In this paper, we provide, for the first time, a comprehensive understanding of LoRa waveform theory in order to quantify its orthogonality. We present LoRa waveform expressions in continuous and discrete time domains, and analyze measures of orthogonality between different LoRa spreading factors (SFs) through cross-correlation functions. The cross-correlation functions are analytically expressed in a general form and they account for diverse configuration parameters (bandwidth, SF, etc.) and different cases of signal displacements (time delay shift, frequency shift, etc.). We quantify their mean and maximum in all time domains. We highlight the impact of the temporal displacement and different bandwidths. The general result is that LoRa modulation is non-orthogonal. Firstly, we observe that for same bandwidths the largest maximum cross-correlation happens for same SF and is equal to 100% due to same symbols; whereas for different bandwidths, the largest maximum cross-correlation is no longer observed at the same SF. Secondly, the maximum cross-correlation is less than 26% between different SFs, is higher for closer SFs and decreases as the difference between SFs increases. After downchirping, the maximum cross-correlation increases and the mean decreases compared to those before downchirping. Moreover, the maximum cross-correlation is insignificantly impacted by the temporal delay which makes it valid to adopt for the performance analysis of both synchronous and asynchronous systems. Finally, we analyze by simulation the bit error probability statistics for different bandwidth ratios and highlight their correlated behaviour with the insights obtained from the maximum cross-correlation expressions.
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