# Calculating SNR

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Persons who are interested in the math behind SNR

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Math, with explanations

SNR is a simple ratio of the total signal to the total noise. For microscope cameras, the equation looks like this:

$SNR=\frac{QE*S}{\sqrt{F_{n}\! ^2*\ QE\ *\ (S+I_{b})+(N_{r}/M)^2}}$

where

• QE = Quantum efficiency
• S = Photons/pixel
• Fn = Noise factor
• Ib = Background
• M = EM gain

From this equation we can see that the total signal is dependent on the QE (or effective QE (eQE) for EM-CCDs) of the camera. Learn more about QE and eQE in our Technical Guide, “Dissecting Camera Specifications: a field guide for biologists.”

The total noise is dependent on the noise factor, background, readout noise, and EM gain.

Signal and Background, S and Ib

Photons falling on the sensor have an average photon flux. The fluctuations in this rate are governed by Poisson statistics and therefore have a standard deviation that is the square root of the number of photons (i.e., photon shot noise). In imaging, there are two sources of photons (and photon shot noise): the signal of interest (S) and the signal from the background (Ib). Limiting the amount of Ib and increasing S is critical to getting images with high SNR.

Quantum Efficiency, QE

The QE of a camera is the wavelength-dependent probability that a photon is converted to a photoelectron. High QE is a fundamental attribute for obtaining high SNR, since QE is a predominant factor in the SNR equation.

EM-CCD Only, M and Fn

EM gain (M) occurs in a voltage-dependent, step-wise manner and the total amount is a combination of the voltage applied and the number of steps in the EM register. EM gain has a statistical distribution and an associated variance which is accounted for by Fn. At typical EM-CCD gains, Fn = √2 ≈ 1.4. All signal in an EM-CCD is subject to this additional noise. Since CCD and CMOS sensors do not have EM gain, Fn = 1 in these three cameras.

Camera Noise, Nr

Read noise (e-) is a statistical expression of the variability within the electronics that converts the charge of the photoelectrons in each pixel to a digital number expressing intensity.

Dark noise (Nd; not shown above) is camera noise that comes from thermally generated electrons and is time- and sensor-temperature dependent. Nd is not presented as a factor here because it is low and exposure times are short enough that it does not contribute significantly to total noise.

Uncorrelated noise is added in quadrature. This means that each noise term must first be squared, and then added to other terms before the total noise can be calculated by taking the square root.

The effect of quadrature is critical; a read noise of 2e- contributes 4e- of noise to the total noise, while a read noise of 4e- contributes 16e-.

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