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.