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Scalability Analysis of a LoRa Network under Imperfect Orthogonality

Aamir Mahmood, Emiliano Sisinni, Lakshmikanth Guntupalli, Raúl Rondón, Syed Ali Hassan, Mikael Gidlund

arXiv:1808.01761v1cs.IT

TL;DR

LoRa scalability is insufficiently characterized when co-SF interference and imperfect inter-SF orthogonality jointly affect dense deployments. The paper develops a stochastic-geometry model for single-cell uplink performance and uses it to assess success, coverage, and scalability under these interference conditions.

  • Problem

    LoRa scalability in dense deployments requires accounting for both same-SF interference and interference from different SFs caused by imperfect orthogonality.

  • Method

    The paper models a multi-annuli single-cell LoRa network with stochastic geometry, representing interference as a Poisson point process while incorporating MAC, PHY, fading, path loss, and regulatory constraints.

  • Results

    The analysis derives SIR distributions for dominant and cumulative co-SF interference and inter-SF interference, with numerical results matching Monte Carlo simulations.

  • Takeaways & Limitations

    Joint co-SF and inter-SF interference gives a more accurate scalability assessment than co-SF-only analysis and supports network dimensioning under reliability constraints.

Abstract

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Low-power wide-area network (LPWAN) technologies are gaining momentum for internet-of-things (IoT) applications since they promise wide coverage to a massive number of battery-operated devices using grant-free medium access. LoRaWAN, with its physical (PHY) layer design and regulatory efforts, has emerged as the widely adopted LPWAN solution. By using chirp spread spectrum modulation with qausi-orthogonal spreading factors (SFs), LoRa PHY offers coverage to wide-area applications while supporting high-density of devices. However, thus far its scalability performance has been inadequately modeled and the effect of interference resulting from the imperfect orthogonality of the SFs has not been considered. In this paper, we present an analytical model of a single-cell LoRa system that accounts for the impact of interference among transmissions over the same SF (co-SF) as well as different SFs (inter-SF). By modeling the interference field as Poisson point process under duty-cycled ALOHA, we derive the signal-to-interference ratio (SIR) distributions for several interference conditions. Results show that, for a duty cycle as low as 0.33%, the network performance under co-SF interference alone is considerably optimistic as the inclusion of inter-SF interference unveils a further drop in the success probability and the coverage probability of approximately 10% and 15%, respectively for 1500 devices in a LoRa channel. Finally, we illustrate how our analysis can characterize the critical device density with respect to cell size for a given reliability target.

I. INTRODUCTION

The paper addresses limited scalability modeling for LoRa networks by accounting for both co-SF interference and imperfect SF orthogonality. It develops analytical tools intended to evaluate performance and support network dimensioning under reliability constraints.

  • Motivation: LoRaWAN targets scalable, energy-efficient long-range connectivity for massive IoT deployments.LPWANs limit bit rates to provide long-range communication, while LoRaWAN supports applications requiring many connected devices.
  • Related Works and Motivation: Inter-SF interference arises because different spreading factors are only quasi-orthogonal, while co-SF interference comes from simultaneous same-SF transmissions.Near-far conditions can make inter-SF interference consequential even when its required SIR protection is low.
  • Contributions: The model combines stochastic-geometry interference analysis with a multi-annuli single-cell LoRa system and realistic MAC- and PHY-layer constraints.The framework models the interference field as a Poisson point process and bases SF allocation on distance-related SNR thresholds.
  • Contributions: 15% further coverage loss is exposed by inter-SF interference for a small number of concurrently transmitting end-devices.The paper evaluates coverage using SIR distributions and reports substantial additional loss beyond co-SF interference alone.
  • Contributions: Coverage-probability contours support cell-size and node-density dimensioning under medium-access and reliability constraints.The paper also studies how three SF-allocation schemes influence overall network performance and formulates an extension toward multi-cell interference modeling.

