noise - What are the various definitions for $rm SNR$, and their associated methods for measuring it?

The definition of $\rm SNR$ seems to be somewhat of a tower of babel in industry. What definitions of $\rm SNR$ are there (feel free to site application), and how exactly can it be measured for that applications?

My specific questions on $\rm SNR$ are:

1. 
   

   How may we measure $\rm SNR$ for a communications system, if we have not yet been able to attain optimal bit sampling timing, and all we have to work with are all the signals in the receiver up to and including the envelope of the I and Q channels? See this post for context.

   
   


2. 
   

   Once we attain optimal bit sampling, and softbits are attained, how best to measure $\rm SNR$ (or $E_bN_0$)? One way I use is: $$ 10\log_{10}\left[ \frac{\textrm{mean}\left\{\lvert s_n \rvert^2\right\}}{\textrm{var}\textrm\{\lvert s_n \rvert\textrm\}}\right], $$

   
   
   

   however I understand this is not suitable for low $\rm SNR$ cases. What other ways exist?

   
   

Answer

It's not practical to come up with a comprehensive list of definitions of signal-to-noise ratio, because different measures are relevant in different applications. Here are a few common measures that I've come across and/or used in the past (you'll find that they are bent toward communications applications):

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  "Pure" SNR: I call this "pure" because it is a literal interpreation of the term "signal to noise ratio." This measure is just the ratio of the power in the signal of interest to the total noise power over the signal's bandwidth. Any noise present outside of the signal bandwidth can be filtered out without removing any signal power, so it can be ignored. This definition isn't often used for digital communications; there are more useful quantities that I reference below. SNR is more relevant for analog modulation.

  
  


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  $\frac{E_b}{N_0}$: This is the most common measure of signal to noise ratio used for digital communications. Referred to in the comments above by Dilip as the "BEND" ratio, Eb/N0 is the ratio of the energy per (information) bit ($E_b$) to the (assumed to be white) noise power spectral density ($N_0$) over the signal bandwidth. This is often used as a figure of merit for digital communications systems because it is normalized by the system's symbol rate and modulation format (since it is the energy per bit). This allows very different system implementations to be compared for efficiency on a level playing field, in an apples-to-apples fashion.

  
  

  In some other specific applications, you might also see $\frac{E_s}{N_0}$ (energy per symbol instead of energy per bit in the numerator) or $\frac{E_c}{N_0}$. I've seen two different meanings for the latter: first, in coded communications systems, you might see this, where $E_c$ stands for the energy per coded bit, in contrast to $E_b$, which is the energy per information bit. Another meaning: in direct-sequence spread spectrum systems, you might see $E_c$ refer to the energy per chip instead.

  
  


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  $\frac{C}{N_0}$: This quantity is often termed "carrier to noise ratio." One place that I've seen this used as a common measure of signal quality is in GPS-related literature. This is very similar to the "pure" definition of SNR, except that the (again, assumed to be white) noise power spectral density value $N_0$ is used in lieu of the total noise power over the signal bandwidth. Since bandwidth is a somewhat squishy term that has many different definitions (there is no one particular measure of what a signal's "bandwidth" is that is accepted to be correct), this ratio is somewhat more precise and less ambiguous than the "pure" SNR metric.