Why is noise the square root of signal?

[Stonius (Markus)]

Can anyone explain why, in simple terms, the noise is always the square root of the signal?

Surely all signals can have varying amounts of noise? Why is there a constant relationship there?

Markus

[Merlin66 (Ken)]

Shot noise, or the random noise in the target "signal" (Read: target image) is a reflection of the randomness of the photon count being received.
If we count the number of photons received over a period of time they will not arrive at the same rate.
Our first count may be 90, second count, 110, 102, 115 etc etc.
This spread of numbers approximates to a Gaussian distribution curve, also called a normal distribution curve.
https://en.wikipedia.org/wiki/Normal_distribution

The possible error +/- (the "noise" in our signal) associated with a 68% probability (1 sigma error) is sqrt(signal)

This has been adopted as an acceptable measure of the random "shot" noise.

[Stonius (Markus)]

Ah, so it's to do with the specific randomness of photons arriving (is this Poisson theory too?).

I've seen it applied to dark noise, but maybe it doesn't apply to read noise as I always see it squared first before being added tot he rest of the terms, then the square root taken, as in;

SNRstack = (Sobj * Csubs)/SQRT(Csubs * (Sobj + Sskyfog + DC + RN^2))

Best,

Markus

[markas (Mark)]

Read Noise specification is an RMS (root mean square) number. Therefore to add it in to total noise it must be squared first, then added to the other noise contributing signals, after which the total noise is calculated by taking the square root of the sum.


Mark

[Merlin66 (Ken)]

Markus,
I took your original question to refer to shot noise...
There many other factors which can influence the total noise....
The read noise component is generally a constant based on the camera electronics - not related to shot noise.

[Merlin66 (Ken)]

Yeah, for convenience it's included in the bracket terms - so needs to be squared first.

The recognised equation for SNR (the "CCD equation") is:
SNR = Nsig/sqrt(Nsig+ Npix(Nsky+Ndark+(Nread)^2))

[Stonius (Markus)]

Just to make sure though - the noise being the square root of the signal applies only to light, right?

The Dark Current Noise is squared in the equation too because thermal noise is still light, even if it's not visible light, so it accumulates in the same fashion.

And as Mark mentioned, RMS is already rooted, so it has to be squared first to end up with the correct contribution to the total noise as the terms are added.

Have I got that right?

Markus

[Merlin66 (Ken)]

The darks have noise similar to the signal so they have a similar Gaussian distribution, so it would not be squared in the brackets.

[JA]

Originally Posted by Stonius

Can anyone explain why, in simple terms, the noise is always the square root of the signal?

Surely all signals can have varying amounts of noise? Why is there a constant relationship there?

Markus

Always is always a big ask, but if we're talking about shot noise ...

In taking the shot we are measuring (sampling) the light intensity from the DSO at each pixel location. Photons from the DSO travel in to the camera, strike the sensor and are converted in to electrons in accordance with the Quantum Efficiency of the sensor. These are "counted" and digitised in to a light intensity at each pixel location in the image at a bit depth according to the type of image JPEG, RAW, etc... by the camera electronics. At each pixel location in the image we are counting the number of events that occur in a given time. <<< That is the classic applicability criterion for the Poisson Distribution - (The probability distribution for a variable that counts the number of events that occur in a given time). When Poisson originally developed it was a distribution of the number of soldiers kicked by horses on a given day (Love the French). Here we measure photons/electrons over a fixed time period.

The Poisson distribution can be modeled as a Normal distribution under certain special conditions, BUT to cut to the chase, your answer lies in the definition of the Standard Deviation (a measure of spread) for the Poisson distribution. It is quite simply equal to the square root of the mean.

So if we take a series of shots and the mean signal intensity = x, then the standard deviation of the signal intensity (which you could think of as variability or noise) will be sqrt(x). That is simply the math for the Poisson Distribution (perhaps see Poisson Distribution for this derivation)

Puting the standard deviation in more usual Normal Distribution terms:
~ 68% of the intensity would be expected to be in a band 1 standard deviation either side of the mean
~95% of the intensity would be expected to be in a band 2 standard deviation either side of the mean
~99.7% of the intensity would be expected to be in a band 3 standard deviation either side of the mean

That variability is what makes all those disgusting dots we don't like.


