Confused about the Signal to Noise Ratio

[otaKu]

Hello everyone!
It seems to me that there are many ways of calculating SNR depending on the type of signal and the nature of problems. I was able to get a decent amount of information for electrical signals but couldn't find resources which discuss the topic in a more generalized manner. I was reading a research paper which calculates the SNR using the method mentioned below:

"where the signal is defined as the height of the signal (power normalized to 1) and the noise is the variance of the noise distribution"

The signal here is simulating the Lorentzian curve of intensity measured by a spectrophotometer and the white noise is added randomly from a Gaussian distribution during the simulation. (Link to the paper: https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-2-1020: the part I am referring to can be found in the last paragraph of page 4 of the PDF version.)
It is unclear to me as to what the author implies with the above-mentioned statement. I get that you need to take the variance of the noise distribution, but the part where he says that the signal to be considered is the height with power normalised to 1 doesn't make complete sense to me. What goes in the numerator? Is it the height of the graph at it's peak? or perhaps the average of the height of the graph for the selected range of frequencies? Also, why are different people using different ways to calculate Signal to Noise Ratio? Most people seem to use the ratio of powers wherever both of these quantities are available, others use variances of the signal and the noise, and then there's this paper which uses the ratio of Intensity(?) of the signal to the variance of the noise. You can ask follow up questions if my message doesn't make sense. I am very new to evaluating performance of signals so I apologise if I get the technical jargon wrong somewhere during the conversation.
Have a nice day!

 

[f95toli]

There is no single definition of SNR; it depends in the context.

Your SNR is equal to 1 when the level of you signal (whatever that is) is equal to the level of your noise (whatever is causing the noise). The signal could be voltage, power, intensity, length, frequency etc (just to mention a few I come across in my work).
This is -as far as I am aware about as close to a "general" definition as you can get. There are of course lots of different definitions of the SNR for specific applications and in some context there is indeed just one -widely accepted- definition, but generally speaking you need to look at the formula used to see what is meant.

 

[anorlunda]

The signal is the part you are interested in. The noise is everything else. Therefore, it is subjective.

 

[Tom.G]

the height with power normalised to 1 doesn't make complete sense to me. What goes in the numerator?

The numerator is the amplitude of the signal you are interested in,
The denominator is the amplitude of the noise.
"Amplitude", as @f95toli noted, can be any of various measurements.
SNR is frequently expressed in decibels (db), which is 10 times the logarithm of the fraction calculated above.

A 'normalized' amplitude is defined by assigning a value of "1" to the maximum amplitude, then expressing related values as a multiplier (which may be less than or more than 1) of the normalized value.

Hope this helps!

Cheers,
Tom

 

[tech99]

The numerator is the amplitude of the signal you are interested in,
The denominator is the amplitude of the noise.
"Amplitude", as @f95toli noted, can be any of various measurements.
SNR is frequently expressed in decibels (db), which is 10 times the logarithm of the fraction calculated above.

A 'normalized' amplitude is defined by assigning a value of "1" to the maximum amplitude, then expressing related values as a multiplier (which may be less than or more than 1) of the normalized value.

Hope this helps!

Cheers,
Tom

Signal to Noise ratio, by convention, refers to power, not amplitude, so the ratio in decibels will be 10 times the log of the power ratio. The power definition removes any issue about the wave shape.
If we use amplitude, there is a problem when the noise is statistical and the signal is sinusoidal. (In any case, for amplitude we need to take 20 times the log of the amplitudes, not 10 times).
With amplitude, there is also a complication where the signal source is not matched to a noisy receiver, because the decibel definition for amplitudes presumes equal impedances,.
It is also common to really mean Signal+Noise / Noise ration, as a pure signal is not usually obtainable in practice.