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LoRa Throughput Analysis with Imperfect Spreading Factor Orthogonality

Antoine Waret, Megumi Kaneko, Alexandre Guitton, Nancy El Rachkidy

arXiv:1803.06534v1cs.NI

TL;DR

LoRa throughput studies often assumed different SFs were perfectly orthogonal, despite co-SF and inter-SF interference. This paper derives uplink-throughput expressions that include LoRa’s reception and capture conditions, then evaluates them under different SF allocations and orthogonality assumptions. The analysis closely approximates simulations, while results show throughput differences across allocations and losses from imperfect orthogonality.

  • Problem

    Many prior LoRa scalability studies assumed perfect orthogonality among SFs, while co-SF and inter-SF interference affect uplink throughput.

    Source: §I · p. 1

  • Method

    The paper derives theoretical uplink-throughput expressions incorporating reception, co-SF capture, and inter-SF capture conditions for SF allocations.

    Source: §I · p. 1

  • Results

    The derived throughput expressions closely approximate simulations; inter-SF interference causes earlier performance decline than perfect orthogonality, and allocation performance varies with device count.

    Source: §IV · p. 4

  • Takeaways & Limitations

    The analysis provides a framework for predicting how SF allocations affect throughput in a single-gateway LoRa uplink.

    Source: §V · p. 4

    The analysis approximates an expression because which condition dominates can depend on j.

    Source: B. Imperfect Orthogonality · p. 3

Abstract

from arXiv · show

LoRa is one of the promising techniques for enabling Low Power Wide Area Networks (LPWANs) for future Internet-of-Things (IoT) devices. Although LoRa allows flexible adaptations of coverage and data rates, it is subject to intrinsic types of interferences: co-SF interferences where end-devices with the same Spreading Factors (SFs) are subject to collisions, and inter-SF interferences where end-devices with different SFs experience collisions. Most current works have considered perfect orthogonality among different SFs. In this work, we provide a theoretical analysis of the achievable LoRa throughput in uplink, where the capture conditions specific to LoRa are included. Results show the accuracy of our analysis despite approximations, and the throughput losses from imperfect SF orthogonality, under different SF allocations. Our analysis will enable the design of specific SF allocation mechanisms, in view of further throughput enhancements.

I. INTRODUCTION

LoRa’s data-rate and coverage trade-off, together with co-SF and inter-SF interference, motivates analyzing uplink throughput beyond the perfect-orthogonality assumption. The paper models two SF-allocation policies in a single-gateway LoRaWAN setting.

I. INTRODUCTION

  • Prior scalability studies often assumed perfect SF orthogonality, while later work found that inter-SF interference can reduce performance, especially at high SFs.Co-SF collisions occur between devices using the same SF; inter-SF collisions occur between devices using different SFs.Source: §I · p. 1
  • The analysis derives uplink throughput accounting for co-SF and inter-SF interference, with successful reception requiring an SNR threshold and both types of SINR capture threshold.It compares perfect and imperfect SF orthogonality under two SF-allocation types.Source: §I · p. 1

II. SYSTEM MODEL

  • The system model considers uniformly distributed end-devices in one cell communicating with a central gateway on one 125 kHz channel under saturated pure-ALOHA traffic.It uses six SFs, from SF7 through SF12.Source: §II · pp. 1–2
  • SF-random assigns each device an SF uniformly, whereas SF-distance assigns SFs according to the device’s distance from the gateway.For SF-distance, devices in an annulus between successive distance thresholds receive the corresponding SF.Source: §II · p. 2

III. PROPOSED THROUGHPUT ANALYSIS

Uplink reception is conditioned on three requirements being met.

III. PROPOSED THROUGHPUT ANALYSIS

  • A packet reaches the gateway successfully only when all three specified reception and capture conditions are fulfilled.The available passage introduces the three-condition requirement but does not state the individual conditions.Source: §III · p. 2

1) Reception condition:

The analysis treats reception and same-SF capture as separate conditions for a successful uplink packet.

