# Shot noise in continuous time

Last time we have moved away from mathematical abstraction of point events by considering pulses of finite duration. Yet the theory we have derived in the earlier post did not fully match the simulated power spectral density. In this post we will see that the discrepancy close to the Nyquist frequency arises due to the observations being made in discrete time.

There is no analytical formulas or any derivation this time, as we have just an interactive app. It differs from the apps from the point process series or the earlier post on shot noise, in that the time series is not discretized prior to calculating power spectral density of the simulated time series. Instead power spectral density is calculated by adding the contributions of each individual pulse using the analytical formula. This way the signal is observed continuously and is not discretized. The signal is only discretized for the time series plot. In all other ways the app below is identical to the one from the previous post.

Observe that in the continuous time with any choice of parameters simulated power spectral density follows analytical prediction reasonably well. Most importantly, the surge in the power for higher frequencies (one not predicted by the analytical formula) disappears. This analysis allows us to conclude that it is merely an observation effect unrelated to the process itself.