link. Unless stated otherwise, simulation parameters are presented in Table I. Without loss of generality, we assume that S-R and R-D links are balanced, i.e.,m 1 =m 2 =m,γ 1 =γ 2 =γ, Ω 1,1 = Ω 2,1 = Ω 1 , and Ω 1,2 = Ω 2,2 = Ω 2 . Note that in all presented results, we consider the SNR per chirp.
In Fig. 2, we study the SER performance versus Ω 2 , for SF ∈ {7, 9, 12}, andm ∈ {2, 4}. From the figure, it can be observed that the error rate performance of the underlying system model improves as SF decreases. It can be further Fig. 2. Average SER versus Ω 2 for m = 2, 4, and SF = 7, 9, 12. noticed that SF has no effect on the achievable diversity order of the system, rather, the diversity order is determined by the fading severity, quantified bym. However, it is noted that the effect of the considered SF value has less impact on the system performance asm value increases. This is due to the fact that asm increases, the LoS link quality improves, and hence, the effect of the channel becomes dominant.
The results obtained in Fig. 3 demonstrate the advantage of LoRa in aerial networks. In particular, Fig. 3 investigates the error rate performance of LoRa-enabled UAV system versus the UAV altitude, H, for Ω 2 ∈ {5, 15} dB. Although it shows that the system performance is highly affected by the quality of the LoS link, it can be noticed that the system performs reliably even at high UAV altitudes. Specifically, it is observed that the system experiences a relatively acceptable error rate when the UAV is flying at altitudes above 1 Km, at all SF values. This further motivates the use of LoRa in aerial networks as an efficient scheme to extend the transmission distance. Finally, obtained results for the no shadowing scenario justify the performance analysis of the considered system under the shadowed Rician channel model.   . Although the considered system model demonstrates an improved performance when SF=7, compared to the SF=12 scenario, it is shown that the error rate behavior follows the same trend for all SF and Ω 1 values. On the other, it can be observed that the LoS component has a dominant impact on the system performance. In particular, when Ω 1 ≥ Ω 2 , the system performance is mainly determined by the LoS average power, whreas for Ω 2 > Ω 1 , the LoS and multipath components impact in the same manner the error rate performance. Furthermore, it is demonstrated in Fig. 4 that for a given Ω 2 , the system's performance enhances proportionally to Ω 1 . Recalling that UAV networks are typically characterized by a strong LoS component, the use of LoRa provides them with highly reliable wireless communication links.

V. CONCLUSION
In this letter, we have proposed an analytical framework in order to investigate the error rate performance of LoRaenabled aerial communications. In particular, we derived a novel closed-form expression for the symbol error rate, under the assumption of shadowed Rician fading. The obtained results confirm the advantages of exploiting LoRa in order to improve the reliability and coverage of UAV-assisted IoT applications.