How is the signal-to-noise ratio of an event-related-potential measured? I often encounter publications where the authors try to "increase the SNR (signal-to-noise ratio) of electroencephalogram (EEG) signals". Some define a numeric value of SNR of EEG signals, but I am confused as to the exact definition of the SNR and how it can be measured for ERPs?

Previously, I have calculated the SNR using a model of noise, for example a Gaussian model. I have also encountered SNR calculations based on a ratio between the mean and the standard deviation of a signal, but both of these approaches seem to be impractical. When I process the EEG signal, I would like to emphasize how the ERP changes in the EEG signal. All of this is strongly related with variance. If the variance approaches to zero, then theory tells me that SNR goes to infinity (a really good signal), but the signal that I will have is useless.

Is there a better way to measure the SNR then I have mentioned?
Based on your comments I interpret your question as:

"(1) What is the definition of the signal-to-noise-ratio (SNR) and (2) how do I determine the SNR for event-related potential (ERP) amplitudes in an EEG signal?".

(1) Signal-to-noise-ratio (SNR) is a term often encountered in electrophysiology (e.g. EEG) and signal processing and can be loosely defined as the ratio of the relevant signal divided by the noise level. The signal in this example is the ERP amplitude, while the noise is the remaining background activity in the EEG that distorts the ERP (unwanted noise). Noise includes hardware noise, movement artifacts by the subjects, random synchronized brain activity and so on. So SNR = signal/noise.

(2) In case of ERP amplitude being the signal you are after, than the noise is the amplitude of the background EEG (SNRERP_amplitude = ERPamplitude / NOISEamplitude). The ERP amplitude can be defined by determining peak amplitude (e.g. relative to baseline). A straightforward (and widely accepted method) to characterize noise amplitude is determining the standard deviation (SD) of the entire EEG epoch (e.g., 500 ms) in which the ERP was recorded (Hu et al., 2010). Then, the SNR becomes ERPamplitude / SDEEGepoch. 

PS: your comment 


  If the variance approaches to zero, then theory tells me that SNR goes to infinity (a really good signal), but the signal that I will have is useless.


is incorrect. The signal is always part of the EEG epoch. Assuming there is a measurable ERP on a flatline background EEG (amplitude=0), than the signal will be the only thing that adds to the noise component. This is counter intuitive, but note that when noise amplitude is defined as, e.g., the SD, than this SD will be very small as it is determined across the entire EEG epoch. Hence, the peak-amplitude of the ERP will be much larger than the SD. In this ideal ERP recording the SNR will be large, but it will never become infinitely large. 

Reference
- Hu et al., NeuroImage (2010); 50(1): 99-111