On the Quasi-Orthogonality of LoRa Modulation

Long Range (LoRa), a low power and wide-area modulation scheme based on chirp spread spectrum, is the most popular and widely adopted Internet of Things (IoT) technique in industry. A notable and interesting property of LoRa modulation is the quasi-orthogonality of signals modulated under different spreading factors (SFs). Unfortunately, in the literature, there has been no analytical effort to establish the theoretical validity of such quasi-orthogonality. This article, for the first time, theoretically tackles the quasi-orthogonality of the LoRa modulation. First, we derive in both continuous- and discrete-time domains the cross-correlation between two nonsynchronized LoRa signals with different SFs, based on which we analyze the quasi-orthogonality of the LoRa modulation and draw some useful engineering insights. Particularly, we analytically show that in the continuous-time domain, the quasi-orthogonality is guaranteed if one of the SFs of the two LoRa signals is large enough; while, in the discrete-time domain, the quasi-orthogonality is ensured if the maximum of the SFs is large enough. Furthermore, for practical values of the SF, the maximum squared magnitudes of the cross-correlation in the continuous- and discrete-time domains are shown to be 1.14% and 1.08%, respectively, compared to their peak values. We demonstrate the validity and accuracy of our analysis through extensive numerical simulations.