Best
JA

[Stonius (Markus)]

So does that mean that on a perfectly calibrated frame, the StdDev will always = SQRT(mean)?

I'm trying to see if that holds true with actual data and whether I can use that principle to see how far I am off a perfect calibration if it doesn't.

M

[ChrisV (Chris)]

As long as you stay in the linear region of the sensor, I assume.
Nice explanation JA! There's tons of stuff out there in nature that is random and can be described by the poisson disn.

[Stonius (Markus)]

Originally Posted by Merlin66

Yeah, for convenience it's included in the bracket terms - so needs to be squared first.

The recognised equation for SNR (the "CCD equation") is:
SNR = Nsig/sqrt(Nsig+ Npix(Nsky+Ndark+(Nread)^2))

Is it? I must be reading it wrong, because it looks to me like here are no 'Signal' terms, they are all Nsig which I assume means 'signal noise'. To make a ratio there must be some mention of the signal, surely?

What is NPix? Pixel noise is new to me. Does it have another name?

Cheers

Markus

[JA]

Originally Posted by Stonius

Is it? I must be reading it wrong, because it looks to me like here are no 'Signal' terms, they are all Nsig which I assume means 'signal noise'. To make a ratio there must be some mention of the signal, surely?

What is NPix? Pixel noise is new to me. Does it have another name?

Cheers

Markus

Well I can't say for sure what may have been intended, but there is certainly another noise term (error) that often goes unmentioned: the quantisation error. It is the error one sees in a digital system associated with assigning a continuous analogue value in to a discrete range of analogue values ( a pigeon-hole) in readiness and suitable for Analogue to Digital Conversion at a given bit depth. It's sort of a round off error which worsens with lower digital word bit depth.

As an example an analogue transducer may return an actual reading of say 1.7320612 Volts, but in being digitised given a possibly limited digital word depth, may only be able to placed in to a discrete range of values (pigeon-hole) of say 1.70 to 1.74 Volts (mid-point 1.72 Volts). The difference between the continuous analogue value (1.7320612 Volts) and the pigeon hole mid-point (1.72 Volts) is 0.0120612 Volts and can be thought of as a sort of rounding-off error associated with pigeon holing the data. It's called the quantisation error in a digital system, but I suppose one could look upon it as a sort of noise, in that like the shot noise for example, it takes us away from the true value of the signal intensity for a given measurement.

Best
JA

[markas (Mark)]

Markus,


'Nsig' is the wanted signal. In the absence of any other noise contribution the equation reduces to Nsig/sqrt(Nsig) =sqrt(Nsig).


For photographic purposes, quantization noise, either in the original sub-frames or properly stacked multiple frames, is negligible compared to the various shot noise components - provided that the camera ADC bit depths are not absurdly low (say <8-10 bits)



Mark

[Merlin66 (Ken)]

Guys,
I used the Gaussian Normal curve to KISS....
the various noise components are Poisson distributions.

The Npix is a typo (sorry!) it should have been npix - "the number of pixels under consideration for the SNR calculation" - Howell p74.
Yes, Nsig is the "total number of photons collected from the object of interest"

[Stonius (Markus)]

I take it all values should be in 16 bit and e-?

M

[Merlin66 (Ken)]

Markus,
Don't know about the 16 bit....
There is another couple of terms which are not included in our simplified SNR calculation...one of which addresses the digitization noise within the a/d converter:
G^2 * Rhof^2
where G is the gain (e/ADU) and Rhof an estimate of the 1 sigma error (Howell gives an approx value of 0.289)

All other numbers are in electrons/ photons. (ADU * Gain)

In a worked example Howell shows the SNR using the shot noise v's using the full SNR equation.
This gave for shot noise SNR=346 and the full equation SNR=342 - less than a 1% difference.....Not worth the effort??????