II. THE LORA SYSTEM

LoRaWAN combines the LoRa PHY modulation scheme with an open protocol stack and star-of-stars networking architecture. Its chirp duration, spreading factors, and ALOHA access determine tradeoffs among throughput, robustness, and collision tolerance.

  • A. LoRaWAN Architecture: LoRa is the proprietary PHY modulation scheme, while LoRaWAN is the open protocol stack developed by the LoRa Alliance.The architecture connects end devices and gateways through a network server and application server.
  • A. LoRaWAN Architecture: The gateway relays LoRa or IP messages, the NetServer manages network resources and authentication, and the AppServer handles admission and encryption.End devices generally focus on event-triggered uplink transmissions.
  • A. LoRaWAN Architecture: ALOHA-based access allows signals using the same or different SFs to overlap in time and frequency.Successful decoding requires the desired signal's SIR to exceed an isolation threshold through the capture effect.
  • B. LoRa PHY layer: Higher SF increases processing gain and reduces the target SNR, but also increases chirp duration and time-on-air.The NetServer adapts bandwidth across 125 and 250 kHz and SF across 7 through 12.
  • B. LoRa PHY layer: Each SIR margin δij specifies the margin required for a packet sent at SFi to decode correctly when the colliding packet uses SFj.The matrix distinguishes co-SF and inter-SF isolation requirements.

III. SYSTEM, SIGNAL AND CHANNEL MODELS

The system model represents a single-gateway LoRa cell with spatially distributed devices, duty-cycled ALOHA activity, distance-based SF regions, and fading path loss. It evaluates a desired uplink under aggregate co-SF and inter-SF interference.

  • A. System Model: Devices are distributed in a disk around the gateway according to a homogeneous PPP with intensity λ.The deployment region has radius R and area A = πR^2, with average device count N̄ = λA.
  • A. System Model: Independent ALOHA transmission decisions and duty cycle α thin the device process into an active-device PPP with intensity αλ.This captures the set of concurrently transmitting devices at a given time.
  • A. System Model: Devices use fixed transmit power and an omnidirectional antenna on the same channel of bandwidth B.These assumptions define the common transmission conditions for the interference model.
  • A. System Model: The cell is divided into K disjoint annuli, with each annulus assigned one SF according to distance-related SNR conditions.Concurrent transmissions within the same annulus create co-SF interference, while transmissions from other annuli create inter-SF interference.
  • B. Signal and Channel Model: The received signal model includes a desired Rayleigh-faded transmission, co-SF and inter-SF interferers, AWGN, and non-singular power-law path loss.The path-loss model uses a critical distance xc to prevent attenuation from diverging as device distance approaches zero.

C. Performance Metrics

The paper characterizes LoRa link performance using SNR/SIR-based success and coverage probabilities, while distinguishing interference-free, co-SF, and inter-SF conditions.

  • Performance Metrics: The CDF of SNR or SIR at threshold τ gives outage probability, while its CCDF gives success probability PX.The relationship is PX = 1 − Po.
  • Performance Metrics: Coverage probability Pc is the probability that a randomly chosen device achieves the target SNR/SIR threshold τ.It is derived from the success probability PX.
  • Performance Metrics: The analysis considers dominant co-SF, cumulative co-SF, and joint cumulative co-SF plus inter-SF interference.These conditions distinguish the effect of same-SF interference from the combined effect of same- and different-SF transmissions.
  • Performance Metrics: In interference-free operation, link performance is determined by an SF-specific SNR threshold.
  • Performance Metrics: The interference-free success expression is independent of device intensities λ and λm, with θSF fixed within an annulus.

V. UPLINK PERFORMANCE ANALYSIS IN POISSON FIELD OF INTERFERERS

The uplink analysis models concurrent LoRa transmissions with stochastic geometry and examines whether the strongest co-SF interferer adequately represents cumulative interference.

