Titel_SS06

Poisson Processes

Filtered Poisson processes may be formulated directly from the simple Poisson process
by attributing random events to the events (occurrences at times ) of the simple Poisson

process. The process:

Nt()

tk

()

1

() (, , )

Nt

kk

k

X tttN



 Y (8.1)

is called a filtered Poisson process when the points are generated by a simple Poisson

process ,

tk

Nt() N(, , )tt Ykk is a response function and are mutually independent random

variables. As

Yk

N(, , )tt Ykk is defined as zero for tt kthe process is initiated at time. A

typical realisation of a filtered Poisson process with rectangular pulses of equal duration a is
illustrated in

tk

Figure 8.1.

x(t)

t

t 1 t 2 t 3 t 4

a a a

y^1 y^2 y 3 y

4

Figure 8.1: Illustration of a realisation of a filtered Poisson process with rectangular pulses.

The mean value and the covariance function of the filtered Poisson process can be obtained
as:

  

0

EXt() 3 N 3( )E t Y d(, , )

 3



(8.2)



0

COV X s X t(), () 3 N 3()E (,, )s Y E N 3(,, )t Y d

 3 (8.3)

As the durations of the pulses a are reduced in the limit to zero the process is referred to as a
Poisson spike process. Due to the non-overlapping events of the Poisson spike process the
covariance function equals to zero for st/.

Consider the realisation of the Poisson spike process X()t illustrated in Figure 8.2.