Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 85

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