  An important modulation technique for Internet of Things (IoT) is the one
proposed by the LoRa allianceTM. 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.
