  LoRa is a chirp spread-spectrum modulation developed for the Internet of
Things. In this work, we examine the performance of LoRa in the presence of
both additive white Gaussian noise and interference from another LoRa user. To
this end, we extend an existing interference model, which assumes perfect
alignment of the signal of interest and the interference, to the more realistic
case where the interfering user is neither chip- nor phase-aligned with the
signal of interest and we derive an expression for the error rate. We show that
the existing aligned interference model overestimates the effect of
interference on the error rate. Moreover, we prove two symmetries in the
interfering signal and we derive low-complexity approximate formulas that can
significantly reduce the complexity of computing the symbol and frame error
rates compared to the complete expression. Finally, we provide numerical
simulations to corroborate the theoretical analysis and to verify the accuracy
of our proposed approximations.