  • Poisson Field Modeling: Stochastic geometry models concurrent-transmission interference as a spatial shot-noise process over Poisson-distributed node locations.The approach sums interfering transmission powers using the path-loss model.
  • Poisson Field Modeling: Duty-cycle constraints significantly reduce concurrently active same-SF devices within an annulus.
  • Dominant Interferer: The dominant co-SF interferer provides a success-probability analysis based on extreme order statistics.
  • Dominant Interferer: Success under the dominant interferer requires the desired signal to be δ times stronger than that interferer.
  • Dominant Interferer: The maximum-interferer CDF is [FXi(z)]^M, where M is Poisson with mean vi = αλai for an annulus.M represents the random number of concurrently transmitting interferers in the annulus.
  • Dominant Interferer: The device-distance distribution within annulus i is 2πx/ai, supporting the path-loss distribution used in the order-statistics analysis.The path-loss PDF is derived from this annular distance distribution.

B. SIR Success - Cumulative Interference

The cumulative-interference analysis derives SIR-based success probabilities for aggregate co-SF interference using Laplace transforms and Poisson point-process functionals.

  • Cumulative Interference: The cumulative-interference analysis measures how aggregate co-SF and inter-SF interference changes performance relative to the dominant-interferer upper bound on outage.
  • Cumulative Co-SF Interference: Under concurrent co-SF interference, success requires the desired SIR to exceed the SF-dependent threshold δii.
  • Cumulative Co-SF Interference: The co-SF success probability is expressed through the Laplace transform LICSF of cumulative interference evaluated at s = δii/(ptl(x1)).The derivation uses the exponential channel-gain distribution and probability generating functionals.
  • Cumulative Co-SF Interference: The derivation averages over the point process and channel gain, then applies the exponential random variable's moment-generating function.
  • Cumulative Co-SF Interference: The probability generating functional of a homogeneous PPP evaluates the spatial expectation underlying cumulative interference.
  • Cumulative Co-SF Interference: A Cartesian-to-polar transformation produces the final spatial integral for the interference-limited success probability.

2) SIR Success under Inter-Spreading Factor Interference:

The inter-SF analysis accounts for interference from quasi-orthogonal spreading factors by assigning SF-pair-dependent SIR margins across separate annuli.

  • SIR Success under Inter-SF Interference: Inter-SF outage occurs when the desired SIR for SFi falls below δij[dB] because of concurrent transmissions using SFj.
  • SIR Success under Inter-SF Interference: For a desired device in annulus i, inter-SF interferers originate from the K \ i other annuli.
  • SIR Success under Inter-SF Interference: Different annuli have different SIR margins against interference from each annulus because the annular point processes are independent.
  • SIR Success under Inter-SF Interference: The inter-SF success probability uses the Laplace transform LIISF of cumulative inter-SF interference at s = δij/(ptl(x1)).LIISF is obtained using the preceding cumulative-interference derivation.

3) SIR outage under Co- and Inter-SF Interference:

The analysis determines success and coverage probabilities under combined co-SF and inter-SF interference using the derived SIR expressions and distance averaging.

  • Combined co-SF and inter-SF success probability is determined from the corresponding analytical expressions for a device in an annulus.
  • Coverage probability is obtained by averaging the analyzed success probabilities over the device’s distance distribution.
  • The resulting outage analysis is evaluated for an average of 1500 end-devices.

VI. RESULTS AND DISCUSSION

Monte Carlo simulations validate the analytical models and show how spreading-factor transitions, interference type, cell size, and device density shape LoRa success and coverage probabilities.

