Environmental Engineering Reference
In-Depth Information
ξ
sn
(
t
)
t
(a)
t
Z
(
)
t
(b)
Figure 2.9. Examples of realizations of (a) WSN
ξ
sn
(
t
), and (b) the homogeneous
compound Poisson process
Z
(
t
). The spikes in (a) are represented as bars of unit
base, instead of delta functions as in Eq. (
2.50
).
The average and the covariance function of the compound Poisson process are
(
Parzen
,
1967
, p. 130)
Z
(
t
)
=
λα
t
,
(2.52)
Z
(
t
)
Z
(
t
)
2
min(
t
t
)
=
2
λα
,
,
(2.53)
respectively.
We can nowobtain the covariance function and themoments ofWSNby considering
that WSN is the formal derivative of the compound Poisson process; as a consequence,
the mean is (
VanMarcke
,
1983
, pp. 110-111)
=
∂
Z
(
t
)
ξ
sn
(
t
)
=
λα,
(2.54)
∂
t
and the covariance is
Z
(
t
)
Z
(
t
)
=−
∂
2
ξ
sn
(
t
)
2
t
)
ξ
sn
(
t
)
=
2
λα
δ
(
t
−
.
(2.55)
∂
(
t
−
t
)
2
2
The variance reads
κ
=
2
λα
δ
(0)
=
2
s
sn
δ
(0), where
s
sn
is the strength of the
2sn
2
[see Eq. (
B2.1-6
)]. Only the mean is finite, whereas all higher-
order moments and cumulants diverge.
noise, i.e.,
s
sn
=
λα
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