Environmental Engineering Reference
In-Depth Information
Appendix A
Power spectrum and correlation
The steady-state pdf
p
(
(
t
) is a key piece of information in
the study of noise-induced phenomena; however, it does not give indications about the
temporal structure of the process. In fact, processes with different temporal evolutions
can share the same pdf. Because some noise-induced phenomena underlie changes
in the temporal behavior of dynamical systems (e.g., the stochastic resonance), it is
useful to introduce two mathematical tools that are commonly used to quantitatively
investigate the temporal structure of a signal, namely the power spectrum and the
autocorrelation function. In this appendix we recall the basic concepts and some
analytical results, referring to specialized textbooks (e.g.,
Papoulis
,
1984
) for a more
comprehensive description. Moreover, in the following discussion we consider signals
in the time domain, though the same results are valid also if the process is sampled in
space, e.g., when transects of spatial fields are studied (see Chapter 5). In this case, the
power spectrum (also known as
structure function
) and the autocorrelation function
are useful tools for investigating the existence of regular patterns in the field.
Let us start from a quite specific case and consider a piecewise continuously
differentiable periodic function
φ
) of a stochastic process
φ
φ
(
t
), with period 2
π
(if the signal has a different
period, it may be mapped to a 2
period through a suitable scaling of time). Fourier
demonstrated that such periodic functions can be written as the superposition of
infinite harmonics (i.e., sinusoidal functions), namely
π
∞
a
0
2
+
φ
(
t
)
=
(
a
k
cos
kt
+
b
k
sin
kt
)
,
(A.1)
k
=
1
where the
k
coefficients are obtainable by minimizing the mean square deviation
between
(
t
) and the summation truncated to the
k
order, and taking advantage of the
fact that the sines and cosines functions form an orthogonal set. It follows that
φ
π
−
π
φ
π
−
π
φ
π
−
π
φ
1
π
1
π
1
π
a
0
=
(
t
)d
t
,
a
k
=
(
t
)cos
kt
d
t
,
b
k
=
(
t
)sin
kt
d
t
,
(A.2)
269
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