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 $\sigma$.