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is the average zero up-crossing frequency of the process (see Eq. 2.95). If
()
x t
is also
narrow banded, such that a zero up crossing and a peak
x
(larger than zero) are
simultaneous events (as shown for the process in Fig. 2.8), then the expected number of
peaks
()
()
x
a
f
aT
f
0
T
>
is
⋅
, while the total number of peaks is
⋅
. Thus
p
p
xp
()
()
f
a
()
x
p
Pr
⎡
xa Pa
⎤
1
≤=
=−
(2.35)
⎣
⎦
p
p
p
f
0
x
a
is given by
from which it follows that the probability density distribution to
()
()
()
⎡
⎤
f
a
df
a
d
d
1
()
()
x
p
x
p
pa
P a
⎢
1
⎥
=
=
−
= −
⋅
p
p
()
da
da
f
0
f
0
da
⎢
⎥
p
p
x
x
p
⎣
⎦
⎡
2
⎤
a
a
1
⎛ ⎞
()
p
p
⎢
⎥
⇒
pa
=
exp
−
(2.36)
⎜ ⎟
p
2
2
⎢
σ
⎥
σ
⎝ ⎠
x
x
⎣
⎦
()
Thus, the probability density
p a
of peaks to a narrow banded Gaussian process is a
p
Rayleigh distribution (see Eq. 2.7). The distribution is illustrated on the right hand side
of Fig. 2.8 (see also Fig. 2.1).
2.4 Extreme values
Fig. 2.9.a shows a collection of
N
short term time series, each a short term realisation of
the fluctuating part
()
()
()
=+
. It is assumed that they
are all stationary and ergodic, and for the validity of the development below it is a
necessary requirement that they are fairly broad banded. From this ensemble of
realisations it may be of particular interest to develop the statistical properties of extreme
values, as illustrated in Fig. 2.9.b. Referring to Eq. 2.33 and Fig. 2.8, an extreme peak
value
x t
of a stochastic variable
X t
xxt
p
ax
=
within each short term realisation occur when
max
1
−
()
⎡
f
a
⎤
T
(2.37)
→
⎣
xp
⎦
