Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 32

and therefore [ (−) ] =− [ () ] , indicating that the phase spectrum, () is
antisymmetrical.
Equation (7) is the time domain Fourier series and equation (8) is the frequency domain
Fourier series. The two equations are an example of a Fourier transform pair, or we can say
f(t) is the inverse Fourier transform of F(n) or C n .

4.3 Application of Fourier series
Fourier series, as we have already seen, is used as an effective tool in the analysis of periodic
functions. We have also noted that any periodic function having a period, T (and satisfying
Dirichlet conditions) can be represented by the infinite series of trigonometric functions.
Thus f(t) is represented by the addition of sinusoidal and cosinusoidal waves whose
frequencies are integral multiples of some fundamental unit of frequency in the signal, f(t).
The frequency, π/T is called the fundamental and its integral multiples are called the
harmonics.
If we plot the Fourier coefficients a n and b n as functions of frequency, we obtain a number of
discrete spectral lines located at fixed spacing of π/T. As T→ ∞ , the spacing of the spectral
lines approaches zero. Analysis of the spectrum of a field gives the energy content of the
field and may represent phenomenal changes in the attributes of the causative agents of the
field. This allows systems to be analyzed in the frequency domain so as to find out the
frequency response of the system from the impulse response and vice-versa. It can be used
as an intermediate step in more elaborate signal processing techniques (e.g. the fast Fourier
transform). Fourier integrals (next section) is useful in the analysis of limited-duration
(transient) signals.

4.4 Fourier integrals
If we look back at equation (6) and decide to put the expressions for a 0 , a n and b n that
followed inside it and use the dummy variable, λ, we can write equation (6) in a more
compact form as

()=

()cos

(9)

()+∑ (−)

We let ω n = nπ/T (the n th angular frequency), ω n-1 = (n-1)π/T and therefore ∆ω = ω n - ω n-1 =
π/T and substituting all these in equation (9), we obtain

( ) =

∆
( ) cos ( − )

( ) + ∑

(10)

If we let T→ ∞ , the following changes will occur in equation (10)
a. The first part of the expression in the RHS will vanish to zero.
b. The increment ∆ω becomes very small and in the limit ∆ω→dω. Thus the digitally
increasing ωn becomes a continuous variable, ω.
c. The summation can be replaced by an equivalent integral with appropriate limits.
Thus equation (10) becomes

()=

(11a)

()cos(−)

This is the Fourier integral. We can further analyze equation (11a) by using the fact that