Environmental Engineering Reference
InDepth Information
as the signal
φ
(
t
) or in the frequency domain through the function
F
(
ω
). Equations
(
A.7
)and(
A.8
)or(
A.9
) relate the process in these two domains.
The Fourier transform is a linear operator, and the result is in general a complex
number, i.e.,
)
e
i
θ
(
ω
)
F
(
ω
)
=
F
R
(
ω
)
+
iF
I
(
ω
)
=
A
(
ω
,
(A.10)
) is the socalled Fourier spectrum [its square
A
2
(
where the amplitude
A
(
ω
ω
)
=
2

) is the phase angle. A number of impor
tant equations relate the time to the frequency domains (
Papoulis
,
1984
). We recall
Parseval's theorem,
∞
−∞

φ
F
(
ω
)

is the energy spectrum] and
θ
(
ω
∞
∞
−∞

1
2
2
d
t
A
2
(
2
d
f
(
t
)

=
ω
)d
ω
=
F
(
f
)

,
(A.11)
π
−∞
expressing the total energy in a signal in time and frequency domains. When the
domain of
φ
−∞
<
<
∞
) the total energy
can be infinite (this is the case, for example, for all periodic signals). In these cases,
the mean power of the signal can be used in place of the total energy,
(
t
) extends over the whole real axis (i.e.,
t
T
1
2
T
2
d
t
lim
T

φ
(
t
)

=
φ
2
.
(A.12)
T
→∞
−
φ
2
usually assumes finite values because of the presence of
T
in the denominator of
Eq. (
A.12
). In this case
A
2
(
ω
) is substituted with the socalled power spectrum (or
power spectral density)
(
t
)
e
i
ω
t
d
t
T
2
1
2
T
P
(
ω
)
=
lim
T
T
φ
.
(A.13)
→∞
−
Therefore
P
(
indicates how much power of the signal is contained in the angular
frequency interval [
ω
)d
ω
ω, ω
+
d
ω
]. Moreover, in many applications the onesided power
spectrum,
P
h
(
), is used. The onesided
power spectrum, often indicated dropping the subscript
h
, is the most adopted tool
for analyzing the structure of a signal in the frequency domain.
The autocovariance function
ω
)
=
P
(
ω
)
+
P
(
−
ω
)

(with 0
≤
ω<
∞
ρ
(
τ
) is another powerful tool for investigating the
ρ
τ
temporal structure of a signal.
(
)isdefinedas
∞
−∞
φ
ρ
(
τ
)
=
(
t
+
τ
)
φ
(
t
)d
t
.
(A.14)
It gives a proxy of the interrelations of the signal
φ
(
t
) at two distinct times,
t
and
t
+
τ
. It depends on the time delay (or lag)
τ
and reflects the memory of the signal:
High values of
) are a symptom of a strong link, whereas low values indicate a
weak link. However, the autocovariance function shows only the linear links in the
signal, whereas the nonlinear ones have to be detected by more complex tools, such
as mutual information (e.g., see
Kantz and Schreiber
,
1997
). Similar to Eq. (
A.12
),
ρ
(
τ
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