Árbók VFÍ/TFÍ - 01.01.1997, Blaðsíða 227

Simulation of stochastic 225
the signal. X(t) is therefore the value at time t of the sum of all such signals that have occurred
or arrived in the semi-closed time interval (0,t].
The impulse response function is typically of the form shown in Fig 1. It is generally
defined as a causal function of time, having backwards-oriented memory, that is, w(t,x,y) = 0
for t<x, or the pulse can not influence the process until it has arrived. Secondly, the shape
function is most often a function of the real time or the time difference only, 0= t-x, whereby
w(t,x,y) = w(0,y). For causal shape functions, the Poisson counting process N(t) as the upper
limit of the sum Eq. (1) can either be replaced by plus infinity (+<») or through a large time T
by N(T), since for arrival times larger than t, w(t,x,y) 0. Thus,
X(t)= £ wCt.Tj.Yj) , T»t (la)
í-o
Usually, the amplitude variable Y is separated from the shape function, that is,
w(t,x,y) = y w(t,x) (2)
This considerably simplifies the statistical moments. For instance, the mean value of the
process is then given by
t
E[X(t)] =E[Y] | w(t ,x)v(x)dx (3)
0
and the second cumulant (s<t) by
'S
[X(t) X(s) ] =E[Y2] Jw(t,x)w(s,x)v(x)dx (4)
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A stationary process X(t), which is amplitude-modulated by applying an envelope func-
tion V|/(t) will render a non-stationary process, i.e. Y(t) = \|/(t)X(t). Consider a homogeneous
compound Poisson process with a constant intensity v0, which is amplitude-modulated at the
time of arrival of each Poisson wave event by the envelope function \)/(t), that is,
Fig. 1. The Impulse Response Function.