I am working towards establishing the error rate of the communication system relying on on-off keying. I have trouble understanding the estimation of bit error rate (BER) performance when the following formula $$\mathrm{BER} \propto \mathrm{erfc}(\sqrt{\mathrm{SNR}}) $$ is used.
Suppose I have a received signal $$x(t) = s(t) + n(t)$$ where $s(t)$ is the original signal and $n(t)$ is AWGN. My signal reads $$ s(t) = \sum_{n}d_n g(t-nT), $$ where $g(t)$ is the modulation pulse (rectangle for simplicity), $d_n$ is either 0 or 1 and $T$ is the width of the pulse.
Defining the SNR as $$\mathrm{SNR} = \frac{P_s}{P_n} = \frac{\mathrm{E}[s(t)]^2}{\mathrm{var}[n(t)]}.$$
However, looking at Wikipedia and signal course lecture materials I was visiting, I noticed that it could also be defined as $$\mathrm{SNR}_\mathrm{alt} = \frac{\mu_x^2}{\sigma_x^2} = \frac{\mathrm{E}[x(t)]^2}{\mathrm{var}[x(t)]}.$$
In practice, it is a nontrivial task to separate $x(t)$ into original $s(t)$ and $n(t)$ so the $\mathrm{SNR}_\mathrm{alt}$ definition is more straightforward for evaluation.
Now, back to my problem. I have tried to do some numerical simulations, generating over a million random bit symbols and assigning them to the rectangular pulse of width $T$. I have found out that the formula $$\mathrm{BER} = \frac{1}{2}\mathrm{erfc}(\sqrt{\frac{T}{2}\mathrm{SNR}}) $$ is a perfect estimation of the BER computed from simulation. However, using the $\mathrm{SNR}_\mathrm{alt}$ instead, the mentioned formula starts to be useless somewhere about $-6$ dB. Rescaling the BER-SNR formula would not help here as the data saturate somewhere around 0.00078.
Can someone please elaborate on this? I am lost in the assumptions and what I can use or not. The formula with SNR seems to be a nice estimation of the performance. But calculating the SNR value without additional processing seems to be impractical.
