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

Search WWH ::

Custom Search