Units of S/N ratio in some papers

[HAYAO]

TL;DR Summary

Several papers showing strange units used for S/N ratio. Are they right?

So this is a paper by Xavier Michalet and Shimon Weiss C. R. Physique 3 (2002) 619-644, showing that S/N ratio is given by equation (2).
This is a fairly famous comprehensive paper on fluorescence microscopy, where other references regarding the same topic show the same equation more or less. Some tutorial-like Nikon's website shows something like this:
https://www.microscopyu.com/tutorials/ccd-signal-to-noise-ratio

I'm not understanding the units here, though. Equation (2) would be in a strange unit of rooted units. Am I missing something? I thought SN ratio should be unitless
I thought that each of the terms in the denominator should be squared as to give a RMS of the combined noise factors.EDIT: I also do not understand why we have same terms on nominator and denominator. S/N ratio should literally be Signal divided by Noise. Why would Signal strength be in the denominator? That's total intensity, not noise.

 

[Gleb1964]

I think, the units supposed to be photons or photo-electrons, or any digital units after voltage converted into digits after analog-digitaln converter. If photoelectron's signal assumed to have a Poisson distribution law, the noise is a squre root from the signal.

 

[HAYAO]

Ah okay, stupid me. Why didn't I think about that. Thanks.

 

[HAYAO]

Sorry for the follow up question. I came upon this article by Andor:

Wait, Dark, Signal, and Spurious Noises are counting noises, and should be subject to Poissonian type noises (Shot noise). Thus the noise should be square root of the various background signal itself. Why is it squared inside the square root like the readout noise? This is different from the description in the OP.

 

[Gleb1964]

Have looked the article you were refer here https://andor.oxinst.com/learning/view/article/emccd-technology-in-spectroscopy
I think you are right about Dark and Signal. I am not sure if a Spurious Noise has the same nature, but may be.
Anyhow, you can chose to use a square root of a number of electrons as a noise or denote noise components by special symbols, like it done in a formula above, if it looks more beautiful. To get a summary noise one need to sum the squares of RMS of noise components under square root. Was that you question?

 

[HAYAO]

Yes, so the total noise level should be squares of RMS of noise components. Readout noise N
_RN is not a counting process, so I can understand having a square on it inside the big root as
N_RN². However, at least dark N_DNand signal noise N_SN arise from Poissoninan counting process, so the noise level is squared root of the dark and noise signal, meaning just N_DNand N_SN instead of N_DN² and N_SN². Therefore, the equation should be:
SNR = QE*P / √(N_RN/G)²+F²(N_DN + N_CLC + N_SN)

At least that's how I understand it. But yeah, it does depend on how these parameters were defined.

 

[Drakkith]

Wait, Dark, Signal, and Spurious Noises are counting noises, and should be subject to Poissonian type noises (Shot noise). Thus the noise should be square root of the various background signal itself. Why is it squared inside the square root like the readout noise? This is different from the description in the OP.

They are taking the convention that $N_{DN} = \sqrt{DN}$ and such. Remember that noise is the square root of the total number of 'events'. So you have to square each one to recover the original number of 'events' for each source, sum them all together to get the total number of 'events', and then take the square root of the whole thing.

For an 'ideal' camera that just has shot noise we get $\frac{P}{\sqrt{P}}$.
If we introduce dark current we get $\frac{P+P_{DC}}{\sqrt{P+P_{DC}}}$.
Note that I didn't square anything in the denominator since I didn't use a separate variable to represent the noise of each signal. Also, I am including dark current in the numerator for reasons that will be explained below.

As the above paragraph alludes to, they are actually missing some things in their formula. Where did we get all of these extra 'events' in the denominator?? If we are counting them on the bottom, shouldn't they be on the top as well? Shouldn't the equation actually be $\frac{QE*P+DN+CIC}{\sqrt{(\frac{N_{RN}}{G})^2+F^2(N_{DN}^2+N_{CIC}^2+N_{SN}^2)}}$?

What they have done, possibly without stating it, is that they have subtracted the dark current and spurious signal (it's not a noise, it's a signal that has noise) from the formula since that is what you would do in real images. You can subtract the dark current and spurious signal from the numerator, but you cannot subtract the noise. That's why those two extra signals are missing from the top. If we wanted to be really precise, we'd need to add in a few more terms to account for the fact that we can't actually know the signals with 100% accuracy and precision. Just subtracting dark current introduces yet more noise, as does subtracting spurious signal since we have to do separate imaging to attain these values, and these images would also contain noise.

Readout noise stands out as the only thing that could be considered 'pure' noise. That is, it isn't a signal that introduces noise, it simply alters the values of all of the signals that passes through the circuit during readout in a way that is independent of the magnitude of each signal. Hence there is no readout signal in the numerator.

Readout noise N
RN is not a counting process, so I can understand having a square on it inside the big root as
NRN2.

Readout noise is just like the other noise terms as far as the formula is concerned (at least in general). From my limited understanding of EMCCD's, the reason it is separated in the formula you linked is:
1. The other noise terms don't care about the gain of the readout circuit.
2. The other noise terms are multiplied by a 'noise factor', which should probably be called a 'gain factor' since it is the result of the amplification effect of EMCCD's. But then that would be confused with the gain of the readout electronics.

I suspect you knew that already, but I wanted to include it for completeness.

For a regular CCD the SNR formula is $\frac{P+P_{DC}+S}{\sqrt{P+P_{DC}+S+RN^2}}$, where $P$ is our target photon signal, $P_{DC}$ is our dark current signal, $S$ is the sum of whatever other signals there might be (background light, spurious signal, lights from a passing aircraft, whatever), and $RN$ is readout noise. If the noise terms get their own variables then I'd need to square them.

 

[Gleb1964]

That is about consideration what should be counted for "Signal". In the most cases that would be incoming photons, but can be only a portion of photons, when using additional signal discrimination, like polarization, spatial, spectral or other variation in the signal.

 

[Drakkith]

That is about consideration what should be counted for "Signal". In the most cases that would be incoming photons, but can be only a portion of photons, when using additional signal discrimination, like polarization, spatial, spectral or other variation in the signal.

Certainly. The formula in Hayao's post isn't 'wrong', it's just a simplification that useful a lot of the time. It's close enough unless you really want to get into the details.