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On the Error Rate of the LoRa Modulation with Interference

Orion Afisiadis, Matthieu Cotting, Andreas Burg, Alexios Balatsoukas-Stimming

arXiv:1905.11252v2eess.SP

TL;DR

Existing same-SF LoRa interference analysis assumes chip and phase alignment, leaving error rates for misaligned interference unaddressed. This paper extends the model and derives error-rate expressions and approximations, finding that the chip-aligned model overestimates SER by approximately 1 dB.

  • Problem

    Existing same-SF LoRa interference modeling assumes perfect alignment, leaving error-rate analysis for misaligned interference unaddressed in a setting relevant to decoding amid same-SF collisions.

    Source: Contributions · p. 2, §IV · p. 4

  • Method

    The paper extends the same-SF interference model to chip- and phase-misaligned signals, derives SER and FER expressions, and develops low-complexity approximations.

    Source: §VIII · p. 10

  • Results

    The chip-aligned model overestimates SER by approximately 1 dB, while proven interference properties reduce SER and FER computation complexity by approximately the symbol length N.

    Source: B. Symbol Error Rate · p. 9, §VIII · p. 10

  • Takeaways & Limitations

    Chip-aligned interference is a worst-case SER scenario, since integer offsets produce larger maximum interference magnitudes than non-integer offsets.

    Source: B. Symbol Error Rate · p. 9

    The proposed SER approximation becomes inaccurate in the low-SNR, AWGN-limited regime, where no single bin dominates the error rate.

    Source: D. Symbol Error Rate Approximation · p. 7

Abstract

from arXiv · show

LoRa is a chirp spread-spectrum modulation developed for the Internet of Things. In this work, we examine the performance of LoRa in the presence of both additive white Gaussian noise and interference from another LoRa user. To this end, we extend an existing interference model, which assumes perfect alignment of the signal of interest and the interference, to the more realistic case where the interfering user is neither chip- nor phase-aligned with the signal of interest and we derive an expression for the error rate. We show that the existing aligned interference model overestimates the effect of interference on the error rate. Moreover, we prove two symmetries in the interfering signal and we derive low-complexity approximate formulas that can significantly reduce the complexity of computing the symbol and frame error rates compared to the complete expression. Finally, we provide numerical simulations to corroborate the theoretical analysis and to verify the accuracy of our proposed approximations.

I. INTRODUCTION … A. Modulation

The paper extends LoRa same-spreading-factor interference analysis to users with neither chip nor phase alignment, deriving symbol- and frame-error-rate expressions and lower-complexity approximations. It situates this work within LoRa’s low-power, long-range IoT use and the risk of collisions under ALOHA-based LoRaWAN access.

I. INTRODUCTION

  • LoRa uses chirp spread-spectrum for low-rate, low-power, long-range communication, while ALOHA-based LoRaWAN access allows collisions that make same-technology interference a concern.Prior work approximated BER with AWGN and same-SF interference, but the paper generalizes that interference model.Source: §I · pp. 1–2

Contributions:

  • The generalized interference model derives SER expressions and approximations, shows non-integer chip misalignment affects SER, and finds the aligned model consistently overestimates it.Two proven same-SF interference symmetries reduce the complexity of exact and approximate SER calculations; the paper also derives a FER approximation for network simulators.Source: Contributions · p. 2

Outline:

  • The paper proceeds from LoRa modulation and demodulation through AWGN SER analysis, generalized same-SF interference, SER complexity reductions, FER approximation, and numerical results.The numerical section reports SER and FER results before the conclusion.Source: Outline · p. 2

II. THE LORA MODULATION

  • Section II briefly summarizes LoRa modulation and demodulation.Source: §II · p. 2

A. Modulation

  • A LoRa symbol uses N = 2^SF chips across N frequency steps, with its chirp increasing in frequency and folding at Nyquist; the analysis assumes sampling frequency fs = B.The stated spreading factors are SF ∈ {7, …, 12}.Source: A. Modulation · p. 2
  • The modulation model assumes a unit-magnitude channel gain and complex AWGN, with noise variance σ^2 = N0/(2N).The channel is time-invariant and frequency-flat.Source: A. Modulation · p. 2

B. Demodulation · III. SYMBOL ERROR RATE UNDER AWGN · A. Distribution of the Decision Metric

LoRa symbols can be demodulated by correlation or an equivalent dechirp-and-DFT method. The AWGN analysis characterizes the decision metrics used to derive symbol error rates and support later SER and FER calculations with interference.

