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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