Optical Communications, 2nd Edition

3.7 SHOT NOISE PROCESSES

The photodetection output current i(t) is described by the shot noise process in Eq. (3.1.4) as

where again k(0, t) is the random count process produced over the volume consisting of the detector area and the time interval (0, t). The count probability over any such volume can be determined from the analysis in Sections 3.4 and 3.5. To investigate the statistical properties of this output current, however, we need additional parameter descriptions beyond just the count process. In particular, we need the statistical properties of the gain parameter g_j in Section 3.6 (if photomultiplication is used), and we need the statistics of the occurrence times {z_j}. These occurrence times of the individual electrons in the shot noise process represent a collection of random locations in time inherently linked to the counting process. In this section, we derive the joint probability density of a given number of such locations in a specified time interval (T₁, T₂) for a Poisson counting process. Specifically, we derive the joint density of the random sequence z = (z₁, z₂,…, z_k) over (T₁, T₂) given k Poisson counts in that interval. We write this conditional probability density as p_z(z₁, z₂,…, z_k|K), and derive its mathematical form as follows. Consider the interval (T₁, T₂) to contain the infinitesimal slots (t₁, t₁ + Δt), (t₂, t₂ + Δt),…, (t_k, t_k + Δt), as shown in ...