B. Demodulation

  • The channel phase shift is fixed within each transmitted packet but is generally uniformly distributed over [0, 2π).Source: B. Demodulation · p. 3
  • Correlation-based demodulation has O(N^2) complexity, while dechirping with an upchirp reference followed by an FFT reduces it to O(N log N).The FFT method selects the frequency-bin index with the maximum magnitude.Source: B. Demodulation · p. 3

III. SYMBOL ERROR RATE UNDER AWGN

  • The AWGN SER analysis supports later calculations of SER and FER with both AWGN and interference; Fig. 2 compares Monte Carlo simulations with approximation (17) across SF 7–12.Source: §III · p. 3, Figure 2 · p. 3

A. Distribution of the Decision Metric

  • With no noise and perfect synchronization, the dechirped DFT has one bin of magnitude N containing all signal energy, while the other N −1 bins have zero energy.With AWGN, every frequency bin contains some energy.Source: A. Distribution of the Decision Metric · p. 3
  • The demodulation metric magnitude follows a Rayleigh distribution for bins other than the transmitted symbol and a Rice distribution for the transmitted-symbol bin.Source: A. Distribution of the Decision Metric · p. 3

B. Symbol Error Rate · C. Symbol Error Rate Approximations

The symbol error rate is obtained by comparing the largest competing decision statistic with the transmitted symbol’s statistic, then evaluating the resulting conditional error probability. The approximations address numerical difficulties at large LoRa modulation sizes and include a Gaussian-based approach and a more concise form.

B. Symbol Error Rate

  • A symbol error occurs exactly when the largest magnitude among the competing statistics exceeds the transmitted symbol’s magnitude.The maximum is formed over the competing symbols, whose statistics are identically distributed.Source: B. Symbol Error Rate · p. 3
  • The conditional symbol error rate is calculated using the distribution of that maximum and the transmitted symbol’s statistic.Source: B. Symbol Error Rate · p. 3
  • The symbol error rate is identical for every transmitted symbol, so the conditional rate also equals the average and, for equiprobable symbols, expected rate.Source: B. Symbol Error Rate · p. 3

C. Symbol Error Rate Approximations

  • Large LoRa modulation sizes can cause numerical problems in evaluating the symbol error rate, motivating efficient approximations derived in.The cited approach uses a Gaussian approximation.Source: C. Symbol Error Rate Approximations · p. 3
  • Additional approximations yield a more concise symbol error rate expression in (18) from the intermediate approximation in (17).Source: C. Symbol Error Rate Approximations · p. 4
  • For large N, an alternative derivation of (18) follows from the convergence of the defined random variable to a Gumbel distribution under the Fisher–Tippett–Gnedenko theorem [28, Section II-B] [33].The passage gives the limiting location parameter using the Euler–Mascheroni constant.Source: C. Symbol Error Rate Approximations · p. 4
  • For supported spreading factors SF ∈ {7, …, 12}, the AWGN symbol error rate is presented with Monte Carlo simulations and approximation (17).Source: C. Symbol Error Rate Approximations · p. 4

IV. SYMBOL ERROR RATE UNDER AWGN AND SAME-SF LORA INTERFERENCE · A. Distribution of the Decision Metric

The section models a gateway decoding a synchronized desired LoRa user amid one unsynchronized same-spreading-factor interferer and AWGN. It characterizes how the interference pattern and its timing and phase affect the demodulation metric.

IV. SYMBOL ERROR RATE UNDER AWGN AND SAME-SF LORA INTERFERENCE

  • Same-SF interference is the dominant concern: its rejection SIR threshold is 0 dB, versus −16 dB for different-SF interference, so the model focuses on same-SF interferers.The section considers a gateway decoding one user's message; the gateway is synchronized to that user, and the model includes one interferer.Source: §IV · p. 4
  • Because LoRaWAN uses non-slotted ALOHA, the interferer is unsynchronized and a desired symbol generally overlaps portions of two distinct interfering symbols.The desired user's gateway synchronization does not align the interfering signal to either the user or gateway.Source: §IV · p. 4
  • The relative timing offset τ is modeled as uniform over [0, N), unlike, which restricts τ to integer chip durations and thus assumes chip alignment.The model partitions the desired symbol's chips into NL1 and NL2 around ⌈τ⌉.Source: §IV · p. 4
  • The interference pattern depends on the interfering symbols sI1 and sI2, channel hI, and time offset τ through the time-domain interference signal yI.The receiver demodulates the resulting signal to obtain the decision metric.Source: §IV · p. 4

A. Distribution of the Decision Metric

  • For a given desired symbol and interference pattern, adding interference changes the mean of the decision-metric distribution; the interference-channel phase θ is fixed per packet but generally uniform in [0, 2π).The passage describes the resulting distribution of |Y′k| without specifying its full expression.Source: A. Distribution of the Decision Metric · p. 5
  • The relative phase is defined as ω = φ − θ, and Fig. 4 depicts the joint effect of AWGN and interference on the signal-of-interest bin value.The figure represents the signal of interest, interference, noise, and bin value Ys.Source: A. Distribution of the Decision Metric · p. 5, Figure 4 · p. 5

B. Symbol Error Rate

The section derives the symbol error rate by conditioning on the interference and phase, then averaging over phase, interferer choices and timing offsets, and the transmitted symbol.