  • The numerical and simulation results match, demonstrating the accuracy of the developed success-probability and coverage models.The simulations use devices distributed according to a homogeneous PPP and evaluate SNR- and SIR-based success over 10^5 realizations.
  • Higher spreading factors create saw-tooth success-probability gains at annulus boundaries because they have lower receiver sensitivities and required SNRs.The trend appears across interference conditions, although it is less prominent than in the interference-free case.
  • The dominant-interferer success probability is an upper bound that becomes less tight at higher-SF annuli as cumulative interference increases.The increasing number of devices with annulus area strengthens aggregate co-SF interference.
  • 15% is the reported maximum success-probability loss from inter-SF interference relative to co-SF interference alone.This loss is considered significant for realistic scalability analysis.
  • Interference dominates success-probability degradation at R = 6 km, whereas noise has greater impact at R = 12 km.With the same average device count, relative interference impact remains the same while noise-related success degrades more for the larger cell.

B. Coverage Probability

Coverage probability remains constant with device density under noise but declines as co-SF and inter-SF interference increase. Including imperfect SF orthogonality produces substantially lower coverage than considering co-SF interference alone and supports cell-dimensioning analysis.

  • Noise-only coverage probability is independent of device density, whereas SIR-based coverage probabilities decrease exponentially as the number of end-devices increases.The decline reflects increasing co-SF and inter-SF interference.
  • The dominant co-SF-interferer model gives an optimistic upper bound compared with aggregate co-SF interference.Its tightness decreases as device density increases.
  • 15% lower coverage probability occurs with joint co-SF and inter-SF interference than with same-SF interference alone for 1500 devices per LoRa channel.Imperfect orthogonality causes the joint-interference coverage probability to decrease faster with device count.
  • Coverage contours relate joint coverage probability to cell size, device count, duty cycle, and transmit power for network dimensioning.They can indicate the number of end-devices supported at a required coverage probability for a given cell radius.
  • At 10% permitted duty cycle, concurrent-transmission interference severely reduces the maximum number of end-devices despite increased cell radius at pt = 27 dBm.

C. SF Allocation Strategies

The study compares equal-interval-based, equal-area-based, and path-loss-based SF allocation schemes. Their success and coverage probabilities differ because annulus geometry and SF selection affect interference and near-far conditions.

  • SF allocation schemes: Three SF allocation strategies are compared: equal-interval-based, equal-area-based, and path-loss-based schemes.EIB uses equal-width annuli, EAB equal-area annuli, and PLB annuli based on path loss and SF-specific SNR thresholds.
  • Experimental setting: The comparison uses a common cell radius R = 9.86 km determined by the PLB strategy.This radius is the maximum distance at which the required SNR for the highest SF is satisfied.
  • Success probability: EIB has mostly higher SNR-based success probability than PLB and EAB, becoming equal at the cell boundary.At the boundary, all schemes use the same SF.
  • Success probability: EAB performance drops up to a certain distance and then improves because narrower outer annuli reduce the pronounced near-far condition.After some annuli, the probability of achieving the co-SF SIR target increases relative to the previous annulus.
  • Joint coverage: EIB maintains higher joint coverage probability than PLB and EAB, while adding a fading margin to PLB brings the schemes' coverage results closer.The fading margin effectively reduces the cell size.

D. Modeling a Multi-Cell LoRa Network

The paper extends its single-cell interference framework toward a multi-cell LoRa model by representing gateways and clustered devices with spatial point processes. The multi-cell success probability combines noise, intra-cell, and inter-cell interference, while evaluating the final inter-cell term is left for future work.

  • Motivation: Practical large-area coverage motivates extending the single-cell analysis to a multi-cell LoRa network.The single-cell coverage contours support dimensioning by cell size and device count but do not directly cover large geographical areas.
  • Spatial model: A Poisson cluster process models gateways as a PPP and devices within each gateway-centered cluster as independent PPPs.
  • Success probability: The multi-cell success probability combines a noise term, intra-cell co-SF and inter-SF interference, and interference from other cells.The final term accounts for inter-cell interference.
  • Limitation: Evaluating the inter-cell expression requires distance distributions and expectations over both devices and gateway clustering.These calculations are left as future work.
  • Analytical framework: The analysis derives SIR distributions for joint co-SF and inter-SF interference and uses them to evaluate coverage and network scalability.
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