On the LoRa Modulation for IoT: Waveform Properties and Spectral Analysis

TLDR: In this paper, the authors analyzed the cross-correlation between the transmitted waveforms and showed that LoRa can be considered approximately an orthogonal modulation only for large $M$.

Abstract: An important modulation technique for Internet of Things (IoT) is the one proposed by the low power long range (LoRa) alliance. In this paper, we analyze the $M$ -ary LoRa modulation in the time and frequency domains. First, we provide the signal description in the time domain, and show that LoRa is a memoryless continuous phase modulation. The cross-correlation between the transmitted waveforms is determined, proving that LoRa can be considered approximately an orthogonal modulation only for large $M$ . Then, we investigate the spectral characteristics of the signal modulated by random data, obtaining a closed-form expression of the spectrum in terms of Fresnel functions. Quite surprisingly, we found that LoRa has both continuous and discrete spectra, with the discrete spectrum containing exactly a fraction $1/M$ of the total signal power.