B. Symbol Error Rate

  • Conditioned on yI and ω, the SER calculation uses the CDF of the maximum interfering bin, based on independent competing-bin magnitudes.The full unconditional SER expression is given in (24).Source: B. Symbol Error Rate · p. 5
  • Averaging the conditional error probability over ω gives the SER conditioned on transmitted symbol s and interference yI.Source: B. Symbol Error Rate · p. 5
  • The SER across packets averages over uniformly distributed interfering symbols and timing offset τ, then averages over uniformly distributed transmitted symbols s.Source: B. Symbol Error Rate · p. 5

V. COMPUTATIONAL COMPLEXITY REDUCTION · A. Interference Patterns · B. Equivalent Interference Patterns

The paper reduces the cost of evaluating the error rate by identifying interference patterns that yield identical frequency-bin magnitude sets. It derives an explicit form for the interference magnitudes and shows how equivalent patterns can be counted once with appropriate weighting.

V. COMPUTATIONAL COMPLEXITY REDUCTION

  • Evaluating (24) requires three sums scaling as N^3 and three numerical integrals for each of the N^3 terms; equivalent patterns offer a way to reduce this cost.The section identifies equivalent interference patterns as the basis for reducing the complexity of evaluating (24).Source: §V · p. 5

A. Interference Patterns

  • The offset τ splits into integer inter-chip misalignment L and non-integer intra-chip misalignment λ, and the interference magnitudes |R_k| are derived explicitly using a geometric-series formula.The derivation starts from the DFT and then uses the geometric series sum formula to express R_k.Source: A. Interference Patterns · pp. 5–6

B. Equivalent Interference Patterns

C. Complexity Reduction

The section reduces error-rate computation by exploiting equivalent interference patterns. These symmetries yield complexity reductions, with further simplifications for integer offsets.

C. Complexity Reduction

  • Only two distinct interference-pattern sets occur for each sI difference, reducing evaluation of (24) by a factor of N/2.The sets are defined for sI = [sI1−sI2]N.Source: C. Complexity Reduction · p. 6
  • For integer offsets, the integrals simplify to sums over integer offsets L; equivalent interference-pattern sets further reduce the expression, which simplifies to the one in in the first case.For the second symmetry, patterns at τ ∈ {N/2, N−1} match patterns at τ ∈ {0, N/2−1}.Source: C. Complexity Reduction · pp. 6–7
  • Patterns with τ ∈ ((N−1)/2, N−1] each match one in [0, (N−1)/2), yielding an additional complexity reduction of approximately 2.The region τ ∈ (N−1, N) is treated separately because the stated equivalence does not cover it.Source: C. Complexity Reduction · pp. 6–7

D. Symbol Error Rate Approximation

The section develops a low-complexity SER approximation by simplifying the interference patterns and focusing on the dominant interference bin. It also identifies when the approximation is unreliable and how to improve it in the low-SNR regime.

D. Symbol Error Rate Approximation

  • The approximation uses the triangle inequality to reduce the interference patterns to a single set and ignores the second integral over τ ∈ (N−1, N).The resulting method targets a simpler evaluation of the SER expression.Source: D. Symbol Error Rate Approximation · p. 7
  • Equivalent interference patterns around τ = (N−1)/2 in the interval (N−1, N) are not used or explicitly proved because they only marginally reduce computational complexity.This observation follows the reasoning of Property 2.Source: D. Symbol Error Rate Approximation · p. 7
  • Assuming interference-induced SER is dominated by max |R_k|, the derivation sets s_I2 = 0 and approximates the magnitude of the selected maximum-interference bin.For τ ∈ [0, (N−1)/2), the maximum of |A_k,2| exceeds that of |A_k,1|, guiding the bin choice.Source: D. Symbol Error Rate Approximation · p. 7
  • For N > 2^7, the probability that the maximum-interference bin coincides with the signal bin is negligible, so the approximation avoids averaging over s and integrating over ω.With the other bins zero-valued, the approximated SER does not depend on s when the bins differ.Source: D. Symbol Error Rate Approximation · p. 7
  • The interference-dominated SER approximates the Rice-distributed |Y′_kmax| with a Gaussian for large location parameters, then evaluates the result numerically over τ using step size ε.The probability expression uses the Q-function.Source: D. Symbol Error Rate Approximation · p. 7
  • The approximation is inaccurate at low SNR, where no single bin dominates; a final SER estimate incorporates the AWGN SER approximation for greater accuracy in that regime.The AWGN SER can be evaluated efficiently using the approximation in (17).Source: D. Symbol Error Rate Approximation · pp. 7–8

