Meaning of AWGN complex and real noise under BPSK Decision Scheme I'm currently studying error correcting codes and decided to replicate some common Bit Error Rate curves under BPSK modulation.

I initially had trouble to convert SNR to a Gaussian $\sigma$ because I was confused why people used diffent $\sigma$ if the channel was real or complex. I initially thought that I could add $\sigma$ for both IQ components. Then I understood from How do I add AWGN to an I and Q representation of a signal? that $\sigma_I = \sigma_Q = \frac{\sigma}{\sqrt(2)}$ for total noise energy consistency.

But after this information regarding how I should add noise to the received signal I don't get how this does not generate a inconsistency under the BPSK decision boundary if I interpret the channel as complex or real.
From what I understand from a given SNR, there is a relationship with the total Gaussian noise $\sigma$ that I should add.


If channel is real (only considering I component), BPSK maps bits to {-1, 1}, noise added in I component is $\sigma_0$ for SNR = 0dB.
If channel is complex (considering IQ components), BPSK maps bits to {(-1, 0), (1, 0)}, noise added for SNR = 0dB has total $\sigma = \sigma_0$, so $\sigma_I = \sigma_Q = \frac{\sigma_0}{\sqrt(2)}$.


Since BPSK should only be concerned with the real values (InPhase values) the Bit Error Rates would be clearly different in both cases.

I guess my lack of understanding must be related to the SNR to $\sigma$ mapping under a real/complex channel. I don't understand how interpreting the problem in 2 dimensions differs from 1 dimension and its relationship with $\sigma$.
It doesn't matter if the channel is complex or real. It's actually always complex, but since you are using BPSK, only the real part matters. How to add it is a mathematical trick. Say you need to generate an SNR of x in linear scale, and your signals have unit energy, then the noise variance must be 1/x. In other words, you don't generate real and complex parts for the noise, because then the SNR will be $1/(2x)$, which is not correct. You generate the real and complex parts only if the signal is complex, at which case the total noise variance gives the desired SNR. Hope this helps.