# Adding Poisson noise to a time series signal

I'm working on a radiation detection problem. I have simulated some measurement count data and want to add an amount of Poisson background noise to achieve a certain SNR. Radiation count measurement (due to the background radiation) is given by the Poisson random variable.

After searching the web, I'm not seeing any clear indication of how to go about it doing this. Is there a similar method as in the case of AWGN? (https://www.mathworks.com/help/comm/ref/awgn.html).

• You need to elaborate on the form of your data. – user28715 Dec 14 '19 at 1:07
• what is an event? – user28715 Dec 14 '19 at 1:41
• Each event is a photon hitting the radiation detector. – pproctor Dec 14 '19 at 18:53
• how does that manifest as a signal? in other words, what does the time series that you want to add noise to, look like? – user28715 Dec 14 '19 at 20:14
• The total number of events (counts) that occurred in a fixed recording interval (e.g. 1s ,5s ,300s...), call it a measurement. In the picture, z is the $n^{th}$ measurement and x is number of counts that occurred in that measurement. Let me know if this image helps clarify or if you'd like to see the actual simulated time series – pproctor Dec 15 '19 at 17:51

If your data $$\mathbf{x}$$ has variance $$\text{var}(\mathbf{x})$$ and you want to add Poisson noise to it with signal to noise ratio of $$\text{SNR}$$ dB, then you can just generate Poisson random numbers with appropriate variance. The variance of the noise $$\mathbf{n}$$ you need is $$\text{var}(\mathbf{n})=\frac{\text{var}(\mathbf{x})}{10^{\text{SNR}/10}}$$. So now you just need to use your Poisson random number generator and then add the noise to get the noisy data: $$\mathbf{x}+\mathbf{n}$$
• This works pretty well when 1$\leq$ var(n) but not in the converse case. I suppose zero mean Gaussian noise for the 1 > var(n) case would be reasonable. – pproctor Dec 16 '19 at 17:54