Time-varying shot noise generation

Question

I'd like to model the behavior of photodiodes, the input is like data bits (0 & 1) where each bit is represented maybe with 100 samples / bit and the bit rate is $B$ bits/sec with period $T_{bit}=\frac{1}{B}$, each sample represents the input optical power.

The required is to generate a time-domain shot noise for input signal based on the well-known shot (Poisson) noise variance $\sigma^2_{I_{shot}}=2qI_{0}\Delta f$, $I_{0}$ is the output current based on the input power via the relation $I_{0}=RP_{in}$ , $R$ is a constant and $\Delta f$ is the noise BW. The input is based on random bits with zero value for 0 bits and $I_0$ value for 1 bits, how to make the noise signal dependent based on the current bit using python? I have thought of a moving average filter that finds the average value for a window $T_{avg}$ but not sure how to relate $T_{avg}$ to $T_{bit}$, and accordingly add the shot noise based on the found average or just add the noise for each simulation sample per bit, is that correct? Is their an easier or a more correct way to do it?

Answer 1

You have two simple cases:

1. Input is 0: then $\sigma^2_{I_{\text{shot}}} =0$, which means it is a constant value. So, in Python, that just means you set the result to = constant, for some value of constant that fits the rest of your needs (variance can't tell us anything about expectations, but =0 might be a good start?)
2. Input is 1: then $\sigma^2_{I_{\text{shot}}}=2qI_0\Delta f$; I don't know what $q$ or $\Delta f$ are, but they must be constants in this context, so essentially, you get shot noise of some constant variance $v^2=2qI_0\Delta f$.

How to generate Poisson-distributed numbers is well-covered in Literature, including Wikipedia, so if you want to understand how it's generated, that's easily researchable. If you want to just to use someone else's code to generate it, well, you can't have looked very far: The popular scipy package contains a poisson generator in its scipy.stats module. It's documented here; just use poisson.rvs with your calculated variance $v^2$ as rate (for Poisson distributions, there's only one parameter, and that's rate, and variance happens to be the rate).

Regarding your update to your question:

The fact that there's "bit periods" underlying doesn't matter to the process – you have sample durations during which you either have variance 0, or the specified variance $v^2$. The fact that your process is poisson says exactly that: the moments of your count solely depend on the duration of your observation, not on what happens elsewhen.

So, your averaging is definitely mislead. I really don't know why you should do that – you have all the tools laid out for you:

0. You know the distribution, Poisson
1. You know the observation length, $T_{bit}/100$
2. You know the rate, because that is identical to variance

So, all you need to do is draw a random number that's Poisson-distributed with the specified rate. That's it.