As the amount of mobile data traffic will rapidly increase during the upcoming years (studies forecast 50 billion Internet of Things (IoT) devices by 2020), new spectrum access strategies adapted to high device densities are ever more crucial. LoRa  is one of the prominent candidates for Low Power Wide Area Networks (LPWANs), providing wide communication coverage with low power consumption, at the expense of data rate. Operating in license-free ISM bands (i.e., 868MHz in Europe), the LoRa PHY layer uses a chirp spread-spectrum modulation where different Spreading Factors (SFs) tune the chirp modulation rates. Lower SFs such as SF7 allow for higher data rates but reduced transmission range, whereas higher SFs such as SF12 provide longer range at lower data rates. On top of the LoRa PHY layer, the higher layers were defined by the LoRa Alliance and referred as LoRaWAN . In particular, the MAC protocol is based on a pure ALOHA access with duty cycle limitations. The LoRaWAN network architecture is a star-like topology where end-devices communicate with gateways over several channels.
Most studies on LoRa scalability so far assumed a perfect orthogonality among SFs, thereby creating virtual channels where multiple users with different SFs could simultaneously operate in the same channel and hence boost the achievable system throughput. Thus, a number of works have considered the effect of co-SF interference only, where end-devices using the same SF on the same channel are subject to collisions 
. In particular, the outage probability of a LoRa system under co-SF interference was analyzed in, where a signal could be captured if its Signal-to-Interference-plus-Noise Ratio (SINR) was higher than 6 dB. As the number of devices increased, it was shown that those co-SF interferences were causing a scalability limit. However, recent studies have pointed out the fact that SFs were not perfectly orthogonal among themselves . Thus, the effect of inter-SF collisions was investigated through computer simulations and/or experiments. Namely,  showed that inter-SF interferences could considerably decrease LoRa performance, especially for high SFs where frames have a greater time on air.
In this work, we propose a theoretical analysis of the achievable throughput on the uplink of a LoRa network, encompassing the effects of co- and inter-SF interferences. To ensure a successful transmission, a packet must thus satisfy three conditions: 1) its SNR is above the reception threshold, 2) its SINR under co-SF interference is above the co-SF capture threshold, and its SINR under inter-SF interference is above the inter-SF capture threshold. Considering two different types of SF allocations, we theoretically derive the achievable throughput expressions for both perfect and imperfect SF orthogonality. Simulation results show the accuracy of our analytical expressions despite the necessary approximations, as well as the impact of the various types of interferences and SF allocations on the overall system performance.
Ii System Model
We consider one cell of radius with one gateway located at its center, as depicted in Fig. 1. There are N
end-devices uniformly distributed within the cell. We denote bythe distance from end-device i to the gateway. Since the goal of our analysis is to derive the achievable rate by LoRa, we assume that all end-devices transmit in a single channel of bandwidth kHz and that they all have packets to transmit. This corresponds to the pure ALOHA access as in LoRaWAN with saturated traffic222Our analysis can be easily applied to multiple channels and duty cycles.. We consider SFs, for , with and , with symbol times . The bit-rate of SF is 
where CR is the coding rate defined as with . Lower SFs allow higher data rate but lower communication range whereas higher SFs provide longer range at the expense of data rate (see Table I).
Two types of SF allocation will be investigated. In the first one, the spreading factors are uniformly distributed, i.e., every end-device has a probability of selecting SF. We refer to this allocation as SF-random. In the second type of allocation referred to as SF-distance, spreading factors are assigned according to the distance . A device located inside the annulus defined by the smaller and larger circle radii and , respectively, has SF. The distance threshold for SF is given by where is the deterministic loss in the path loss model as in , the carrier frequency and the path loss exponent. is the receiver sensitivity of SF (see Table I). All nodes transmit at equal power . We assigned to and the origin of the cell and its radius respectively, i.e., and . The ranges for each SF are given in Table I. The probability of selecting SF for the SF-distance allocation is then given by , where is the pdf of the position of an end-device in the cell at distance from the gateway. For uniform distribution of devices within a cell of radius , we get .
The instantaneous SNR of end-device is defined as , where is the channel gain between end-device and the gateway (for Rayleigh fading, ). [dBm] is the AWGN power and NF, the receiver noise figure.
Based on , it is assumed that in the event of a collision between frames of different SFs, one signal is received successfully if its SINR is higher than its “InterSF capture threshold” in Table I. Moreover, if there are several signals with equal SFs transmitting on the same frequency simultaneously, the gateway is able to successfully receive one of them if its SINR is higher than 6 dB, for any SF . Therefore, both types of interferences will be considered.
|SF||Bit-rate [kb/s]||Receiver Sensitivity [dBm]||Reception thresh. [dB]||InterSF capture thresh. [dB]||Dist. thresh. [km]|
Iii Proposed Throughput Analysis
An end-device’s packet transmission in uplink is successfully received at the gateway if the three following conditions are fulfilled:
1) Reception condition:
Signal power must be above the SF-specific threshold ,
which is the probability that a received signal from end-device at a distance from the gateway has a SNR above the threshold (Table I).