VI. FRAME ERROR RATE UNDER AWGN AND SAME-SF INTERFERENCE

The section derives an approximate uncoded frame error rate (FER) for LoRa under AWGN and same-SF interference by accounting for the shared frame offsets and averaging over relative frame positions. It treats partially interfered symbols as fully interfered and applies to the rate-4/5 LoRa mode, which provides error detection but no error correction.

VI. FRAME ERROR RATE UNDER AWGN AND SAME-SF INTERFERENCE

  • SER cannot directly yield FER because symbols in a frame share the same time and phase offsets, so the section derives a frame-level expression.The uncoded FER expression applies to the LoRa mode with rate-4/5 coding, which detects errors but does not correct them.Source: §VI · p. 8
  • The derivation assumes perfect synchronization for the desired frame and an equal-length interfering frame, an assumption made for clarity and generalizable to other lengths.Because of the frames’ relative time offset, only part of the desired frame is interfered with.Source: §VI · p. 8
  • FER averages over random relative frame positions; the model approximates the partially interfered symbol as fully interfered, and Fi is uniform over {1, …, F}.For a frame of F symbols, Fi symbols are treated as interfered and F − Fi as affected only by AWGN.Source: §VI · p. 8
  • The FER approximation averages the conditional frame error rate over the possible numbers of interfered symbols, yielding the final expression.The number of interfered symbols, Fi, is uniformly distributed over {1, …, F} under the equal-frame-length assumption.Source: §VI · p. 8

VII. RESULTS … VIII. CONCLUSION

The paper analyzes LoRa symbol and frame error rates under same-SF interference that is neither chip- nor phase-aligned with the signal of interest. Its results show that the chip-aligned model overestimates interference effects, while the proposed low-complexity approximations closely match the full analysis.

A. Choice of Discretization Step

  • Using discretization step ǫ = 1/5, rather than smaller steps beyond 1/3, is chosen because further accuracy gains are small while evaluation complexity remains low.The evaluation complexity scales as 1/ǫ relative to the approximation in; Monte Carlo simulations use ǫ = 1/10 as a comparison baseline.Source: A. Choice of Discretization Step · p. 8

B. Symbol Error Rate

  • At an SIR of 3 dB, same-SF interference strongly affects SER, but relative phase offset ω has little apparent effect.The chip-aligned model is pessimistic because integer time offsets produce larger maximum interference magnitudes than non-integer offsets.Source: B. Symbol Error Rate · p. 9, Figure 6 · p. 9, Figure 7 · p. 9

VII. RESULTS

B. Symbol Error Rate

  • The unified SNR-SIR framework allows reception at SIR below 6 dB when SNR is sufficiently high, replacing a hard SIR threshold with a trade-off.The paper notes that extending this framework to multiple same-SF interferers requires further study.Source: B. Symbol Error Rate · pp. 9–10, Figure 8 · p. 9

C. Frame Error Rate

  • Longer frames require a larger SNR increase under the same interference, and stricter target FER requirements further increase the required SNR.The comparison uses SF = 7, frame lengths F = 10, 20, and 30, and target FERs of 10^-1 and 10^-2.Source: C. Frame Error Rate · p. 10

VIII. CONCLUSION

  • The proposed model derives SER and FER for interference with neither chip nor phase alignment, and two proved interference properties reduce computation complexity by approximately the symbol length N.The paper also derives low-complexity SER and FER approximations and reports that ignoring non-integer time offsets overestimates error rate by 1 dB.Source: §VIII · p. 10

APPENDIX

The appendix proves equivalence and symmetry properties of the interfering-signal patterns, including a special case in which all patterns are equivalent.

Proof of Proposition 1

  • For fixed sI1 and sI2, cyclic shifts yield N−(sI1−sI2) equivalent interference patterns in YI1.The proof establishes the corresponding cardinality |YI1| = N−(sI1−sI2).Source: APPENDIX · pp. 10–11
  • Patterns are equivalent within each shift class but differ across classes; when λ = 0 (integer τ), all patterns in YI1 and YI2 are equivalent.Source: APPENDIX · p. 11

Proof of Proposition 2

  • The mapping k′ = −k −(N −1) + [sI1 + sI2]N preserves |Rk| for every k from 0 to N−1.Source: APPENDIX · p. 11
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