Supplementary data A
Fourier analysis is the standard method for the analysis of various signals in many
applications from different fields of science and life. In physical chemistry, it is commonly
used in the Fourier transform spectroscopy - a measurement technique based on
measurements of the electromagnetic radiation. Besides, it is often used for characterization
of time-series (for example, from kinetic measurements by potentiometric or
spectrophotometric technique) [1, 2]. Especially, if some periodicity exists in time series,
Fourier analysis is powerful method for the identification of important frequencies or periods
(reciprocal frequencies) of oscillations. The signal which is originally given as a function of
time, X(t), is transformed to the complex function of the frequency, Y(f):

Y( f ) 
 X (t )e
2 ift
dt
(1)

The new function Y(f) is known as Fourier transform of the function X(t). The exponential
function of the complex variable is equivalent to the trigonometric form:
e2 ift  cos(2 ft )  i sin( 2 ft )
(2)
Hence, the Fourier transformation decomposes the analyzed signal to the contributions of
harmonic functions with different frequencies f.
The Fourier transform can be used to determine the frequency content of a signal X(t) if it is
known for -∞ < t < +∞. However, in practice, X(t) is known for only a finite length of time T,
and hence the so-called finite Fourier transform is used:
T
Y ( f )   X (t )e 2 ift dt
(3)
0
The finite Fourier transform provides a mechanism for representing a signal as the sum of
simple sine and cosine functions. These functions correspond to discrete lines in the
frequency spectrum.
A time series is usually collected over a finite time T and, moreover, it consists of a discrete
number of points obtained at a chosen sampling frequency. These data can be represented as a
sum of sine and cosine functions of time t with the period being an integer submultiple of T.
The Fourier transform of these discrete finite extent data is obtained by using the discrete
Fourier transform (DFT). In this case, discrete summation is used instead of continuous
integration. Results of a Fourier analysis are often presented as the power spectrum –
normalized square of the magnitude of the Fourier transformation.
1
P( f ) 
T
T
 X (t )e
0
2
2 ift
dt 
1
2
Y( f )
T
(4)
The power spectrum gives result in real number set and therefore it is often more suitable for
data representation.
A special case of the DFT is the fast Fourier transform (FFT). The fast Fourier transform
(FFT) is essentially an efficient computational scheme that takes advantage of certain
symmetry properties in the cosine and sine functions at their points of evaluation in order to
achieve speed over conventional methods. If the number of data points is N, then the FFT
requires N log2N operations, whereas conventional techniques require N2 operations. The
development of FFT is attributed to Cooley and Tukey (1965) [3]. There are many
commercial software packages available for determining the FFT of a given signal. In this
paper the 1D fast Fourier transform based on the FFTW library was used.
[1] Nayfeh AH, Balachandran B (2004) Applied Nonlinear Dynamics - Analytical,
Computational, and Experimental Methods, WILEY-VCH Verlag GmbH & Co. KGaA,
Weinheim.
[2] Hilborn RC (2004) Chaos and Nonlinear Dynamics – An introduction for Scientists and
Engineers, Second Edition, Oxford University Press, Oxford.
[3] Cooley TW, Tukey JW (1965) Math Comput 19: 297- 301