1) Reception condition:

  • Reception requires the received signal’s SNR to exceed the threshold specific to its SF.The passage describes this as a probability for a device at distance d_i from the gateway.Source: 1) Reception condition · p. 2

2) Co-SF capture condition:

3) Inter-SF capture condition:

The imperfect-orthogonality analysis accounts for interference between different SFs when evaluating uplink throughput.

3) Inter-SF capture condition:

A. Perfect Orthogonality

Under perfect SF orthogonality, the success probability is evaluated without inter-SF interference. It accounts for the number of devices using SFm and the joint reception and co-SF capture probability.

A. Perfect Orthogonality

1) For j = 1:

When j = 1, the device has no co-SF interferers, so success depends on reception. The analysis models its SNR and evaluates the reception probability numerically when needed.

1) For j = 1:

  • With j = 1, the device has no co-SF interference, so only the reception condition must be satisfied.Source: 1) For j = 1 · p. 3
  • The reception calculation models SNR γi as an exponential random variable with a mean determined by the stated assumptions.Source: 1) For j = 1 · p. 3
  • The reception threshold q_SFm is specific to SFm, and the resulting integral is evaluated numerically because it has no closed form.Source: 1) For j = 1 · p. 3

2) For j ≥2:

For j ≥ 2, co-SF capture implies the reception condition under the stated thresholds. The analysis evaluates co-SF capture using the j − 1 same-SF interferers and derives the SF-distance expression.

2) For j ≥2:

  • For all SFs, q_SFm ≤ 1 while q_coSFm = 4 (6 dB); therefore, satisfying co-SF capture also satisfies reception.Source: 2) For j ≥2 · p. 3
  • With j devices at SFm, the co-SF capture calculation accounts for j − 1 interferers using the same SF and distance boundaries.Source: 2) For j ≥2 · p. 3
  • The co-SF capture probability is developed by marginalizing over random instantaneous and average SNR variables, using independence across user channels.Source: 2) For j ≥2 · p. 3
  • For α = 4, the primitive function of I(ri) is given, and the resulting expression is written for SF-distance allocation with perfect SF orthogonality.Source: 2) For j ≥2 · p. 3

B. Imperfect Orthogonality

With imperfect orthogonality, successful transmission must satisfy reception, co-SF capture, and inter-SF capture conditions. The analysis treats the cases j = 1 and j ≥ 2 and approximates their joint probability when an exact expression is difficult.

B. Imperfect Orthogonality

IV. NUMERICAL RESULTS

Under imperfect orthogonality, throughput depends on SF allocation and device count, while the derived expressions closely match simulations.

IV. NUMERICAL RESULTS

  • Evaluation uses fc = 868 MHz, BW = 125 kHz, P0 = 14 dBm, α = 4, and R = 1 km.The path-loss exponent is set as in for an urban setting.Source: §IV · p. 4
  • Up to 50% system-throughput loss occurs under imperfect SF orthogonality; expressions closely match simulations, with a small discrepancy around five devices for imperfect SF-distance.As device count increases, co-SF interference still leads to a scalability limit.Source: §IV · p. 4
  • For fewer than 25 devices, SF-distance achieves up to 100% gain; at greater N, SF-random performs slightly better.The comparison is for imperfect orthogonality; Fig. 3 shows throughput performance for both allocations at R = 1 km and α = 4.Source: Figure 3 · p. 4, §IV · p. 4

V. CONCLUSION

The paper derives a theoretical throughput expression for a single-gateway LoRa uplink and incorporates reception, co-SF capture, and inter-SF capture conditions. Its results show effects from imperfect orthogonality and SF allocation, supporting tailored allocation design based on predicted performance.

V. CONCLUSION

  • The analysis derives a theoretical throughput expression for the uplink of a single-gateway LoRa network.The model is based on the LoRa physical layer.Source: §V · p. 4
  • Successful-frame conditions include the SNR reception level, SINR for co-SF capture, and SINR for inter-SF capture.Source: §V · p. 4
  • Results show non-negligible effects from imperfect SF orthogonality and SF allocation; the framework predicts allocation impacts to support tailored designs.Allocation choices are to be tailored to environments and requirements.Source: §V · p. 4
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