Statistical properties of registers with multiple stages

Equations 4.5 and 4.8 can be extended to registers with multiple gain stages as a geometric series. For a register of $s$ gain stages described by model 1, summing the series for Equation 4.5 gives:

$$\sigma_{out}^{2}=\mu^{2s}\sigma_{in}^{2} + \mu^{s-1}\left <m_{in} \right >\left( \mu^{s} - 1\right)$$ (4.9)

For a register of $s$ gain stages described by model 2, Equation 4.8 gives:

$$\sigma_{out}^{2}=\mu^{2s}\sigma_{in}^{2} + \mu^{s-1}\left <m_{in} \right >\left( \mu^{s} - 1\right) \left(2 - \mu\right)$$ (4.10)

If the inputs are Poisson-noise limited, $\sigma_{in}^{2}=\left <m_{in} \right >$ so for model 1 we have:

$$\begin{array}{ll} \par\sigma_{out}^{2} & = \mu^{2s}\left <m... ...\right >\left (\mu^{s+1} + \mu^{s} - 1\right ) \par\end{array}$$ (4.11)

For a register of $s$ gain stages described by model 2, we have:

$$\begin{array}{ll} \par\sigma_{out}^{2} & = \mu^{2s}\left <m... ..._{in} \right >\left (2\mu^{s} + \mu - 2\right) \par\end{array}$$ (4.12)

If we remember that the gain $\mu=\alpha+1$ then Equation 4.12 is clearly consistent with Robbins & Hadwen (2003).

If the stage gain $\mu \left (t \right )$ varies as a function of time (e.g. due to voltage fluctuations), then the overall gain applied to the signal from one pixel in the imaging array will be the product of the $\mu \left (t \right )$ values at each stage while the signal is passed along the serial multiplication register. The large number of gain stages will suppress the effects of high frequency fluctuations in the gain voltage, and only long term drifts in the gain would be expected to affect the output signal.