Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 38

()=

()

(18)

()= ()

(19)

Note again that f(t) is real time domain signal, while F(ω), the amplitude spectrum is a
complex function.

5.3 Fourier spectrum
A time function, f(t) such as gravity field may be transform into another function, F(ω),
where the amplitude of all frequency components present in f(t) and their corresponding
phases are expressed as function of frequency. The two transform relations are already
given in equations (18) and (19).
The complex function, F ( ω ), is called the Fourier spectrum and its modulus and arguments
as earlier explained are called amplitude and phase spectra respectively. The cosine
transform part of F ( ω )[ = a ( ω ) + ib ( ω )], a ( ω ) is called the co-spectrum and the sine transform
part, b ( ω ) is called the quadrature spectrum.

5.4 Power spectrum
If is the mean power of a real function, f(t) whose period is T, then (Thompson 1982)

=lim →

(())

(20)

Where (f(t)) 2 is termed the instantaneous energy and the complete integration in equation
(20) is the total (mean) energy of the function.
We have already noted that for two Fourier pairs, f 1 (t)↔F 1 (ω) and f 2 (t)↔F 2 (ω), then

f 1 (t).f 2 (t)↔

∗ ()∗ ()

and that

=

( () )

| () |

[Parseval's theorem].

The power spectrum |()| and its total energy E T are then related by

E T =

=

| () | | () |

,

where the power spectrum |()| is a real quantity.

5.5 Spectral windows and their uses
When we were discussing the convolution theorem, we noted that we might run into
convolution operations in truncating lengthy data (signal) by use of window functions.
Data windowing can be viewed as the truncation of an infinitely long function, f(t). A box-
car function, w(t) = 1, -T<t<T and w(t) = 0 elsewhere, has a value 1 over the required length
(2T) and zero elsewhere. The function, w(t), a time window can be used to truncate f(t) and
the truncated time function, f tr (t) = f(t).w(t). Using the convolution theorem, the Fourier
transform, F tr (ω) of the truncated function, f tr (t) is given by ()=

()∗() , where
F(ω) and W(ω) are the Fourier transforms of f(t) and w(t) respectively. We can compute the