2) Co-SF capture condition:
The co-SF SINR of end-device is defined as
where is the number of end-devices with the same SF as end-device . A signal is successfully received if its is above . Therefore, the second condition is given by
3) Inter-SF capture condition:
A transmission is successful if is higher than the threshold , thus the third condition is given as
Therefore, the uplink throughput can be expressed as
where is the bit-rate of SF and , the probability of a successful transmission.
Next, we analyze the throughput under perfect and imperfect SF-orthogonality, for the SF-distance case.
Iii-a Perfect Orthogonality
We assume first that SFs are perfectly orthogonal, i.e., no end-device suffers inter-SF interferences. Hence, the probability of a successful transmission is given by
where denotes the total number of end-devices at SF, is the probability of having end-devices among at SF and is the joint probability for reception condition and co-SF capture.
the end-device is not subject to co-SF interferences, thus only the reception condition needs to be satisfied,
First, we determine for the SF-distance case. Given our assumptions, the SNR
is modeled as an exponential random variable with mean. Therefore,
where is the specific threshold of SF. Defining , where is the path-loss constant, we can now rewrite,
Although this integral cannot be expressed in closed form, it can be efficiently determined by numerical methods.
both the reception and co-SF conditions must be fulfilled. As 1 in linear for all SFs whereas (6 dB) for all SFs as explained in Section II, if co-SF capture is satisfied, so is the reception condition, hence
In case of co-SF interferences, there are -1 interferers, , which is developed using random instantaneous SNR variables and random average SNR (position) variables as
Marginalizing over and making the change of variable , we get by independency of user channels,
We define . Nodes are uniformly distributed within the cell, therefore . As a result, the expression becomes,
In particular, for , the primitive function of is
In case of co-SF interferences, the interferers are the end-devices with the same SF as end-device , thus with the same distance boundaries. The expression of becomes
Iii-B Imperfect Orthogonality
In reality, spreading factors are not perfectly orthogonal, so all three capture conditions are to be satisfied to achieve a successful transmission. Thus, becomes
where is the joint probability for reception condition, capture co-SF and capture inter-SF.
the end-device is only subject to inter-SF interferences and the reception condition. As shown in Table I, the inter-SF condition is dominant over the reception one, especially as the number of end-devices increases. Thus,
The expression of is similar to , only with different thresholds and number of interferers. When dealing with inter-SF interferences, one must consider the end-devices within the cell that are not in the annulus corresponding to SF , i.e., the end-devices with a different spreading factor. If is the number of end-devices at SF, then there are end-devices with other spreading factors. Therefore,
Here, , where denotes the whole cell area excluding the area corresponding to SF. Using (10), we obtain
all capture conditions are to be considered. As in the perfect orthogonality case, the reception condition derives from the two others, thus
Because of the difficulty to find an exact expression of (16), as one condition might be dominant over the other depending on , we approximate it as
Finally, (13) can be written for SF-distance allocation with imperfect orthogonality as,
Due to lack of space, the analytic throughput expressions for the SF-random case are not given, as they can be obtained similarly as for SF-distance.
Iv Numerical Results
System throughput was evaluated for perfect/imperfect orthogonality as well as for both types of allocations, to assess the validity of our analytical expressions. The main parameters are MHz, kHz and the end-devices’ transmit power dBm. The path loss exponent was set to as in  (urban) and km.
Fig. 2 shows throughput performance obtained by simulations and by our theoretical derivations under SF-distance allocation for both perfect and imperfect orthogonality against varying numbers of end-devices transmitting simultaneously. We observe that our derived throughput expressions approach almost perfectly the simulations results with only a small interstice for small values of (around 5 end-devices) in the imperfect orthogonality case. Therefore, despite approximations, we obtain accurate throughput expressions for both orthogonality cases. Next, we can see that inter-SF interferences cause an early decrease of performance compared to the perfect orthogonality case. However, as the number of devices increases, co-SF interferences always lead to a scalability limit. These results show the impact of imperfect SF orthogonality over the system throughput, up to loss.
Next, the throughput performance for both types of allocations with imperfect orthogonality are shown in Fig. 3. First, we can see that our analysis provides a good approximation of the achievable throughput under SF-random allocation. Then, we notice that for a small amount of end-devices (less than 25), better throughput efficiency is achieved in SF-distance allocation case (up to gain), since devices are more likely to satisfy the specific threshold . On the other hand, SF-random allocation performs slightly better for greater values of . Given the higher density of co-SF end-devices under SF-distance, these results suggest that even a simple SF-random policy provides a higher throughput as the number of end-devices increases. This is because the SF-random allocation favors the case where a limited number of devices randomly choose a small SF, by decreasing their collision probability. Moreover, small SFs lead to larger throughput than large SFs. Thus, our analysis will be useful to devise new allocation policies under various conditions and environments.
We have considered the uplink of a single gateway LPWAN based on LoRa physical layer, for which a theoretical throughput expression was derived. Unlike most previous works, our analytical expression encompasses all three conditions required for successful frame transmission: SNR reception level, SINR level for co-SF capture, and SINR level for inter-SF capture. Results have shown the non-negligible impact of SFs’ imperfect orthogonality, as well as the drastic effects of SF allocations on the overall throughput. Our analytic framework hence provides a precious tool for designing tailored SF allocations depending on environments and requirements, by predicting their impact on system performance.
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