Environmental Engineering Reference

In-Depth Information

Box 3.3: Methods for detecting stochastic resonance

The power spectrum
S
(

) is defined as the squared absolute value of the Fourier

transform of a signal and indicates the energy contained in each of the frequencies

ω

ω

present in a given signal (see Appendix
A
). When all frequencies give the same

contribution to the signal, the spectrum is flat. This case corresponds to a white noise.

Conversely, peaks in the spectrum are a clear symptom that the corresponding

frequencies dominate the signal. This property is particularly suitable for the

quantification of stochastic resonance, as it provides a way to extract from the signal the

periodicity of the transitions induced by cooperation between noise and deterministic

periodic forcing.

The importance of the peak associated with the frequency

ω
p
of the periodic forcing

can be better quantified in terms of the signal-to-noise ratio (SNR),

ω
p
)

S
N
(

S
(

SNR

=

ω
p
)
,

(B3.3-1)

where the numerator indicates the magnitude of the largest peak in the power spectrum

and the denominator is the value of
S
corresponding to the smooth background of the

power spectrum at the same frequency. The smooth background can be obtained by a

suitable moving average of the spectrum, in which the spike corresponding to

ω
p
has

been preventively eliminated. The rationale underlying the definition of the SNR is that

the power spectrum can be written as the sum of a background plus a number of peaks at

integer multiples of the driving frequency

ω
p
. In other words, the SNR quantifies the

importance of the peak with respect to the background noise.

We remark that, because some of the frequently used techniques in time-series

analysis are based on power-spectrum analysis, a rich body of literature exists on the

technical issues emerging in the evaluation of power spectra. These issues include,

among others, the effect of the sampling time and the use of suitable weighting windows

to reduce biased evaluations (e.g.,
Papoulis
,
1984
). This explains why the power

spectrum has been the first and most used approach to the study of stochastic resonance,

and several theoretical results are available for this method. We refer the interested

reader to
Gammaitoni et al.
(
1998
) and
Wellens et al.
(
2004
) for a review of these

methods.

The analysis of the distribution
p
(
t
c
) of the crossing times
t
c
across a suitable

threshold is another approach frequently used to quantify the effect of stochastic

resonance (
Wellens et al.
,
2004
). For example, consider the bistable system discussed in

Subsection
3.3.1
. In this case, stochastic resonance regularizes the transitions across the

potential barrier at

0. Thus it is sensible to choose this value as the threshold for the

crossing-time analysis. In fact, when stochastic resonance does not occur the transitions

are random and the distribution of crossing times is exponential with a mean equal to

φ
=

[Eq. (
3.60
)]; in contrast, as stochastic resonance emerges, the transitions become

more regular and the crossing-time distribution departs from the exponential

distribution. In particular, when

t
c

t
c
π/ω
p
, a maximum appears at time
t
c
=

t
c

,

indicating how in these conditions the dynamical system

φ

(
t
) has a remarkably high

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