Power spectrum and correlation - Noise-induced Phenomena in the Environmental Sciences - page 273

mean and uniform distribution in the interval [ 0.5, 0.5], the second row corresponds

to a deterministic periodic signal given by the sum of three sinusoids,

−

φ ( t ) = sin(0 . 1 t ) + sin(0 . 15 t ) + sin(0 . 5 t ); (A.18)

finally, in the third case the signal is obtained as the sum of ( A.18 ) and white noise.
The spectrum of the white noise is flat, showing that no particular frequency
dominates the signal. This is confirmed also by the autocorrelation function that falls
to negligible values for any time lag different from zero. In contrast, in the case of the
periodic deterministic signal, the power spectrum has three spikes corresponding to
the frequency of the sinusoids (i.e., 0 ). Notice that also

the autocorrelation shows the existence of regular waves in the signal - in particular,
it has the same period as the original signal - but its interpretation is more difficult.
Finally, in the third case we can appreciate the importance of analyzing signals in
both the time and frequency domains. In fact, the time series appears quite irregular and
the periodic carrier wave is not easily detectable by eye. Similarly, the autocorrelation
function does not give clear indications of the periodicity embedded in the signal.
In contrast, the power spectrum allows us to recognize the existence of periodic
components and to detect their frequency.

. 1 / 2 π ,0 . 15 / 2 π ,and0 . 5 / 2 π