Shot noise

In the last few posts we have looked at spectral densities of a select point processes. In those posts we have represented events as points on the time axis. Such approach is just an approximation, which is valid when the duration of events is negligible, but it is not always so. When the events have finite duration, but the process is otherwise identical to the Poisson process, we obtain a peculiar type of noise known as the shot noise.

Power spectral density of a signal with fixed rectangular pulses

So, if the signal can represented as

\begin{equation} I(t) = \sum_k A_k(t-t_k) , \end{equation}

with \( A_k(s) \) being the \( k \)-th pulse profile. Then the Fourier transform of the signal is given by

\begin{equation} \mathcal{F}\left\{ I(t) \right\} = \sum_k e^{-2 \pi \mathrm{i} f t_k} \mathcal{F}\left\{A_k(t) \right\} . \end{equation}

If \( A_k(s) \) is a rectangular pulse with fixed duration \( \theta \) and fixed unit height, then the Fourier transform of the pulse is no longer trivial (in comparison with an earlier post)

\begin{equation} \mathcal{F}\left\{ A_k(s) \right\} = \int_0^\theta e^{-2 \pi \mathrm{i} f u} \mathrm{d} u = \frac{\mathrm{i}}{2 \pi f} \left(e^{-2\pi\mathrm{i} f \theta} - 1 \right) . \end{equation}

The Fourier transform of the signal is then given by

\begin{equation} \mathcal{F}\left\{ I(t) \right\} = \frac{\mathrm{i}}{2 \pi f} \left(e^{-2\pi\mathrm{i} f \theta} - 1 \right) \cdot \sqrt{\frac{2}{T}} \sum_k e^{-2 \pi \mathrm{i} f t_k} . \end{equation}

The power spectral density of the signal is then given by

\begin{equation} S(f) = \lim_{T\rightarrow\infty} \left\langle \frac{2}{T} \cdot \left\lvert \frac{\mathrm{i}}{2 \pi f} \left(e^{-2\pi\mathrm{i} f \theta} - 1 \right) \right\rvert^2 \cdot \left\lvert \sum_k e^{-2 \pi \mathrm{i} f t_k} \right\rvert^2 \right\rangle . \end{equation}

As we have dealt with the most of the complications of the above in an earlier post, we can now just deal with

\begin{equation} \left\lvert \frac{\mathrm{i}}{2 \pi f} \left(e^{-2\pi\mathrm{i} f \theta} - 1 \right) \right\rvert^2 = \frac{1}{2 \pi^2 f^2} \left[1 - \cos\left(2 \pi f \theta \right) \right]. \end{equation}

Then, for a finite signal, we have

\begin{equation} S(f) \approx \frac{3 N}{T \pi^2 f^2} \left[1 - \cos\left(2 \pi f \theta \right) \right] . \end{equation}

Alternatively one could rewrite the expression as a square of the sinc function. Thus the tell-tale of the shot noise in the spectral domain is its oscillating power spectral density. Notably the first "dip" can be found at \( f_1 = \theta^{-1} \), later ones are observed at the integer multiples of \( f_1 \).

Interactive app

Interactive app below generates a signal with (possibly overlapping) rectangular pulses. The duration of the pulses \( \theta \) is fixed, and you may change this value as desired. The height of the pulses is fixed to \( 1 \), as the power spectral density dependence on the height of the pulses is trivial.

Note that the pulses may overlap, because the inter-event time \( \tau \) (it is sampled from an exponential distribution with rate \( \lambda \)) in this case is treated as the duration between the starting times of the pulses. So if a particular inter-event time is smaller than the fixed duration of the pulse, i.e., \( \tau_i < \theta \), then multiple pulses could overlap. To avoid excessive overlaps, which slow down the simulation by a lot, we have limited the possible rates, so that \( \lambda \leq \frac{1}{\theta} \). In the earlier point process model this was not possible, because the pulses had negligible duration.