Mathematical Characterization and Simulation of LoRa


LoRa is a recently developed physical layer modulation technique characterized by low power consumption and long range communication capabilities. Due to the organic relevance of these features in the Internet of Things, LoRa has gained significant traction in the contemporary telecom industry. LoRa has been rigorously analyzed with regard to its functionalities and applications. However, due to its proprietary nature, precise descriptions of the underlying modulation waveforms are not publicly available. In this paper, we first derive a mathematical characterization of LoRa with regard to its chirp frequency and phase. Our analysis shows that a single matched filter is sufficient to decode all LoRa data symbols. Further, when the output of the matched filter is sampled at a rate equal to the transmission bandwidth, a constant frequency output is produced, which directly corresponds to the transmitted symbol value. For a visual understanding of LoRa, we present detailed simulations of its waveforms at different stages of the communication pipeline. In particular, the error performance simulation clearly demonstrates that LoRa can receive and decode data packets even when the SNR is below 0 dB.

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  1. 1.

  2. 2.

    Sforza, F., Communications system, US Patent 8406275.

  3. 3.

    Winkler, M., (1962). “Chirp signals for communications”, in IEEE WESCON Convention Record, Vol. 14,(2).

  4. 4.


  5. 5.

    Goursaud, C., & Gorce, J. M. (2015). “Dedicated networks for IoT: PHY / MAC state of the art and challenges”. EAI Endorsed Transactions on Internet Things, 150597.

  6. 6.

    Georgiou, O., & Raza, U. (2017). “Low Power Wide Area Network Analysis: Can LoRa Scale?”, in IEEE Wireless Communications Letters, Vol. 6, No. 2, April 2017.

  7. 7.

    Vangelista, L., Zanella, A., Zorzi, M. (2015). “Long-Range IoT Technologies: The Dawn of LoRa”, In: Proceeding of the 1st EAI Intl. Conf. on Future access enablers of ubiquitous and intelligent infrastructures, September 2015.

  8. 8.

    Bor, M., Roedig, U., Voigt, T., Alonso, J.M., (2016). “Do LoRa Low-Power Wide-Area Networks Scale?”, In: Proc. of the 19th ACM Intl. Conf. on Modeling, Analysis and Simulation of Wireless and Mobile Systems, November 2016.

  9. 9.

    Knight, M., & Seeber, B. (2016). “Decoding LoRa: Realizing a Modern LPWAN with SDR”, In: Proc. of the 6th GNU Radio Conf.

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Correspondence to Tulasi R. Haritsa.

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Haritsa, T.R., Yashu, B.M., Kumar, U.V. et al. Mathematical Characterization and Simulation of LoRa. Wireless Pers Commun (2020).

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  • LoRa
  • Chirp waveform
  • Chirp spread spectrum modulation
  • Matched filter
  • Internet of things