In the previous blog the focus was on binning (charge, voltage and digital domain) in the case the readout noise was dominating over the photon-shot noise. In other words, for the case of small signals or low light levels. This time, the situation for a shot-noise limited condition is considered. And actually the story can be very short : it does not matter when or where the binning is done, in all cases the result is exactly the same.

For the charge domain : if *n x n* charge packets are added together, each with *m* electrons, then after binning the final charge packet holds :

*n x n x m* electrons.

The photon-shot noise in each individual charge packet was

*sqrt(m)* electrons,

so the SNR for every individual charge packet was

*SNR = m/sqrt(m) = sqrt(m)*.

After binning the total photon-shot noise is equal to

*sqrt(n x n x m)* electrons

and the SNR will be equal to :

*SNR = n x n x m/sqrt(n x n x m) = sqrt (n x n x m)*.

After binning in the charge domain, the increase in SNR will be

*sqrt(n x n x m)/sqrt(m) = n !*

For the voltage domain or digital domain : if *n x n* signals are added together, each with *m* electrons, then before binning the output signal would have been *k x m* V or DN, with *k* being the conversion gain from input (charge) to output (Volts or Digital Numbers). Then after binning the final signal will be

*n x n x k x m* V or DN.

The photon-shot noise of each individual signal before binning was

*k x* *sqrt(m)* V or DN,

so the SNR for the individual signal before binning was

*SNR = (k x m)/(k x sqrt(m)) = sqrt(m)*.

After binning the total photon-shot noise is equal to

*k x sqrt(n x n x m)*

and the SNR will be equal to :

*SNR = (n x n x k x m)/(k x sqrt(n x n x m)) = sqrt (n x n x m)*.

After binning in the voltage of digital domain, the increase in SNR will be

*sqrt(n x n x m)/sqrt(m) = n !*

Conclusion : if there is enough light so that the performance of the sensor or the camera is shot-noise limited, it does not matter how the binning is realized, charge domain, voltage domain or digital domain. The increase of the SNR after binning is always equal to a factor *n*, being the kernel size in the case of *n x n* binned pixels.

Albert, 31-05-2016.