On the paper: "Analysis of Subthreshold Current Reset Noise in Image Sensors" which can be publicly found here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4883354/ the author is using a method to calculate the Ensemble Average of Squares of a given noisy function of time. I'm not able to understand how the author passes from equation 19) to equation 20).
For reference, equation 19) is the solution to a differential equation that boils down to:
$$ 19) \ \ \ \ v(t)=\int_0^t dt'\ \frac{1+\frac{t'}{\tau}}{1+\frac{t}{\tau}}i_n(t')\ +\ v(0)\frac{1}{1+\frac{t}{\tau}} $$
Where:
- $i_n(t)$ is a Poisson Process with $<i_n(t_1)i_n(t_2)>=qI(t)\delta(t_1-t_2)$
- $I(t)$ has an exponential form but I don't think it is used in the calculation
Then it goes on saying that by squaring and averaging eq 19 the result is:
$$ 20) \ \ \ \ <v^2(t)> \ = k \left( 1+\frac{1}{(1+\frac{t}{\tau})^2}\right)\ +\ <v^2(0)>\frac{1}{(1+\frac{t}{\tau})^2} $$
How did he get to eq 20)?
He uses this method three times throughout the paper. When he uses it the first time between equations 9) and 10) it is clear to me. How is this method called? Is there any reference that explains it in clearer details?