Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 30

avoid aliasing effect, we must make the sampling frequency sufficiently high to ensure
making the Nyquist frequency at least equal to the cutoff frequency of the original signal.

4. Fourier analysis

Fourier analysis is the theory of the representation of a function of a real variable by means
of a series of sines and cosines. The discussion of Fourier analysis starts with a statement of
Fourier theorem. Fourier (1768 - 1830) stated without proof and used in developing a
solution of the heat equation, the so-called Fourier theorem.

4.1 Fourier theorem
Any single-valued function, f(x) defined in the interval [-π, π] may be represented over this
interval by the trigonometric series, i.e.

()=

+∑ ( cos+ sin)

(1)

The expansion coefficients, a n and b n are determined by use of Euler's formulas:

=

) , =

( ) cos(=0,1,2,3… ( ) sin(=1,2,3…

)

Fourier investigated many special cases of the above theorem, but he was unable to develop
a logical proof of it.
Dirichlet (1805 - 1859) in 1829 formulated the restrictions under which the theorem is
mathematically valid. These restrictions are normally called Dirichlet conditions, and are
summarized as follows: that for the interval [-π, π], the function, f(x) must (1) be single-
valued, (2) be bounded, (3) have at least a finite number of maxima and minima, (4) have
only a finite number of discontinuities: piece-wise continuous and (5) periodic, i.e. f(x + 2π)
= f(x). For values of x outside of [-π, π], the series in equation (1) above converges to f(x) at
values of x for which f(x) is continuous and to

[ ( +0 ) +(−0) ] at points of discontinuity.
The quantities f(x+0) and f(x-0) refer to the limits from the right and left respectively of the point
of discontinuity. The coefficients are still given by the Euler's formulas.

4.2 Different forms of Fourier series
According to the Fourier theorem, a function such as f(t), which satisfies Dirichlet's
conditions can be represented by the following infinite series, the Fourier series, as

()=

+∑ ( cos+ sin)

(2)

Where the constants a 0 , a n and b n are given by

=

( )

=

( ) cos(=1,2,3…)

(3)

=

()nt(=1,2,3…)

These coefficients ( a 0 , a n and b n ) are called Fourier coefficients and the determination of their
values is called Fourier or harmonic analysis.