Orthogonality Analysis in LoRa Uplink Satellite Communications Affected by Doppler Effect

This paper provides, for the first time, the general analytical expressions for the Long-Range (LoRa) waveform and cross-correlation in both continuous and discrete time domains under the Doppler effect in satellite communication. We characterize the Doppler effect and propose the concept of the shared visibility window for a three-dimensional (3D) communication system model with two ground devices toward one satellite. Our analysis covers cross-correlation results with varying spreading factors (SF) and bandwidths for both no-Doppler and with-Doppler cases. We find that the maximum cross-correlation with different SFs and the mean cross-correlation are immune to the Doppler effect. However, the maximum cross-correlation with the same SFs is only immune to high Doppler shift, with its value fluctuating between 0.6 and 1 under the condition of high Doppler rate. We propose the explanation strategy for revealing the underlying mechanism behind this fluctuation by highlighting the relationship between transmission start time and cross-correlation. We provide a parameter analysis for orbit height, inclination angle, and ground device distance. Additionally, we conduct the bit error rate (BER) simulation for LoRa signals and observe worse performance under same-SF interference and the high Doppler shift. We observe that increasing the signal-to-noise ratio (SNR) or the signal-tointerference ratio (SIR) improves the BER only when the Doppler effect is below a frequency threshold. Notably, under the Doppler effect, the performance behaviors of BER no longer align with those of maximum cross-correlation. Finally, our results lead to two sets of recommendations: 1) To mitigate the Doppler impact on cross-correlation, we recommend utilizing low SFs, high orbit height, short ground device distance, and the transmission start time with high Doppler shift; 2) To mitigate the Doppler impact on BER, we recommend employing low SFs, high bandwidth, and transmission start time with high Doppler rate. These conflicting recommendations regarding transmission start time highlight the necessity of Doppler shift compensation techniques to help operate LoRa in space properly