Relation of SNR and BER

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.

The alternative definition is not equivalent to $$P_s/P_n$$. It is a different definition, applicable in different contexts; it is not a replacement.
• Right. It's usually very difficult to estimate $P_s$ and $P_n$ from $x(t)$ alone. Having said that, I'm not sure what you're trying to do: if what you want is to simulate the BER, you get to define $P_s$ and $P_n$. If what you want is to estimate the SNR given an experimental BER, then you can estimate it using your simulated model (i.e. knowing the BER, you can look up the SNR from the plot, or use the formula).