# Adding shot noise in frequency domain

I have a current signal represented by an array in python, which I would like to add shot noise to. Shot noise is defined as $$σ_i = \sqrt{2qI \Delta f}$$ where $q$ is the elementary charge, $I$ the current and $\Delta f$ is the bandwidth of interest. I guess the simplest way to add the noise would be to to add it in the time domain, using the np.random.normal function in python and simply adding it element-wise to the signal array. The frequency is needed to calculate the shot noise, so I would calculate $\Delta f = \frac{1}{2\Delta t}$ where $\Delta t$ is the time between two consecutive samples in the array (correct me if I'm wrong!).

My question is, how can I add this noise in the frequency domain? I would expect to be able to express the shot noise in $\frac{A}{\sqrt{Hz}}$, and therefore make the magnitude independent of the frequency. This post explains the relationship of the magnitude of white noise in time and frequency domain, but if I use that simple equation ($\frac{\sigma^2 M}{2}$ ) it doesn't become independent on the frequency.

• @JasonR: When calculating the noise of an amplifier, for example, the noise is usually expressed as $frac{A}{\sqrt{Hz}}$ or $frac{V}{\sqrt{Hz}}$. The total noise in the time domain can then be calculated by integrating over the desired frequency range. – Uffe Oct 8 '13 at 10:32
• Oops, I was too slow. Here it comes again: @JasonR: When calculating the noise of an amplifier, as an example, the noise is usually expressed as $\frac{A}{\sqrt{Hz}}$ or $\frac{V}{\sqrt{Hz}}$. The total noise in the time domain can then be calculated by integrating over the desired frequency range. I would expect to be able to do the same here, but when I use the equation from the link ($\frac{\sigma^2 M}{2}$) in combination with the shot noise definition, I get a frequency domain variance of $2qI{\Delta f}^2 M$. – Uffe Oct 8 '13 at 13:04