  It has been shown that the coherent detection of LoRa signals only provides
marginal gains of around 0.7 dB on the additive white Gaussian noise (AWGN)
channel. However, ALOHA-based massive Internet of Things systems, including
LoRa, often operate in the interference-limited regime. Therefore, in this
work, we examine the performance of the LoRa modulation with coherent detection
in the presence of interference from another LoRa user with the same spreading
factor. We derive rigorous symbol- and frame error rate expressions as well as
bounds and approximations for evaluating the error rates. The error rates
predicted by these approximations are compared against error rates found by
Monte Carlo simulations and shown to be very accurate. We also compare the
performance of LoRa with coherent and non-coherent receivers and we show that
the coherent detection of LoRa is significantly more beneficial in interference
scenarios than in the presence of only AWGN. For example, we show that coherent
detection leads to a 2.5 dB gain over the standard non-coherent detection for a
signal-to-interference ratio (SIR) of 3 dB and up to a 10 dB gain for an SIR of
0 dB.
