Expression of the mean and variance of spectroscopic Shot Noise

[fab13]

TL;DR Summary

I saw different expression for the variance of Shot Noise for spectroscopic probe in spectroscopic survey. I would like to conclude about this quantity.

Hello,

I would like to know the right expression for the expression of variance of Shot noise in spectroscopic probe.

Sometimes, I saw $\sigma_{SN,sp}^{2} = 1/n_{sp}$ with $n_{sp}$ the average density of galaxies, whereas my tutor tells me that $\sigma_{SN,sp}^{2} = 1/n_{sp}^{2}$ , so I would like to know the right expression and a demonstration if possible.

For example, I show you a slide, from Nico Hamaus, a pseudo or correct demonstration of spectroscopic Shot Noise :

and the infered expression for Shot Noise spectroscopic :

As you can see on this second slide, we have for the variance : $\sigma_{sn}^{2}=1/\bar{n}_{sp}$ (and not squared like my tutor told me).

Any clarification is welcome (ideally, a demonstration like on these slides would be great). There are not a lot of documentation on the net for this quantity "Shot noise spectroscopic" and its mean, variance.

Best regards

 

[hutchphd]

I would say not squared. At low N the distribution is a time sequence Poisson and at high N it becomes Gaussian.

 

[fab13]

Thanks a lot. Could you demonstrate it please with simple equations or reasoning ? Indeed, in my case, this is not a stochastic process (I mean depending of time) but the inverse of density of galaxies (squared or not but a not squared gives better results for the rest of my computations).

By the way, could I apply in this case (for low N) the equality between the variance and the expectation if this is a Poisson process ?

Regards

 

[hutchphd]

Did you look at the link?

 

[fab13]

I tried to find an equivalent expression on the part of your link like :

" Detectors :

The flux signal that is incident on a detector is calculated as follows, in units of photons:
$P=\frac{\Phi \Delta t}{\frac{h c}{\lambda}}$
c is the speed of light, and $h$ is the Planck constant. Following Poisson statistics, the shot noise is calculated as the square root of the signal:
$S=\sqrt{P}$ "

or on this other part :

" For large numbers, the Poisson distribution approaches a normal distribution about its mean, and the elementary events (photons, electrons, etc.) are no longer individually observed, typically making shot noise in actual observations indistinguishable from true Gaussian noise. Since the standard deviation of shot noise is equal to the square root of the average number of events $N$, the signal-to-noise ratio (SNR) is given by:
$\mathrm{NR}=\frac{N}{\sqrt{N}}=\sqrt{N}$
Thus when $N$ is very large, the signal-to-noise ratio is very large as well, and any relative "

But I can't manage to do the link with my formula : $\sigma_{SN,sp}^{2}=\dfrac{1}{\bar{n}_{sp}}$,

if you could see how to proceed to justify this variance expression...

Regards

 

[hutchphd]

I commmend this to you:
https://en.wikipedia.org/wiki/Signal-to-noise_ratio
In this case my definition of signal to noise would just be 1/variance ..(i.e. 1/ (noise/signal). I see there are other definitions which may the heart of your disagreements. The power (which often goes as the square of the signal) will obviously get rid of the root. I think that is not correct here.

 

[fab13]

@hutchphd

Thanks for your quick answer. If this would be only mine, I would take $\text{Variance}(Shot\,noise) = 1/n_{gal}$ with $n_\text{gal}$ the average density of galaxies.

But my tutor disagreess. Actually, the formula that puts the mess is the following one :

I am only interested in the term A=B, that is to say $\Delta_{ij}^{GG}$.

So we have from eq(138) : $\Delta_{ij}^{GG} = \sqrt{\dfrac{2}{(2\ell+1)f_{sky}\Delta\ell}}[C_ij^{GG}+N_{ij}^{GG}]$

I don't understand why the standard deviation for shot noise in eq(138) is defined as $N_{ij}^{GG}=\dfrac{1}{n_{i}}\delta_{ij}^{K}$ and not $N_{ij}^{GG}=\dfrac{1}{\sqrt{n_{i}}}\delta_{ij}^{K}$.

Such way, I could have the $\text{variance}(P_{shot}) = \dfrac{1}{n_{gal}}$.

One has to not forget that $\Delta_{ij}^{GG}$ is a standard deviation and not a variance.

Do you understand my issue ?

Any help is welcome