Orthogonality Analysis in LoRa Uplink Satellite Communications Affected by Doppler Effect

This paper provides, for the first time, 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 propose the concept and formulas of the shared visibility window for satellites toward two ground devices. Our analysis covers cross-correlation results with varying spreading factors (SF) for no-Doppler and with-Doppler cases. We find 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 high Doppler rate. We interpret this fluctuation by introducing the relationship between transmission start time and cross-correlation. We provide a parameter analysis for orbit height, ground device distance, and inclination angle. Additionally, we analyze the bit error rate (BER) for LoRa signals and observe worse performance under high Doppler shift or interference with same SF. Increasing the SNR or the SIR improves the BER only when Doppler effect is below a frequency threshold. Notably, under Doppler effect, the performance behaviors of BER no longer align with those of maximum cross-correlation. Finally, our results lead to two recommendations: 1) To mitigate 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 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.